Photoemission intensity and the exact representation

The term exact in this document means that no R-A approximation is made, i.e., that, even if a R-A expansion is used, it is carried to all orders, since it will in principle converge to the exact result. We note in this context that Brouder and Sébilleau have shown that the R-A representation carried to all orders turns out to be an accurate, stable and efficient way to calculate the exact propagator.

For simplicity in the formulas of photoemission intensity, we do not include effects due to inelastic scattering or vibrational motion, even though it is clear that these effects are essential for a quantitative description of experimental data. Both of these effects tend to damp out scattering from atoms further from the emitter, thus causing multiple-scattering paths to be effectively reduced in length. Both of these effects are, however, included in our MSCD programs and in some of the calculations presented in this document. Using the exact propagator,

(3)

the multiple scattering photoemission intensity can be expressed as:

(4)

where is the photoemission intensity from electronic subshell (n_i,l_i), as detected with wave vector k in the (q, f) direction; (n_i,l_i,m_i) are the quantum numbers of the initial core orbital (n_i = 1, 2, 3, 4, ... = K,L,M,N,... shells, respectively, l_i = 0, 1, 2, 3, ... = s, p, d, f, ... subshells, and m_i is the magnetic quantum number); L_f = (l_f,m_f) are angular momentum quantum numbers of the final state. The dipole selection rules imply that l_f - l_i = ±1, m_f - m_i = 0, where we for simplicity restrict ourselves to linear polarized incident light, although this can be generalized in a straightforward way to other polarizations, such as circular or elliptical. is the exact nth-order multiple scattering Green's function for a scattering path from the emitter at R₀ º R_emitter via scatterers at R₁, R₂, ... , R_n-1 to the detector at R_n º R_detector º R_d. The quantities m_lf,c and d_lf,c are the amplitude and phase of the dipole matrix element into a given final state, and are related to the short-range central potential of the ionized atom; that is, the long-range Coulomb field is neglected due to the assumed screening of the core hole near a solid surface. The quantities m_lf,c and d_lf,c are calculated from <Y_Ekin,lf| e·r| f_nili>, with Y_Ekin,lf is the final continuum state of the photoelectron at a kinetic energy E_kin which propagates in direction r, f_nili is the initial core orbital L_i = (l_i, m_i) from which the photoelectron is emitted, and e is the radiation polarization vector. The summations run over all emitters, all final states L_f = (l_f, m_f) and over all combinations of order n, the number of scatterers (number of atoms = n+1) in a given scattering path from single scattering (n = 1) to the highest order considered n = n_max (typically we select n_max = 8 or higher, corresponding to 7 or more scattering events).

Now we choose the z direction to be parallel to the e vector, to simplify the matrix element evaluation. The expression for the matrix element then becomes:

(5)

where R_Ekin,lf(r) is the radial part of the continuum orbital at l_f, R_nili(r) is the radial part of the initial core orbital with quantum numbers n_i and l_i, and Y_lm(q,f) are the relevant spherical harmonics.

In the convergent separable representation of the propagator derived by Rehr and Albers, the exact propagator G_LL' is re-written as

(6)

where a new combination index l = (m,n) is introduced, such that, for an exact representation, m = -l_max to l_max, and n = 0 to |m|. However, this expansion converges relatively quickly, and can usually be truncated without significant loss of accuracy, as we shall show later. The quantities and have the following forms:

	(7)
	(8)
with
	(9)
	(10)

Here z = 1/(i|r|), C_l(z) is the degree-l polynomial factor of the spherical Hankel function, is a matrix which rotates the bond direction r onto the z axis, and W _r represents the Euler angles for this rotation. The matrix transforms the spherical harmonics as

	(11)
Substituting Eq. (6) into Eq. (3), one thus obtains the exact equivalent form
	(12)

This is the Rehr-Albers separable representation formula for curved-wave multiple-scattering, which is a direct analog of the plane-wave approximation or the point-scattering approximation.

In Eq. (12), the scattering-amplitude matrices F_{lj,l j-1}(r_j, r_j-1) at each site are defined in the partial-wave expansion as

(13)

where r' and r are the interatomic vectors leading from and to the site, as illustrated in Fig. 2. The sum on L runs over both l and m quantum numbers.

