2 Synchrotron-Radiation-Based Energy-Domain Mössbauer …
67
Fig. 2.6 A schematic
diagram of parameters used
for the detailed expression of
SR-based Mössbauer spectra
Transmitter
Scatterer
SR
zt
z
dz
z s
Scattering
to Detector
E s (w, w s , z) = E 0s exp
−
μ es z
2
exp
−i
μ ns z
2(2(w − w s ) + i)
.
(2.2)
Here, w s denotes the nuclear resonant energy of the scatterer, and z denotes the
depth to which SR penetrates as indicated in Fig. 2.6. The other variables denote
similarly to those of the transmitter, although the suffix “t” should be changed to “s,”
which denotes the scatterer. Equation (2.2) is expressed as a function of w, w s , and
z because w s is controlled by the velocity transducer and w and z are used for the
calculation in the following equations. This field was absorbed by the scatterer at the
depth z, as described in the following step, and thus, z is less than the thickness of
the scatterer z s . Because the energy reference substance is assumed at the scatterer,
hyperfine splitting is absent, and the summation of the nuclear transition index m
is not described. Subsequently, we consider the scattering process at the scatterer.
Here, it is enough to consider the w s -dependent processes, which are as follows: (1)
the emission following the recoilless nuclear resonant absorption of radiation and
(2) the scattering of the radiation due to the photoelectric absorption. Here, we call
the former process as channel A and the latter as channel C according to [26]. In
channel A, SR penetrates the transmitter and scatterer until the depth z and is then
resonantly absorbed. In this channel, the scattering field E A (w, w s , z) satisfies
E A (w, w s , z) ∝
1
w − w s + i/2
E t (w)E s (w, w s , z).
Now, we can obtain the detected intensity I A (w s ) as follows:
I A (w s ) = C A
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )E A (w, w s , z)
2
= C A
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )
w − w s + i/2
E t (w)E s (w, w s , z)
2
,
(2.3)
where C A denotes a proportionality factor. This expression includes (1) the Fourier
transformation of E A (w, w s , z) to obtain the time dependence of E A and the square
67
Fig. 2.6 A schematic
diagram of parameters used
for the detailed expression of
SR-based Mössbauer spectra
Transmitter
Scatterer
SR
zt
z
dz
z s
Scattering
to Detector
E s (w, w s , z) = E 0s exp
−
μ es z
2
exp
−i
μ ns z
2(2(w − w s ) + i)
.
(2.2)
Here, w s denotes the nuclear resonant energy of the scatterer, and z denotes the
depth to which SR penetrates as indicated in Fig. 2.6. The other variables denote
similarly to those of the transmitter, although the suffix “t” should be changed to “s,”
which denotes the scatterer. Equation (2.2) is expressed as a function of w, w s , and
z because w s is controlled by the velocity transducer and w and z are used for the
calculation in the following equations. This field was absorbed by the scatterer at the
depth z, as described in the following step, and thus, z is less than the thickness of
the scatterer z s . Because the energy reference substance is assumed at the scatterer,
hyperfine splitting is absent, and the summation of the nuclear transition index m
is not described. Subsequently, we consider the scattering process at the scatterer.
Here, it is enough to consider the w s -dependent processes, which are as follows: (1)
the emission following the recoilless nuclear resonant absorption of radiation and
(2) the scattering of the radiation due to the photoelectric absorption. Here, we call
the former process as channel A and the latter as channel C according to [26]. In
channel A, SR penetrates the transmitter and scatterer until the depth z and is then
resonantly absorbed. In this channel, the scattering field E A (w, w s , z) satisfies
E A (w, w s , z) ∝
1
w − w s + i/2
E t (w)E s (w, w s , z).
Now, we can obtain the detected intensity I A (w s ) as follows:
I A (w s ) = C A
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )E A (w, w s , z)
2
= C A
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )
w − w s + i/2
E t (w)E s (w, w s , z)
2
,
(2.3)
where C A denotes a proportionality factor. This expression includes (1) the Fourier
transformation of E A (w, w s , z) to obtain the time dependence of E A and the square