X-Ray Fluorescence Analytical Techniques - CNSTN : Centre ...
X-Ray Fluorescence Analytical Techniques - CNSTN : Centre ...
X-Ray Fluorescence Analytical Techniques - CNSTN : Centre ...
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X-<strong>Ray</strong> <strong>Fluorescence</strong> <strong>Analytical</strong><br />
<strong>Techniques</strong><br />
Moussa Bounakhla<br />
&<br />
Mounia Tahri<br />
CNESTEN
CONTENT<br />
SECTION I: Basic in X-<strong>Ray</strong> <strong>Fluorescence</strong><br />
I. History of X-<strong>Ray</strong> <strong>Fluorescence</strong><br />
II. Introduction<br />
III. Physics of X-<strong>Ray</strong>s<br />
III.1 Electromagnetic Radiation, Quanta<br />
III.2 Properties of X-<strong>Ray</strong>s<br />
III.3 The Origin of X-<strong>Ray</strong>s<br />
III.4 Bohr’s Atomic Model<br />
III.5 Nomenclature<br />
III.6 X-<strong>Ray</strong> Emission<br />
III.6.1 Continuum<br />
III.6.2 Characteristic Emission<br />
III.7 Interactions of X-<strong>Ray</strong> with Matter<br />
III.7.1 Photoelectric Absorption<br />
III.7.2 Compton Effect<br />
III.7.3 <strong>Ray</strong>leigh Scattering (Elastic Scattering)<br />
III.7.4 Competitive Interactions<br />
III.8 <strong>Fluorescence</strong> Yield<br />
IV. X-<strong>Ray</strong> Production Sources<br />
IV.1 X-<strong>Ray</strong> Tubes<br />
IV.1.1 Side-window Tubes<br />
IV.1.2 End-window Tubes<br />
IV.2 Radioisotope Sources<br />
SECTION II: Energy Dispersive X-<strong>Ray</strong> <strong>Fluorescence</strong> (ED-XRF)<br />
I. Introduction<br />
II. Instrumentation<br />
II.1 Excitation Mode<br />
II.1.1 Direct Tube Excitation<br />
II.1.2 Secondary Target Excitation<br />
II.1.3 Radio-Isotopic Excitation<br />
II.2 Detectors<br />
II.3 Pulse Height Analysis
II.4 Energy Resolution<br />
III. Spectrum Evaluation<br />
IV. Detector Artefacts<br />
IV.1 Escape Peaks<br />
IV.2 Compton Edge<br />
IV.3 Resulting Spectral Background<br />
V. The Approach to Quantification in EDXRF Analysis<br />
V.1 Thin Samples Technique<br />
V.2 Intermediate Thickness Samples<br />
V.3 Infinitely Thick Samples<br />
SECTION III: Total Reflexion X-<strong>Ray</strong> <strong>Fluorescence</strong> (TXRF)<br />
I. Introduction<br />
II. Advantages of TXRF<br />
III. Principle of Total Reflection X-<strong>Ray</strong> <strong>Fluorescence</strong> Analysis<br />
IV. Instrumentation<br />
IV.1 Excitation Sources for TXRF<br />
IV.2 Sample Reflectors<br />
IV.3 Detectors<br />
V. Quantification<br />
VI. Influence on Detection Limits<br />
VII. General Sample Preparation<br />
VIII. Application of TXRF<br />
SECTION IV: Wavelength Dispersive X-<strong>Ray</strong> <strong>Fluorescence</strong> (WD-XRF)<br />
I. Introduction<br />
II. Principle of WD-XRF<br />
II.1 Collimator Masks<br />
II.2 Collimator<br />
II.3 Analyzing Crystals<br />
II.3.1 Bragg’s Law<br />
II.3.2 Reflections of Higher Orders<br />
II.3.3 Crystal Types<br />
II.3.4 Dispersion, Line Separation<br />
II.3.5 Synthetic Multilayers<br />
II.4 Detectors
II.4.1 Gas Proportional Counter<br />
II.4.2 Scintillation Counters<br />
II.4.3 Pulse Height Analysis (PHA), Pulse Height Distribution<br />
III. Points of Comparison between ED-XRF and WD-XRF<br />
SECTION V: Sample Preparation<br />
I. Solids<br />
II. Powders and Briquets<br />
III. Fused Materials<br />
IV. Filters and Ions-Exchange Resins<br />
V. Thin Films<br />
VI. Liquids<br />
SECTION VI: Quantitative Analysis<br />
I. Detection Limits<br />
II. Disturbing Effects<br />
II.1 Interelement Radiation<br />
II.2 Matrix Effects<br />
II.2.1 Absorption Effect<br />
II.2.2 Enhancement Effect<br />
II.3 Particle-Size Effects<br />
II.4 Mineralogical Effects<br />
II.5 Surface Effects<br />
III. Mathematical Models<br />
III.1 Sherman Equation<br />
III.2 Empirical Alpha Models<br />
III.3 Fundamental Parameters Method<br />
III.4 Fundamental Alphas<br />
III.5 Semi-Quantitative Analysis<br />
Exercises and Solutions
Module:<br />
Title: X-<strong>Ray</strong> <strong>Fluorescence</strong> <strong>Analytical</strong> <strong>Techniques</strong><br />
Learning objective:<br />
Make potential users proficient in the use of X-<strong>Ray</strong> <strong>Fluorescence</strong> <strong>Analytical</strong> <strong>Techniques</strong><br />
Target public:<br />
Potential users including students in science and technology<br />
Profile:<br />
Senior technicians, Students, Teachers, Researchers and <strong>Analytical</strong> Specialists in Scientific<br />
Fields.<br />
Qualifications:<br />
University education related to application of sciences and technology<br />
English and French literacy
SECTION I<br />
BASIC IN X-RAY FLUORESCENCE<br />
I. History of X-<strong>Ray</strong> <strong>Fluorescence</strong><br />
The history of X-ray fluorescence dates back to the accidental discovery of X-rays in<br />
1895 by the German physicist Wilhelm Conrad Roentgen. While studying cathode rays in a<br />
high-voltage, gaseous-discharge tube, Roentgen observed that even though the experimental<br />
tube was encased in a black cardboard box the barium-platinocyanide screen, which was lying<br />
adjacent to the experiment, emitted fluorescent light whenever the tube was in operation.<br />
Roentgen's discovery of X-rays and their possible use in analytical chemistry went unnoticed<br />
until 1913. In 1913, H.G.J. Mosley showed the relationship between atomic number (Z) and<br />
the reciprocal of the wavelength (1/λ) for each spectral series of emission lines for each<br />
element. Today this relationship is expressed as:<br />
2<br />
c / λ = a ( Z−<br />
s)<br />
; (I.1)<br />
where:<br />
a is a proportionality constant,<br />
s is a constant dependent on a periodic series.<br />
Mosley was also responsible for the construction of the early X-ray spectrometer. His<br />
design centered around a cold cathode tube where the air within the tube provided the<br />
electrons and the analyte which served as the tube target. The major problem experienced laid<br />
in the inefficiency of using electrons to create x-rays; nearly 99% of the energy was lost as<br />
heat.<br />
In the same year, the Bragg brothers built their first X-ray analytical device. Their<br />
device was based around a pinhole and slit collimator. Like Mosley's instrument, the Braggs<br />
ran into difficulty in maintaining efficiency. Progress in XRF spectroscopy continued in 1922<br />
when Hadding investigated using XRF spectrometry to analyse mineral samples. Three years<br />
later, Coster and Nishina put forward the idea of replacing electrons with X-ray photons to<br />
excite secondary X-ray radiation resulting in the generation of an X-ray spectrum. This<br />
technique was attempted by Glocker and Schrieber, who in 1928 published Quantitative<br />
Roentgen Spectrum Analysis by Means of Cold Excitation of the Spectrum in Ann. Physics.<br />
Progress appeared to be at a standstill until 1948, when Friedman and Birks built the<br />
first XRF spectrometer. Their device was built around a diffractometer, with a Geiger counter<br />
for a detection device and proved comparatively sensitive for much of the atomic number<br />
range. It might be noted that XRF spectrometers have progressed to the point where elements<br />
ranging from Beryllium to Uranium can be analysed.<br />
Although the earliest commercial XRF devices used simple air path conditions,<br />
machines were soon developed utilizing helium or vacuum paths, permitting the detection of<br />
lighter elements. In the 1960’s, XRF devices began to use lithium fluoride crystals for<br />
diffraction and chromium or rhodium target X-ray tubes to excite longer wavelengths. This<br />
development was quickly followed by that of multichannel spectrometers for the simultaneous<br />
measurement of many elements. By the mid 60’s computer controlled XRF devices were<br />
coming into use. In 1970, the lithium drifted silicon detector (Si(Li)) was created, providing
very high resolution and X-ray photon separation without the use of an analysing crystal. An<br />
XRF device was even included on the Apollo 15 and 16 missions.<br />
Meanwhile, Schwenke and co-workers have fine tuned a procedure known as total<br />
reflection X-ray fluorescence (TXRF), which is now used extensively for trace analysis. In<br />
TXRF, a Si(Li) detector is positioned almost on top of a thin film of sample, many times<br />
positioned on a quarts plate. The primary radiation enters the sample at an angle that is only<br />
slightly smaller than the critical angle for reflection. This significantly lowers the background<br />
scattering and fluorescence, permitting the detection of concentrations of only a few tenths of<br />
a ppb.<br />
II. Introduction<br />
X-ray <strong>Fluorescence</strong> (XRF) Spectroscopy involves measuring the intensity of X-rays<br />
emitted from a specimen as a function of energy or wavelength. The energies of large<br />
intensity lines are characteristic of atoms of the specimen. The intensities of observed lines for<br />
a given atom vary as the amount of that atom present in the specimen. Qualitative analysis<br />
involves identifying atoms present in a specimen by associating observed characteristic lines<br />
with their atoms. Quantitative analysis involves determining the amount of each atom present<br />
in the specimen from the intensity of measured characteristic X-ray lines.<br />
The emission of characteristic atomic X-ray photons occurs when a vacancy in an inner<br />
electron state is formed, and an outer orbit electron makes a transition to that vacant state. The<br />
energy of the emitted photon is equal to the difference in electron energy levels of the<br />
transition. As the electron energy levels are characteristic of the atom, the energy of the<br />
emitted photon is characteristic of the atom. Molecular bonds generally occur between outer<br />
electrons of a molecule leaving inner electron states unperturbed. As X-ray fluorescence<br />
involves transitions to inner electron states, the energy of characteristic X-ray radiation is<br />
usually unaffected by molecular chemistry. This makes XRF a powerful tool of chemical<br />
analysis in all kinds of materials.<br />
In a liquid, fluoresced X-rays are usually little affected by other atoms in the liquid and<br />
line intensities are usually directly proportional to the amount of that atom present in the<br />
liquid. In a solid, atoms of the specimen both absorb and enhance characteristic X-ray<br />
radiation. These interactions are termed 'matrix effects' and much of quantitative analysis with<br />
XRF spectroscopy is concerned with correcting for these effects.<br />
While the principles are the same, a variety of instrumentation is used for performing Xray<br />
fluorescence spectroscopy. There are two basic classes of instruments: Wavelength<br />
Dispersive and Energy Dispersive. Wavelength Dispersive spectrometers measure X-ray<br />
intensity as a function of Wavelength while Energy Dispersive spectrometers measure X-ray<br />
intensity as a function of energy.<br />
An extremely important aspect of X-ray fluorescence spectroscopy is the method by<br />
which the inner orbital vacancy is created. Bombarding the sample with high energy X-rays is<br />
one method. Bombarding with high-energy electrons and protons are other approaches. An<br />
incident photon beam experiences a photon absorption interaction with the specimen while<br />
electron and proton beams primarily experience a Coulomb interaction with the specimen.<br />
X-ray tubes accelerate high-energy electrons at a target within the tube that is then<br />
caused to fluoresce X-rays. The resulting X-ray beam includes a continuum and characteristic<br />
lines of the tube target. Radioactive sources can also be used to generate X-ray, electron (beta
emitters), and proton (alpha emitters) beams. X-ray tubes can generate a high power X-ray<br />
beam, but the radiation is not monochromatic. Radioactive sources produce monochromatic<br />
beams, but of comparatively lower power. Proton-Induced X-ray Emission (PIXE) utilises a<br />
beam of protons. Wavelength Dispersive Spectrometry (WDS) generally utilises an X-ray<br />
tube as does Energy Dispersive X-ray Spectrometry (EDX). Instruments such as the electron<br />
microprobe and electron microscope directly bombard the sample with high-energy electrons<br />
to eject inner orbital electrons (EDS). Note that the charged particle beam approaches require<br />
the specimen to be electrically conductive.<br />
III. Physics of X-<strong>Ray</strong>s<br />
III.1 Electromagnetic Radiation, Quanta<br />
X rays are electromagnetic radiation. All X-rays represent a very energetic portion of<br />
the electromagnetic spectrum (Table 1) and have short wavelengths of about 0.1 to 100<br />
angstroms (Å). They are bounded by ultraviolet light at long wavelengths and gamma rays at<br />
short wavelengths X-rays in the range from 50 to 100 Å are termed soft X-rays because they<br />
have lower energies and are easily absorbed.<br />
Table I.1: Energy and names of various wavelength range.<br />
Energy range (eV) Wavelength range Name<br />
< 10 -7 cm to km Radio Waves (short, medium, long waves)<br />
< 10 -3 mm to cm Micro Wave<br />
< 10 -3 mm to mm Infra Red<br />
0.0017 – 0.0033 380 to 750 nm Visible Light<br />
0.033 – 0.1 10 to 380 nm Ultra Violet<br />
0.11 - 100 0.01 to 12 nm X-<strong>Ray</strong>s<br />
10 - 5000 0.0002 to 0.12 nm Gamma Radiation<br />
The range of interest for X-ray is approximately from 0.1 to 100 Å. Although,<br />
angstroms are used throughout these notes, they are not accepted as SI unit. Wavelengths<br />
should be expressed in nanometers (nm), which are 10 -9 meters (1 Å = 10 -10 m), but most texts<br />
and articles on microprobe analysis retain the use of the angstroms. Another commonly used<br />
unit is the micron, which more correctly should be termed micrometer (µm); a micrometer is<br />
10 4 Å.<br />
The relationship between the wavelength of electromagnetic radiation and its<br />
corpuscular energy (E) is derived as follows. For all electromagnetic radiation:<br />
E = h ν ; (I.2)<br />
where:<br />
h is the Planck constant (6.62 10 -24 J.s);<br />
ν is the frequency expressed in Hertz.<br />
For all wavelengths,<br />
ν = c / λ ; (I.3)
where:<br />
c = speed of light (2.99782 10 8 m/s);<br />
λ= wavelength (Å).<br />
Thus:<br />
E = hc / λ = 1.<br />
9863610<br />
−24<br />
/ λ ; (I.4)<br />
where E is in Joule and λ in meters.<br />
The conversion to angstroms and electron volts (1 eV = 1.6021 10-19 Joule) yields the<br />
Duane-Hunt equation:<br />
E(<br />
eV)<br />
o<br />
= 12.<br />
396/<br />
λ ( A)<br />
. (I.5)<br />
Note the inversion relationship. Short wavelengths correspond to high energies and long<br />
wavelengths to low energies. Energies for the range of X-ray wavelengths are 124 keV (0.1<br />
Å) to 124 eV (100 Å). The magnitudes of X-ray energies suggested to early workers that Xrays<br />
are produced from within an atom. Those produced from a material consist of two<br />
distinct superimposed components: continuum (or white) radiation, which has a continuous<br />
distribution of intensities over all wavelengths, and characteristic radiation, which occurs as a<br />
peak of variable intensity at discrete wavelengths.<br />
III.2 Properties of X-<strong>Ray</strong>s<br />
A general summary of the properties of X-rays is presented below:<br />
• Invisible;<br />
• Propagate with velocity of light (3.10 8 m/s)<br />
• Unaffected by electrical and magnetic fields;<br />
• Differentially absorbed in passing through matter of varying composition, density and<br />
thickness;<br />
• Reflected, diffracted, refracted and polarized;<br />
• Capable of ionising gases;<br />
• Capable of affecting electrical properties of solids and liquids;<br />
• Capable of blackening a photographic plate;<br />
• Able to liberate photoelectron. And recoils electrons<br />
• Emitted in a continuous spectrum;<br />
• Emitted also with a line spectrum characteristic of the chemical element;<br />
• Found to have absorption spectra characteristic of the chemical element.<br />
III.3 The Origin of X-<strong>Ray</strong>s<br />
An electron can be ejected from its atomic orbital by the absorption of a light wave<br />
(photon) of sufficient energy. The energy of the photon (hν) must be greater than the energy<br />
with which the electron is bound to the nucleus of the atom. When an inner orbital electron is<br />
ejected from an atom, an electron from a higher energy level orbital will transfer into the<br />
vacant lower energy orbital (Figure I.1). During this transition a photon may be emitted from<br />
the atom. To understand the processes in the atomic shell, we must take a look at the Bohr’s<br />
atomic model.
The energy of the emitted photon will be equal to the difference in energies between the<br />
two orbitals occupied by the electron making the transition. Due to the fact that the energy<br />
difference between two specific orbital shells, in a given element, is always the same (i.e.,<br />
characteristic of a particular element), the photon emitted when an electron moves between<br />
these two levels will always have the same energy. Therefore, by determining the energy<br />
(wavelength) of the X-ray light (photons) emitted by a particular element, it is possible to<br />
determine the identity of that element.<br />
Figure I.1: A pictorial representation of X-ray fluorescence using a generic atom and<br />
generic energy levels. This picture uses the Bohr model of atomic structure<br />
and is not to scale.<br />
III.4 Bohr’s Atomic Model<br />
Bohr’s atomic model describes the structure of an atom as an atomic nucleus<br />
surrounded by electron shells (Figure I.2). The positively charged nucleus is surrounded by<br />
electrons that move within defined areas (shell). The differences in the strength of the<br />
electron’s bonds to the atomic nucleus are very clear depending on the area or level they<br />
occupy, i.e., they vary in their energy. When we talk about this, we refer to energy levels or<br />
energy shells. This means that a clearly defined minimum amount of energy is required to<br />
release an electron of the innermost shell from the atom. To release an electron of the second<br />
innermost shell from the atom, a clearly defined minimum amount of energy is required that<br />
is lower that that needed to release an innermost electron. An electron’s bond to an atom is<br />
weaker the further away it is from the atom’s nucleus. The minimum amount of energy<br />
required .to release an electron from an atom, and thus the energy with which it is bound to<br />
the atom, is also referred to as the binding energy of the electron to the atom.
