A fully rigorous method for including inelastic attenuation is so far not available, and thus we use the common phenomenological approach of an exponential decay of the amplitude of each component of the photoelectron wave with the distance traveled in the solid before escaping through the surface, called electron inelastic mean free path (IMFP). If the distance traveled along a given path is a and IMFP is l, then the exponential decay factor for the amplitude of this path is exp(-a/(2l)). Considering the inelastic scattering and vibrational effects, we only need to rewrite eqn. (18) as follows:

(26)

where a' is the internuclear distance of the bond vector leading from the site in a single scattering process, see Figure 2. s_c² is the thermal mean square relative displacement, which will be discussed later.

The inelastic mean free path can be obtained from theory and certain types of experiments. Tanuma, Powell and Penn have found an empirical formula, called TPP-2 formula, to calculate IMFP for 50-2000 eV electrons, based on the assumption that the Born approximation is valid and on the neglect of vertex corrections, self-consistency, exchange and correlation. The TPP-2 formula is

(27)

where l is the IMFP (in Å), E is the electron energy (in eV), E_p=28.821(N_v r/M)^1/2 is the free-electron plasmon energy (in eV), r is the density of the bulk (in g.cm^-3), N_v is the number of valence electrons per atom (for elements) or molecule (for compounds) and M is the atomic or molecular weight. The terms b, g, C and D are parameters given by

(28)

and E_g is the bandgap energy (in eV) for non-conductors, and equals zero for conductors. The relationship of electron energy E and wave number k is

(29)

Although there is no known physical basis, there is another relatively simple and convenient means for expressing IMFP dependence on electron energy, which was proposed by Wagner, Davis and Riggs.

(30)

where k and m are material-dependent parameters. They found that m ranged from 0.54 to 0.81. Generally, similar results have been reported by others. Tables 2 and 3 list the empirical parameters k and m for 27 elements and 15 inorganic compounds over the electron energy range 500-2000 eV. It appears from Tables 2 and 3 that m=0.75±0.03 (one standard deviation) is a reasonable approximation for this group of materials over 50-2000 eV.

There is no generally applicable yet accurate model for including both anisotropic and correlated thermal vibrational effects in multiple scattering calculations. We here follow Kaduwela, Friedman and Fadley and adopt a correlated vibrational factor which is expected to depend on the distance between the present scatterer and the previous scatterer. With the definition of the effective mean square displacement with thermal averaging, the equivalent correlated Debye-Waller-type attenuation factor is given by

(31)

where s_c² is the mean square relative displacement (MSRD),

(32)

where M_s is the substrate or average atom atomic mass, k_B is the Boltzmann constant, q_D is the effective or average atom Debye temperature, t=q_D/T, T is the sample temperature in K, |R_j-R_j-1| is the internuclear distance of the bond vector leading from the site in a single scattering process, i.e. a' in Figure 3, q_D=w_D/v is the associated Debye wave vector, v is the velocity of sound which is taken as constant in the Debye approximation, w_D is the cutoff frequency determined by (6q²v³N/V) ^1/3, N is the number of acoustic phonon modes in volume V with the wave vector less than q_D, which equals the number of primitive cells, or usually the number of atoms in volume V. So we have

(33)

This Debye-Waller attenuation factor has been adopted in equation (32). Accounting for the surface atomic vibration is not as straightforward. The relation between the MSRD and different atomic masses has been given by Allen, Alldredge and Wette,

(34)

Correlating eqn (34) with (32), an effective surface atomic mass is introduced such that

(35)

where M_{surface-effective}=M_surface if T/q_D<<1 or M_{surface-effective}=M_bulk if T/q_D>1. For T/q_D ~ 1, M_{surface-effective} is allowed to vary between the surface and bulk atomic masses.

A photoelectron with energy E within the jellium medium will have energy E-E₀ in the vacuum far from the surface. This loss of kinetic energy E₀ may be related to a potential barrier whose total height is V₀, defined as inner potential. The only energy relevant for the scattering problem is the electron's kinetic energy when it encounters a scattering potential. In photoemission, the scattered electron is detected, and the inner potential represents the physical kinetic energy lost when the electron travels from the scattering potential edge to the detector. This inner potential is approximately the sum of the work function and the valence band-width.

The inner potential barrier will alter the photoelectron path. We adopt a planar step barrier of height V₀ just outside the last row of ion cores. This is the usual first-order model for the surface barrier, introduced for both low energy photoemission and low-energy electron diffraction. The important consequence of this model is a prediction that the emerging photoelectron will be refracted in a direction away from the surface normal in the manner of optical paths with

(36)

where V₀ is the inner potential, E_in and E_out are the electron kinetic energy inside and outside the sample surface, k_in and k_out the wave number inside and outside the surface, q_in and q_out the photoelectron directions before and after refraction at the surface away from the normal direction, see Figure 4. Substituting eqn (29) into eqn (36), we obtain the inner potential correction formula

(37)

Presently there exist no generally acknowledged theoretical or experimental inner potential data for any kind of element. Hence it is treated as an adjustable parameter and fit to experiment.

Figure 4. Inner potential correction.

The experimental apparatus for measuring photoemission intensities has a small but finite angular resolution characterized by half the angle subtended by the aperture at the source, q in Figure 5. For small apertures, q is the radius of the aperture projected on a unit sphere so that the detected area is q². The instrumental angular averaging due to this finite aperture of the detector is done by summing the photoelectron intensities over a grid of points on a circular aperture centered on the nominal emission direction as defined by wave vector k. We calculate photoemission intensities I_a, I_b, I_c, I_d and I_e for five different directions (0,0) (q,0) (q,q/2) (q,q) (q,3q/2) and (q,2q), see Figure 5, and assume that the average intensity in each sector is the average of the intensities at the three triangle corners. Then, for example, in sector abc, the average intensity will be I_abc=(I_a+I_b+I_c)/3. For five point calculations, we obtain the average intensity for a finite aperture of the detector

(38)

Figure 5. Instrumental angular averaging; q is the half aperture angle.