Multipole Calculations with FEFF8

["John J. Rehr" <jjr@leonardo.phys.washington.edu>]

Dear Feff-users,

Matt Newville raised the question of how to separate various channels
of x-ray absorption in FEFF8 polarization dependent calculations:

Matt wrote: 
>I'm not sure whether this is a 'user' or 'developer' question,
>but I'd like to extract the EXAFS (yes, E-XAFS) contribution
>from the different orbitals for L_III edges, especially for
>studying the polarization dependence of L-edges.  I see
>calculated polarization dependencies that I don't fully
>understand, and want to be able to isolate the different terms.
>How can I do this?  Ideally, I'd like to be able to make
>separate feff.dat files for l->l+1, l->l-1, cross terms, and
>quadrapole contributions.

This question seems appropriate both for feff-users and feff-developers.
The separation could not be done simply (i.e., without hard wired coding)
with previous released versions of FEFF (8.1 and earlier). However we have
introduced a new MULTIPOLE card to make this easier. The card is being
tested in recent experimental versions of FEFF8, which we hope soon to release
as FEFF8.2. This code is presently available only to feff-developers
and collaborators, and is nearly ready for release. However, updates to
the document and license information  are not yet finished.

Example applications of the MULTIPOLE card below illustrate how one can
separate quadrupolar and various channels of dipolar transitions:

* MULTIPOLE  le2 [l2lp]  
*  le2 specifies the maximum included multipoles:
*  le2=0 E1 electric dipole only (default)
*  le2=1 M1 add magnetic dipole
*  le2=2 E2 add electric quadrupole
*  l2lp specifies which l-->l+/-1 channels are included 
*  examples:

  MULTIPOLE 0  0 ! E1 only; both l-->l+/-1 channels
  MULTIPOLE 0  1 ! E1 only; l-->l+1 transitions only
  MULTIPOLE 0 -1 ! E1 only; l-->l-1 transitions only
  MULTIPOLE 1  0 ! E1 and M1; this only works with l2lp=0
  MULTIPOLE 2  0 ! E1, E2, and E1-E2 cross terms; only works with l2lp=0

Notes: To compare different channels quantitatively you should
scale xmu.dat files by the normalization factor (i.e., absolute crosssection
above 50 eV) which is reported in the header of the  xmu.dat file.
These normalization factors must be used in linear combinations
to separate the various contributions.

For the quadrupolar contribution, MULTIPOLE 2 0 yields polarization averaged
results, unless one specifies the direction of the incident x-ray beam,
with the ELLIPTICITY card (see below).

Two interesting linear combinations that one can make are:

1) Isolate quadrupolar contribution. Requires 2 calculations of xmu.dat,
   MULTIPOLE 2  0 [mu(E1+E2], and  MULTIPOLE 0  0 [mu(E1)]
   mu(E2)+ mu(E2,E1) = mu(E2+E1) - mu(E1)

   This procedure looks like it has an unavoidable cross-term mu(E1-E2),
   but typically this contribution is small or cancels out. Exceptions
   are very unsymmetric crystals; e.g. XNCD in LiIO3 is due to mu(E2,E1).
   In cases where the cross-term is not negligible, one has to do additional
   calculations with the ELLIPTICITY card. Note that the cross term
   changes sign if one reverses the direction of the x-rays (kx,ky,kz),
   while the pure E1 and E2 terms stay the same. Thus, e.g., for x-rays
   incident along the z axis, one needs calculations with

  ELLIPTICITY 0  0. 0.  1
  ELLIPTICITY 0  0. 0. -1

   mu(E2,E1) = (mu(E2+E1,z) - mu(E2+E1,-z)) / 2
   
   We have discussed the quadrupolar transitions in our recent publication
   [Ankudinov, Rehr, PRB 62, 2437 (2000)] which treats both calculations
   of the elastic scattering amplitude f' and XNCD in LiIO3.

2) Dipole cross term for polarized XAS: see e.g. the FEFF7 reference 
   [Ankudinov and Rehr PRB 56, R1712 (1997)]

   mu(cross) = mu(E1) - mu(E1,l-->l+1) - mu(E1, l-->l-1)
   Thus it is necessary to combine 3 'xmu.dat' files to get this cross term.

 For a discussion of the experimental extraction of the cross term at the Cd L3 edge
 and it's importance for EXAFS, see Le Fevre, Magnan & Chandesris, PRB 54, 2830 (1996).

Other cards which may be important in these calculations are the following:

a) x-ray k-vector
*  ELLIPTICITY  ell  kx  ky  kz
*   specifies direction of propagation of x-rays  (kx,ky,kz):
*   ell is the mean ellipticity (0 linear +1 rhc and -1 lhc).

  This is necessary for the correct quadrupolar contribution, but not for the
polarization average signal.

b) polarization direction and/or spectroscopy
*  POLARIZATION  x  y  z
*  specifies polarization direction

  or with the

*   XMCD  (or XNCD)
*   specifies type of type of measurement or spectroscopy
*   XMCD card for circular (or XNCD linear) polarizations
  In the case of XMCD, the xmu.dat file will contain the circular dichroism signal.

c) spin dependence
* SPIN ispin  sx  sy  sz
*  specifies type of spin calculation and spin vector (default along z)

   This card must be used for spin-dependent systems (e.g. XMCD).
   There are 2 possible approaches for including spin in XMCD, set by the ispin
   variable nspx=1 or nspx=2.  With nspx=1 spin-flip processes (not spin-orbit
   interaction) are neglected and calculations must be done separately for spin-up
   and spin-down; the program 'spin.f' (see the Appendix of the FEFF8 document)
   can be used to combine the two 'xmu.dat' files. With nspx=2 the full spin
   dependence is included, including spin-flip.  More accurate calculations can
   be done with nspx=2, but the memory requirements and calculation time can be
   about 4 times longer due to a doubling of the matrix size.  The output 'xmu.dat'
   contains the total XMCD from spin up and down signals.

The above description will be included in the revised FEFF8.2 document.  We would
appreciate any comments/ciritque/suggestions on the current scheme of cards used
in FEFF8 to specify information about x-ray: incidence, polarization, ellipticity,
etc.  How confusing is it? and how one could simplify it or change the description
to make it more clear?

Yours sincerely,
Alexei Ankudinov and John J. Rehr.