Theory of High-Energy Photoemission

The angular distribution of emitted photoelectrons is theoretically studied for high-energy X-rays where electric quadrupole and magnetic dipole transitions are taken into account in addition to the electric dipole transition. We apply irreducible tensor expansion of photon ﬁeld instead of usual power series expansion. The present numerical calculations show that the conventional power series expansion works so well up to quite high energy for the core excitation. We also discuss the recoil eﬀects of nuclei associated with photoemission excited by high-energy X-rays, which gives rise to the energy shift and the peak broadening. For practical calculation we furthermore introduce Debye approximation for phonon spectra, which brings about the simple free atom recoil energy shift. In the simple Debye approximation we can expect no contribution to the broadening. [DOI: 10.1380/ejssnt.2005.373]


I. INTRODUCTION
So far most of XPS studies have been devoted to surface science by use of photoelectrons with small mean free path (∼0.5nm) at ∼100eV. [1] On the other hand, highenergy XPS has been employed to study bulk electronic structures, which raises some problems about breakdown of the electric dipole approximation and recoil effects.
The former problem has been studied in photoemission from free atoms such as Ne, Ar and Kr from experimental [2] and theoretical [3] sides. For small polyatomic molecules Langhoff et al. derive a new theoretical expression including the lowest-order nondipole retardation terms. [4] About the latter problem, Varga et al. have observed experimental results on the energy shifts and broadening by use of a high-energy-resolution spectrometer from different (C, Si, Ni and Au) surfaces in quite high energy region (1-5 keV). [5] Werner et al. have also observed prominent recoil shifts in quasi-elastic electron reflection spectra for energies between 50 and 3400 eV. [6] They found that the simple classical approximation works rather well as pointed out by Laser and Seah. [7] Domcke and Cederbaum investigated the recoil effects for diatomic molecules. [8] Their interests were focused on valence electron excitations. As far as we know the recoil effects in photoemission processes has rarely been studied.
In this work, the angular distribution of emitted photoelectrons is theoretically studied for high-energy X-rays where electric quadrupole and magnetic dipole transitions are taken into account in addition to the electric dipole transition. For multipole expansion of photon field, power series expansion is commonly used, exp(iqx) ≈ 1 + iqx + · · · . [9] Whereas it is known that this expansion causes mixture of various multipoles. On the contrary irreducible tensor gives proper expansion. We apply irreducible tensor expansion of photon field instead of usual power series expansion. Numerical calculations for noble gases demonstrate the importance of nondipole contribu-tion in discussing the angular distribution.
We also discuss the recoil effects of nuclei associated with photoemission excited by high-energy X-rays, which give rise to the energy shift. In this case the atomic displacement after the core-hole production also plays some important roles. For practical calculation we furthermore introduce Debye approximation for phonon spectra, which gives rise to the simple free atom recoil energy shift for the high-energy photoemissions. Numerical calculations are carried out for Li, Be and C (graphite and diamond) powders. In general the interference between the recoil and the Franck-Condon processes can contribute to the broadening, but in the simple Debye approximation we can expect no contribution to the broadening.

II. NONDIPOLE EFFECTS
At first we discuss the breakdown of the electric dipole transition excited by high-energy X-rays.

A. Photoemission Theory
Photoemission intensity from a core φ c (r)(= R lc (r)Y Lc (r)) measuring photoelectron with momentum k (energy ε k = k 2 /2) is given for the excitation by photon field q (energy ω q = cq) [10] as where ψ − k is the photoelectron wave function, E 0 and E * 0 are the energy of the target before and after the excitation. The incident X-rays are linearly polarized in the z-direction, and propagating to positive x-direction. Site T-matrix expansion of ψ − k yields [11] where Z 1 , Z 2 , Z 3 , · · · are direct term (suffering no elastic scattering), single and double scattering terms, and so on.
Let first investigate the atomic term Z 1 in detail: where δ A l is phase shift of the X-ray absorbing atom A, and R l is the radial part of l-th partial wave of the photoelectrons.
For soft X-ray photoemission, qr c 1 is satisfied, where r c is the size of the core function φ c . In this case we can safely use electric dipole approximation exp(iqx) ≈ 1. On the other hand qr c 1 is not satisfied for hard X-ray photoemission: In this case we have to pay special care for the analyses of X-ray photoemission angular distribution.
We carefully study the effect of the operator ∆ on the deep core function φ c from which a photoelectron is excited, Two different spherical harmonics are combined to one by use of the Gaunt integrals G( In the first term of eq. (4) we can pick up the electric dipole term (l = 1) Substitution j 0 (qr) + j 2 (qr) 1, which correspond to the lowest order term in the power series expansion, yields conventional electric dipole approximation. The first order term j 1 (qr) gives rise to the next order contribution. In eq. (5) l is restricted to 0 or 2, but the monopole term vanishes because Y 10 (q) = 0 (q x). The quadrupole term is finite and is given by Higher order terms of electric multipole moments can be obtained in the same way, whereas no magnetic multipole term can be produced from the first term of the large parenthesis in eq. (4).
The operation of the second term of the large parenthesis in eq. (4) has no influence on the spherically symmetric core functions because L ± Y 00 = 0, whereas it has finite contribution to the photoemission from nonspherical core functions. We should note that this term can give magnetic dipole transition operator. The most important contribution from j 0 is just electric dipole operator The next important contribution from j 1 (qr) is magnetic dipole term and given by