Figure I.2: Bohr’s atomic model, shell model.<br />
The binding energy of an electron in an atom is established mainly by determining the<br />
incident. It is for this reason that the term absorption edge is very often found in literature.<br />
Energy level = binding energy = absorption edge<br />
The individual shells are labelled with the letters K, L; M; N, …, the innermost shell<br />
being the K-shell, the second innermost the L-shell etc. the K-shell is occupied by 2 electrons;<br />
the L-shell has three sub-levels and can contain up to a total of 8 electrons. The M-shell has<br />
five sub-levels and can contain up to 18 electrons.<br />
III.5 Nomenclature<br />
The production of X-rays involves transitions of the orbital electrons at atoms in the<br />
target between allowed orbits or energy states, associated with ionization of the inner atomic<br />
shell. The permissible transitions that electrons can undergo from initial to final state are<br />
specified by three quantum selection rules:<br />
1. The change in n must be ≥ 1 (∆n ≠ 1);<br />
2. The change in l only can be ±1;<br />
3. The change in j can only be ±1 or 0.<br />
When an electron is ejected from the K-shell by electron bombardment or by the<br />
absorption of a photon, the atom becomes ionized. If this electron vacancy is filled by an<br />
electron coming from an L shell, the transition is accompanied by the emission of an X-ray<br />
line known as K line; this process leaves a vacancy in the L shell. On the other hand, the<br />
vacancy in the L shell might be filled by an electron coming from the M shell that is<br />
accompanied by the emission of an L line (Figure I.3). The terminology of energy levels and<br />
X-ray lines are showed in Figure I.4.
III.6 X-<strong>Ray</strong> Emission<br />
Figure I.3: Schematic illustration of production of K and L lines.<br />
Figure I.4: X-ray line labelling.<br />
X-rays re generated from the disturbance of the electron orbitals of atoms. This may be<br />
accomplished in several ways, the most common of which is to bombard a target element with<br />
high energy electrons, X-rays or accelerated charged particles. The first two are frequently<br />
used in X-ray spectrometry, either directly or indirectly. Electron bombardment results in both<br />
a continuum of X-ray energies and radiation that is characteristic of the target elements.<br />
Because both types of radiation will be encountered in X-ray spectrometry, each will be<br />
discussed.<br />
III.6.1 Continuum<br />
Continuum X-rays are produced when electrons or high energy charged particles lose<br />
energy in passing through the Coulomb field of a nucleus. In this interaction, the radiant<br />
energy (photon) lost by the electron is called Bremsstrahlung (Figure I.5). The emission of<br />
continuous X-rays finds a simple explanation in terms of classic electromagnetic theory, since<br />
according to this; the acceleration of charged particles should be accompanied by emission of<br />
radiation. In the case of high energy electrons striking a target, they must be rapidly
decelerated as they penetrate the material of target, and such a high negative acceleration<br />
should produce a pulse of radiation.<br />
Figure I.5: On the left, the classical model showing the production of Bremsstrahlung.<br />
On the right, the Continuum X-ray emission spectrum.<br />
The probability of radiative energy loss (Bremsstrahlung) is roughly proportional to<br />
q 2 z 2 T/M0 2 , where q is the particle charge in units of the electron charge e, Z is the atomic<br />
number of the target material, T is the particle kinetic energy, and M0 is the rest mass of the<br />
particle. Because of the fact that protons and heavier particles have large masses, compared to<br />
the electron mass, they irradiate relatively little, e.g., the intensity of continuous X-rays<br />
generated by protons is about four orders of magnitude lower than the generated by electrons.<br />
The ratio of energy lost by Bremsstrahlung to that lost by ionization can be<br />
approximated by:<br />
2<br />
⎛ m0<br />
⎞ ZT<br />
⎜<br />
M ⎟<br />
2<br />
⎝ 0 ⎠ 1600 m0<br />
c<br />
where m0 is the rest of the electron.<br />
III.6.2 Characteristic Emission<br />
, (I.6)<br />
The purpose of X-ray fluorescence is to determine chemical elements both qualitatively<br />
and quantitatively by measuring their characteristic radiation. To do this, the chemical<br />
elements in a sample must be caused emit X-rays. As characteristic X-rays only rise in the<br />
transition of atomic shell electron to lower, vacant energy levels of the atom, a method must<br />
be applied that is suitable for releasing electrons from the innermost shell of an atom. This<br />
involves adding to the inner electrons amounts of energy that are higher than the energy<br />
bonding them to the atom.<br />
This can be done in a number ways:<br />
• Irradiation using elementary particles of sufficient energy (electrons, protons, a-particles…)<br />
that transfer the energy necessary for release to the atomic shell electrons during collision<br />
processes.<br />
• Irradiation using X- or gamma rays from radionuclides.<br />
• Irradiation using X-rays from an X-ray tube.
III.7 X-<strong>Ray</strong> Interactions with Matter<br />
When X-rays are directed into an object, some of the photons interact with the particles<br />
of the matter and their energy can be absorbed or scattered. This absorption and scattering is<br />
called attenuation. Other photons travel completely through the object without interacting<br />
with any of the materials particles. The number of photons transmitted through a material<br />
depends on the thickness, density and atomic number of the material, and the energy of the<br />
individual photons.<br />
Even when they have the same energy, photons travel different distances within a<br />
material simply based on the probability of their encounter with one or more of the particles<br />
of the matter and the type of encounter that occur. Since the probability of an encounter<br />
increases with the distances travelled, the umber of photons reaching a specific point within<br />
the matter decreases exponentially with distance travelled (Figure I.6).<br />
Figure I.6: Exponential attenuation of photon energy with distance travelled in the<br />
material.<br />
The formula that describes this curve is:<br />
−µ<br />
x<br />
0 e<br />
I = I (Beer-Lambert law); (I.7)<br />
where:<br />
I0 is the initial intensity of photons;<br />
µ is the linear absorption coefficient;<br />
X is the distance travelled.<br />
The linear absorption coefficient has the dimension [1/cm] and is depend on the energy<br />
or the wavelength of the X-ray quants and the special density ρ (in [g/cm 3 ]) of the material<br />
that was passed through.<br />
It is not the linear absorption coefficient that is specific to the absorptive properties of<br />
the element, but the coefficient applicable to the density ρ of the material that was passed<br />
through:<br />
µ/ρ = mass attenuation coefficient.
The mass attenuation coefficient has the dimension [cm 2 /g] and only depends on the<br />
atomic number of the absorber element and the energy, or wavelength, of the X-ray quants.<br />
The mass attenuation coefficient accounts for the various interactions and is therefore<br />
composed of here major components:<br />
µ E)<br />
= τ(<br />
E)<br />
+ σ ( E)<br />
+ σ ( E)<br />
; (I.8)<br />
( coh inc<br />
τ(E) is the photoelectric mass absorption coefficient;<br />
σcoh(E) is the coherent mass absorption coefficient;<br />
σinc(E) is the incoherent mass absorption coefficient.<br />
III.7.1 Photoelectric Absorption<br />
In the photoelectric interaction, a photon transfers all its energy to an electron located in<br />
one of the atomic shells (Figure I.7). The electron is ejected from the atom by this energy and<br />
begins to pass through the surrounding matter. The electron rapidly loses its energy and<br />
moves only a relatively short distance from its original location. The photon’s energy is,<br />
therefore, deposited in the matter close to the site of the photoelectric interaction. The energy<br />
transfer is a two-step process. The photoelectric interaction in which the photon transfers its<br />
energy to the electron is the first step. The depositing of the energy in the surrounding matter<br />
by the electron is the second step.<br />
Photoelectric interactions usually occur with electrons that are firmly bound to the atom,<br />
that is, those with a relatively high binding energy. Photoelectric interactions are most<br />
probable when the electron binding energy is only slightly less than the energy of the photon.<br />
If the binding energy is more than the energy of the photon, a photoelectric interaction cannot<br />
occur. This interaction is possible only when the photon has sufficient energy to overcome the<br />
binding energy and remove the electron from the atom. The probability of photoelectric<br />
interactions occurring is also dependent on the atomic number of the material. An explanation<br />
for the increase, the binding energies move closer to the photon energy. The general<br />
relationship is that the probability of photoelectric interactions is proportional to Z 3 . In<br />
general, the conditions that increase the probability of photoelectric interactions are low<br />
photon energies and high atomic number materials.<br />
This process is often the major contributor of the absorption X-rays, and is the mode of<br />
excitation of the X-rays spectra emitted by elements in samples. Primarily as a result of the<br />
photoelectric process, the mass absorption coefficient decreases steadily with increasing<br />
energy of the incident X-radiation. There are sharp discontinuities at which the photoelectric<br />
process is especially efficient. Energies at which these discontinuities occur are called<br />
absorption edges (Figure I.8).
Figure I.7: Schematic description of photoelectric principle.<br />
Figure I.8: Absorption edges for different shells.<br />
The Figure I.8 supplies the following:<br />
• The overall progression of the coefficient decreases as energy increases, i.e. the higher the<br />
energy of the X-ray quants, the less they are absorbed.<br />
• The rapid changes in the mass attenuation coefficient reveal the binding energies of the<br />
electrons in the appropriate shells. If an X-ray quant has a level of energy that is equivalent<br />
to the binding energy of an atomic shell electron in an appropriate shell, it is then able to<br />
transfer all its energy to this electron and displace it from the atom. In this case, absorption<br />
increases sharply. Quants whose energy is only slightly below the absorption edge are<br />
absorbed far less rapidly.<br />
III.7.2 Compton Effect<br />
Also known as incoherent scattering, Compton effect is the interaction of a photon with<br />
a free electron that is considered to be at the rest. The weak binding of electrons to atoms may
e neglected provided that momentum transferred to the electron greatly exceeds the<br />
momentum of the electron in the bound state. Figure I.9 shows the Compton effect<br />
schematically.<br />
Relativistic energy and momentum are conserved in this process and the scattered X-ray<br />
photon has less energy and therefore a longer wavelength than the incident photon. Compton<br />
scattering is important for low atomic number specimens.<br />
The change in wavelength of the scattered photon is given by:<br />
c c<br />
h<br />
− = λ′ − λo<br />
= ( 1 − cosθ)<br />
. (I.9)<br />
ν′ νo<br />
moc<br />
Theta is the scattering angle of the scattered photon.<br />
Figure I.9: Compton effect.<br />
III.7.3 <strong>Ray</strong>leigh Scattering (Elastic Scattering)<br />
Elastic scattering is a process by which photons are scattered by bound atomic electrons<br />
and in which the atom is neither ionized nor excited. The incident photons are scattered with<br />
unchanged energy and with a definite phase relation between incoming and scattered waves<br />
(Figure I.10). The intensity of the radiation scattered by an atom is determined by summing<br />
the amplitudes of the radiation coherently scattered by each of the electrons bound in the<br />
atom. It should be emphasized that coherence extends only over the Z electrons of individual<br />
atoms. The interference is always constructive, provided the phase change over the diameter<br />
of the atom is less than one-half a wavelength. <strong>Ray</strong>leigh scattering occurs mostly at the low<br />
energies and for high Z materials.
Figure I.10: Coherent scattering of an X-ray by an atom.<br />
III.7.4 Competitive Interactions<br />
The energy at which interactions change from predominantly photoelectric to Compton<br />
is a function of the atomic number of the material. The Figure I.11 shows this crossover<br />
energy for several different materials. At the lower photons energies, photoelectric<br />
interactions are much more predominant than Compton. Over most of the energy range, the<br />
probability of both decreases with increased energy. However, the decrease in photoelectric<br />
interactions is much greater. This because the photoelectric rate changes in proportion to<br />
(1/E 3 ), whereas Compton interactions are much less energy dependent.<br />
Figure I.11: Comparison of Photoelectric and Compton interaction rates for different<br />
materials and photon energies.
In higher atomic number materials, photoelectric interactions are more probable, in<br />
general, and they predominate up to higher photon energy levels. The conditions that cause<br />
photoelectric interactions to predominate over Compton are the same conditions that enhance<br />
photoelectric interactions, hat is, low photon energies and materials with high atomic<br />
numbers.<br />
III.8 <strong>Fluorescence</strong> Yield<br />
When an electron is ejected from an atomic orbital by the photoelectric process, there<br />
two possible results: X-ray emission, or Auger electron ejection (Figure I.12). One of these<br />
two events occurs for each excited atom, but not both. Therefore, Auger electron production is<br />
a process which is competitive with X-ray photon emission from excited atoms in a sample.<br />
The faction of the excited atoms which emits X-rays is called the Fluorescent yield. This<br />
value is a property of the element and the X-ray line under consideration. Figure I.13 shows a<br />
plot of X-ray fluorescent yield versus atomic number of the elements for the K and L lines. It<br />
is an unfortunate fact that low atomic number elements also have low fluorescent yield.<br />
Figure I.12: The excitation energy from the inner atom is transferred to one of the outer<br />
electrons causing it to be ejected from the atom (Auger electron).<br />
Figure I.13: Fluorescent yield versus atomic number for K and L lines.
IV. X-<strong>Ray</strong> Production Sources<br />
IV.1 X-<strong>Ray</strong> Tubes<br />
A variety of radiation sources of sufficient energy, emitting ether particles, γ-rays, or Xrays,<br />
are potential candidates as sources for exciting the elements of interest in a sample to<br />
emit characteristic radiation. The use of sample excitation by electrons is used in electron<br />
probe micro-analysis (EPMA), and excitation by charged particles, like protons, is achieved in<br />
articles-induced X-ray emission (PIXE). Most XRF analyzers have an X-ray tube for sample<br />
excitation.<br />
All modern X-ray tubes owe their existence to Coolidge’s hot-cathode X-ray tube<br />
(Coolidge 1913). It consists essentially of a vacuum sealed glass tube containing a tungsten<br />
filament for the production for electrons, an anode and a beryllium window. From variety of<br />
modifications, two geometries have emerged as the most suitable for all practical purposes:<br />
the end-window tube and the side-window tube, both having their own merits and limitations.<br />
The general requirements are as follows:<br />
1. Sufficient photon flux over a wide spectral range, with increasing emphasis on the intensity<br />
of the low-energy continuum. The actual intense interest in low-Z element analysis<br />
certainly activated research in this direction.<br />
2. Good stability of the photon flux (< 0.1 % at least). Short-term stability is an absolute<br />
requirement for obtaining acceptable precision.<br />
3. Tunable tube potential allowing the creation of the most effective excitation conditions for<br />
each element, because the intensity of the analyte lines varies considerably with excitation<br />
conditions.<br />
4. Freedom from two many interfering lines from the characteristic spectrum of the anode<br />
(scatter peaks).<br />
All X-ray tubes work on the same principle: accelerating electrons in an electrical field<br />
and decelerated them in a suitable anode material. The region of the electron beam in which<br />
this takes place must be evacuated in order to prevent collisions with gas molecules. Hence<br />
there is a vacuum within housing. The X-rays escape from the housing at a special point that<br />
is particularly transparent with a thin beryllium window.<br />
An X-ray tube emits the characteristic radiation of the anode material, in addition to the<br />
Bremsstrahlung radiation; a typical spectrum obtained with an X-ray tube of Rh anode<br />
material is shown in Figure I.14.<br />
The main differences between tube types are in the polarity of the anode and cathode<br />
and the arrangement of the exit window.
Figure I.14: A Bremsstrahlung (Continuum) with characteristic radiation of the anode<br />
material (Rh as example).<br />
IV.1.1 Side-window Tubes<br />
In side-window tubes, a negative high voltage is applied to the cathode. The electrons<br />
emanate from the heated cathode and are accelerated in the direction of the anode. The anode<br />
is set on zero voltage and thus has no difference in potential to the surrounding housing<br />
material and the laterally mounted beryllium exit window (Figure I.15).<br />
Figure I.15: The principle of the side-window tube.<br />
For physical reasons, a proportion of the electrons are always scattered on the surface of<br />
the anode. The extent to which these back-scattering electrons arise depends, amongst other<br />
factors, on the anode material and can be as much as 40 %. In the side-window tube, these<br />
back-scattering electrons contributes to the heating up of the surrounding material, especially<br />
the exit window, the exit window must withstand high levels of thermal stress any cannot be<br />
selected with just any thickness. The minimum usable thickness of a beryllium window for<br />
side-window tubes is 300 µm. this causes an excessively high absorption of the low-energy<br />
characteristic L radiation of the anode material in the exit window and thus a restriction of the<br />
excitation of lighter elements in a sample.
IV.1.2 End-window Tubes<br />
The distinguishing feature of the end-window tubes is that the anode has a positive high<br />
voltage and the beryllium exit window is located on the front end of the housing (Figure I.16).<br />
Figure I.16: The principle of the end-window tube.<br />
The cathode is set around the anode in a ring (anular cathode) and is set at zero voltage.<br />
The electrons emanate from the heated cathode and are accelerated towards the electrical field<br />
lines on the anode. Due to the fact that there is a difference in potential between the positively<br />
charged anode the surrounding material, including the beryllium window, the back-scattering<br />
electrons are guided back to the anode and thus do not contribute to the rise in the exit<br />
window’s temperature. The beryllium window remains cold and can therefore be thinner than<br />
the side-window tube. Windows are used with a thickness of 125 mm and 75 mm. this<br />
provide a prerequisite for exciting light elements with the characteristic L radiation of the<br />
anode material (e.g. rhodium).<br />
Due to the high voltage applied, non-conductive, de-ionised water must be used for<br />
cooling. Instruments with end-window tubes are therefore equipped with a closed, internal<br />
circulation system containing de-ionised water that cools the tube head as well.<br />
End-window tubes have been implemented by all renowned manufacturers of<br />
wavelength dispersive X-ray fluorescence spectrometers since the early 80’s.<br />
IV.2 Radioisotope Sources<br />
Radioisotopes are commonly used because of their stability and small size when<br />
continuous and monochromatic sources are required. Safety regulations require that X-ray<br />
emission from these sources is limited to about 10 7 photons s -1 steradian -1 compared with 10 12<br />
or 10 13 photons for X-ray tubes; the difference is only partly compensated for by the small<br />
size of the source, which allows very compact source-specimen-detector assemblies to be<br />
constructed that are very convenient due to their portability. On the other hand, the low<br />
intensities preclude crystal dispersion so that these sources are used almost exclusively in<br />
energy dispersion techniques. Separation of analytical lines is sometimes done with selective
filters but more often with pulse height analysers in combination with high resolution Si(Li)<br />
semiconductor detectors.<br />
Radioisotope XRF systems are often tailored to a specific but limited range of<br />
applications. They are simpler and often considerably less expensive than analysis systems<br />
based on X-ray tubes, but these attributes are often gained at the expense and flexibility.<br />
Radioisotope excitation is preferred to X-ray tubes when simplicity, ruggedness, reliability<br />
and cost of equipment are important; when a minimum size, weight and power consumption<br />
are necessary; when a very constant and predictable X-ray output is required; and when the<br />
use of high energy X-rays is advantageous. Radioisotope systems, especially those involving<br />
scintillation or proportional detectors, must be carefully matched to the specific application.<br />
The activity of radioisotopes is specified in terms of the rate of disintegration of the<br />
radioactive atoms, i.e. decays per second or Becquerels (Bq) (the Becquerel replaces the non-<br />
SI unit, the Curie (Ci), which equals 3.7 10 7 Becquerels). The activity decreases with the time<br />
from Ao to A(t) after an elapsed time t:<br />
( t)<br />
= A exp(<br />
−0.<br />
693t<br />
/ T ) ,<br />
A o<br />
1/<br />
2<br />
where T1/2 is the half-life of the radioisotope. The source decays to half of its original<br />
emission rate after the time equal to its half-life has passed. The radioisotope source has<br />
usually to be replaced after several half-lives. Several sources are listed in Table I.2.<br />
Table I.2: Typical radioisotope sources used for XRF.<br />
Isotope Fe-55 Cm-244 Cd-109 Am-241 Co-57<br />
Energy (keV) 5.9 14.3 - 18.3 22.88 59.5 122<br />
Elements (K-lines) Al - V Ti - Br Fe - Mo Ru - Er Ba - U<br />
Elements (L-lines) Br - I I - Pb Yb - Pu None None<br />
An important property of a given radioisotope is the type of its decay and the spectrum<br />
of the electromagnetic radiation accompanying the nuclear disintegration. The basic<br />
radioactive decays are:<br />
1. α decay: when a radioactive nucleus emits a helium nucleus (α particle) consisting of two<br />
protons and two neutrons. The energy spectrum of alpha particles is linear because of the<br />
quantization of the energy levels of nuclei.<br />
2. β + decay: when one of the protons is transformed into a neutron, emitting a positron (β +<br />
particle) and a neutrino. The energy spectrum of positrons is also continuous.<br />
3. K capture: when a nucleus captures one of the K-shell electrons, the final result also being<br />
the proton-into-neutron transformations, as in the case of the β + decay.<br />
In addition to the above nuclear transformations (decays), resulting in transformations<br />
of original nuclei into nuclei of other elements, the following accompanying processes may<br />
also occur:<br />
1. Emission of gamma radiation: occurring when the resulting nucleus is not in its ground<br />
state. The existing energy surplus can be either emitted in the form of electromagnetic<br />
radiation or transferred to the atomic shell electrons (internal conversion). Sometimes a<br />
nucleus reaches its ground state through subsequent intermediate (compound) states. In<br />
such cases, every decay may be accompanied by several photons and/or internal conversion<br />
electrons, with energies being equal to the energy differences between the individual
compound states of the nucleus. An example for such a cascade transition to the ground<br />
state is given in Figure I.17.<br />
Figure I.17: Decay scheme showing the principal transitions in Am-241, Fe-55 and Cd-<br />
109.<br />
2. Internal conversion: when the excitation energy of the nucleus is given up to one of the<br />
atomic electrons, which is then ejected from the atom with the kinetic energy: Ee = En – Eb<br />
; where En is the excitation energy of the nucleus and Eb is the binding energy of the<br />
electron in a given atomic shell. The quantitative description of this phenomenon uses the<br />
concept of the internal conversion coefficient, defined as the ration of the number on<br />
internal conversion electrons to the number of gamma photons emitted during the same<br />
time interval. The internal conversion coefficient increases strongly as the atomic number<br />
increases and conversion is a competitive process with respect to the emission of gamma<br />
radiation, just as the Auger effect is competitive with respect to the emission of X-rays.<br />
3. Emission of X-rays: resulting from filling the holes in the atomic shells with electrons from<br />
higher levels. The holes in the atomic shells are due both to K-capture and to internal<br />
conversion.
