An Introduction to the Ion-Optics of Magnet Spectrometers

An Introduction to the Ion-Optics of
Magnet Spectrometers
U. Tokyo, RIKEN
The 14 th RIBF Nuclear Physics Seminar
Series of Three Lectures
CNS, University of Tokyo
February 27, 2006
Georg P. Berg
University of Notre Dame

The Lecture
Series
1 st Lecture: 2/27/06, 10:30 am: Formalism of ion-optics optics and design
of a complete system
2 nd Lecture: 2/27/06, 1:30 pm: Ion-optical optical elements, design, systems
3 rd Lecture: 2/27/06, 3:00 pm: Experiments with dispersion matched
high resolution spectrometers

1st Lecture
1 st Lecture: 2/27/06, 10:30 – 11:30 am: Formalism and ion-optical optical elements
• Motivation, references, remarks (4 - 8)
• The driving forces (9)
• Definitions & first order formalism (10 - 16)
• Phase space ellipse, emittance, measurement (17 - 27)
• Higher orders, Taylor expansion (28 - 30)
• Solving the equation of motion (31)
• Design and construction of a Fragment Separator (32 - 43)

Motivation
• Manipulate charged particles (β +/− , ions, like p,d,α, ΗΙ)
• Beam lines systems
• Magnetic & electric analysis/ separation (Experiments!)
• (Acceleration of ions, not covered in these lectures)

Who needs ion-optics optics ?
• A group of accelerator physicists are using it to build machines
to provide high energetic ion beam for research and applications!
• Many physicists using accelerators, beam lines and magnet system
(or their data) needs some knowledge of ion-optics.
• This lecture series is an introduction to the last group and I will
discuss some applications used by nuclear physicists.

Introductory remarks
• Introduction for physicists Focus on ion-optical definitions, and tools that
are useful for physicist working with spectrometers.
• Light optics can hardly be discussed without lenses & optical instruments
ion-optics requires knowledge of ion-optical elements.
• Analogy between Light Optics and Ion-Optics is useful but limited.
• Ion-optical & magnet design tools needed to understand electro-magnet systems.
Ion-optics is not even 100 years old and (still) less intuitive than optics developed
since several hundred years
Historical remarks:
Rutherford, 1911, discovery of atomic nucleus
Galileo
Telescope 1609
Optics in Siderus Nuncius 1610

Basic tools of the trade
• Geometry, drawing tools, CAD drafting program (e.g. AutoCad)
• Linear Algebra (Matrix calculations), first order ion-optics (e.g. TRANSPORT)
• Higher order ion-optics code to solve equation of motion, e.g. COSY Infinity,
GIOS, RAYTRACE (historic)
• Electro-magnetic field program (solution of Maxwell’s Equations),
(e.g. finite element (FE) codes, 2d & 3d: POISSON, TOSCA, MagNet)
• Properties of incoming charged particles and design function of electro-magnetic
facility, beam, reaction products (e.g. kinematic codes, charge distributions of
heavy ions, energy losses in targets, detectors, etc, e.g. LISE++ for fragment
separators)
• Many other specialized programs, e.g. for accelerator design (e.g. synchrotrons,
cyclotrons) or gas-filled systems, are not covered in this lecure series.

Literature
• Optics of Charged Particles, Hermann Wollnik, Academic Press, Orlando, 1987
• The Optics of Charged Particle Beams, David.C Carey, Harwood Academic
Publishers, New York 1987
• Accelerator Physics, S.Y. Lee, World Scientific Publishing, Singapore, 1999
• TRANSPORT, A Computer Program for Designing Charged Particle Beam
Transport Systems, K.L. Brown, D.C. Carey, Ch. Iselin, F. Rotacker,
Report CERN 80-04, Geneva, 1980
• Computer-Aided Design in Magnetics, D.A. Lowther, P. Silvester, Springer 1985

Ions in static or quasi-static static electro-magnetic fields
Lorentz Force
For ion acceleration electric forces are used.
(1)
q = electric charge
B = magn. induction
E = electric field
v = velocity
For momentum analysis the magnetic force is preferred because the
force is always perpendicular to B. Therefore v, p and E are constant.
Force in magnetic dipole B = const: p = q B ρ
p = mv = momentum
Dipole field B
perpendicular
.
to paper plane
Object (size x 0 )
Radius
ρ
p+δp
p
x
Note: Dispersion δx/δp
used in magnetic analysis,
e.g. Spectrometers, magn.
Separators,
ρ = bending radius
Βρ = magn. rigidity
General rule:
Scaling of magnetic system
in the linear region results
in the same ion-optics