B. Photoemission Angular Distribution from Spherical Core Function
In this case the second term of the large parenthesis in eq. (4) can be neglected. The direct photoemission amplitude is thus simply given by where the radial integrals ρ s (1), ρ s (2) depend on q and k Within the single-atom approximation where the scatterings from neighboring atoms are neglected, the photoemission angular distribution is given by The first term shows wellknown electric dipole photoemission angular distribution, the second and the third terms show the electric quadrupole angular distribution, and their interference term. The angular distribution of the photoelectrons emitted by unpolarized X-ray is easily obtained by use of the averaging procedure described in the previous paper [12] whereθ is the angle measured from the incident photon propagation direction. The above equation is also written in the Cooper's formula eq. (17) [3] where β = 2, δ = 0 and Figure 1 shows the photon energy ω q dependence of the parameter γ calculated in the present work. The parameter γ is one of the important measure of the relative contribution of the E2 transition, i.e. nonzero γ indicates the breakdown of the electric dipole approximation. As expected the photoemission from the shallow (large r c ) cores gives the larger value than that from the deep (small r c ) cores. For example, we observe that γ 1.4 for Ne 1s at ω q = 5keV, and 0.8 for Ar 1s excitation, because Ar 1s core is much smaller than Ne 1s. Figure 2 compares the calculated γ parameters with the experimental ones for Ar 1s [13] and Ne 2s [14] photoemission. We observe good agreement.
The irreducible tensor correction is quite small (< 1%) in the energy range and all atoms considered here, since j l (qr c ) behaves in the limit qr c 1 , In the usual power series expansion the first term of above equation is only taken into account, and the irreducible tensor correction is in the order of (qr c ) 2 in the K-shell excitation, which is in the order octupole transition. Typically these correction is vanishingly small.

C. Photoemission Angular Distribution from Nonspherical Core Function
In this case we should take ∆ E1 defined by eq. (8) in addition to ∆ E1 even in the electric dipole transition approximation. The direct photoemission amplitude Z 1 from a deep p or d orbital is given by From now on our attention is focused on the photoemission from a deep p orbital in this subsection. The photoemission intensity from a deep p orbital is thus given by eq. (16), where the scattering from neigh- boring atoms are neglected The E1 · E2 and E1 · M 1 interference terms (the second and the third terms) are the most important correction terms to the electric dipole term (the first term). The sum 1 + βP 2 (cos θ) + δ + γ cos 2 θ sin θ cos φ (17) which is well-known result in atomic physics. [3] In the case of photoemission from nonspherical core the parameter β is important to discuss the angular distribution, and the relative contribution of the E2 transition is estimated by the parameter ζ(= γ + 3δ). Figure 3 shows the photon energy dependence of the parameter ζ calculated in the present work, which shows monotonically increasing function of the photon energy. The correction due to irreducible tensor is less than 1% in this calculation, thus we can safely use the power series expansion again.

