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26 2. Theoretical background
I. Photoemission cross-sections Due to the large energy difference between exciting X-
rays and the binding energy of the initial state electrons in the solid, in particular for valence
band photoemission, the photoionization cross-section σ for hard X-ray excitation is small.
For an individual orbital (nlj), it is continuously reduced as
⎧
⎪⎨ (E
σ nlj (hν) ∝ kin ) −7 /2 , for s subshells, and
⎪⎩ (E kin ) −9 /2 , for p, d, and f subshells. 100 (2.25)
The significant decrease of atomic cross-sections for hard X-ray photoelectron excitation is
shown in Fig. 2.15 (inset).
Figure 2.15.: Photoemission
cross-sections for hard
X-ray excitation. Au
valence levels in HAX-
PES deviate significantly
from soft X-ray photoemission.
85 The inset
exemplifies the drop of
photoemission crosssection
with excitation
energy for Al. 101
II. Forward scattering and Debye-Waller factors for photoemission Only unscattered or
coherently elastically scattered photoelectrons contribute to an analyzable spectrum. Elastic
scattering events can deflect the photoelectrons which are emitted in direction of the surface
normal away from that direction. For high excitation energies (hν = 5–15 keV), Thompson
and Fadley (1984) have shown 102 that the forward scattering (i. e. within 90 ◦ in direction of
the original emission direction) becomes significantly narrower for photoemission from surface
layers (Fig. 2.16). This reduces the directional anisotropy of the photoelectrons which is
advantageous for angular-resolved PES. Moreover, the narrow emission plume of photoelectrons
can be better approximated by a simple expression for the electron’s effective attenuation
length, and thus allows one to predict information depths of a HAXPES experiment
more accurately. 86 The photoemission process inside a solid can be thought of as a scattering
process of the photoexcited ion inside the crystal. 80 Thus, the photoelectrons reflect the effect
of the lattice vibrations of the crystal in a way similar to that in X-ray diffraction or neutron
scattering, which is described by the Debye-Waller factor. The Debye-Waller factors for photoemission
intensities are strongly dependent on temperature and on the excitation energy:
an increase of temperature leads to a decreased intensity of coherently scattered photoelectrons.
A calculated example is shown in Fig. 2.17, from which we clearly see that low sample
temperatures are advantageous, and for larger X-ray excitation energies the Debye-Waller
factor significantly drops. In practice, at a fixed excitation energy, the cooling of the sample
will significantly increase the analyzable intensity (i. e. unscattered or coherently scattered
photoelectrons).
The effective attenuation length, λ ∗ , is introduced by eq. (2.26) on page 28.