SECTION II<br />
ENERGY DISPERSIVE X-RAY FLUORESCENCE<br />
(ED-XRF)<br />
I. Introduction<br />
In Energy Dispersive X-<strong>Ray</strong> <strong>Fluorescence</strong> spectrometry (ED-XRF), the identification of<br />
characteristic lines is performed using detectors that directly measure the energy of the<br />
photons. In the simplest case an electron is ejected from an atom of the detector material by<br />
photoabsorption. The loss of energy of this just created primary electron results in a shower of<br />
electron-ion pairs in the case of a proportional counter, optical excitations in the case of<br />
scintillation counter, or showers of electron-hole pairs in a semiconductor detector. The<br />
resulting detector signal is proportional to the energy of the incident photon, in contrast to<br />
wavelength dispersion in which the Bragg reflecting properties of a crystal are used to<br />
disperse X-rays at different reflection angles according to their wavelengths. Although energy<br />
dispersive detectors generally exhibit poorer energy resolution than wavelength dispersive<br />
analyzers, they are capable of detecting simultaneously a wide range of energies.<br />
The most frequently used detector in EDXRF is the silicon semiconductor detector,<br />
which nowadays can have excellent energy resolution. The two other types of detectors,<br />
mentioned above, with their poorer energy resolution are limited to special cases where<br />
certain features of semiconductors are not acceptable. Also the germanium semiconductor<br />
detector with its comparable characteristics has a major drawback for conventional XRF:<br />
inherently the escape peaks of intense lines can obscure other lines of interest.<br />
II. Instrumentation<br />
An ED-XRF system consists of several basic functional components, as shown in<br />
Figure II.1: an –ray excitation source, sample chamber, Si(Li) detector, preamplifier, main<br />
amplifier and mutlichannel pulse height analyzer. The properties and performances of an ED-<br />
XRF system differ upon the electronics and the enhancements from the computer.<br />
Figure II.1: Typical ED-XRF detection arrangement.
II.1 Excitation Mode<br />
II.1.1 Direct Tube Excitation<br />
Because of the simplicity of the instrument and the availability of a high photon output<br />
flux by using direct tube excitation, the X-ray fluorescence spectrometer equipped with an Xray<br />
tube as direct excitation source is gaining more and more attention from manufactures as<br />
well as from analytical chemists. The spectrometer is more compact and cheaper compared to<br />
secondary target systems. Of course, the drawback is still the less flexible selection of<br />
excitation energy. However, by using an appropriate filter between tube and sample, one can<br />
obtain an optimal excitation. The understanding of the process of continuum excitation and<br />
the possibility to obtain a good estimate of the continuum excitation spectrum originating<br />
from the tube has minimized the problems associated with quantization, so that very<br />
satisfactory quantitative analysis can be carried out. The most popular X-ray tube used in<br />
direct excitation ED spectrometer is the side window tube for reasons of simplicity and safety.<br />
With direct tube excitation, low powered X-ray tubes (< 100 W) can be used. These air cooled<br />
tubes are very compact, less expensive, and only require compact, light, inexpensive, highly<br />
regulated solid state power supplies. In a WD spectrometer, on the other hand, high-power<br />
tubes (3-4 kW) are essential to compensate for the losses in the crystal and collimator. With<br />
the low-power tubes used in ED spectrometer, better excitation of light elements (i.e. low-Z<br />
element), analysis of smaller samples, small spot analysis, and compact systems can be<br />
obtained. The use of X-ray tubes with a multi-element anode having a thin layer of low-Z<br />
element (e.g. Cr) sputtered onto a heavy element target (e.g. Mo) has been reported.<br />
Optimized excitation can be obtained by operating the multi-element anode tube at different<br />
voltages to switch between the excitation by the light element and the heavy element targets.<br />
II.1.2 Secondary Target Excitation<br />
The principle of secondary target excitation was developed to avoid the intense<br />
Bremsstrahlung continuum from the X-ray tube by using a target between tube and sample<br />
(Figure II.2).<br />
Figure II.2: Schematic illustration of secondary target excitation.<br />
The ratio of the intensity of the characteristic lines to that the continuum in secondary<br />
target excitation is much higher than that in direct tube excitation because the continuum part<br />
of the excitation spectrum of the secondary target is generated only by scattering. One can<br />
excite various elements efficiently by selecting a secondary target that has characteristic lines
just above the absorption edges of the elements of interest in the sample. Therefore, secondary<br />
target excitation has some obvious advantages over direct tube excitation: its flexibility for<br />
getting an optimized and near monochromatic excitation providing a better selectivity and an<br />
improved sensitivity. However, to compensate for the intensity losses that occur at the<br />
secondary scatterer, a high-powered tube as used in WD spectrometers is required; making<br />
the whole system more sophisticated and expensive compared to direct tube excitation setups.<br />
II.1.3 Radio-Isotopic Excitation<br />
Radio-isotopic sources are simple, cheap and quasi-monochromatic excitation sources.<br />
They are very suitable sources when combined with a solid state detector for in situ analysis<br />
(Figure II.3).<br />
Figure II.3: Geometry of an EDXRF spectrometer with annular source excitation.<br />
A variety of about 30 commercially available radio-isotopic materials can be chosen for<br />
an optimal excitation. The X-rays and/or γ-rays emitted from these radio-isotopic sources<br />
cover a wide range (10 – 60 keV) of excitation energies. With a high energy source like 241<br />
Am, K lines instead L lines can be used for quantification in the case of analyzing high-Z rare<br />
earth elements, with considerably less matrix effects and spectrum overlaps. Sometimes the<br />
same idea as in the secondary target excitation is used to avoid non-photon radiation. A<br />
proper design of excitation-detection geometry can improve greatly the sensitivity and<br />
accuracy of the XRF analysis with such excitation source. The disadvantages of using radioisotopic<br />
sources however lie in their low photon output, intensity decay and storage problems.<br />
II.2 Detectors<br />
The selective determination of elements in a mixture, using X-ray spectrometry,<br />
depends upon resolving the spectral lines emitted by the various elements into separate<br />
components. This process requires some form of energy sorting or wavelength dispersing<br />
device. In the case of wavelength dispersive X-ray spectrometers, this is accomplished by the<br />
analyzing crystal, which requires mechanical movement to select each desired wavelength<br />
according to Bragg’s Law. Optionally, several fixed-crystal channels may be used for<br />
simultaneous measurements. In contrast, energy dispersive X-ray spectrometry is based upon<br />
the ability of the detector to create signals proportional to the X-ray photon energy, therefore,<br />
mechanical devices, such as analyzing crystals, are not required. Several types of detectors<br />
have been employed, including silicon, germanium and mercuric iodide.<br />
The solid state, lithium-drifted silicon detector, Si(Li), was developed and applied to Xray<br />
detection in the 1960’s. By the early 1970’s, this detector was firmly established in the<br />
field of X-ray spectrometry, and was applied as an X-ray detection system for scanning
Electron Microscopy (SEM) as well as X-ray spectrometry. The principal advantage of the<br />
Si(Li) detector is its excellent resolution. Figure II.4 shows a diagram of a Si(Li) detector.<br />
Figure II.4: Cross section of an Si(Li) detector crystal with p-i-n structure and the<br />
production of electron-hole pair.<br />
Si(Li) detector can be considered as a layered structure in which a lithium-drifted active<br />
region separates a p-type entry side from an n-type side. Under reversed bias of approximately<br />
600 V, the active region acts as an insulator with an electric field gradient throughout its<br />
volume. When an X-ray photon enters the active region of the detector, photoionization<br />
occurs with an electron-hole pair created for each 3.8 eV of photon energy. Ideally, the<br />
detector should completely collect the charge created by each photon entry, and result in a<br />
response for only that energy. In reality, some background counts appear because of the<br />
energy loss in the detector. Although these are kept to a minimum by engineering, incomplete<br />
charge collection in the detector is a contributor to background counts. In the X-ray<br />
spectrometric, important region of 1 – 20 keV, silicon detectors have excellent efficiency for<br />
conversion of X-ray photon energy into charge. Some of the photon energy may be lost by<br />
photoelectric absorption of the incident X-ray, creating an excited Si atom which relaxes to<br />
yield an Si Kα X-ray. This X-ray may escape from the detector, resulting in an energy loss<br />
equivalent to the photon energy; in the case of Si Kα, this is 1.74 keV. Therefore, an escape<br />
peak 1.74 keV lower than the true photon energy of the detected X-ray may be observed for<br />
intense peaks. For Si(Li) detectors, these are usually a few tenths of one percent, and never<br />
more than 2%, of the intensity of the main peak. The escape peak intensity relative to the<br />
main peak is energy dependent, but not count rate dependent. For precise quantitative<br />
determinations, the spectroscopist must be aware of the possibility of interference by escape<br />
peaks.<br />
Resolution of an energy dispersive X-ray spectrometer is normally expressed as the Full<br />
Width at Half Maximum amplitude (FWHM) of the Mn X-ray at 5.9 keV. The resolution will<br />
be somewhat count rate dependent. Commercial spectrometers are supplied routinely with<br />
detectors which display approximately 145 eV (FWHM @ 5.9 keV). The resolution of the<br />
system is a result of both electronic noise and statistical variations in conversion of the photon
energy. Electronic noise is minimized by cooling the detector, and the associated preamplifier,<br />
with liquid nitrogen (Figure II.5). In many cases, half of the peak width is a result of<br />
electronic noise.<br />
II.3 Pulse Height Analysis<br />
Figure II.5: The Si(Li) detector schematic.<br />
The X-ray spectrum of the sample is obtained by processing the energy distribution of<br />
X-ray photons which enter the detector. A single event of one X-ray photon entering the<br />
detector causes photoionization and produces a charge proportional to the photon energy.<br />
Numerous electrical sequences must take place before this charge can be converted to a data<br />
point in the spectrum. It is not necessary for the spectroscopist to have a detailed knowledge<br />
of the electronics; however, it is important to have an understanding of their functional use.<br />
When an X-ray photons enters the Si(Li) detector, it is converted into an electrical<br />
charge which is coupled to a Field Effect Transistor (FET). The FET, and the rest of the<br />
associated electronics which make up the preamplifier, produce an output proportional to the<br />
energy of the X-ray photon. Using a pulsed optical preamplifier, this output is in the form of a<br />
step signal. Because photons vary in both energy and number per unit time, the output signal,<br />
due to successive photons being emitted by a multielement sample, resembles a staircase with<br />
various step heights and time spacing. When the output reaches a predetermined level, the<br />
detector and the FET circuitry is reset to its starting level, and the process repeated.<br />
The preamplifier stage integrates each detector charge signal to generate a voltage step<br />
proportional to the charge. This is then amplified and shaped in a series of integrating and<br />
differentiating stages. Owing to the finite pulse-shaping time, in the range of microseconds,<br />
the system will not accept any other incoming signals in the meanwhile (dead time), but<br />
extend its measuring time instead. In a further step the height of these signals is digitized as a<br />
channel number (analog-to-digital converter, ADC), stored to a memory (multichannel<br />
analyzed, MCA) and finally displayed as a spectrum, where the number of counts reflects the<br />
respective intensity. In a more modern approach, the output signals of the preamplifier are<br />
digitized directly, which can increase the throughput of the system significantly.<br />
For high count rates there is an increasing probability that two photons of, for example,<br />
a very intense line, are absorbed in the detector crystal within such a short time interval that<br />
their charges are not collected as two individual signals with a certain energy, but rather as a<br />
single signal with twice the energy (sum peak).
II.4 Energy Resolution<br />
The energy resolution of the EDXRF spectrometer determines the ability of a given<br />
system to resolve characteristic X-rays from multiple-element samples and is normally<br />
defined as the full width at half maximum (FWHM) of the pulse-height distribution measured<br />
for a monoenergetic X-ray. A conventional choise of X-ray energy is 5.9 keV, corresponding<br />
to the Kα energy of Mn. Figure II.6 shows a typical pulse-height spectrum of Mn-Kα X-rays<br />
simultaneously with a calibrated pulser. The purpose of the pulser measurement is to monitor<br />
the resolution of the electronic system independent of any peak broadening due to the detector<br />
itself. Typical state-of the art detectors Si(Li) and Ge(HP) achieve 130 to 170 eV, but depends<br />
strongly on the size of the crystal. The smaller the crystal, the better is the resolution.<br />
Figure II.6: Mn-Kα spectrum and calibrated pulser.<br />
The instrumental energy resolution of a semiconductor detector spectrometer is a<br />
function of 2 independent factors:<br />
( E ) ( E ) ( E )<br />
2 2 2<br />
∆ total = ∆ det + ∆ elec . (II.1)<br />
The FWHM of the X-ray line (∆Etotal) is described as the convolution of a contribution<br />
due to the detector processes (∆Edet), which is determined by the statistics of the free charge<br />
production processes together with a component associated with limitations in the electronic<br />
pulse processing (∆Eelec).<br />
The average number of electron hole pairs produced by an incident photon can be<br />
calculated as the photon energy divided by the mean energy required for the production of a<br />
single electron-hole pair. If the fluctuation in this average were governed by Poisson statistics,<br />
the variance would be n . In semiconductor devices the details of the energy loss process are<br />
such that the individual events are not strictly independent and a departure from Poisson<br />
behaviour is observed. This is considered by the addition of the FANO-Factor in the<br />
expression for the detector contribution to the FWHM:<br />
( ) ( 2.35)<br />
2 2<br />
∆ Edet = Eε F ; (II.2)
E is the photon energy, ε is the average energy required to produce a free electron-hole pair, F<br />
is the FANO factor and 2.35 converts the root means square deviation to FWHM. For an<br />
equivalent energy, the detector contribution to the resolution is 28 % less for the case of Ge<br />
compared to Si.<br />
The contribution to resolution associated with electronic noise (∆Eelec) is the result of<br />
random fluctuations in thermally generated leakage currents within the detector and in the<br />
early stages of the amplifier components.<br />
III. Spectrum Evaluation<br />
Spectrum evaluation in energy dispersive XRF is certainly more critical than in WD-<br />
XRF, because of the relatively low resolution of the solid-state detectors employed. The aim<br />
is the extraction of the analytically relevant information (net number of counts under a peak)<br />
from experimental spectra.<br />
In EDXRF, the characteristic radiation of a particular line can be described in an<br />
adequate first-order approximation by a Gaussian function (detector response function). The<br />
spectral background results a variety of processes: for photon excitation, the main<br />
contribution is the incoherently scattered primary radiation and therefore depends on the<br />
shape of the excitation spectrum and on the sample composition. For particle-induced X-ray<br />
emission and electron excitation, the background observed is mainly due to Bremsstrahlung.<br />
The most straightforward method to obtain the net data area under a line of interest<br />
consists of interpolating the background under the peak and summing the backgroundcorrected<br />
channel contents in a window over the peak. In practice, this approach is limited by<br />
the curvature of the background and by the presence of other peaks and can therefore not be<br />
used as a general tool for spectrum processing in EDXRF. An example of overlapping peaks<br />
is the analysis of lead and arsenic simultaneously present in a sample (Figure II.7).<br />
Figure II.7: A spectrum of As K, overlapped with a Pb L line spectrum, both excited by<br />
a Mo X-ray tube, under identical conditions. The energy of AsKα1,2 (10.53<br />
keV) and Pb Lα1,2 (10.55 keV) cannot be separated by an Si(Li) detector.
A widely used method is non-linear least squares fitting of the spectral data with an<br />
analytical function. This algebraic function, including all important parameters, such as the<br />
net areas of the fluorescent lines, their energy, resolution, etc., is used as a model for the<br />
measured spectrum. It will consist of the contribution from all peaks (modified Gaussian<br />
peaks, with corrections for low-energy tailing, escape peaks, etc.) within a certain region of<br />
interest and the background (described by, for example, linear or exponential polynomials).<br />
The optimum values of the parameters are those for which the difference between the model<br />
and the measured spectrum is minimal. Unfortunately some of these parameters are nonlinear,<br />
which places some importance on the minimization procedure (usually the Marquardt<br />
algorithm is used).<br />
In another frequently used approach the discrete deconvolutions of a spectrum with a<br />
so-called top-hat filter suppresses the low-frequency component, i.e. the slowly varying<br />
background. A severe distortion of the peaks is introduced. But applying this filter to both the<br />
unknown spectrum and well defined, experimentally obtained, reference spectra, a multiple<br />
linear least-squares fitting to the filtered spectra will result in the net peak areas of interest. A<br />
disadvantage of this method is that reference and unknown spectra should be acquired under<br />
preferably identical conditions; especially, energy calibration changes of<br />
more than only few e can generate large systematic errors.<br />
IV. Detector Artefacts<br />
IV.1 Escape Peaks<br />
A photon is detected in the Si(Li) diode primarily by ionizing the K-shell of a Si atom<br />
besides the interaction of elastic and inelastic scattering. Subsequently a Si-Kα photon is<br />
produced. If the Si-Kα is absorbed within the active volume of the detector, the resulting<br />
amplitude pulse will have the amplitude which is proportional to the original photon energy.<br />
If the Si-Kα photon escapes the active volume of the detector, the event will be recorded at an<br />
energy which is too low by an amount equal to the energy of the escaping photon. Thus, a<br />
peak with an energy E-E(Si-Kα) will occur in the spectrum. Figure II.8 shows this<br />
schematically.