Definition of BEAM for mathematical formulation of ion-optics
optics
What is a beam, what shapes it,
how do we know its properties ?
• Beam parameters, the long list
• Beam rays and distributions
• Beam line elements, paraxial lin. approx.
higher orders in spectrometers
• System of diagnostic instruments
Not to forget: Atomic charge Q
Number of particles n

Code TRANSPORT:
(x, Θ, y, Φ, 1, dp/p)
(1, 2, 3, 4, 5, 6 )
Convenient “easy to use” program
for beam lines with paraxial beams
Not defined in the figure are:
dp/p = rel. momentum
l = beam pulse length
Defining a RAY
central ray
Ion-optical
element
All parameters are relative
to “central ray”
Code: COSY Infinity:
(x, a, y, b, l, δ K , δ m , δ z )
Needed for complex ion-optical systems including several
charge states
different masses
velocities (e.g. Wien Filter)
higher order corrections
Not defined in the figure are:
δ K = dK/K = rel. energy
δ m = dm/m = rel. energy
δ z = dq/q = rel. charge change
a = p x /p 0
b = p y /p 0
All parameters are relative
to “central ray” properties
Note: Notations in the Literature is not consistent! Sorry, neither will I be.

TRANSPORT
Coordinate System
Ray at initial
Location 1

6x6 Matrix
representing
optic element
(first order)
Transport of a ray
(2)
Ray after
element at
Location t
Ray at initial
Location 0
Note: We are not building
“random” optical elements.
Many matrix elements = 0
because of symmetries, e.g.
mid-plane symmetry

TRANSPORT matrices of a
Drift and a Quadrupole
For reference of TRANSPORT code and formalism:
K.L. Brown, F. Rothacker, D.C. Carey, and Ch. Iselin,
TRANSPORT: A computer program for designing
charged particle beam transport systems, SLAC-91,
Rev. 2, UC-28 (I/A), also: CERN 80-04 Super Proton
Synchrotron Division, 18 March 1980, Geneva,
Manual plus Appendices available on Webpage:
ftp://ftp.psi.ch/psi/transport.beam/CERN-80-04/
David. C. Carey, The optics of Charged Particle Beams,
1987, Hardwood Academic Publ. GmbH, Chur Switzerland

6x6 Matrix
representing
first optic element
(usually a Drift)
Transport of a ray
though a system of
beam line elements
x n = R n R n-1 …R 0 x 0
(3)
Ray at final
Location n
Ray at initial
Location 0
(e.g. a target)
Complete system is represented by
one Matrix R system = R n R n-1 …R 0
(4)

Wollnik, p. 16
(x|a) = 0
Point-to-point focusing
(a|x) = 0 “Telescope”
Parallel-to-parallel focusing
Geometrical interpretation
of some TRANSPORT
matrix elements
Achromatic system:
R 16 = R 26 = 0
(x|x) = 0
Parallel-to-point focusing
(a|a) = 0
Point-to-parallel focusing
Focusing Function
(x|a) Wollnik
= dx/dΘ physical meaning
= (x|Θ) RAYTRACE
= R 12 TRANSPORT

Defining a BEAM Ellipse Area = π(det σ) 1/2
is called “Phase space”
Emittance:
ε = √⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ σ 11 σ 22 -(σ 12 ) 2 (5)
2 dimensional cut x-Θ is shown
Emittance ε is constant for
fixed energy & conservative
forces (Liouville’s Theorem)
Warning:
This is a mathematical abstraction of a beam:
It is your responsibility to verify it applies to your beam
Note: ε shrinks (increases) with
acceleration (deceleration); Dissipative
forces: ε increases in gases; electron,
stochastic, laser cooling
Attention:
Space charge effects occur when the particle density is high,
so that particles repel each other

Equivalence
of Transport of
ONE Ray ⇔ Ellipse
Defining the σ Matrix representing a Beam

