List of Formulas
- The list of formulas used in FEFF
The corresponding soubroutines are quoted in the square bracket.
Parameter of angular momentum and quantum number kappa
The quantum number κ is a function of the orbital angular momentum quantum number l and the total angular momentum quantum number j.
κ = lΘ(l − j) − (l + 1)Θ(j − l)
where Θ(x) is equal to one if x is greater than zero and is equal to zero if x is less than zero. N.B. In the above equation the argument of Θ(x) can never equal zero since l is integral and j is half-integral.
The above equation is equivalent to:
Density of States calculations
if (msapp != 1)
subroutine setkap(ihole, kinit, linit)
The contribution to the XAS from a given site and orbital angular momentum .
. where is the smoothly varying atomic background contributions and is the fine-structure or XAFS spectrum. A prime to denote final-state quantities calculated in the presence of a screened core hole.
Adds the contributions from each path and absorber, including Debye-Waller factors. Writes down main output: chi.dat and xmu.dat cchi(E) = S02 * TrG(E)
Solving Dirac equations [afovrg]
The spin-orbit eigenfunction (eigenstates of )
Matrix Elements for Electromagnetic Multipole Transitions (Grant) [xmult]
, A = upexp(iω.r), | ω | = ΔE / c where up is a polarization vector, and ω is the propagation vector.
Simplify the problem by taking ω to define the z-axis , so that we can write the polarization vector , , where ξ0,ξ1,ξ − 1 are the usual spherical basis vectors.
The decomposition of eik.r in terms of the irreducible tensors , where the angles involved are spherical polar coordinates with ez as polar axis.
The vector operator α is a tensor of order 1. The product of two tensors can be decomposed in terms of the irreducible tensors. , where
In the form of Wigner-Eckart theorem:
In the dipole approximation it is enough to keep only one term (l = 0, L = 1).
Note: the usual notation of atomic physics .
subroutine xmult(k, kp, ls, lb, xm1, xm2)
See Grant eq. 6.30. calculate the factors
xm1, xm2 both either real or pure imaginary
corresponds to the upper(lower) component of Dirac spinor.
- calculate xm1 (β = 1 in eq.6.30)
- calculate xm2 (β = − 1 in eq.6.30)
Performs radial integration for multipole matrix element or central atom absorption [radint]
if ifl = 2
subroutine xrci ( mult, xm, dgc0, dpc0, p, q, bf, value)
- output: value
r-dependent multipole matrix element (before r-integration)
dgc0*q* (xm(2)*bf(0) + xm(4)*bf(2)) + dpc0*p * (xm(1)*bf(0) + xm(3)*bf(2))
for double radial integral (use the irregular components of the final state)
Radial Integration Routine
- output: da
Modified to use complex p and q. SIZ 4/91
integration by the method of simpson of (dp+dq)*dr**m from 0 to r=dr(np)
dpas=exponential step; for r in the neighborhood of zero (dp+dq)=cte*r**da
Wigner 3j symbol [cwig3j]
function cwig3j (j1,j2,j3,m1,m2,ient)
- output: wigner 3j coefficient for integers (ient=1) or semiintegers (ient=2) other arguments should be multiplied by ient
Composition relation for the spherical harmonics
6j symbols [sixj]
9j symbols [ninej]