IV.2 Compton Edge<br />
Figure II.8: Escape effect.<br />
At low energy of the spectrum lies the Compton shoulder. This rise in the background is<br />
caused by high-energy photons incoherently scattered from the front side of the detector<br />
crystal, leaving only a small fraction of their energy with the recoiled Compton electron in the<br />
detector (Figure II.9). The energy at which the Compton edge occurs is given by the formular<br />
beneath. Both the detector resolution and multiple scattering tend to smear out this sharp<br />
edge.<br />
IV.3 Resulting Spectral Background<br />
Figure II.9: Compton edge.<br />
Figure II.10 shows the spectrum obtained by monochromatic excitation of 17.5 keV and<br />
a Si(Li) detector. The width of the coherent scatter peak reflects the detector resolution at 7.4<br />
keV. The incoherent peak is much broader due to the range of scattering angles included<br />
about the nominal 90° scattering angle. The low energy tail on the incoherent peak extending<br />
down to about 10 keV is primarily due to multiple Compton scattering in the specimen.<br />
The major background represented by the cross-hatched area, is due to incomplete<br />
charge collection in the Si(Li) detector. This occurs when a portion of the positive and<br />
negative charges produced in the detector by the 16.8 and 17.4 keV photons recombine before<br />
they are collected. The result is a pulse of abnormally low amplitude recorded at a lower than<br />
normal energy. The intensity of background due to incomplete charge collection is a function<br />
of detector quality and X-ray energy.
Figure II.10: Background contribution in an EDX spectrometer with monochromatic<br />
17.5 keV excitation.<br />
V. The Approach to Quantification in EDXRF Analysis<br />
The approach to quantification in EDXRF analysis is usually different for thin,<br />
intermediate thickness and infinitely thick samples.<br />
V.1 Thin Samples Technique<br />
If a homogeneous sample to be analysed has a very small mass per unit area (or<br />
thickness), the detected intensity of characteristic X-rays, Ithin, of the ith element is simply<br />
given by:<br />
with<br />
and<br />
Ithin = Sm i i , (II.3)<br />
⎛ 1 ⎞<br />
G<br />
⎜ ⎟<br />
Si = Io( Eo) ε( Ei) τi( Eo) ωipi 1<br />
sin φ ⎜ ⎟<br />
⎜ j ⎟<br />
⎝ i ⎠<br />
, (II.4)<br />
mi =µ i m.<br />
(II.5)<br />
Where G is the geometry factor; φ is the effective incidence angle for primary radiation; Io(Eo)<br />
is the intensity of primary photons of energy Eo (monochromatic excitation), ε(Ei) is the<br />
detector efficiency for recording the photons of energy Ei; τi(Eo) is the photoelectric mass<br />
absorption coefficient for the ith element at the energy o, in cm 2 .g -1 ; ji is the jump ratio; mi<br />
and µi are the mass per unit area and the weight fraction of the ith element, respectively; and<br />
m is the total mass per unit area of a given sample.
The relative error resulting from applying equation (II.3) instead of the exact equation<br />
does not exceed 5% when the total mass per unit area is lower than:<br />
0.1<br />
( ) ( )<br />
µ E cos ecφ+µ E cos ecψ<br />
o i<br />
; (II.6)<br />
where µ(Eo) and µ(Ei) are the total mass attenuation coefficients for the whole specimen at the<br />
energy of primary radiation (Eo) and the energy of characteristic X-rays of the ith element<br />
(Ei), respectively; φ is the effective angle of incidence of the primary exciting beam; and ψ is<br />
the effective take-off angle of characteristic X-rays. The total mass attenuation coefficient<br />
µ(E) for the whole specimen at the energy E is given by the mixture rule:<br />
n<br />
µ ( E) = ∑ Wj µ j ( E)<br />
, (II.7)<br />
j=<br />
1<br />
where Wj and µj(E) are the weight fraction and the mass attenuation coefficient of the jth<br />
element present in the sample, respectively, and n is the total number of the elements in the<br />
sample. A major feature of the thin sample technique is that the intensity of characteristic Xrays,<br />
Ithin, depends linearly on the concentration of the ith element; it is equivalent to the fact<br />
that the so-called matrix effects can safely be neglected.<br />
The values of the constant Si (called the sensitivity factors), which are necessary to<br />
convert the measured intensity of the characteristic X-rays into mass concentrations, can be<br />
determined either experimentally as the slope of the straight calibration line for the ith<br />
element obtained on the basis of thin homogeneous standard samples or semi-empirically<br />
based on both the experimentally determined (G/sin φ)IoEo value and the relevant fundamental<br />
parameters (τi(Eo), ωi, ρi and ji). Also the detector efficiency ε(Ei) can b determined either<br />
experimentally or theoretically based on the parameters of a given detector. In multi-element<br />
XRF analysis, the calibration process can be greatly simplified because the elemental<br />
sensitivities Si vary as a smooth function with atomic number.<br />
Various homogeneous standard samples are now commercially available from several<br />
manufactures. In many cases, one can also produce synthetic laboratory standard according to<br />
the actual needs and possibilities; for example, by precipitating known quantities of elements<br />
in solution, and filtering off as a thin layer or on a filter membrane.<br />
V.2 Intermediate Thickness Samples<br />
Intermediate thickness samples are defined as those samples whose masses per unit area<br />
fulfil following inequality:<br />
mthin < m< mthick<br />
, (II.8)<br />
where mthick is the mass of the so-called infinitely thick or saturated sample, above which<br />
practically no further increase in the intensity of the characteristic radiation will be observed<br />
as the sample thickness is increased, given by:<br />
mthick<br />
4.61<br />
= . (II.9)<br />
µ E cos ecφ+µ E cos ecψ<br />
( ) ( )<br />
o i<br />
Intermediate thickness samples can be preferable to thick specimens because less<br />
material is required, remaining uncertainties in the knowledge of the mass attenuation<br />
coefficients have a smaller effect on the analysis results, the sensitivity is more favourable for<br />
low-Z elements, and secondary enhancement effects are less important. In practice, samples<br />
of intermediate thickness are used when the investigated material is scarce and does not allow
preparation of a thick sample, and when preparation of a thin sample is difficult or even<br />
impossible. Such cases might occur in the analysis of biological and environmental<br />
specimens.<br />
In recent years a number of approaches have been developed for quantitation in XRF<br />
analysis of intermediate thickness samples. Some of them are based on the emissiontransmission<br />
(E-T) method. The original version of the E-T method requires the<br />
measurements of the specific X-ray intensities from the sample alone and from the sample<br />
and a certain target positioned behind it in a fixed geometry. Alternative correction<br />
procedures are based on the use of scattered primary radiation which suffers similar matrix<br />
absorption as fluorescent peaks and behaves similarly with instrumental variations. The<br />
scattered radiation peaks also provide the only direct spectral measure of the total or average<br />
matrix of the analysed materials when these contain large quantities of light elements such as<br />
carbon, nitrogen and oxygen, usually observed by their characteristic X-ray peaks.<br />
For homogeneous, intermediate thickness samples, the mass per unit area of the element<br />
i, mi can be calculated from the following equation:<br />
Ii<br />
mit Si<br />
= , (II.10)<br />
where Ii is the measured intensity of the characteristic X-rays of the ith element and t is the<br />
absorption factor, given by:<br />
1−exp { − [ µ ( Eo) cosecφ+µ ( Ei) cos ecψ] m}<br />
t =<br />
. (II.11)<br />
⎡µ ( E ) cos ecφ+µ ( E ) cos ecψ⎤m ⎣ o i ⎦<br />
In the E-T method, the t-factor, representing the combined attenuation of both the<br />
primary and fluorescent radiations in the whole specimen, is determined individually for each<br />
sample. This is done by measuring the X-ray intensities with and without the specimen from a<br />
thick multi-element target located at a position adjacent to the back of the specimen, as shown<br />
in Figure II.11. If (Ii)S, (Ii)T and (Ii)o are the intensities, after background correction, from the<br />
sample alone, from the sample plus target and from the target alone, respectively, then the<br />
combined fraction of the exciting and fluorescent radiations transmitted through the total<br />
sample thickness is expressed by:<br />
( Ii) − ( I<br />
T i)<br />
S<br />
exp{ − ⎡<br />
⎣µ ( Eo) cos ecφ+µ ( Ei) cos ecψ ⎤<br />
⎦ m} = = T.<br />
(II.12)<br />
( I )<br />
Since the parameter T is determined experimentally, the following equation for the tfactor<br />
is obtained:<br />
1 − T<br />
t =<br />
−lnT<br />
. (II.13)<br />
It is necessary to emphasize that the E-T method can only be applied in quantitative<br />
XRF analysis of homogeneous samples of masses per unit area smaller than the critical value<br />
mcrit, defined as:<br />
−lnTcrit<br />
mcrit<br />
= , (II.14)<br />
µ ( Eo) cos ecφ+µ ( Ei) cos ecψ<br />
where Tcrit is the critical transmission factor, Equation (II.12) is equal, in practice, to 0.1 or<br />
0.05. A number of more complex and more versatile versions of the E-T method have been<br />
developed.<br />
i o
Figure II.11: Schematic diagram of experimental procedure used in the emissiontransmission<br />
method.<br />
V.3 Infinitely Thick Samples<br />
Samples exceeding the thickness mthick given in Equation (II.9) can be considered<br />
‘infinitely thick’ or ‘saturated’ with respect to X-ray absorption. In this case, the exponential<br />
term in the nominator of the X-ray absorption factor given by Equation (II.11) can be<br />
neglected, giving:<br />
1<br />
t =<br />
⎡<br />
⎣µ ( Eo) cos ecφ+µ ( Ei) cos ecψ⎤ ⎦m<br />
and Equation (II.10) can be written as:<br />
, (II.15)<br />
Sm i i SC i i<br />
mi<br />
= =<br />
⎡<br />
⎣µ ( Eo) cos ecφ+µ ( Eo) cos ecψ⎤ ⎦m<br />
µ ( Eo) cos ecφ+µ ( Eo) cos ecψ<br />
. (II.16)<br />
Hence, the characteristic X-ray intensity is directly proportional to the elemental<br />
concentration and not dependent on the sample thickness. Thus, knowledge of the sample<br />
thickness is no longer relevant. Various ways have been developed to cope with matrix effects<br />
in infinitely thick samples.
I. Introduction<br />
SECTION III<br />
Total Reflexion X-<strong>Ray</strong> <strong>Fluorescence</strong><br />
(TXRF)<br />
The phenomenon of total reflection of X-rays had been discovered by Compton (1923).<br />
He found that the reflectivity of a flat target strongly increased below a critical angle of only<br />
0.1°. In 1971, Yoneda and Horiuchi (1971) first took advantage to this effect for X-ray<br />
fluorescence (XRF). They proposed the analysis of a small amount of material deposited on a<br />
flat totally reflecting support. This idea was subsequently implemented in the so-called total<br />
reflection X-ray fluorescence (TXRF) analysis which has spread out worldwide. It is now<br />
recognized analytical tool with high sensitivity and low detection limits, down to the<br />
femtogram range.<br />
Total reflection X-ray fluorescence (TXRF) has become increasingly popular in micro<br />
and trace elemental analysis. It is being used in geology, biology, materials science, medicine,<br />
forensics, archaeology, art history, and more. Unlike the high incident angles (~ 40 °) used in<br />
traditional XRF, TXRF involves very low incident angles. These low angles allow the X-rays<br />
to undergo total reflection. This minimizes the adsorption of the X-rays and greatly enhances<br />
the lower limits of detection. The fluorescent X-rays illuminating from the sample are then<br />
discriminated using an energy dispersive detector.<br />
II. Advantages of TXRF<br />
• Background reduced.<br />
• Double excitation of sample by both the primary and the reflected beam.<br />
• No matrix effects.<br />
• A single internal standard greatly simplifies quantitative analyses.<br />
• Calibration and quantification independent from any sample matrix.<br />
• Simultaneous multi-element ultra-trace analysis.<br />
• Several different sample types and applications.<br />
• Minimal quantity of sample required for the measurement (5 ml).<br />
• Unique microanalytical applications for liquid and solid samples.<br />
• Excellent detection limits (ppt or pg) for all elements from sodium to plutonium.<br />
• Excellent dynamic range from ppt to percent.<br />
• Possibility to analyse the sample directly without chemical pretreatment.<br />
• No memory effects.<br />
• Non destructive analysis.<br />
• Low running cost.<br />
The background is reduced because most of the incident beam is reflected, only a small<br />
part (described by the transmission coefficient T = 1 – R, R is the Reflection coefficient)<br />
penetrates into the reflector causing background. The line intensity is enhanced by about a
factor of 2, because also the reflected beam contributes to sample excitation. Figure III.1<br />
shows both effects as function of the angle of incidence.<br />
Figure III.1: Effect on spectral line and background of total reflection.<br />
III. Principle of Total Reflection X-<strong>Ray</strong> <strong>Fluorescence</strong> Analysis<br />
Total reflection X-ray fluorescence analysis (TXRF) is basically an energy dispersive<br />
analytical technique in special excitation geometry (Figure III.2). This geometry is achieved<br />
by adjusting the sample carrier, not inclined under 45° to the incident beam, as for standard<br />
EDXRF, but with angles of about 1 mrad (0.06°) to the primary beam. The incident beam thus<br />
impinges at angles below the critical angle of (external) total reflection for X-rays onto the<br />
surface of a plane smooth polished reflector.<br />
Figure III.2: Scheme of total reflection X-ray fluorescence (TXRF).<br />
Usually a liquid sample, with a volume of only 1 – 100 µL, is pipetted in the center of<br />
this surface and the droplet will cover an area of a few millimetres in diameter. As result of<br />
the drying process where the liquid part of the sample is evaporated, the residual is irregularly<br />
distributed on the reflector (within the above stated diameter), forming a very thin sample.<br />
The simplified equation (valid above the highest K absorption edge of the reflector<br />
material) for the critical angle of total reflection ϕcrit (in mrad) depends on the energy E (in<br />
keV) of the incident photons and the density ρ (in g/cm 3 ) of the reflector material:
20.<br />
3<br />
ϕcrit<br />
= ρ . (III.1)<br />
E<br />
For example, for incident Mo Kα (17.5 keV) radiation and quartz glass as reflector, the<br />
critical angle calculates as 1.7 mrad (= 0.1°).<br />
The preferred types of samples are either aqueous or acidic solutions (Figure III.3).<br />
With special sample preparation techniques, the pg/g concentration level can be reached.<br />
There are no corrections for absorption or secondary excitation necessary due to the sample<br />
formation in a very thin layer. In any case the addition of an internal standard of known<br />
concentration is essential for the quantification (typical elements, preferably not present in the<br />
sample are Co, Ga, Ge, Y …). Rewardingly, the calibration curves are linear over several<br />
orders of magnitude and therefore the calculations for converting the measured intensities to<br />
concentrations are simple and can be based on experimentally or theoretically determined<br />
relative sensitivity curves Srel(Z) as a function of the atomic number Z for all elements in<br />
respect to the internal standard element. The concentration wi of an element i can be<br />
calculated by:<br />
n 1<br />
w = w . (III.2)<br />
i<br />
i<br />
n st Srel<br />
st<br />
Note that nst and wst are the intensity and the concentration of the internal standard element.<br />
Figure III.3: Spectrum of a 3 µL mineral water sample, spiked with 1 ng/µL Ga as<br />
internal standard element. Excitation in TXRF geometry with a multilayer<br />
monochromator by a Mo X-ray tube (50 kV, 10 mA, 1000 s measuring time).<br />
The angular dependence of intensities in the regime of total reflection can be used to<br />
investigate surface impurities, thin near-surface layers, and even molecules absorbed on flat<br />
surfaces. From these angle-dependent intensity profiles the composition, thickness and<br />
density of layers can be obtained. It is the low penetration depth of the primary beam at total<br />
reflection that enables also the non-destructive in-depth examination of concentration profiles<br />
in the range of 1 – 500 nm.
IV. Instrumentation<br />
The major components of a TXRF spectrometer are shown in Figure III.4.<br />
Figure III.4: Major components of a TXRF spectrometer.<br />
IV.1 Excitation Sources for TXRF<br />
The usual excitation source for TXRF is a high power diffraction X-ray tube with a Mo<br />
anode with an electrical power of 2 – 3 kW. This type of X-ray tube is also available with Cr,<br />
Cu, Ag and W targets. The line focus of the anode has to be used so that the emitted brilliance<br />
is in correlation with the slit collimation necessary to produce a narrow beam with the<br />
divergence less than the critical angles involved. A higher photon flux on the sample can be<br />
achieved by using rotating anodes, which can stand up to 18 kW. In all cases, the focal size of<br />
the electron beam on the anode is a line with the dimensions of 0.4 × 8 mm 2 (fine focus) or<br />
0.4 × 12 mm 2 (long fine focus). The emission of the X-rays is observed under the angle of 6°<br />
to the anode surface, so that the width of the focus is reduced optically by the projection with<br />
sin6° (= 0.1) to 0.04 mm.<br />
The emitted spectrum consists of the continuum (Bremsstrahlung) and superimposed are<br />
the characteristic lines of the anode material (e.g, Mo Kα and Mo Kβ) (see Figure III.5).
