Appendix H: XANES
Appendix H.1: Cross Sections
In the EXAFS Chap. 7 we blithely assumed that the interaction of X-rays with matter
was primarily electric dipole in nature . . .
Following Brouder, we write a more general interaction between a particle with
charge q, mass m, gyromagnetic ratio g, and spin s
!
, in a potential V( r
!
) submitted to
an external electromagnetic field Φ r
!
, t
, A
!
r
!
, t
as:
H ¼
1
2m
Àiħ∇
! À qA
!
h
i 2
þ V r
!
þ qΦ À
gq
2m
s
! Á B
!
which he expands and simplifies as:
H ¼ H 0 þ
iħq
m
A
! Á ∇
! þ
iħq
m
∇
! Á A
! þ
q
2
2m
A
! 2 þ qΦ À
gq
2m
s
! Á B
!
where:
H 0 ¼ À
ħ
2
2m
∇
! 2
þ V r
!
Assuming a plane wave electromagnetic field in the Coulomb gauge allows him
Φ ¼ 0 and ∇
! Á A
! ¼ 0, leading to:
H ¼ H 0 þ
iħq
m
A
! Á ∇
! þ
q
2
2m
A
! 2 À
gq
2m
© Springer Nature Switzerland AG 2020
S. P. Cramer, X-Ray Spectroscopy with Synchrotron Radiation, Biological and Medical
Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-28551-7
337
Appendix H.1: Cross Sections
In the EXAFS Chap. 7 we blithely assumed that the interaction of X-rays with matter
was primarily electric dipole in nature . . .
Following Brouder, we write a more general interaction between a particle with
charge q, mass m, gyromagnetic ratio g, and spin s
!
, in a potential V( r
!
) submitted to
an external electromagnetic field Φ r
!
, t
, A
!
r
!
, t
as:
H ¼
1
2m
Àiħ∇
! À qA
!
h
i 2
þ V r
!
þ qΦ À
gq
2m
s
! Á B
!
which he expands and simplifies as:
H ¼ H 0 þ
iħq
m
A
! Á ∇
! þ
iħq
m
∇
! Á A
! þ
q
2
2m
A
! 2 þ qΦ À
gq
2m
s
! Á B
!
where:
H 0 ¼ À
ħ
2
2m
∇
! 2
þ V r
!
Assuming a plane wave electromagnetic field in the Coulomb gauge allows him
Φ ¼ 0 and ∇
! Á A
! ¼ 0, leading to:
H ¼ H 0 þ
iħq
m
A
! Á ∇
! þ
q
2
2m
A
! 2 À
gq
2m
© Springer Nature Switzerland AG 2020
S. P. Cramer, X-Ray Spectroscopy with Synchrotron Radiation, Biological and Medical
Physics, Biomedical Engineering, https://doi.org/10.1007/978-3-030-28551-7
337