The 2-dimensional 2
case ( x, Θ )
Ellipse Area = π(det σ) 1/2
Emittance ε = (det σ) 1/2 is constant
for fixed energy & conservative
forces (Liouville’s Theorem)
Note: ε shrinks (increases) with
acceleration (deceleration);
Dissipative forces: ε increases in
gases; electron, stochastic, laser
cooling
σ = ⎛σ 11 σ 21 ⎫
⎝σ 21 σ 22 ⎭
σ −1 = 1/ε 2 ⎛σ 22 −σ 21 ⎫
⎝−σ 21 σ 11 ⎭
Exercise 1:
Show that:
2 dimensional cut x-Θ is shown
Real, pos. definite
symmetric σ Matrix
σσ −1 = ⎛1 0 ⎫
⎝ 0 1⎭
Inverse Matrix
= I (Unity Matrix)
2-dim. Coord. vectors
(point in phase space)
X =
⎛x ⎫
X T = (x Θ)
⎝Θ⎭
Ellipse in Matrix notation:
X T σ −1 X = 1
(6)
Exercise 2: Show that Matrix notation
is equivalent to known Ellipse equation:
σ 22 x 2 −2σ 21 x Θ + σ 11 Θ 2 = ε 2

Courant-Snyder Notation
In their famous “Theory of the Alternating Synchrotron” Courant and Snyder used a
Different notation of the σ Matrix Elements, that are used in the Accelerator Literature.
For you r future venture into accelerator physics here is the relationship between the
σ matrix and the betatron amplitude functions α, β, γ or Courant Snyder parameters
σ =
⎛σ 11 σ 21 ⎫
⎝σ 21 σ 22 ⎭
= ε
⎛ β −α ⎫
⎝−α γ ⎭

Transport of 6-dim 6
σ Matrix
Consider the 6-dim. ray vector in TRANSPORT: X = (x, Θ, y, Φ, l, dp/p)
Ray X 0 from location 0 is transported by a 6 x 6 Matrix R to location 1 by: X 1 = RX 0
(7)
Note: R maybe a matrix representing a complex system (3) is : R = R n R n-1 …R 0
Ellipsoid in Matrix notation (6), generized to e.g. 6-dim. using σ Matrix: X 0
T
σ 0
−1
X 0 = 1
Inserting Unity Matrix I = RR -1 in equ. (6) it follows X 0
T
(R T R T-1 ) σ 0
−1
(R -1 R) X 0 = 1
from which we derive (RX 0 ) T (Rσ 0 R T ) -1 (RX 0 ) = 1
(6)
(8)
The equation of the new ellipsoid after transformation becomes X 1
T
σ 1
−1
X 1 = 1
(9)
where σ 1 = Rσ 0 R T (10)
Conclusion: Knowing the TRANSPORT matrix R that transports one
ray through an ion-optical system using (7) we can now also transport
the phase space ellipse describing the initial beam using (10)

The transport of rays and phase ellipses
in a Drift and focusing Quadrupole, , Lens
Focus
Focus
2.
Lens 2
3.
Lens 3
Matching of emittance and acceptance

Increase of Emittance ε
due to degrader
Focus
for back-of-the-envelop discussions!
A degrader / target increases the
emittance ε due to multiple scattering.
The emittance growth is minimal when
the degrader in positioned in a focus
As can be seen from the schematic
drawing of the horizontal x-Theta
Phase space.

Emittance ε measurement
by moving viewer method
The emittance ε is an important
parameter of a beam. It can be
determined by measurements
of x max
at 3 locations
Viewer V1 V2 V3
Beam
⎯⏐⎯⎯⎯⏐⎯⎯⎯⎯⏐⎯→
L 1 L 2
L = Distances between viewers
( beam profile monitors)
σ 11 + 2 L 1 σ 12 + L 2
(x max (V2)) 2 = 1 σ 22
(11)
(x max (V3)) 2 = σ 11 + 2 (L 1 + L 2 )σ 12 + (L 1 + L 2 ) 2 σ 22
where σ 11 = (x max (V1)) 2
Emittance:
ε = √ ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ σ 11 σ 22 -(σ 12 ) 2
(12)
Discuss practical aspects
No ellipse no ε? Phase space!