III. RECOIL EFFECTS IN HIGH-ENERGY PHOTOEMISSION
In the analyses of high-energy photoemission spectra, it is important to take the recoil effect into account. The recoiled atomic motion can excite the phonons around the X-ray absorbing atom. We collectively designate phonon states as v = (v 1 , v 2 , · · · ). At first we use a single-site approximation which works so well in high-energy photoemission analyses excited from randomly oriented systems like polycrystals. [10] In the single-site approximation, the photoemission amplitude M (k) c measuring photoelectrons with momentum k is given for the excitation by photon field with momentum q (energy ω q ) by where |φ − Ak is now written as In the above equation the outgoing wave φ A k (r − R A ) can be expanded in terms of spherical waves for the spherical potential v A .
So far we have neglected the nuclear motions, and treated R A as just a parameter. In the case we go beyond the static model, we should calculate the photoemission intensity incorporating the vibrational transition due to nuclear recoil, where b s (b † s ) designates the phonon annihilation (creation) operator for the normal mode s. We should does not depend on R A . In order to calculate the phonon effects, we write A is the equilibrium position of the X-ray absorbing atom A, and In the case of the core excitation the displacement of the equilibrium position is expected to be quite small, because the core electrons do not take part in chemical bond. [15] We thus use linear approximation for the core-hole vibra- which has eigenstates |v where H v is the harmonic ground-state vibrational Hamiltonian. From now on we use Q = q−k, and try to rewrite eq. (20) in a correlation function formula for the energy where A = iQ · u A . Next we take average over the vibrational states in thermal equilibrium by use of the density matrix ρ 0 = exp(−βH v )/Tr exp(−βH v ), Correlation function D(Q; t) is defined as where X = Tr(ρ 0 X) Our main task is thus reduced to the calculation of the correlation function D(Q; t). We can evaluate the moments of the spectral function D(Q; ω) from the correlation function D(Q; t); they are related by [16] From eq. (26) we can calculate n-th order moment The peak broadening is thus written by [17] (∆ω) where α(ω), C(ω) and H(ω) are spectral function due to the recoil, Franck-Condon and their interference effects. The peak broadening due to the nuclear displacement does not depend on the kinetic energy of photoelectrons, and the interference term is in the order of Q. To explicitly calculate the energy shift and the broadening, we further introduce a practical approximation, i.e. Debye approximation. To simplify the calculation, we use onedimensional model for the lattice vibrations. In these approximations, the spectral functions C(ω) and H(ω) are explicitly given for ω < ω D where δu is the nuclear displacement after the core-hole production, d is the lattice constant. In the high frequency region ω > ω D , both of them are null, We thus have the energy shift caused by the recoil from phonon excitations because the interference term has no contribution in these very simple models. This term is coincident with the energy shift caused by the recoil of a free atom in the photoemission processes. For Li 1s excitations by X-ray photons with 10 keV, the energy shift is about −0.95 eV from eq. (31), whereas −0.72 eV for Be 1s, −0.53 eV for C 1s excitation. These recoil shifts are comparable with typical chemical shifts due to the difference in chemical states.
On the other hand, the energy broadening due to the recoil and the Franck-Condon effects is given by use of eq. (28), where θ is the Debye temperature: The interference has no contribution. Figure 4 shows temperature dependence of the peak broadening (∆ω) 2 due to the recoil and the Franck-Condon processes, for Li, Be and C (graphite and diamond) 1s excitations by 10 keV X-ray photons. We assume that δu = 0.25 a.u. for all systems. The peak broadening of Li and graphite (soft systems with small ω D ) are sensitive to temperature compared with Be and diamond (hard systems with large ω D ). Next we assume that δu = 0 a.u. for all systems, so that the second term of eq. (32) vanishes. Comparing Fig. 5 with Fig. 4, we observe the small difference; i.e. Franck-Condon effect (the second term of eq. (32)) plays a minor role.

IV. CONCLUSION
We study the nondipole effects of photoelectrons excited by high-energy X-rays. In this case the electric quadrupole and the magnetic dipole transitions are taken into account in addition to the electric dipole transition. We apply irreducible tensor expansion of photon field instead of usual power series expansion, and verify the correctness of the power series expansion. The relative irre-ducible tensor correction is in the order of (qr c ) 2 , which is the same order as the octupole transition. In the energy range in this work, the power series expansion is good enough.
We also study the recoil effects of photoelectrons excited by high-energy X-rays. In this case the atomic displacement after the core-hole production can also play some important roles. For practical calculations we furthermore introduce the Debye approximation for phonon spectra, which gives rise to the simple free atom recoil energy shift. Numerical calculations are carried out for 1s excitation from Li, Be and C (graphite and diamond) powders, which show the broadenings depending on temperature and crystal structure, in contrast to the energy shift.
We have only studied photoemission from core levels. Of course the two effects considered here play some different roles in high-energy photoemission from extended states, which will be discussed in near future.