Fig. 2. A scattering event leading from atom a to atom c via atom b; r' = ka' and r = ka are dimensionless interatomic vectors leading from and to the site in question, with a' and a the corresponding vectors and k the wave number (k = |k|, where k is the wave vector), while b is the angle between the interatomic vectors r' and r. The quantity Fll' (r,r') is the effective scattering amplitude via the Rehr-Albers approximation.

The R-A representation, Eq. (13), is thus an exact formula if we take all the possible l = (m,n) values into account. But in practice, noting the asymptotic form F_ll' µ (r) ^-(2n+m) ·(r')^{-(2 n'+m')} for large r' and r, we can safely truncate F_ll' at different approximation orders. Table 1 lists, as a function of the R-A approximation order, the dimensions of the scattering-amplitude matrices, and their possible (m,n) values. For most real cases that we have encountered, it was found that the second order ((6x6) matrices) is adequate to simulate experimental curves, and this will be further investigated in later sections.

The advantage of the R-A representation is that the approximation leads to smaller matrix sizes, resulting in much reduced computation times. In the exact formalism, Eq. (3), the propagator matrix G_LL' has the dimensions (l_max+1)² by (l_max+1)², where l_max can be estimated as described previously; for a typical muffin-tin radius R_mt of 1.5 Å this yields matrix sizes of (36x36) to (441x441) from low to high energies. By contrast, in most cases, the R-A representation requires matrix sizes of only (6x6), although we discuss below some cases where going up to (15x15) might be required for ultimate quantitative accuracy.

In equation (11), we defined the rotation W_r as a sequence of rotations with three Euler angles a, b, and g. There are, however, several conventions in existence for choosing the so-called Euler angles. We here adopt the convention used by A. Messiah, which differs slightly from the one generally adopted in the theory of the gyroscope.

The general displacement of a rigid body due to a rotation about a fixed point may be obtained by performing three Euler rotations about two of three mutually perpendicular axes fixed in the body. We assume a right-handed frame of axes, and define a positive rotation about a given axis to be one which would carry a right-handed screw in the positive direction along that axis. Thus a rotation about the z-axis which carried the x-axis into the original position of the y-axis would be considered to be positive. The rotations are to be performed successively in the following order (see Fig. 4):

1. a rotation by a (0 =< a <= 2p) about the z-axis, bringing the frame of axes and the body together from the initial position S into the position S'.
2. a rotation by b (0 =< b <= p) about the y-axis of the frame S' with its body, resulting in a new position S''.
3. a rotation by g (-p =< g <= p) about the z-axis of the frame S'' with its body. The position of this axis depends on the previous rotations a and b. The final position of the frame is symbolized by S'''.

Fig. 3. Definition of Euler angles.

Using this definition, we have the rotation matrices

(14)

Here is an real and unitary matrix, which has the Wigner formula

(15)

Equation (15) is only valid for (m' >= |m|). For other combinations of m and m', we can use the following symmetry properties:

(16)

For the case l=1, we have

(17)

In this expression, the successive lines correspond to m=1,0,-1; the columns are arranged in the same order from left to right.

Substituting equations (7) and (8) into equation (13), we can re-express the scattering-amplitude matrix F_ll' (r,r') as

(18)

Here, we introduced the composite rotation matrix

	(19)
and
	(20)

where is a rotation taking bond vector r into z, and is a rotation taking z into bond vector r'. So we define the composite rotation matrix which takes r into z and then z into r'. By choosing the photon polarization direction as the z direction, we have W_r = (0,q,p-f), W_r'^-1 = (f'-p, -q',0). Let W_rr' º (a, b, g), then

(21)

Now we can derive the composite rotation Euler angles (a, b, g) using this equation for the case l=1. Substituting equations (14) and (17) into equation (21) for l=1:

(22)

From R₂₂(W_rr') = S R_2m(0,q, q-f) R_m2(f'- q,-q',0) we obtain

(23)

Here we can get the conditions for b=0 or q, as follows:

b=0 only when (i) q'=q, f'=f or (ii) q'=q=0 or (iii) q'=q= q, and

b=q only when (i) q'+q =q, f'- f=q or (ii) q'=q, q=0 or (iii) q'=0, q =q

Evaluating the matrix elements R₂₂(W_rr'), R_-10(W_rr'), R_0-1(W_rr'), R_-1-1(W_rr'), and R_-11(W_rr'), assuming g=0 and using the above conditions when b=0 or q, we finally obtain the following expression for the Euler angles (abg) of the composite rotation,

	(24)
with
	(25)