Brouder eventually goes on to argue that the A
! 2
and s
! Á B
!
terms are negligible.
From time-dependent perturbation theory, the transition probability per unit time for
a perturbation W(t) ¼ Wexp(Àiωt) + W
à exp(iωt) is written as:
w ¼
2π
ħ
X
f
f j W j i
2
δ E f À E i À ħω
À
Á
and he writes the electromagnetic wave as:
A
!
r
!
, t
¼ A 0 b ε exp i k
! Á r
! À ωt
i
þ A
Ã
0 b ε
à exp Ài k
! Á r
! À ωt
i
h
h
Following a Taylor series expansion of the electromagnetic plane wave, in which
only dipole and quadrupole terms are retained, gets the absorption cross section σ as:
σ ω
ð Þ ¼ 4π
2
αħω
X
i f
f jb ε Á r
!
ji
2 þ 1=4
ð
Þ f jb ε Á r
! k
! Á r
!
ji
2
&
'
δ E f À E i À ħω
À
Á
This is a formula that is useful for describing the XANES cross section (Chap. 7),
and only the first term needs to be kept for most EXAFS analyses (Chap. 6). The same
matrix elements also enter the description of RIXS (Chap. 8). But, let’s back up.
It turns out that the A
! 2
and s
! Á B
!
terms are not always negligible. In fact, the A
! 2
term gives rise to X-ray Raman scattering and inelastic X-ray scattering that make up
the second half of Chap. 8. The s
! Á B
!
magnetic dipole term is often the major
contributor to the nuclear cross sections that are the subject of Chaps. 9 and 10. So, in
a broader perspective we can consider the interaction of X-rays with matter as
governed by three terms that become significant under different conditions:
H interaction ¼
iħq
m
A
! Á ∇
!
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
XAS, XES,
&RIXS
þ
q
2
2m
A
! 2 À
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
XRS&IXS
q
2
2m
s
! Á B
!
|fflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl ffl}
SMS, SRPAC,
&IXS
Appendix H.2: XANES Software
Links to Software for Calculation of XANES Spectra [609]
ADF
https://www.scm.com/product/adf/
[610]
CTM4XAS
http://www.anorg.chem.uu.nl/CTM4XAS/software.html
[330]
FEFF
http://monalisa.phys.washington.edu/index-feffproject.html
[228]
ORCA
https://orcaforum.cec.mpg.de
[272]
QuantumEspresso
http://www.quantum-espresso.org
[611]
StoBe
http://www.fhi-berlin.mpg.de/KHsoftware/StoBe/
[273]
Wien2K
http://www.wien2k.at
http://susi.theochem.tuwien.ac.at
[282]
338
Appendix H: XANES
! 2
and s
! Á B
!
terms are negligible.
From time-dependent perturbation theory, the transition probability per unit time for
a perturbation W(t) ¼ Wexp(Àiωt) + W
à exp(iωt) is written as:
w ¼
2π
ħ
X
f
f j W j i
2
δ E f À E i À ħω
À
Á
and he writes the electromagnetic wave as:
A
!
r
!
, t
¼ A 0 b ε exp i k
! Á r
! À ωt
i
þ A
Ã
0 b ε
à exp Ài k
! Á r
! À ωt
i
h
h
Following a Taylor series expansion of the electromagnetic plane wave, in which
only dipole and quadrupole terms are retained, gets the absorption cross section σ as:
σ ω
ð Þ ¼ 4π
2
αħω
X
i f
f jb ε Á r
!
ji
2 þ 1=4
ð
Þ f jb ε Á r
! k
! Á r
!
ji
2
&
'
δ E f À E i À ħω
À
Á
This is a formula that is useful for describing the XANES cross section (Chap. 7),
and only the first term needs to be kept for most EXAFS analyses (Chap. 6). The same
matrix elements also enter the description of RIXS (Chap. 8). But, let’s back up.
It turns out that the A
! 2
and s
! Á B
!
terms are not always negligible. In fact, the A
! 2
term gives rise to X-ray Raman scattering and inelastic X-ray scattering that make up
the second half of Chap. 8. The s
! Á B
!
magnetic dipole term is often the major
contributor to the nuclear cross sections that are the subject of Chaps. 9 and 10. So, in
a broader perspective we can consider the interaction of X-rays with matter as
governed by three terms that become significant under different conditions:
H interaction ¼
iħq
m
A
! Á ∇
!
|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}
XAS, XES,
&RIXS
þ
q
2
2m
A
! 2 À
|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
XRS&IXS
q
2
2m
s
! Á B
!
|fflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl ffl}
SMS, SRPAC,
&IXS
Appendix H.2: XANES Software
Links to Software for Calculation of XANES Spectra [609]
ADF
https://www.scm.com/product/adf/
[610]
CTM4XAS
http://www.anorg.chem.uu.nl/CTM4XAS/software.html
[330]
FEFF
http://monalisa.phys.washington.edu/index-feffproject.html
[228]
ORCA
https://orcaforum.cec.mpg.de
[272]
QuantumEspresso
http://www.quantum-espresso.org
[611]
StoBe
http://www.fhi-berlin.mpg.de/KHsoftware/StoBe/
[273]
Wien2K
http://www.wien2k.at
http://susi.theochem.tuwien.ac.at
[282]
338
Appendix H: XANES