Figure III.5: Measured primary spectrum of a fine-focus Mo diffraction X-ray tube (45<br />
kV acceleration voltage) as typically used for TXRF. The characteristic Mo<br />
Kα and Mo Kβ lines are superimposed on the Bremsstrahlung background.<br />
Monochromators also can modify the primary radiation and they are usually set to the<br />
angry of the most intense characteristic line of the anode material. For a Mo-anode X-ray tube<br />
Mo Kα or for a W-anode W-Lβ are selected, but a part of the continuum can be<br />
monochromatized as well. Commonly used crystal monochromators have the disadvantage of<br />
a very narrow energy band transmitted (usually in the range of few electron volts), whereas<br />
synthetic multilayer structures are characterized by higher ∆E/E and reflectivities of up to 75<br />
% for premium quality materials.<br />
IV.2 Sample Reflectors<br />
For the trace analysis of granular residues, a carrier with high reflectivity that serves as<br />
a totally reflecting sample support is required. Therefore, the mean roughness should be in the<br />
range of only a few nanometers and the overall flatness should be typically be less than λ/20<br />
(λ = 589 nm, the mean wavelength of the visible light). Furthermore, reflectors should be free<br />
of impurities so that the black spectrum should be free from contamination peaks and the<br />
carrier material must not have fluorescence peaks in the spectral region of interest. In<br />
addition, the carrier material must be chemically inert (also against strong chemicals, which<br />
are often used for the sample preparation), easy to be cleaned for repeated use.<br />
Some of the reflector materials that are in use are: quartz glass (most common), silicon,<br />
germanium, glassy carbon, niobium, boron nitride and (as cheapest material) Plexiglas. The<br />
requirements for the reflector are: no interfering fluorescence or diffraction lines, high purity,<br />
chemical resistance, hardness, machineability for polishing and an acceptable price. The<br />
surface must be flat and the mean roughness in the range of nanometers. Usually, they are<br />
disk shaped, with 30 mm diameter and 3 – 5 mm thickness, but also squares of 30 mm side<br />
length and rectangular types are in use.<br />
IV.3 Detectors<br />
Total reflection XRF is an energy dispersive XRF method, the radiation is measured<br />
mainly by Si(Li) detectors. A good detector offers a high energy resolution [Full Width at<br />
Half Maximum (FWHM) in the range of 140 eV at 5.89 keV], intrinsic efficiency close to 1<br />
for the X-ray lines of interest, symmetric peak shapes, and low contribution to the<br />
background. Primarily, incomplete charge collection at the electrodes leads to low-energy<br />
tailing. The detector escape effect creates escape peaks and thus an increased background in<br />
certain spectral regions. An inherent advantage of semiconductor detectors is the possibility of<br />
bringing the detector crystal very close to the sample, which results in a large solid angle.<br />
Light elements emit fluorescent lines in the range from 100 to 1000 eV. The usually used Be<br />
entrance window would completely absorb them, so new window materials, offering better<br />
transmission characteristics, are used instead.<br />
V. Quantification
One of the inherent advantages in TXRF is that one deals with thin samples so the<br />
simple conversion of fluorescent intensities I into concentration data C is applicable, as there<br />
is a linear correlation between I and C. After establishing a calibration curve either from<br />
known multielement standards or by using the fundamental parameters to calculate the<br />
calibration curve theoretically, the conversion of I into C can be immediately performed. The<br />
addition of one element as internal standard of known concentration into the sample is<br />
recommended to improve the accuracy of the results, because in this case geometric and<br />
volumetric errors will cancel. The simple relation to calculate the concentration of the<br />
unknown is given:<br />
I S<br />
= C ; (III.3)<br />
x std<br />
Cx ⋅ ⋅<br />
Sx<br />
Istd<br />
std<br />
Cx: Concentration of unknown, Cstd: Concentration of Standard,<br />
Ix: Intensity of standard, Istd: Intensity of Standard,<br />
Sx: Sensitivity of unknown, Sst: Sensitivity of Standard.<br />
A sample is “thin” if its thickness does not exceed the critical thickness, which about 4<br />
µm for organic tissue, 0.7 µm for mineral powders, and 0.01 µm for metallic smears. Under<br />
the assumption that the matrix absorption for the analyte differs only slightly from that of the<br />
internal standard element, these values can be generally be higher by a factor of 40 – 400. For<br />
the calculation of these values, the standing-wave field was not taken into account. This effect<br />
and the sample self-absorption can lead to contradictory requirements for the sample<br />
thickness.<br />
VI. Influence on Detection Limits<br />
The advantages of excitation in total reflection geometry are listed:<br />
1. Efficient excitation by both, the primary and the reflected beam- the fluorescent signal is<br />
doubled compared to standard excitation geometries 45° incident - 45° emission angle.<br />
2. The spectral background caused by scattering on the substrate is reduced because the<br />
primary radiation scarcely penetrates into the reflector substrate (high reflectivity, low<br />
transmission into the material). The scatter contribution from the sample itself is a<br />
minimum because of the 90 degree condition between incident and scattered radiation<br />
towards the detector.<br />
3. The detector is mounted closely to the sample of small amounts are required. The samples<br />
must be prepared in aware that thin film approximation is applicable. Therefore no<br />
absorption occurs and a linear correlation between intensity and concentration of the<br />
element is valid.<br />
4. Simultaneous multielement determination is possible due to the use of energy dispersive<br />
detectors.<br />
Due to the argument 1 and 2 automatically the peak to background ratio is increased<br />
compared to standard XRF.<br />
Improvements in the detection limits can be expected if the physical parameters<br />
influencing the minimum detection limits are optimised. The generally accepted definition is:
3 IB<br />
m min = ⋅ ; (III.4)<br />
S t<br />
S: sensitivity (cps/ng);<br />
Ib: background intensity (cps);<br />
t: live time (s).<br />
The sensitivity depends upon:<br />
1. The intensity, respectively the brightness of the primary radiation;<br />
2. The distances source to sample and sample to detector;<br />
3. The detector active area;<br />
4. The electronics in use to process the incoming countrate;<br />
5. The insertion devices in the primary beam path to modify the spectral distribution of the<br />
exciting radiation, improving the background but reducing the primary intensity.<br />
The background intensity depends upon:<br />
1. The primary intensity, its spectral distribution and the respective scattering cross<br />
sections for the sample;<br />
2. The sample mass (for thin film samples);<br />
3. The geometric form and the material of the substrate and the reflection coefficient<br />
which is practically 1.<br />
4. Measuring in air leads to scatter of the primary radiation and contribution from<br />
characteristic radiation of air (Ar and Kr);<br />
5. The actual adjusted angle of incidence;<br />
6. Solid angle of the detector;<br />
The live time t should be chosen according to practical and economical reasons.<br />
From the physical point of view, the sensitivity S can be influenced drastically by the<br />
proper choice of the source and its intensity, energy and beam size.<br />
VII. General Sample Preparation<br />
The detection limits obtained for a special sample depend very much on the sample<br />
preparation. Figure III.6 gives an overview of various common methods for sample<br />
preparation in TXRF, depending on the kind of sample to be analysed. Of course, one has to<br />
be aware that sample preparation can cause loss of elements as well as contamination by other<br />
elements and the sample taken for analysis must represent the whole specimen; therefore,<br />
homogenisation might be required.
Figure III.6: Sample preparation methods in TXRF.<br />
Solid samples can be crushed and then ground to a fine powder of micrometer grain<br />
size. This powder can be mixed with a liquid to produce a suspension, which can be pipetted<br />
after adding an internal standard on the sample reflector. The pulverized sample can also be<br />
dissolved in a suitable solvent, and after adding the internal standard, an aliquot is pipetted on<br />
the sample reflector and dried (Figure III.7).
Figure III.7: Preparation steps for the TXRF analysis of liquids.<br />
For the decomposition of biological and environmental materials, various methods have<br />
been utilized (e.g. with a low-temperature oxygen plasma asher, followed by dissolving the<br />
ash in an acid). The most popular method of decomposition of biological and environmental<br />
samples like plants, tissue, sediments, and so forth is the wet digestion in Teflon vessels<br />
(Teflon bombs) with acids like HNO3, HF, HNO3+HCl, HNO3+H2O2; and so forth, in<br />
different proportions. Using the hydrofluoric acid might be a problem if quartz glass reflectors<br />
are used. The use of microwave oven for heating the Teflon bomb reduces the time of<br />
digestion to less than 1 h.<br />
The volume of some sample solutions or any sample containing water can be reduced<br />
by freeze-drying. The sample is frozen and the solvent is evaporated under vacuum<br />
conditions. The dried residue can be dissolved in small volume of acid or wet digested.<br />
It is also possible to extract traces of certain elements by phase separation. To a given<br />
volume of sample water solutions at appropriate pH and spiked with internal standard, an<br />
organic solvent is added and mixed thoroughly. Then, the two phases are separated. The<br />
traces of metal ions stay in the organic phase, whereas the matrix elements are left in the<br />
inorganic solution. The organic liquid can be directly pipetted onto the reflector. Also, the<br />
separation of traces by adding a chelating agent and precipitating the metal ions is commonly<br />
used technique. The metal complexes are filtered through a membrane filter and dissolved in a<br />
suitable organic solvent.<br />
VIII. Application of TXRF<br />
Three main advantages characterize TXRF: simultaneous multielement capability, low<br />
detection limits for many elements, and small sample volume. Additional advantages are the<br />
absence of matrix effects, easy calibration, fast analysis, and comparatively low costs. Table
III.1 gives an overview of various kinds of sample that have been already analysed with<br />
TXRF. Generally all kinds of aqueous or acidic liquids where the liquid matrix can be<br />
evaporated, leaving a small amount on a sample reflector, can be analysed. Oils, alcohols,<br />
whole blood, and blood serum can be analysed after special treatment.<br />
Table III.1: Applications of TXRF.<br />
Environment<br />
Water Rain, river, sea, drinking, waste.<br />
Air Aerosols, airborne particles, dust, fly ash<br />
Soil Sediments, sewage sludge<br />
Plant material Algae, hay, leaves, lichen, moss, needles, roots, wood<br />
Foodstuff<br />
Fish, flour, fruits, crab, mussel, mushrooms, nuts, vegetables,<br />
wine, tea<br />
Various Coal, peat<br />
Medicine/Biology/Pharmacology<br />
Body fluids Blood, serum, urine, amniotic fluid<br />
Tissue Hair, kidney, liver, lung, nails, stomach, colon<br />
Various Enzymes, polysaccharides, glucose, proteins, cosmetics, biofilms<br />
Industrial/Technical<br />
Surface analysis Water surfaces<br />
Implanted ions<br />
Thin films<br />
Oil Crude oil, fuel oil, grease<br />
Chemicals Acids, bases, salts, solvents<br />
Fusion/Fission research Transmutational elements Al + Cu, Iodine in water<br />
Geology/Mineralogy<br />
Ores, rocks, minerals, rare earth elements<br />
Fine arts/Archaeology/Forensic<br />
Pigments, painting, varnish<br />
Bronzes, pottery, jewellery<br />
Textile fibres, glass, cognac, dollar bills, gunshot residue, drugs, tapes, sperm, finger prints
I. Introduction<br />
SECTION IV<br />
WAVELENGTH DISPERSIVE X-RAY<br />
FLUORESCENCE (WD-XRF)<br />
Wavelength Dispersive X-<strong>Ray</strong> <strong>Fluorescence</strong> Spectrometry (WD-XRF) is the oldest<br />
method of measurement of X-rays, introduced commercially in the 1950’s. This name is<br />
descriptive in that the radiation emitted from the sample is collimated with a Soller collimator,<br />
and then impinges upon an analyzing crystal. The crystal diffracts the radiation to different<br />
extents, according to Bragg’s law, depending upon the wavelength or energy of the Xradiation.<br />
This angular dispersion of the radiation permits the sequential or simultaneous<br />
detection of X-rays emitted by elements in the sample. Simultaneous instruments normally<br />
contain several sets of analyzing crystals and detectors; one is adjusted for each desired<br />
analyte in the sample. These instruments tend to be very expensive, but efficient for the<br />
routine determination or preselected elements.<br />
WD-XRF is a technique that has become indispensable when fast, accurate elemental<br />
analysis is needed, as when controlling a melt in a steel works or the raw mix at a cement<br />
plant. One reason for its popularity in these applications is that its ease of use, and the<br />
ruggedness of the equipment, allows quality results to be obtained in plant conditions by<br />
operators without advanced analytical skills. Furthermore, its inherent precision, speed, and<br />
simplicity of sample preparation can often eliminate many of the problems encountered with<br />
solution based methods like ICP or Atomic Absorption spectroscopy.<br />
WD-XRF spectrometers are usually larger and more expensive than other<br />
spectrometers. Because the analyzing crystal d-spacing determines wavelength sensitivity,<br />
they are usually more sensitive than other spectrometers. To overcome losses in X-ray optics<br />
of the WD-XRF spectrometers and to maximize primary radiation intensity, X-ray tubes are<br />
usually employed. The sample is usually held under vacuum to reduce contamination and<br />
avoid absorption of light element characteristic radiation in air.<br />
Typical uses of WD-XRF include the analysis of oils and fuel, plastics, rubber and<br />
textiles, pharmaceutical products, foodstuffs, cosmetics and body care products, fertilizers,<br />
minerals, ores, rocks, sands, slags, cements, heat-resistant materials glass, ceramics,<br />
semiconductor wafers; the determination of coatings on paper, film, polyester and metals; the<br />
sorting or compositional analysis of metal alloys, glass and polymeric materials; and the<br />
monitoring of soil contamination, solid waste, effluent, cleaning fluids, sediments and air<br />
filters.<br />
II. Principle of WD-XRF<br />
WD-XRF spectrometers measure X-ray intensity as a function of wavelength. This is<br />
done by passing radiation emanating from the specimen through an analyzing diffraction
crystal mounted on a 2θ goniometer. By Bragg’s Law, the angle between the sample and<br />
detector yields the wavelength of the radiation:<br />
2dsinθ= nλ;<br />
(IV.1)<br />
where:<br />
d is the d-spacing of the analyzing crystal,<br />
θ is half the angle between the detector and the sample,<br />
n is the order of diffraction.<br />
The analyzing crystal must be oriented so that the crystal diffraction plane is directed in<br />
the appropriate direction. Figure IV.1 shows a simplified schematic of the WD-XRF<br />
spectrometer. A scintillation or flow-proportional detector usually measures the fluoresced<br />
radiation. The heights of the resulting pulses are proportional to energy so using a pulse<br />
height analyzer (PHA), scattered or undesired diffraction-order X-rays can be ejected. The Xray<br />
beam is usually collimated before and after the analyzing crystal.<br />
Each of the components showed in the Figure IV.1 were be described in the following<br />
sections.<br />
II.1 Collimator Masks<br />
Figure IV.1: Schematic description of WD-XRF principle.<br />
The collimator masks are situated between the sample and collimator and serve the<br />
purpose of cutting out the radiation coming from the edge of the cup aperture (Figure IV.2).<br />
The size of the mask is generally adapted to suit of the cup aperture being used.<br />
The masks perform one of the two functions: background reduction and improved<br />
fluorescence (Figure IV.3).
Figure IV.2: Use of Cu 200 µm filter for cutting off the radiation coming from Rh X-ray<br />
tube.<br />
Figure IV.3: Use of Al 100 µm filter for improvement of the ratio peak/background.<br />
II.2 Collimator<br />
Collimators consist of a row of parallel slats (Figure IV.4) and select a parallel beam of<br />
X-rays coming from the sample and striking the crystal. The spaces between the slats<br />
determine the degree of parallelism and thus the angle resolution of the collimator.<br />
A 0.077° collimator is adequate for high resolution measurement parameters.<br />
Collimators with low resolution (e.g. 1.5 -2.0°) are advantageous for light elements such as<br />
Be, B and C (Figure IV.5). Using a collimator with a low resolution increases then intensity<br />
significantly. This enables intensity to be increased without a loss in angle resolution when<br />
analyzing light elements.
Figure IV.4: Collimators with different angles of resolution.<br />
Figure IV.5: Example of the influence of collimator resolution on the intensity of a light<br />
element.<br />
II.3 The Analyzing Crystals<br />
II.3.1 Bragg’s Law<br />
Crystals consist of a periodic arrangement of atoms (molecules) that form the crystal<br />
lattice. In such an arrangement of particles you generally find numerous planes running in<br />
different directions through the lattice points (= atoms, molecules), and not only horizontally<br />
and vertically but also diagonally. These are called lattice planes. All of the planes parallel to<br />
a lattice plane are also lattice planes and are at a defined distance from each other. This<br />
distance is called the lattice plane distance “d”.<br />
When parallel X-ray light strikes a lattice plane, every particle within it acts as a<br />
scattering centre and emits a secondary wave. All of the secondary waves combine to form a<br />
reflected wave. The same occurs on the parallel lattice planes for only very little of the X-ray<br />
wave is absorbed within the lattice plane distance “d”. All these reflected waves interfere with
each other. If the amplification condition “phase difference = a whole multiple of the<br />
wavelength” (∆λ = nλ) is not precisely met, the reflected wave will interfere such that<br />
cancellation occurs. All that remains is the wavelength for which the amplification condition<br />
is met precisely. For a defined wavelength and a defined lattice plane distance, this is only<br />
given with a specific angle, the Bragg angle (Figure IV.6).<br />
Figure IV.6: Bragg’s Law.<br />
Under amplification conditions, parallel, coherent X-ray light (1,2) falls on a crystal<br />
with a lattice plane distanced ‘d’ and is scattered below the angle θ (1′,2′). The proportion of<br />
the beam that is scattered on the second plane has a difference of ‘ACB’ to the proportion of<br />
the beam that was scattered at the first plane.<br />
The amplification condition is fulfilled when the phase difference is a whole multiple of<br />
the wavelength λ. This results in Bragg’s Law:<br />
2dsinθ= nλ;<br />
(IV.2)<br />
n = 1, 2, 3… Reflection order.<br />
On the basis of Bragg’s Law, by measuring the angle θ, you can determine either the<br />
wavelength λ, and thus chemical elements, if the lattice plane distance ‘d’ is known or, if the<br />
wavelength λ is known, the lattice plane –value distance ‘d’ and thus the crystalline structure.<br />
This provides the basis for two measuring techniques for the quantitative and qualitative<br />
determination of chemical elements (XRF) and crystalline structures (molecules, XRD),<br />
depending on whether the wavelength λ or the 2d-vale is identified by measuring the angle θ<br />
(Table IV.1).<br />
Table IV.1: Wavelength dispersive X-ray techniques.<br />
Known Sought Measured Method Instrument type<br />
d λ θ X-ray fluorescence Spectrometer<br />
λ d θ X-ray diffraction Diffractometer<br />
In X-ray diffraction (XRD) the sample is excited with monochromatic radiation of a<br />
known wavelength (λ) in order to evaluate the lattice plane distance as per Bragg’s equation.<br />
In XRF, the ‘d’-value of the analyzer crystal is known and we can solve Bragg’s<br />
equation for the element characteristic wavelength (λ).
II.3.2 Reflections of Higher Orders<br />
Figures IV.7a and IV.7b illustrate the reflections of the first and second order of one<br />
wavelength below the different angles θ1 and θ2. Here, the total reflection is made up of the<br />
various reflection orders (1, 2 …, n). The higher the reflection order, the lower the intensity of<br />
the reflected proportion of radiation generally is. How great the maximum detectable order is<br />
depends on the wavelength, the type of crystal used and the angular range of the spectrometer.<br />
Figure IV.7a: First order reflection: λ = 2 d sin θ1.<br />
Figure IV.7b: Second order reflection: 2λ = 2 d sin θ2.<br />
It can be seen from Bragg’s equation that the product of reflection orders ‘n = 1; 2; ..’<br />
and wavelength ‘λ’ for greater orders, and shorter wavelengths ‘λ* < λ’ that satisfy the<br />
condition ‘λ* = λ/n’, give the same result.<br />
Accordingly, radiation with one half, one third, one quarter etc. of the appropriate<br />
wavelength (using the same type of the crystal) is reflected below the identical angle θ:<br />
1λ= 2( λ /2) = 3( λ /3) = 4( λ /4) =KK<br />
As the radiation with one half of the wavelength has twice the energy, the radiation with<br />
one third of the wavelength three times the energy etc., peaks of twice, three times the energy<br />
etc. can occur in the pulse height spectrum (= energy spectrum) as long as appropriate<br />
radiation sources (elements) exist (Figure …..).<br />
Figure IV.8 shows the pulse height distribution of the flow counter using the example of<br />
the element hafnium (Hf) in a sample with a high proportion of zircon. The Zr Kα1 peak has<br />
twice the energy of the Hf Lα1 peak and appears, when the Hf Lα1 peak is set, at the same<br />
angle in the pulse height spectrum.
