Physics Surface-extended X-ray absorption fine structure

1 physics

1.1 basics
1.2 theory of exafs
1.3 incorporation of exafs-auger

physics
basics

the absorption of x-ray photon atom excites core level electron, generating core hole. generates spherical electron wave excited atom center. wave propagates outwards , scattered off neighbouring atoms , turned towards central ionized atom. oscillatory component of photoabsorption originates coupling of reflected wave initial state via dipole operator mfs in (1). fourier transform of oscillations gives information spacing of neighboring atoms , chemical environment. phase information carried on oscillations in auger signal because transition time in auger emission of same order of magnitude average time photoelectron in energy range of interest. thus, proper choice of absorption edge , characteristic auger transition, measurement of variation of intensity in particular auger line function of incident photon energy measure of photoabsorption cross section.

this excitation triggers various decay mechanisms. these can of radiative (fluorescence) or nonradiative (auger , coster–kronig) nature. intensity ratio between auger electron , x-ray emissions depends on atomic number z. yield of auger electrons decreases increasing z.

theory of exafs

the cross section of photoabsorption given fermi s golden rule, which, in dipole approximation, given as

p
=



2
π

ℏ



∑

f



|


m

f
s




|


2


δ
(

e

i


+
ℏ
ω
−

e

f


)
,


{\displaystyle p={\frac {2\pi }{\hbar }}\sum _{f}|m_{fs}|^{2}\delta (e_{i}+\hbar \omega -e_{f}),}









m

f
s


=
⟨
f

|

e

ϵ

⋅

r


|

i
⟩
,


{\displaystyle m_{fs}=\langle f|e\mathbf {\epsilon } \cdot \mathbf {r} |i\rangle ,}

where initial state, energy ei, consists of atomic core , fermi sea, , incident radiation field, final state, ƒ energy eƒ (larger fermi level), consists of core hole , excited electron. ε polarization vector of electric field, e electron charge, , ħω x-ray photon energy. photoabsorption signal contains peak when core level excitation neared. followed oscillatory component originates coupling of part of electron wave upon scattering medium turned towards central ionized atom, couples initial state via dipole operator, mi.

assuming single-scattering , small-atom approximation krj >> 1, rj distance central excited atom jth shell of neighbors , k photoelectrons wave vector,

k
=


1
ℏ




[
2
m
(
ℏ
(
ω
−

ω

t


)
+

v

o


)
]


,


{\displaystyle k={\frac {1}{\hbar }}{\sqrt {[2m(\hbar (\omega -\omega _{t})+v_{o})]}},}

where ħωt absorption edge energy , vo inner potential of solid associated exchange , correlation, following expression oscillatory component of photoabsorption cross section (for k-shell excitation) obtained:

χ
(
k
)
=

k

−
1



|

f
(
k
,
π
)

|


∑

j


 

w

j


sin
⁡
[
2
k

r

j


+
α
(
k
)
]
exp
⁡
(
−
γ

r

j


−
2

σ

j


2



k

2


)
,


{\displaystyle \chi (k)=k^{-1}|f(k,\pi )|\sum _{j}\ w_{j}\sin[2kr_{j}+\alpha (k)]\exp(-\gamma r_{j}-2\sigma _{j}^{2}k^{2}),}

where atomic scattering factor in partial wave expansion partial wave phase-shifts δl given by

f
(
k
,
θ
)
=
(
1

/

k
)

∑

l
=
0


∞


 
(
2
l
+
1
)
[
exp
⁡
(
2
i

δ

l


(
k
)
)
−
1
]

p

l


(
cos
⁡
θ
)
.


{\displaystyle f(k,\theta )=(1/k)\sum _{l=0}^{\infty }\ (2l+1)[\exp(2i\delta _{l}(k))-1]p_{l}(\cos \theta ).}

pl(x) lth legendre polynomial, γ attenuation coefficient, exp(−2σik) debye–waller factor , weight wj given in terms of number of atoms in jth shell , distance as

w

j


=



n

j



r

j


2




.


{\displaystyle w_{j}={\frac {n_{j}}{r_{j}^{2}}}.}

the above equation χ(k) forms basis of direct, fourier transform, method of analysis has been applied analysis of exafs data.

incorporation of exafs-auger

the number of electrons arriving @ detector energy of characteristic wαxy auger line (where wα absorption edge core-level of element α, incident x-ray line has been tuned) can written as

n

t


=

n


w

α


x
y


(
ℏ
ω
)
+

n

b


(
ℏ
ω
)
,


{\displaystyle n_{t}=n_{w_{\alpha }xy}(\hbar \omega )+n_{b}(\hbar \omega ),}

where nb(ħω) background signal ,




n


w

α


x
y


(
ℏ
ω
)


{\displaystyle n_{w_{\alpha }xy}(\hbar \omega )}

auger signal interested in, where

n


w

α


x
y


(
ℏ
ω
)
=
(
4
π

)

−
1



ψ


w

α


x
y


[
1
−
κ
]

∫

Ω



∫

0


∞


 

ρ

α


(
z
)



p


w

α




(
ℏ
ω
;
z
)
exp
⁡

[



−
z


λ
(

w

α


x
y
)



cos
⁡
θ
]

 
d
z
d
Ω
,


{\displaystyle n_{w_{\alpha }xy}(\hbar \omega )=(4\pi )^{-1}\psi _{w_{\alpha }xy}[1-\kappa ]\int _{\omega }\int _{0}^{\infty }\ \rho _{\alpha }(z)\,\,p_{w_{\alpha }}(\hbar \omega ;z)\exp \left[{\frac {-z}{\lambda (w_{\alpha }xy)}}\cos \theta \right]\ dzd\omega ,}

where




ψ


w

α


x
y




{\displaystyle \psi _{w_{\alpha }xy}}

probability excited atom decay via wαxy auger transition, ρα(z) atomic concentration of element α @ depth z, λ(wαxy) mean free path wαxy auger electron, θ angle escaping auger electron makes surface normal , κ photon emission probability dictated atomic number. photoabsorption probability,




p


w

α




(
ℏ
ω
;
z
)


{\displaystyle p_{w_{\alpha }}(\hbar \omega ;z)}

term dependent on photon energy, oscillations in function of energy give rise similar oscillations in




n


w

α


x
y


(
ℏ
ω
)


{\displaystyle n_{w_{\alpha }xy}(\hbar \omega )}

.