2 Synchrotron-Radiation-Based Energy-Domain Mössbauer …
89
value of the energy difference between γ-rays from the two emitters is important
for the beating pattern on the time spectrum owing to the factor cos(δ Et/) =
cos(−δ Et/) in eq. (2.15). In actual experiments, we do not detect signals in the
period, where the velocity is not constant owing to the change of sign of the velocity.
We also note that following time distributions have to be treated as an incoherent
broadening of the time spectrum: (1) the time resolution of the detector (in this study
~1 ns) and (2) the distribution of the arrival time of SR owing to the spatial distribution
of the electrons in one bunch (approximately 50 ps (FWHM) in BL09XU of SPring8). As an experimental time spectrum, we obtain ¯
I exp (t) = I (t) ⊗ D(t) + B, where
D(t) is the total incoherent distribution function and B is a background constant
noise.
2.4.3.2 Quasielastic Scattering Using Time-Domain Interferometry
of Single-Line Mössbauer Gamma Rays
Next, we consider QEGS experimental setup shown in the lower figure of Fig. 2.18a
and derive the expression of the QEGS time spectrum. Here, we introduce the sample
response function g(q, t). It cannot be immediately assumed that E 0 (t) ∝ δ(t)
because we do not know the timescale of the sample response [121]. Only the case that
the sample shows longer time response than T that we can assume E 0 (t) ∝ δ(t).
In this subsection, we assume this specific case of E 0 (t) ∝ δ(t) because it makes the
discussion simpler and instructive. The effect of the finite time width of the incident
SR on the QEGS time spectrum is discussed in Sect. 2.4.4 based on the discussion
of this subsection.
The electric field amplitude after emitter 1 is R 1 (t) ⊗ E 0 (t). We define t s as a time
when the Rayleigh scattering process of the prompt SR pulse occurs in the sample.
The sample response generally depends on both t s and t. Conversely, the nuclear
response is independent of t s [119, 122]. The electric field amplitude after scattering
by the sample is written as g(q, t s + t)E A (t), where we used the time response
function of the sample g(q, t s + t) defined in Sect. 2.4.2 [115]. After transmitting
emitter 2, the total electric field amplitude E tot (q, t s + t) at the angle corresponding
to q is
E tot (q, t s + t) ∝ R 2 (t) ⊗ {g(q, t s + t)[R 1 (t) ⊗ E 0 (t)]}.
(2.16)
By assuming E 0 (t) ∝ δ(t), E tot (q, t s + t) is written as
E tot (q, t s + t) ∝
∞
∫
0
R 1
t
g
q, t s + t
R 2
t − t
dt
.
(2.17)
Neglecting the RC effect, we obtain the electric field amplitude of γ-rays
E(q, t s + t) as
E(q, t s + t) ∝ g(q, t s + t)G(t)e
iδ Et/
+ g(q, t s )G(t).
(2.18)
89
value of the energy difference between γ-rays from the two emitters is important
for the beating pattern on the time spectrum owing to the factor cos(δ Et/) =
cos(−δ Et/) in eq. (2.15). In actual experiments, we do not detect signals in the
period, where the velocity is not constant owing to the change of sign of the velocity.
We also note that following time distributions have to be treated as an incoherent
broadening of the time spectrum: (1) the time resolution of the detector (in this study
~1 ns) and (2) the distribution of the arrival time of SR owing to the spatial distribution
of the electrons in one bunch (approximately 50 ps (FWHM) in BL09XU of SPring8). As an experimental time spectrum, we obtain ¯
I exp (t) = I (t) ⊗ D(t) + B, where
D(t) is the total incoherent distribution function and B is a background constant
noise.
2.4.3.2 Quasielastic Scattering Using Time-Domain Interferometry
of Single-Line Mössbauer Gamma Rays
Next, we consider QEGS experimental setup shown in the lower figure of Fig. 2.18a
and derive the expression of the QEGS time spectrum. Here, we introduce the sample
response function g(q, t). It cannot be immediately assumed that E 0 (t) ∝ δ(t)
because we do not know the timescale of the sample response [121]. Only the case that
the sample shows longer time response than T that we can assume E 0 (t) ∝ δ(t).
In this subsection, we assume this specific case of E 0 (t) ∝ δ(t) because it makes the
discussion simpler and instructive. The effect of the finite time width of the incident
SR on the QEGS time spectrum is discussed in Sect. 2.4.4 based on the discussion
of this subsection.
The electric field amplitude after emitter 1 is R 1 (t) ⊗ E 0 (t). We define t s as a time
when the Rayleigh scattering process of the prompt SR pulse occurs in the sample.
The sample response generally depends on both t s and t. Conversely, the nuclear
response is independent of t s [119, 122]. The electric field amplitude after scattering
by the sample is written as g(q, t s + t)E A (t), where we used the time response
function of the sample g(q, t s + t) defined in Sect. 2.4.2 [115]. After transmitting
emitter 2, the total electric field amplitude E tot (q, t s + t) at the angle corresponding
to q is
E tot (q, t s + t) ∝ R 2 (t) ⊗ {g(q, t s + t)[R 1 (t) ⊗ E 0 (t)]}.
(2.16)
By assuming E 0 (t) ∝ δ(t), E tot (q, t s + t) is written as
E tot (q, t s + t) ∝
∞
∫
0
R 1
t
g
q, t s + t
R 2
t − t
dt
.
(2.17)
Neglecting the RC effect, we obtain the electric field amplitude of γ-rays
E(q, t s + t) as
E(q, t s + t) ∝ g(q, t s + t)G(t)e
iδ Et/
+ g(q, t s )G(t).
(2.18)