II.3.3 Crystal Types<br />
Figure IV.8: Second order reflection (n = 2).<br />
The wavelength dispersive X-ray fluorescence technique can detect every element<br />
above the atomic number 4 (Be). The wavelengths cover the range of values of four<br />
magnitudes: 0.01 – 11.3 nm. As the angle θ can theoretically only be between 0° and 90° (in<br />
practice 2° to 75°), sinθ an only accept values between 0 and +1. When Bragg’s equation is<br />
applied:<br />
nλ<br />
0< = sinθ
Ge<br />
InSb<br />
PET<br />
AdP<br />
TIAP<br />
OVO-55<br />
OVO-N<br />
OVO-C<br />
OVO-B<br />
Germanium<br />
Indiumantimonide<br />
Pentaerythite<br />
Ammoniumdihydrogenphosphate<br />
Thalliumhydrogenphthalate<br />
Multilayer [W/Si]<br />
Multilayer [Ni/BN]<br />
Multilayer [V/C]<br />
Multilayer [Mo/B4C]<br />
II.3.4 Dispersion, Line Separation<br />
P, S, Cl<br />
Si<br />
Al – Ti<br />
Mg<br />
F, Na<br />
O – Si (C)<br />
N<br />
C<br />
B (Be)<br />
0.653<br />
0.7481<br />
0.874<br />
1.0648<br />
2.5760<br />
5.5<br />
11<br />
12<br />
20<br />
The extent of the change in angle ∆θ upon changing the wavelength by the amount ∆λ<br />
(thus: ∆θ/∆λ) is called “dispersion”. The greater the dispersion, the better is the separation of<br />
two adjacent or overlapping peaks. Resolution is determined by the dispersion as well as by<br />
surface quality and the purity of the crystal.<br />
Mathematically, the dispersion can be obtained from the differentiation of the Bragg<br />
equation:<br />
∆θ n<br />
=<br />
∆λ 2dcosθ . (IV.4)<br />
It can be seen from this equation that the dispersion (or peak separation) increases as the<br />
lattice plane distance ‘d’ declines.<br />
II.3.5 Synthetic Multilayers<br />
Multilayers are not natural crystals but artificially produced ‘layer analyzers’. The<br />
lattice plane distances ‘d’ are produced by applying thin layers of two materials in alternation<br />
on to a substrate (Figure IV.9). Multilayers are characterized by high reflectivity and a<br />
somewhat reduced resolution. For the analysis of light elements the multilayer technique<br />
presents an almost revolutionary improvement for numerous applications in comparison to<br />
natural crystals with large lattice plane distances (e.g. RbAp, PbST, KAP).<br />
Figure IV.9: Diffraction in the layers (here: Si/W) of a multilayer.
II.4 Detectors<br />
When measuring X-ray, use is made of their ability to ionize atoms and molecules, i.e.<br />
to displace electrons from their bonds by energy transference. In suitable detector materials,<br />
pulses whose strengths are proportional to the energy of the respective X-ray quants are<br />
produced by the effect of X-ray. The information about the X-ray quarts energy is contained<br />
in the registration of the pulse height. The number of X-ray quants per unit of time, e.g. pulses<br />
per second (cps = counts per second, KCps = kilocounts per second), is called their intensity<br />
and contains in a first approximation the information about the concentration of the emitting<br />
in the sample. Two main types of detectors are used in wavelength dispersive X-ray<br />
fluorescence spectrometers: the gas proportional counter and the scintillation counter.<br />
II.4.1 Gas Proportional Counter<br />
The gas proportional counter comprises a cylindrical metallic tube in the middle of<br />
which a thin wire (counting wire) is mounted. This tube is filled with a suitable gas (e.g. Ar+<br />
10% CH4). A positive high voltage (+U) is applied the wire. The tube has a lateral aperture or<br />
window that is sealed with a material permeable to X-ray quants (Figure IV.10).<br />
Figure IV.10: A gas proportional counter.<br />
An X-ray quant penetrates the window into the counter’s gas chamber where it is<br />
absorbed by ionizing the gas atoms and molecules. The resultant positive ions move to the<br />
cathode (tube), the free electrons to the anode, the wire. The number of electron-ion pairs<br />
created is proportional to the energy of the X-ray quant. To produce an electron-ion pair,<br />
approx. 0.03 keV are necessary, i.e. the radiation of the element boron (0.185 keV) produces<br />
approx. 6 pairs and the K-alpha radiation of molybdenum (17.5 keV) produces approx. 583<br />
pairs. Due to the cylinder geometric arrangement, the primary electrons created in this way<br />
see an increasing electrical field on route to the wire. The high voltage in the counting tube is<br />
now set so high that the electrons can obtain enough energy from the electrical field in the<br />
vicinity of the wire to ionize additional gas particles. An individual electron can thus create up<br />
to 10.000 secondary electron-ion pairs.<br />
The secondary ions moving towards the cathode produce measurable signal. Without<br />
this process of gas amplification, signals from boron, for example, with 6 or molybdenum<br />
with 583 pairs of charges would not be able to be measured as they would not be sufficiently<br />
discernible from the electronic noise. As amplification is adjustable via high voltage in the<br />
counting tube and is set higher for measuring boron than for measuring molybdenum. The<br />
subsequent pulse electronics supply pulses of voltage whose height depends, amongst other<br />
factors, on the energy of the X-ray quants.
II.4.2 Scintillation Counters<br />
The scintillation counter, “SC”, used in XRF comprises a sodium iodide crystal in<br />
which thallium atoms are homogeneously distributed ‘NaI(Tl)’. The density of the crystal is<br />
sufficiently high to absorb all the XRF high energy quants. The energy of the pervading X-ray<br />
quants is transferred step by step to the crystal atoms that then radiate light and cumulatively<br />
produce a flash. The amount of light in this scintillation flash is proportional to the energy that<br />
the X-ray quant has passed to the crystal. The resulting light strikes a photocathode from<br />
which electrons can be detached very easily. These electrons are accelerated in a<br />
photomultiplier and, within an arrangement of dynodes, produce so-called secondary<br />
electrons giving a measurable signal once they have become a veritable avalanche (Figure<br />
IV.11). The height of the pulse of voltage produced is, as in the case of the gas proportional<br />
counter, proportional to the energy of the detected X-ray quant.<br />
Figure IV.11: Scintillation counter including photomultiplier.<br />
II.4.3 Pulse Height Analysis (PHA), Pulse Height Distribution<br />
If the number of the measured pulses (intensity) dependent on the pulse height is<br />
displayed in a graph, we have the ‘pulse height spectrum’. Synonymous terms are: ‘pulse<br />
height analysis’ or ‘pulse height distribution’. As the height of the pulses of voltage is<br />
proportional to the X-ray quants energy, it is also referred to as the energy spectrum of the<br />
counter (Figure IV.12a and IV.12b). The pulse height is given in volts, scale divisions or in<br />
‘%’ (and could be started in keV after appropriate calibration). The “%”-scale is defined in<br />
such a way that the peak to be to be analyzed appears at 100 %.
Figure IV.12a: Pulse height distribution<br />
(S) Gas proportional counter.<br />
Figure IV.12b: Pulse height distribution<br />
(Fe) Scintillation counter.<br />
If argon is used as the counting gas component in gas proportional counters, an<br />
additional peak, the escape peak (Figure IV.13), appears when X-ray energies are irradiated<br />
that are higher than the absorption edge of argon.<br />
Figure IV.13: Pulse height distribution (Fe) with escape peak.<br />
The escape peak arises as follows:<br />
The incident X-ray quant passes its energy to the counting gas thereby displaying a K electron<br />
from an argon atom. The Ar atom can now emit an Ar Kα1,2 X-ray quant with an energy of 3<br />
keV. If this Ar-fluorescence escapes from the counter then only the incident energy minus 3<br />
keV remains for the measured signal. A second peak, the escape peak that is always 3 keV<br />
below the incident energy, appears in the pulse height distribution.
When using other counting gases (Ne, Kr, Xe) instead of argon, the escape peaks appear with<br />
an energy difference below the incident energy that is equivalent to the appropriate emitted<br />
fluorescence radiation (Kr, Xe). Using neon as the counting gas component produces no<br />
recognizable escape peak as the Ne K-radiation, with energy of 0.85 keV, is almost<br />
completely absorbed in the counter. Also, the energy difference to the incident of 0.85 keV<br />
and the fluorescence yield are very small.<br />
III. Points of Comparison between ED-XRF and WD-XRF<br />
1. Resolution: it describes the width of the spectra peaks. The lower the resolution number<br />
the more easily an elemental line is distinguished from the nearby X-ray line intensities.<br />
a. The resolution of the WD-XRF system is dependant on the crystal and optics design,<br />
particularly collimation, spacing and positional reproducibility. The effective resolution<br />
of a WD-XRF system may vary from 20 eV in an inexpensive benchtop to 5 eV or less<br />
in a laboratory instrument. The resolution is not detector dependant.<br />
b. The resolution of ED-XRF system is dependent on the resolution of the detector. This<br />
can vary from 150 V or less for a liquid nitrogen cooled Si(Li) detector, 150 – 220 eV<br />
for various solid state detectors, or 600 eV or more for gas filled proportional counter.<br />
Advantage of WD-XRF: High resolution means fewer spectral overlaps and lower<br />
background intensities.<br />
Advantage of ED-WRF: WD-XRF crystal and optics are expensive, and are one more failure<br />
mode.<br />
2. Spectral overlaps: Spectral deconvolutions are necessary for determining net intensities<br />
when two spectral lines overlap because the resolution is too high for them to be measured<br />
independently.<br />
a. With a WD-XRF instrument with very high resolution (low number of eV) spectral<br />
overlap corrections are not required for a vast majority of elements and applications.<br />
The gross intensities for each element can be determined in a single acquisition.<br />
b. The ED-XRF analyzer is designed to detect a group of elements all at once. The some<br />
type of deconvolutions method must b used to correct for spectral overlaps. Overlaps are<br />
less of a problem with 150 eV resolution systems, but are significant when compared to<br />
WD-XRF. Spectral overlaps become more problematic at lower resolutions.<br />
Advantage WD-XRF: Spectral deconvolutions routines introduce error due to counting<br />
statistics for every overlap correction onto every other element being corrected for. This can<br />
double or triple the error.<br />
3. Background: The background radiation is one limiting factor for determining detection<br />
limits, repeatability, and reproducibility.<br />
a. Since a WD-XRF instrument usually uses direct radiation flux the background in the<br />
region of interest is directly related to the amount of continuum radiation within the<br />
region of interest the width is determined by the resolution.<br />
b. The ED-XRF instrument uses filters and/or targets to reduce the amount of continuum<br />
radiation in the region of interest which is also resolution dependant, while producing a<br />
higher intensity X-ray peak to excite the element of interest.
Even, WD-XRF has the advantage due to the resolution. If a peak is one tenth as wide it has<br />
one tenth the background.<br />
ED-XRF counters with filters and targets that can reduce the background intensities by a<br />
factor of ten or more.<br />
4. Excitation Efficiency: Usually expressed in PPM per count-per-second (cps) or similar<br />
units, this is the other main factor for determining detection limits, repeatability, and<br />
reproducibility. The relative excitation efficiency is improved by having more source x-rays<br />
closer to but above the absorption edge energy for the element of interest.<br />
a. WDXRF generally uses direct unaltered x-ray excitation, which contains a continuum<br />
of energies with most of them not optimal for exciting the element of interest.<br />
b. EDXRF analyzers may use filter to reduce the continuum energies at the elemental<br />
lines, and effectively increasing the percentage of X-rays above the element absorption<br />
edge. Filters may also be used to give a filter fluorescence line immediately above the<br />
absorption edge, to further improve excitation efficiency. Secondary targets provide an<br />
almost monochromatic line source that can be optimized for the element of interest to<br />
achieve optimal excitation efficiency.
SECTION V<br />
SAMPLE PREPARATION<br />
XRF analysis is a physical method which directly analyses almost all chemical elements<br />
of the periodic system in solids, powders or liquids. These materials may be solids such as<br />
lass, ceramics, metal, rocks, coal, plastic or liquids, like petrol, oils, paints, solutions or blood.<br />
With XRF spectrometer both very small concentrations of very few ppm and very high<br />
concentrations of up to 100 % can directly be analyzed without any dilution process.<br />
Therefore XRF analysis is a very universal analysis method, which, based on simple and fast<br />
sample preparation, has been widely accepted and has found a large number of users in the<br />
field of research and above all in industry.<br />
The quality of sample preparation in X-ray fluorescence analysis is at least as important<br />
as the quality of measurements.<br />
An ideal sample would be prepared so that it is:<br />
• Representative of the material;<br />
• Homogeneous;<br />
• Thick enough to meet the requirements of an infinitely thick sample;<br />
• Without surface irregularities;<br />
• Composed of small enough particles for the wavelengths to be measured.<br />
The care taken to determine the best method of sample preparation for a given material,<br />
and the careful adherence to that method, will often determine the quality of results obtained.<br />
It is safe to say that sample preparation is likely the single most important step in an analysis.<br />
A wide variety of sample types may be analyzed by X-ray spectrometer; hence, a wide variety<br />
of sample preparation techniques is required (Figure V.1).<br />
Samples are often classified into two types based upon their thickness as measured by<br />
the attenuation of X-rays. Infinitely thick samples are those which completely attenuate Xrays<br />
emitted from the back side of the sample before they emerge from the sample. Further<br />
increase in the thickness yields no increase in observed X-ray intensity. Clearly, the critical<br />
value for infinite thickness will depend upon the energy of the emitted X-radiation and the<br />
mass absorption coefficient of the sample matrix for those X-rays.<br />
On the other hand, a thin sample has been defined as one in which m⋅µm ≤ 0.1, where m<br />
is the mass per unit area (g/cm 2 ) and µm is the sum of the mass absorption coefficient for the<br />
incident and emitted X-radiation. Although there are many advantages to thin samples, it is<br />
rarely feasible to prepare them for routine samples. Many samples fall between these two<br />
cases and require extreme care in preparation.
I. Solids<br />
Figure V.1: Type of samples analyzed by XRF spectrometry.<br />
Solid samples will be defined single, bulk materials, as opposed to powders, filings, or<br />
turnings. In many cases, solid samples may be machinated to the shape and dimensions of the<br />
sample holder. Care should be taken that the processing does not contaminate the sample<br />
surface to be used for analysis. In other cases, small parts and pieces must be analysed as<br />
received. Often, it is found useful to make a wax mold of the part which will fit into the<br />
sample holder. Using the mold as a positioning aid, other identical samples may be<br />
reproducibly placed in the spectrometer. This technique is especially useful for small<br />
manufactured parts.<br />
Samples taken from unfinished, bulk material will often require surface preparation<br />
prior to quantitative analysis. Surface finishing may be accomplished by use of a polishing<br />
wheel, steel wool, or belt grinders with subsequent polishing using increasingly fine<br />
abrasives. Surface roughness less than 100 micrometers is usually sufficient for X-ray<br />
energies above approximately 5 keV, whereas surface roughness of less than 20–40<br />
micrometers is required for energies down to approximately 2 keV.<br />
II. Powders and Briquets<br />
Powder samples may be received as powders, or prepared from pulverized bulk material<br />
which is too inhomogeneous for direct analysis. Typical bulk samples which are pulverized<br />
prior to analysis are ores, minerals, refractory materials, and freeze-dried biological tissues.<br />
Powders may be presented as such to the spectrometer, or pressed into pellets or briquets.<br />
Also, they may be fused with flux such as lithium tetraborate. The fused product may be
eground and pressed or cast as a disk. For precise quantitative determinations, loose powders<br />
are rarely acceptable, especially when low energy X-rays are detected. Pressed briquets are<br />
much more reliable.<br />
Briquets or pressed powders yield better precision than powder samples and are<br />
relatively simple and economical to prepare. In many cases all that is needed is a hydraulic<br />
press and a suitable die. In the simplest case, the die diameter should be the same as the<br />
sample holder so that the pressed briquets will fit directly into the holder. The amount of<br />
pressure required to press a briquet which yields maximum intensity depends upon the sample<br />
matrix, the energy of the X-ray to be used, and the initial particle size of the sample.<br />
Therefore, prior grinding of the sample to a small particle size (< 100 micrometers) is<br />
advisable. In cases of materials which will not cohere to form stable briquets, a binding agent<br />
may be required. A wide, variety of binding agents have been used such as: powdered<br />
cellulose, detergent powders, starch, stearic acid, boric acid, lithium carbonate, polyvinyl<br />
alcohol and commercial binders.<br />
III. Fused Materials<br />
Fusion of materials with a flux may be done for several reasons. Some refractory<br />
materials cannot be dissolved, ground into fine powders, or otherwise put in a suitable,<br />
homogeneous form for X-ray spectrometric analysis. Other samples may have compositions<br />
which lead to severe interelement effects, and dilution in the flux will reduce these. The fused<br />
product cast into a glass button, provides a stable, homogeneous sample well suited for X-ray<br />
measurements. The most severe disadvantages to fusion techniques are the time and material<br />
costs involved, and the dilution of the elements which can result in an order of magnitude<br />
reduction in X-ray intensity. However, when other methods of sample preparation fail, fusion<br />
will often provide the required results.<br />
More common are the glass-forming fusions with lithium borate, lithium tetraborate or<br />
sodium tetraborate. Flux to sample ratios range from 1:1 to 10:1. The lithium fluxes have<br />
lower mass absorption coefficients and, therefore, less effect on the intensity of the low<br />
energy X-rays. Lithium carbonate may be added to render acidic samples more soluble in the<br />
flux, whereas lithium fluoride has the same effect on basic sample. Lithium carbonate can<br />
also reduce the fusion temperature. Oxidants such as sodium nitrate, potassium chlorate or<br />
others may be added to sulfides and other mixtures to prevent loss of these elements.<br />
IV. Filters and Ions-Exchange Resins<br />
Various filters, ion-exchange resin beads, and ion-exchange resin impregnated filter<br />
papers have become important sampling substrates for samples for X-ray spectrometric<br />
analysis. Filter materials may be composed of filter paper, membrane filters (i.e., Nuclepore,<br />
Millipore), glass fiber filters, and others. Filters are used in a variety of applications.<br />
One of the most widely used applications of filters is in the collection of aerosol<br />
samples from the atmosphere. Loadings of several milligrams of sample on the filter may<br />
correspond to sampling several hundred cubic meters of atmosphere. Many elements may be<br />
determined directly o these filters by X-ray spectrometric analysis.
Filters may also be used for non-aerosol atmospheric components such as reactive<br />
gases. Filter materials may be impregnated with a reagent reactive to the gas which will trap it<br />
chemically.<br />
V. Thin Films<br />
Thin film samples are ideal for X-ray spectrometric analysis. The X-ray intensity of an<br />
infinitely thin sample is proportional to the mass of the element on the film, and the spectral<br />
intensities are free of interelement and mass absorption coefficient effects. However, in<br />
practice, perfect thin film samples are difficult to encounter. Powder samples of sufficiently<br />
small and homogeneous particle size may be distributed on an adhesive surface such as<br />
Scotch tape, or placed between two drum tight layers on Mylar film mounted on a sample<br />
cup.<br />
More important thin film types are platings and coatings on various substrates. Analysis<br />
of these sample types is of increasing importance for the electronics industry.<br />
VI. Liquids<br />
Although X-ray spectrometry is relatively unique in its ability to perform qualitative and<br />
quantitative elemental determinations on solid samples, liquids may also be analyzed. The<br />
design of X-ray spectrometric instrumentation using what has become known as inverted<br />
optics, in which the specimen is above the X-ray source and detector, facilitates the use of<br />
liquid samples. This convenient geometry demands caution in the preparation of liquid<br />
samples so that spills, leaking sample cups, and other accidents do not damage the source or<br />
detector.<br />
Liquid samples have the excellent advantage that quantitative standards are easily<br />
prepared. However, they have the disadvantage that because solvents are usually composed of<br />
low atomic number elements, the <strong>Ray</strong>leigh and Compton scatter intensity is high; this<br />
increases background and leads to high limits of detection. Fortunately, the problems can be<br />
minimized by use of suitable primary tube filters. These reduce the scattered X-radiation in<br />
the analytically useful region. Care must be taken with liquids which contain suspended<br />
solids. If the suspension settles during the measurement time, the X-ray intensity of the<br />
contents of the sediment will be enhanced. The X-ray intensity from solution components or<br />
homogeneous suspension may decrease as a result of sediment absorption, which leads to<br />
erroneous results. This possibility is tested by taking repetitive measurements for short time<br />
periods, beginning immediately after a sample is prepared. Any observed increase or decrease<br />
in intensity with time is an indication of segregation in the sample. In these cases, an additive<br />
may be used which stabilizes the suspension, or the suspended content may be collected on a<br />
filter for analysis. Liquid samples should be considered as viable candidates for X-ray<br />
spectrometric analysis.