Emittance ε measurement
by tuning a quadrupole
Lee, p. 55
The emittance ε can also be
determined with a quadrupole
and a viewer at distance L.
x max =
σ 11 (1 + σ 12 L/ σ 11 − L g) + (εL) 2 /σ 22
(13)
g =
dBz/dx ∗ l
Bρ
(Quadr. field strength
l = eff. field length)
(14)
L = Distance between quadrupole and
beam profile monitor
Take minimum 3 measurements of
x max (g) and determine Emittance ε

IUCF, K600 Spectrometer
Diagnostics in focal plane
of spectrometer
Typical in focal plane of
Modern Spectrometers:
Two position sensitive
Detectors:
Horizontal: X1, X2
Vertical: Y1, Y2
Fast plastic scintillators:
Particle identification
Time-of-Flight
Measurement with IUCF K600
Spectrometer illustrates from top
to bottom: focus near, downstream
and upstream of
X1 detector, respectively

Kinematic affect in K600
spectrum with impurities

Higher order beam aberrations
Example Octupole
(S-shape in x-Θ plane
3 rays in focal plane
1.
2.
3.
Detector X1 X2
T 126
T 1222
1.
2.
3.
Other Example:
Sextupole T 122
C-shape in x-Θ plot

Effect of 2 nd order aberrations
Saturation of
Grand Raiden
at 450 MeV
(3He,t)

,l
(15)
Taylor expansion
Note: Several notations are in use for
6 dim. ray vector & matrix elements.
R nm
= (n|m)
TRANSPORT RAYTRACE
Notation
Linear (1 st order)TRANSPORT Matrix R nm
Remarks:
• Midplane symmetry of magnets
reason for many matrix element = 0
• Linear approx. for “well” designed
magnets and paraxial beams
• TRANSPORT code calculates 2 nd order
by including Tmno elements explicitly
• TRANSPORT formalism is not suitable
to calculate higher order ( >2 ).

(15)
Solving the equations
of Motion
Methods of solving the equation of motion:
1) Determine the TRANSPORT matrix.
2) Code RAYTRACE slices the system in small
sections along the z-axis and integrates numerically
the particle ray through the system.
3) Code COSY Infinity uses Differential Algebraic
techniques to arbitrary orders using matrix
representation for fast calculations

Design of a Fragment Separator Define the purpose and design parameters
This step in the design determines what the instrument will be able to do:
Particle type, energy, acceptances
Now you need a good “concept”

Schematic Layout
of an
Achromatic Magnet
System

An Achromatic
Energy Selection System
(ESS in Proton Therapy
Facility)
R 16 = R 26 = 0

Achromatic magnet separator with Wedge
↓ “Wedge” = 0.1 mm “Si=detector”
A
^
E-loss in WEDGE ΔE ~ Z 2 /v 2
Isotopes with different Z have
different velocities v
Therefore A/Z selection in B
B
Figure from Experimental Techniques at NSCL, MSU, Th. Baumann, 8/2/2002
Effect of “Wedge” ⇒
Note:
For large dp/p) the
degrader should be
Wedge-shaped to
restore achromaticity
effected by degrader
with constant thickness

A1900 MSU/NSCL Fragment Separator
Ref. B.Sherrill, MSU

Design of a Fragment Separator 1 st order envelop of beam line & magnet system
Calculated with
TRANSPORT
This study determines the ion-optics & required fields:
Dispersion, envelops with dp=0, 2.6%
Horz. & vert. Good-field regions, Field Lengths, Strengths

TRIμP an achromatic secondary beam separator
SH = Slits
Q = Quadrupoles
B = Dipoles
Section B
Wedge/ΔE-Si
Design parameters
Section A
ΔE-Si detector
meter

TRIμP ion-optics
optics
This study determines the final performance including higher orders:
Details of rays, focal plane image
Write “Specifications” for manufacturer

TRIμP
Magnet Installation &Alignment
December 2003

Intermediate, dispersive Focal
Plane
BEAM

Commissioning
( TRIμP Production of clean radioactive beam of 21 Na)
Production of 21 Na via H( 21 Ne,n) 21 Na with
21
Ne 7+ beam at 43MeV/nucleon using
theTRIμP Separator, KVI Groningen.

The First Users at the
TRIμP Separator
Precision measurement of GTbranching
ratio in 21 Na b + decay
21
Na
b +
0.0004%
94.98%
5.02%
352 keV
21
Ne
g
Lynda Achouri (See proposal to this PAC)
Jean-Claude Angélique
both LPC, CAEN, France

End Lecture 1