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- 4 -
where
()
2
2
M
++
0
qkk
and
()
2
2
M
+
0
kk
are the kinetic energies of the atoms before
and after the collision. The momentum conservation and energy conservation are valid in such an
expression. In addition to the two terms in Eq. (3), which describe the energy shift and the
Doppler broadening of the elastic peak, the third term in above equation treats the negative
shifting of the peak position as due to the anisotropic vibration of the target atoms.
For we can also write the recoil energy as,
(
)
2
2
22
r
k
E
M
M
+
=−
'
qk
, (5)
by keeping the same form as in Eq.(3) but replacing the ideal
q
with the measured
'q
,
0
k
is then given by,
(
)
(
)
2'2
2
2
2
qq
q
⎡
⎤
=
−−+−⋅
⎣
⎦
'
0
q
kqqk
. (6)
Therefore,
0
k is oppositional to q and its effect is to reduce the value of q . In bracket of
above equation, the first term represents the position shift of the peaks, which has been introduced
in the present Monte Carlo simulation model. The second term is a minor Doppler broadening
term for the shifting. In our simulation model, we have reasonably neglected this minor and
symmetrical broadening term. The peak position shift is thus taken as,
(
)
22
0
'2kqq q=− .
To be more clearly, we rewrite Eq. (4) as follows:
()
2
0
1
1
sin 2 2
r
k
q
E
PMM
θ
⎡⎤
⋅
≈− ⋅ ⋅ +
⎢⎥
⎢⎥
⎣⎦
qk
. (7)
The first term can thus be reasonably taken as the most probable value of the recoil loss energy
for anisotropic vibrating atoms and the second term is for the Doppler broadening. It is shown that
the most probable value of the recoil energy loss corresponding to the peak position of the spectra
will tend to be smaller than
0r
E
, as indicated by the experimental facts. Here, we have neglected
the final state effects in above discussion for it appears to be only a minor part for the deviation of
the elastic peak positions [21].
In our Monte Carlo simulation, it is convenient to use the scalar form for each scattering event
along electron trajectories:
()
()()
0
21
1 1 cos cos cos cos sin sin cos
sin 2
r
k
mM
EE
MP mE
ε
θ
ϑθϑθϑϕ
θ
⎧⎫
⎡⎤
⎪⎪
=−⋅ −+ − −
⎨⎢ ⎥ ⎬
⎢⎥
⎪⎪
⎣⎦
⎩⎭
,
(8)
where
ε
is the kinetic energy of the isotropic vibration of the target atoms, whose value
follows the Maxwell-Boltzmann thermal energy distribution with the mean kinetic energy
32
k
E
kT=
,
ϑ
and
ϕ
are the polar and azimuthal angles characterizing the velocity direction
of an atom related to the moving direction of an incident electron.
2.2. Electron elastic scattering
Monte Carlo simulation is based on the tracing of electron trajectories made of joining of
randomly sampled of electron scattering events [17, 22]. For the treatment of electron elastic
scattering, the Mott cross section [23] with the Thomas-Fermi-Dirac atomic potential [24] is
employed,
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