SECTION VI<br />
QUANTITATIVE ANALYSIS<br />
The observed photon rate from an analyte element in a specimen is a function of many<br />
factors including the concentration (weight fraction) of the element, the matrix<br />
(accompanying element), the specimen type (bulk, powder, liquid, thin film structure, etc.),<br />
size, preparation, geometrical set-up, spectral distribution of the exciting radiation, and the<br />
detection system. Theoretical as well as empirical approaches are used to determine<br />
concentrations from florescent intensities.<br />
The theoretical methods are based on mathematical models for the excitation of atoms<br />
and subsequent relaxation processes, the absorption of radiation within the specimen, and the<br />
inter-element effects. Compared to a real set-up, simplifying assumption are made, for<br />
example that the specimen is perfectly flat and homogeneous, and that the incident primary<br />
beam is parallel.<br />
An alternative is the empirical parameter method, which employs relatively simple<br />
mathematical descriptions of the relationship between photon counts and concentration<br />
(calibration curves). The general principle is that the ideal calibration curve is assumed to be a<br />
linear function, which is obtained from the (non-linear) experimental relationship by applying<br />
a number of corrections. The coefficients, by which the extent of the various corrections is<br />
introduced, are called empirical parameters. They are determined experimentally from<br />
calibration standards or sometimes by theoretical approaches.<br />
Many times in both methods count-rate ratios (relative intensities) rather than absolute<br />
counts are used. By building ratios of the ratios of the count rate from a line of an element<br />
with the one from the same element in another specimen (which can be a standard, containing<br />
this element, or a pure element), many geometrical factors, the detection efficiency and the<br />
absolute intensity level of the primary radiation cancel.<br />
I. Detection Limits<br />
The limit of detection is the lowest concentration level that can be determined to be<br />
statistically significant from an analytical blank. The limit of detection is expressed as a<br />
concentration or an amount and is derived from the smallest measured value x.<br />
The standard deviation s for counting of photons emitted at complementary random<br />
intervals of time t obeys for an average number of accumulated counts N the equations:<br />
σ N = N , (VI.1)<br />
following the Poisson statistics.<br />
Figure VI.1 shows evaluation of background.
Figure VI.1: Schematic view of net peak and background definition.<br />
Assuming a confidence level of 95 % the total deviation is given by 2σ. The standard<br />
deviation from peak and background is assumed to be equal:<br />
3⋅ NB 3⋅ IB⋅t 3 I<br />
LLD = ⋅ m = ⋅ m = ⋅ B<br />
NN IN ⋅t<br />
S t<br />
2 2 2<br />
σ T = σ P +σ B = 2σ<br />
B<br />
2 2<br />
LLD = 2σ T = 2 2σB ≈3 σ B = 3 NB<br />
with S = INm, (VI.2)<br />
generally normalized to 1000 s measuring time.<br />
The detection limit is a method to compare the power of different analytical methods of<br />
trace element determination. Generally detection limits are derived from single element<br />
samples, so no line interference is considered. They are idealized and extrapolated values and<br />
one has to multiply these values by about a factor of 3 to come to the minimum measurable<br />
amount in real sample, but these values are good to compare different analytical methods for<br />
trace element determination.<br />
Table VI.1 shows a comparison of detection limits in µg/cm 2 for WD-XRF of<br />
Nuclepore filters and for secondary target EDXRF of Teflon filters.<br />
Table VI.1: Comparison of Detection Limits for WDXRF and EDXRF of filter<br />
materials.<br />
Detection Limit Detection Limit<br />
Element<br />
Element<br />
WDXRF EDXRF<br />
WDXRF EDXRF<br />
Na 310 Ge 3<br />
Mg 60 As 4<br />
Al 6.7 130 Se 2<br />
Si 16 45 Br 16 2<br />
P 8.2 Rb 3<br />
S 5.6 15 Sr 20 3<br />
Cl 2.2 13 Zr 8<br />
K 3.1 6 Mo 5<br />
Ca 1.2 5 Ag 5<br />
Ti 4.3 30 Cd 7.5 (Lα) 6<br />
V 1.6 20 In 6
Cr 1.2 16 Sn 4.8 (Lα) 8<br />
Mn 2.6 12 Sb 7.1 (Lβ) 8<br />
Fe 2.2 12 Te 10<br />
Co 6.0 I 13<br />
Ni 6.0 5 Cs 24<br />
Cu 4.8 6 Ba 5.1 (Lα) 40<br />
Zn 5 Hg 7 (Lα)<br />
Ga 4 Pb 21 (Mα) 8 (Lα)<br />
II. Disturbing Effects<br />
II.1 Interelement Radiation<br />
By the term “interelements” we mean those elements in the sample which become<br />
excited together with the wanted element under the influence of the primary radiation of the<br />
source. The fluorescent radiation of interelements may disturb X-ray fluorescence<br />
determination of the element of interest, or even make it completely impossible. These<br />
disturbing effects can be classified as follows:<br />
1. The K-series peak of the wanted element partially overlaps the K-series peaks of<br />
interelements. Remember that if the difference between the atomic numbers of two<br />
elements is lower than 3 (∆Z < 3), their characteristic peaks can be separated only by means<br />
of a solid-state detector (Si(Li)) or appropriate absorption-edge filters.<br />
2. The K-series peak of the wanted element overlaps the L-series peaks of some of the<br />
interelements. As an example, let us consider zinc determination (ZnKα = 8.64 keV) in a<br />
sample which also contains tungsten (WLα = 8.39 keV). From the energy values of the<br />
ZnKα and WLα lines it is evident that the complete separation of two peaks may be a hard<br />
task even if a solid-state detector is used.<br />
3. The characteristic peak of the wanted overlaps the escape peaks of interelements. An<br />
example of such a situation is the determination of vanadium (VKα = 17.47 keV) in a<br />
sample which also contains tungsten (WLα = 8.39 keV).<br />
It follows from these examples that the choice of optimum measurement conditions for<br />
X-ray fluorescence determination of a particular element in a given material requires<br />
information concerning the latter’s chemical composition and expected concentration ranges<br />
of all the sample constituents in every analysed sample.<br />
II.2 Matrix Effects<br />
Generally speaking, the matrix effects in X-ray fluorescence analysis result from the<br />
influence of the variations of chemical compositions of the sample matrix on the fluorescent<br />
intensity of the wanted element. These effects can manifest themselves either via a difference<br />
in the absorption of both the primary and fluorescence radiations in samples of different<br />
matrix composition (absorption effect) or via an increase of the radiation intensity<br />
(enhancement effect) due to the fluorescence radiation of some of the interelements. These<br />
two phenomena will be discussed in detail in subsequent sections. The elements whose<br />
varying concentrations in the analysed samples lead to the effects mentioned above will be<br />
called disturbing elements.
Matrix effects constitute in most cases the major and the least easily removable sources<br />
of errors in X-ray fluorescence analysis, both in wavelength dispersive techniques and in<br />
energy dispersive ones.<br />
II.2.1 Absorption Effect<br />
This effect occurs when the variations in the matrix chemical composition result in<br />
changes of the mean absorption coefficients of both the primary radiation of the source and<br />
the fluorescence radiation of the wanted element. Note that the primary radiation beam<br />
penetrating into a sample undergoes attenuation due to photoelectric absorption which may<br />
occur not only in the atoms of the wanted element but also in the atoms of all the other matrix<br />
constituents. The net attenuation of the primary radiation in a matrix consisting of n elements<br />
as a function of the mean mass absorption coefficient m µ o , expressed by the formula:<br />
m<br />
n<br />
m<br />
µ o = ∑ µ oi wi;<br />
(VI.3)<br />
i=<br />
1<br />
where m µ oi are the mass absorption coefficients for the primary radiation of all matrix<br />
constituents and wi are their concentrations expressed as weight fractions.<br />
The absorption effects occurring in a matrix may either decrease or increase the<br />
intensity of the fluorescence radiation of the element under determination, depending on<br />
whether the matrix composition changes diminish or augment the mass absorption coefficient.<br />
A strong decrease of the fluorescence radiation of the wanted element will be observed if the<br />
concentrations of disturbing elements of slightly lower atomic numbers become larger.<br />
II.2.2 Enhancement Effect<br />
This effect involves an extra excitation of the atoms of the wanted element by the<br />
fluorescence radiation of some of the matrix elements, which in this case become interferants.<br />
An example of this phenomenon is illustrated in Figure VI.2.<br />
Figure VI.2: Schematic illustration of secondary excitation in sample containing Cr, Fe<br />
and Ni.
The mechanism of the enhancement effect involves retransmission of the energy of the<br />
primary radiation of the source in the form of secondary (fluorescence) radiation of<br />
interelements. The energy of this secondary radiation is just slightly higher than the<br />
absorption edge of the wanted element (Figure VI.3), the latter will be excited more efficiency<br />
than by the primary radiation of the source, whose energy is higher than that of the secondary<br />
radiation and, consequently, further from the absorption edge.<br />
Figure VI.3: Mass absorption coefficient for chromium as a function of X-ray energy.<br />
Note the strong absorption of Fe X-rays in chromium.<br />
It can be seen that enhancement effects constitute a cascade of events, ach involving<br />
excitation of lighter atoms by the fluorescence radiation of heavier ones. Consequently, the<br />
intensity of the fluorescence radiation of the wanted element in a sample which also contains<br />
some heavier elements will depend on their atomic numbers and concentrations.<br />
Quantitative theoretical analysis of the enhancement effects is a much more difficult<br />
problem than the analogous analysis of absorption effects. The contributions to the<br />
enhancement of a given X-ray line due to individual matrix elements cannot be expressed<br />
quantitatively by some appropriate coefficients, which might add together in a simple way as<br />
the mass absorption coefficients did. This phenomenon has been considered in theoretical<br />
terms by such author as Gillam and Heal (1952), Sherman (1955, 1959), Shiraiwa and Fujino<br />
(1966), and Lubecki (1970). The total intensity of the fluorescence radiation of an element<br />
excited within the sample both by the primary radiation of the source and by the fluorescence<br />
X-rays of one of the matrix elements may be expressed, in the most general manner, by the<br />
formula:<br />
( 1 )<br />
o<br />
I f = I f + S ; (VI.4)<br />
o where I f is the intensity of the fluorescence radiation of the atoms excited by the primary<br />
radiation of the source and S is the enhancement factor which depends, among other things,<br />
on the atomic number and weight fraction of the matrix element.<br />
II.3 Particle-Size Effects<br />
Particle size effects (grain-size effects, granulation effects) involve the dependence of<br />
the intensity of the secondary radiation from heterogeneous sample on the size of individual<br />
sample grains (particles) (Figure VI.4). These effects can only be removed by adequate
homogenization of the analysed material which is most easily done by fusing the sample or<br />
taking it into solution. Such a preparation of samples, however, makes the whole analysis<br />
complicated and time consuming.<br />
Figure IV.4: Schematic illustration of particle size effects.<br />
The direction of the changes in the fluorescent intensity which follow given changes in<br />
particle size depends on the ratio of the absorption coefficients for this fluorescent radiation of<br />
the particles containing the excited element (fluorescent particles) and of the matrix particles.<br />
For a rather weakly absorbing matrix, the fluorescent intensity decreases with the particles<br />
size. In strongly absorbing matrices, this relationship is just the opposite (Figure IV.5).<br />
Figure IV.5: Variation in the fluorescent X-ray intensities versus diameter of the sample<br />
grains; 1- weak absorption of fluorescent radiation in light matrix, 2- strong<br />
absorption of fluorescent radiation in matrix containing heavy elements.<br />
II.4 Mineralogical Effects<br />
These effects are caused by the influence, on the fluorescence intensity of the wanted<br />
element, of the type of mineral in which this element occurs (Figure VI.6). These phenomena<br />
have been reported by, among other, Campbell and Thatcher (1960) for fluorescence<br />
determination of calcium in such samples as carbonate, tungstate, or phosphate. A similar<br />
effect has been reported by Bernstein (1962, 1963) in the analysis of copper ores, where the<br />
copper was in the form of chalcopyrite (CuFeS2) and connelite (CuS).
Mineralogical effect were discussed by Claisse (1957a, 1957b), who tried to explain<br />
them starting from an hypothesis according to which the fluorescence intensity would depend<br />
on the interatomic distances separating the excited atoms in the crystal lattice. This hypothesis<br />
might be corrected in those cases where any changes in the interatomic distances lead to<br />
marked variations of the crystal density, and consequently, of the linear absorption coefficient<br />
for a given radiation in the crystal. It seems, however, that the main reason for the<br />
mineralogical effects is simply the different absorption of the fluorescent radiation in the<br />
particles of minerals of different chemical composition.<br />
II.5 Surface Effects<br />
Figure VI.6: Schematic illustration of mineralogical effects.<br />
In X-ray fluorescence analysis of solid samples, such as alloys, one observes some<br />
effects due to surface irregularities. These effects are caused by the influence of the surface<br />
coarseness (or surface finish) on the intensity of the detected fluorescent radiation. The<br />
coarseness of the surface may be defined quantitatively by the dimensions of the microirregularities,<br />
i.e., micro-protuberances and micro-cavities which occur at the sample surface.<br />
These surface effects have been studied by, among others, Gunn (1961) and Michaelis and<br />
Kilday (1962). Their occurrence may be attributed to the shielding (absorption) effects taking<br />
place in individual protuberances at the sample surface with the fluorescence radiation<br />
emerging from the sample at different angles. According to the authors cited above, the<br />
surface irregularity effects may be of importance where both the primary and secondary<br />
radiation beams are collimated.<br />
The magnitude of the surface irregularity effects should depend on the energy of<br />
fluorescence radiation and on the chemical composition of the sample (i.e., the sample<br />
absorption coefficient). Such dependence has actually been established.<br />
III. Mathematical Models<br />
III.1 Sherman Equation<br />
Use of X-ray fluorescence to determine chemical composition of unknown specimens<br />
became more common in the following decade. With this came the need to better understand<br />
X-ray absorption and enhancement. Sherman derived a more specific equation for the
fluoresced X-ray intensity from a multi-element specimen subjected to a monochromatic nondivergent<br />
incident radiation of energy E that only accounted for primary absorption:<br />
S Ω Cg i i Ei Ii i E<br />
Ii<br />
=<br />
4πsinψ1µ i<br />
+<br />
sin ψ1 sin ψ2<br />
κ ( , ) µ ( )<br />
, (VI.5)<br />
( E)<br />
µ ( E )<br />
where:<br />
Ii: Intensity of observed characteristic line of element i.<br />
E: Energy of incident radiation.<br />
Ei: Energy of the characteristic line of element i being measured.<br />
S: Irradiated surface area of specimen.<br />
Ci: Concentration of element i in the specimen.<br />
gi: Proportionality constant for characteristic line of element i.<br />
ψ1: Angle between the specimen surface and the incident x-rays.<br />
ψ2: Angle between the specimen surface and the detector.<br />
Ω: Solid angle subtended by the detector.<br />
κ(Ei,Ii): Response of instrument at energy Ei of characteristic line energy of element i.<br />
µi(E): Mass absorption coefficient of element i at incident energy E.<br />
µ(E): Total absorption coefficient of specimen at incident energy E.<br />
µ(Ei): Total absorption coefficient of specimen at characteristic line energy of element i.<br />
Also note that:<br />
( E) ∑C<br />
( E)<br />
µ = j µ j . (VI.6)<br />
j<br />
Sherman later developed his theory to express the emitted X-ray intensity from a multielement<br />
specimen subjected to a polychromatic radiation source. Sherman’s theory was then<br />
further refined by Shiraiwa and Fujino:<br />
⎧<br />
⎪<br />
J 1 E<br />
i − S Ω ⎪ max<br />
τi<br />
( E)<br />
Ji−1<br />
Ii = κ ( Ei, Ii) Cipiω( )<br />
4 sin i ⎨ ∫ Io E dE + Cjpjωj∑ Jiπ ψ1 ⎪ Eiedge µ ( E)<br />
µ ( Ei<br />
)<br />
j 2Ji<br />
+<br />
⎩<br />
⎪ sin ψ1 sin ψ2<br />
Emax τi( Ej) τ j ( E) ⎡ ⎛<br />
sin 1 ( E)<br />
⎞ ⎛<br />
sin 2<br />
( Ei<br />
) ⎞⎤<br />
⎫<br />
ψ µ<br />
ψ µ<br />
⎪<br />
∫<br />
• ⎢ ln ⎜1+ ⎟+ ln ⎜1+ ⎟⎥<br />
dE⎬<br />
Eiedge µ ( E) µ ( Ei<br />
) ⎢µ ( E) ⎜ µ ( Ej) sin ψ ⎟ µ<br />
1 ( Ei)<br />
⎜ µ ( Ej)<br />
sin ψ ⎟⎥<br />
+ ⎢ 2<br />
⎣ ⎝ ⎠ ⎝ ⎠⎥⎦<br />
⎪<br />
⎭<br />
sin ψ1 sin ψ2<br />
, (VI.7)<br />
where;<br />
Ji: Jump ratio of the photoelectric mass absorption coefficient at the absorption edge for the<br />
line of element i being measured.<br />
ωi: Fluorescent yield for the line of element i being measured.<br />
Io(E): Intensity of incident radiation at energy E.<br />
τi(E): Mass photoabsorption coefficient of element i at incident energy E.<br />
τi(Ei): Mass photoabsorption coefficient of element i at energy Ei of characteristic line<br />
energy of element i.<br />
pi: Transition probability of observed line of element i.<br />
Ei edge: Energy of the absorption edge of the characteristic line of element i.<br />
Emax: Maximum energy of the incident radiation.
In general this equation is referred to as the Sherman equation. The sum over j is the<br />
sum over all characteristic lines of all elements strong enough to excite the observed line of<br />
element i. The first term in the above equation represents the primary absorption of the<br />
incident and characteristic line of element i in the specimen. The second term represents the<br />
secondary enhancement of the characteristic line of element i by all other characteristic lines<br />
fluoresced by the specimen. Though not included here, Shiraiwa and Fujino also gave an<br />
expression for the tertiary enhancement of the observed line of element i by all other<br />
characteristic lines in the sample.<br />
The Sherman equation set the stage for modern X-ray fluorescence spectroscopy. In the<br />
early days of modern x-ray fluorescence spectroscopy, the computing power required to<br />
determine the integrals of equation (VI.7) was not readily available to spectroscopy<br />
laboratories. Thus efforts turned to empirical approximations to the above equations. Using<br />
Sherman’s first equation for an incident beam of monochromatic radiation (Equation (VI.5)),<br />
Beattie and Brissey showed that by taking the ratio between counts measured for element i in<br />
an unknown to the counts measured from a pure specimen of element i, a system of<br />
simultaneous equations was created:<br />
Ci ⎛ ⎞<br />
⎜<br />
µ i<br />
+ ⎟<br />
⎜ sin ψ1 sin ψ2<br />
⎟<br />
Ri = Ci + ∑ Cj ⎜ ⎟=<br />
Ci + ∑ AijCj , (VI.8)<br />
j≠i ⎜ µ i ( E)<br />
µ i( Ei)<br />
j≠i + ⎟<br />
⎜ sin ψ1 sin ψ ⎟<br />
2 ⎟<br />
⎝ ⎠<br />
∑ C j<br />
j<br />
= 1.<br />
(VI.9)<br />
( E)<br />
µ i( E j)<br />
By measuring the intensity ratios Ri for a set of standards of known composition, this<br />
system of equations could be solved for each element in the substance to determine the<br />
constants Aij.<br />
The appearance of Ci on both sides of Equation (VI.9) meant that the system of equations had<br />
to be solved numerically. Observing that concentration was roughly proportional to measured<br />
x-ray intensity, Lucas-Tooth and Pyne rearranged the above equation to yield:<br />
⎛ ⎞<br />
Ci = ai + biIi⎜1+ ∑ kijI j ⎟<br />
⎝ j ⎠<br />
. (VI.10)<br />
Once the parameters ai, bi, and kij had been determined using a set of standards, the<br />
concentration could be determined directly from this equation for each element in the<br />
specimen. Though limited in accuracy away from concentrations of the standards used to<br />
determine the coefficients, this method was highly attractive in the days before inexpensive<br />
laboratory computers.<br />
III.2 Empirical Alpha Models<br />
A problem with the formulation of Beatie and Brissey was that the system of equations<br />
had no constant terms and so was over-determined. In the mid sixties, LaChance and Traill<br />
made the rather obvious observation that if Equation (VI.9) is substituted into Equation<br />
(VI.8), the over-determination of the system of equations was removed. This equation became<br />
the basis for the empirical alphas equations that followed:<br />
⎛ ⎞<br />
Ci = Ri⎜1+ ∑ αij<br />
Cj<br />
⎟<br />
⎝ j≠i ⎠<br />
. (VI.11)
Later attempts were made to find an empirical equation that more accurately accounted<br />
for the real relationship between measured X-ray intensity and specimen concentration.<br />
Claisse and Quintin took the original Sherman equation (Equation VI.5) and modeled for<br />
polychromatic incident radiation by taking the superposition of mass absorption coefficients<br />
at multiple energies.<br />
Though there is no direct theoretical support of this, it was generally found that the<br />
LaChance and Traill equation accounted for minor enhancement of X-ray intensities with<br />
negative alpha coefficients. Rasberry and Heinrich observed that strongly enhancing elements<br />
in binary mixtures yielded a concentration/intensity plot that did not follow the hyperbolic<br />
dependence of the LaChance and Traill equation. This led them to propose a modified form of<br />
the equation where a new term was to be used in place of the LaChance and Traill alpha<br />
coefficient for analytes causing significant secondary enhancement:<br />
⎛ ∑ βij<br />
C j ⎞<br />
i≠j Ci = R<br />
⎜<br />
i 1 ijC ⎟<br />
∑<br />
⎜<br />
+ α j +<br />
i≠j 1 C ⎟<br />
⎜ + i ⎟<br />
⎝ ⎠<br />
. (VI.12)<br />
By the middle of the seventies, other forms of the alpha correction models had been<br />
proposed. Most notable are the equations of Tertian who, observing that alpha coefficients are<br />
more properly not constant with specimen composition, proposed forms of the LaChance and<br />
Traill, and Rasberry and Heinrich equations utilizing alpha coefficients that were linear<br />
functions of element concentration Ci. Later, Tertian also showed that for a binary system, his<br />
modified form of the Rasberry and Heinrich equation reduced to the Claisse and Quintin<br />
equation.<br />
III.3 Fundamental Parameters Method<br />
Sherman’s equation (Equation VI.7) expresses the intensity of a characteristic X-ray<br />
fluoresced from an element contained in a specimen of known composition. By determining<br />
the concentrations of elements required to produce the measured set of intensities the<br />
composition of a specimen can be determined. The direct use of Sherman’s equation is termed<br />
‘the fundamental parameters method’. Instrument and measurement geometry effects are<br />
removed by measuring characteristic line intensities emanating from standards of known<br />
composition. Since this equation accounts for all absorption and enhancement, in theory only<br />
one standard is required for each element. It should be noted that the standard should also<br />
account for reflection from the surface of the specimen. As such, the surface texture of the<br />
standard should be similar to that of the unknown.<br />
Equation (VI.7) requires a knowledge of all elements contained in the specimen, the<br />
values of the total mass absorption and mass photoabsorption coefficients of each of these<br />
elements, and the step ratios of the mass photoabsorption coefficients at the absorption edges<br />
of the measured characteristic lines. A knowledge of the incident X-ray tube intensity<br />
distribution is also required. To account for secondary enhancement in the specimen, a<br />
knowledge of shell fluorescent yields and line transition probabilities are required.<br />
Criss and Birks were among the first to utilize the full fundamental parameters method.<br />
They were able to obtain uncertainties in concentrations for nickel and iron-base alloys<br />
between 0.1% and 1.7%. Aside from the requirement for significant computing power to<br />
evaluate the above integrals, the method is limited by the accuracy of the fundamental<br />
parameters themselves, and how well the tube spectrum is known. Determining a tube spectral<br />
distribution is no trivial matter. Due to the intensity of the primary radiation, direct<br />
measurement is not feasible. A common approach was to measure the reflected distribution
from sugar, but then this involved properties of reflection. In the original Criss and Birks<br />
paper, the measured spectra of Gilfrich and Birks were utilized. Later developments either<br />
continued to use the spectra of Gilfrich and Birks, allowed user-entered spectra which usually<br />
implied the use of the spectra of Gilfrich and Birks, or utilized Kramer’s Law to generate the<br />
spectrum.<br />
The need for computing power sufficient to evaluate the above integral and the lack of<br />
good knowledge of the tube spectrum led a number of authors to the use of an effective<br />
incident wavelength in place of the actual tube spectral distribution. Comparisons between<br />
fundamental parameters software packages utilizing effective wavelength and tube spectral<br />
distributions have demonstrated the shortcomings of this approach.<br />
The strength of the fundamental parameters method is that only one standard is<br />
required. Since the method predicts the degree of correction for a given composition, a single<br />
standard should be sufficient for all ranges of composition of an unknown specimen.<br />
Empirical alpha models of correction require significantly more standards, and these<br />
standards need to be of similar composition to the unknown being analyzed. Early<br />
developments of the fundamental parameters method noted that most of the fundamental<br />
parameters drop out for a pure substance. Taking the ratio between X-ray intensities measured<br />
from the unknown specimen to those measured from pure substances allowed the most direct<br />
use of the fundamental parameters method. As noted by Sherman himself, and later by Criss,<br />
Birks and Gilfrich, this tends to increase the reliance on the fundamental parameters that are<br />
known to be in error by as much as 10%. The degree of correction (and so the error in<br />
correction) is reduced by using standards similar in composition to the unknown.<br />
III.4 Fundamental Alphas<br />
The strength of the fundamental parameters method is that it is theoretically exact, and<br />
requires relatively few standards. Aside from the need for accurate fundamental parameters<br />
and a knowledge of the X-ray tube spectrum, the fundamental parameters method is<br />
numerically intensive, and so could take a significant amount of time to compute the<br />
composition of a specimen on early laboratory mini-computers. To take advantage of the few<br />
number of standards required by the fundamental parameters method and the relatively small<br />
computing resource needed for the empirical alphas methods, the hybrid fundamental alphas<br />
method came into being. These methods use the fundamental parameters method on larger<br />
computing facilities to compute empirical coefficients that are later used in traditional<br />
empirical alphas equations on a smaller laboratory computer.<br />
There are different methods used to compute theoretical alpha coefficients. One<br />
approach involves computing synthetic standards using the fundamental parameters method,<br />
and then computing the empirical alphas using standard regression techniques. Another<br />
approach is to compute the empirical alpha coefficients directly from Sherman’s equation for<br />
binary systems. Rousseau proposed a new empirical alphas equation that can be more directly<br />
related to Sherman’s equation:<br />
1+<br />
∑C<br />
α<br />
j<br />
Ci = Ri<br />
1+<br />
∑C<br />
ρ<br />
j<br />
j ij<br />
j ij<br />
where ρij are another set of alpha coefficients.<br />
; (VI.13)<br />
As noted by LaChance, the fundamental parameters method and theoretical alphas of<br />
the fundamental alphas method rely on inherently different concepts. This means that the
flexibility gained by the fundamental alphas approach implies a loss of the ability to define<br />
those coefficients explicitly from theory.<br />
It is our opinion that with the increase in laboratory computing power available by the<br />
mid 1990’s, the need for compromise with the fundamental parameters method has vanished.<br />
III.5 Semi-Quantitative Analysis<br />
A throw-back to the early days of XRF spectroscopy is semi-quantitative analysis. This<br />
method of matrix correction involved simply computing the concentration of an element from<br />
the product of the unknown to standard intensity ratio with the concentration of the element in<br />
the standard. More sophisticated so-called quantitative analysis methods utilized polynomials<br />
and peak counts corrected for background. These approaches are both mathematically and<br />
theoretically simplistic, and with well-designed modern XRF software, are completely<br />
unnecessary.
I. (The Photon Concept)<br />
EXERCICES<br />
An FM radio station of frequency 107.7 MHz puts out a signal of 50,00 W. How many<br />
photons per second are emitted?<br />
II. (X-<strong>Ray</strong> Production)<br />
III.<br />
IV.<br />
18 kV accelerating voltage is applied across an X-ray tube. Calculate:<br />
i. The velocity of the fastest electron striking the target,<br />
ii. The minimum wavelength in the continuous spectrum of X-rays produced.<br />
Mass of electron = 9 x 10 -31 kg;<br />
Charge on electron = 1.6 x 10 -19 C;<br />
h = 6.6 x 10 -34 Js;<br />
c = 3 x 10 8 m/s.<br />
An X-ray tube is operated with an anode potential of 10 kV and an anode current of 15<br />
mA. Calculate<br />
i. The number of electrons hitting the anode per second,<br />
ii. The rate of production of heat at the anode stating any assumptions made and<br />
iii. The frequency of the emitted X-ray photon of maximum energy.<br />
e = 1.6 x 10 -19 C h = 6.6 x 10 -34 J.s<br />
(a) What is the threshold frequency for the photoelectric effect on lithium (We = 2.9<br />
eV)?<br />
(b) What is the stopping potential if the wavelength of the incident light is 400 nm?<br />
V. (The Photoelectric Effect)<br />
A television tube operates at 20,000 V. What is λmin for the continuous x-ray spectrum<br />
produced when the electrons hit the phosphor?<br />
VI. (The Compton Effect)
A photon having 40 keV scatters from a free electron at rest. What is the maximum<br />
energy that the electron can obtain?<br />
VII. (Pair Production)<br />
How much photon energy would be required to produce a proton-antiproton pair?<br />
Where could such a high-energy photon come from?
I.<br />
Solution<br />
SOLUTIONS<br />
An FM radio station of frequency 107.7 MHz puts out a signal of 50.00 W. How many<br />
photons per second are emitted?<br />
The radio waves put out by the radio station are just EM waves. Therefore, we need to<br />
calculate the energy per photon from the wave frequency (ν = 107.7 MHz), and then<br />
determine the number of photons from the signal intensity of 50,000 Watts.<br />
The energy of each photon is:<br />
E=hν<br />
−34<br />
6<br />
= 6.63× 10 × 107.7× 10 J.s/s<br />
-26<br />
=7.14× 10 J<br />
The intensity is just I = NE, where N is the number of photons. Therefore, the number of<br />
photons is:<br />
I<br />
N<br />
E<br />
50.00<br />
W/J<br />
26<br />
7.14 10 −<br />
=<br />
=<br />
×<br />
29<br />
N = 7.00× 10 photons/s<br />
This is a large number of photons per second, and is expected because the station power is<br />
quite large.<br />
II.<br />
18 kV accelerating voltage is applied across an X-ray tube. Calculate:<br />
iii.The velocity of the fastest electron striking the target,<br />
iv.The minimum wavelength in the continuous spectrum of X-rays produced.<br />
Mass of electron = 9 x 10 -31 kg;<br />
Charge on electron = 1.6 x 10 -19 C;<br />
h = 6.6 x 10 -34 Js;<br />
c = 3 x 10 8 m/s.<br />
Solution<br />
V = 18 x 10 3 V<br />
me = 9 x 10 -31 kg<br />
e = 1.6 x 10 -19 C<br />
h = 6.6 x 10 -34 Js
c = 3 x 10 8 m/s<br />
(i)<br />
1 2<br />
mV = eV<br />
2<br />
⇒ V = =<br />
2eV −19<br />
3<br />
2× 1.6× 10 × 18× 10<br />
m<br />
−31<br />
9× 10<br />
− 19+ 3+ 31−1 2 16 10 2<br />
⇒ V = × × ×<br />
⇒ V = 64× 10<br />
⇒ V = 8× 10 m ⋅<br />
(ii)<br />
min<br />
7<br />
14<br />
−1<br />
s<br />
hc<br />
eV =<br />
λ min<br />
hc<br />
⇒ λ min =<br />
eV<br />
−34<br />
8<br />
6.63× 10 × 3× 10<br />
⇒ λ min =<br />
−19<br />
3<br />
1.16× 10 × 18× 10<br />
⇒ λ ≈<br />
III.<br />
-11<br />
6.9 x 10 m<br />
An X-ray tube is operated with an anode potential of 10 kV and an anode current of 15 mA.<br />
Calculate<br />
i.the number of electrons hitting the anode per second,<br />
ii.the rate of production of heat at the anode stating any assumptions made,<br />
iii.the frequency of the emitted X-ray photon of maximum energy.<br />
e = 1.6 x 10 -19 C<br />
h = 6.6 x 10 -34 Js<br />
Solution<br />
I = 15 x 10 -3 A<br />
V = 10 x 10 3 V<br />
e = 1.6 x 10 -19 C<br />
h = 6.6 x 10 -34 Js<br />
(i)
charge<br />
= 15× 10 Cs<br />
second<br />
−3<br />
-1<br />
charge<br />
−19<br />
= 1.6× 10 C<br />
electron<br />
−3<br />
electron 15× 10<br />
⇒ =<br />
second −19<br />
1.6× 10<br />
electron<br />
16<br />
⇒ = 9.375× 10 / second<br />
second<br />
(ii)<br />
Power = V × I<br />
-3 -3<br />
Power = 10×10 × 15×10<br />
Power = 150W<br />
Entire range of wavelengths produced gives heat.<br />
(iii)<br />
hfmax = eV<br />
eV<br />
⇒ fmax<br />
=<br />
h<br />
−19<br />
3<br />
1.6× 10 × 10× 10<br />
⇒ fmax<br />
=<br />
−34<br />
6.63× 10<br />
18<br />
⇒ fmax = 2.4× 10 Hz<br />
IV.<br />
(a) What is the threshold frequency for the photoelectric effect on lithium (We = 2.9 eV)?<br />
The lithium emitter has a work function of We = 2.9 eV. The important thing to remember is<br />
to not get confused with the units of energy (1 eV = 1.6 x 10 -19 Joules).<br />
The threshold frequency gives the photon energy (hν) that equals the work function of the<br />
emitter (We):<br />
We 2.9 eV<br />
ν o = =<br />
h −15<br />
4.136× 10 eV. s<br />
14<br />
ν o = 7.0× 10 Hz<br />
Note the value of Planck's constant in eV.s was used instead of J.s to ensure that the units are<br />
correct.
(b) What is the stopping potential if the wavelength of the incident light is 400 nm?<br />
The stopping potential is calculated using Einstein's Photoelectric Equation for the light<br />
wavelength λ = 400 nm. The photon energy for the incident wavelength is:<br />
hc 1240 eV. nm<br />
E = hν = = = 3.10 eV<br />
λ 400 nm<br />
Note that a standard value of hc = 1240 eV.nm was used. Get used to using certain<br />
combinations of fundamental constants, and manipulating the value of these constants in<br />
different units. This will save a lot of time doing calculations, and will ensure that answer<br />
comes out with the correct units.<br />
The stopping potential energy is then:<br />
eVs = hν− We = 3.10 − 2.9eV<br />
eVs = 0.20 eV<br />
Therefore, the stopping potential is VS = 0.20 Volts (and is negative by definition).<br />
V.<br />
A television tube operates at 20,000 V. What is λmin for the continuous x-ray spectrum<br />
produced when the electrons hit the phosphor?<br />
The minimum emission wavelength for X-ray generation is given by the Duane-Hunt Rule.<br />
By accelerating the electron through a region of total potential V = 20,000 volts, we have<br />
given it a kinetic energy of eV = 20,000 eV = 20 keV. Therefore:<br />
VI.<br />
hc 1240 eV. nm<br />
λ min = =<br />
eV 20.000eV<br />
λ min = 0.062 nm<br />
A photon having 40 keV scatters from a free electron at rest.What is the maximum energy that<br />
the electron can obtain?<br />
The photon scattering off the electron undergoes a Compton shift. It transfers some of its<br />
energy and momentum to the electron and emerges with a reduced energy and momentum<br />
(and therefore increased wavelength λ'). The electron kinetic energy is then equal to the<br />
difference between the photon energies (by conservation of energy).<br />
The electron will have a maximum kinetic energy when the Compton shift is largest. The<br />
maximum Compton shift occurs when cosθ is a minimum (cosθ = -1). The Compton shift is<br />
then given by:
h<br />
∆λ = ( 1 − cosθ ) = 2λ c = 0.0486 Angstroms<br />
moc Where λC is the Compton wavelength (λC = 0.0243 Angstroms).<br />
The incident photon wavelength is:<br />
hc 1240 eV. nm<br />
λ= = = 0.031nm = 0.31 A<br />
E 40000eV<br />
The scattered photon wavelength is then λ'= λ + ∆λ = 0.31+ 0.0286 A = 0.3386 A. The<br />
equivalent photon energy then:<br />
hc 1240 eV. nm<br />
E′ = = = 36.621eV ≈36.6<br />
keV<br />
λ′<br />
0.03386nm<br />
Therefore, the maximum electron kinetic energy is:<br />
VII.<br />
Ke = E − E′ = 40.0 − 36.6keV = 3.4keV<br />
How much photon energy would be required to produce a proton-antiproton pair? Where<br />
could such a high-energy photon come from?<br />
According to conservation of energy, the minimum photon energy required to produce a<br />
particle-antiparticle pair is just equal to twice the rest mass energy of the particle:<br />
2<br />
E = 2 mpc = 2× 938.3 MeV = 1.877 GeV<br />
This energy could be obtained in a particle accelerator or a synchrotron.