90
M. Seto et al.
Similarly, to the NFS case, the theoretical equation of the γ-ray time spectrum is written as I
q, t s + t
∝
G
t
2
g
q, t s + t
2 +
g
q, t s
2 + g
q, t s +
t
g
∗
q, t s
e
iδ Et/
+ g
∗
q, t s + t
g
q, t s
e
−iδ Et/
. The experimentally observed
¯
I (q, t) is obtained by averaging I (q, t s + t) by t s over a long measurement time as
¯
I (q, t) ∝ |G(t)|
2
|g(q, t s + t)|
2
+
|g(q, t s )|
2
+
g(q, t s + t)g
∗
(q, t s )
e
iδ Et/
+
g
∗
(q, t s + t)g(q, t s )
e
−iδ Et/
(2.19)
where · · · indicates averaging by t s over a long measurement time. It follows that
S(q, 0) =
|g(q, t s + t)|
2
=
|g(q, t s )|
2
. In classic mechanical cases, it follows that
S(q, t) = g
∗
(q, t s + t)g(q, t s ) = g(q, t s + t)g
∗
(q, t s ) suggesting S(q, t) can be
treated as a real number [115, 119, 122, 123]. We define S
(q, t) as the intermediate
scattering function normalized by S(q, 0) as
S
(q, t) = g(q, t s + t)g(q, t s )/
|g(q, t s + t)|
2
,
(2.20)
we obtain
¯
I (q, t) ∝ |G(t)|
2 S(q)
1 + S
(q, t) cos(δ Et/)
.
(2.21)
When a single exponential relaxation is assumed for S
(q, t), we can write
S
(q, t) ∝ exp{−t/τ }, where τ is a relaxation time. Often, there is an intrinsic
relaxation in S
(q, t) even when standard samples with no detectable dynamics are
measured [115]. To express it, we introduce a relaxation function F
int
(t). By using
this factor, the time spectrum is given by
¯
I (q, t) ∝ |G(t)|
2
1 + F
int
(t)S
(q, t)cos(δ Et/)
.
(2.22)
The measured time spectrum ¯
I exp (q, t) is written as ¯
I exp (q, t) = ¯
I (q, t)⊗D(t)+B
as described in Sect. 2.4.3.1.
We calculated NFS and QEGS time spectra, as shown in Fig. 2.18c, where we
used the following conditions: the effective thickness of each emitter T e = 10, v
= 20 mm/s, and τ = 0.5τ 0 . The corresponding γ-ray energy spectra are shown
in Fig. 2.18b. The figure shows that the time-spectrum shape changes, reflecting
dynamics as the disappearance of the quantum beat.
In Fig. 2.18d, experimentally obtained time spectra by (I) NFS and QEGS studies
on o-terphenyl at (II) 270 K, (III) 280 K, and (IV) 290 K are shown. It can be
confirmed that the quantum beat disappears by heating the sample. Least squares
fittings using eq. (2.22) were successfully performed considering the time resolution
D(t) and background. The obtained relaxation time was confirmed to be consistent
with previous results [124].
M. Seto et al.
Similarly, to the NFS case, the theoretical equation of the γ-ray time spectrum is written as I
q, t s + t
∝
G
t
2
g
q, t s + t
2 +
g
q, t s
2 + g
q, t s +
t
g
∗
q, t s
e
iδ Et/
+ g
∗
q, t s + t
g
q, t s
e
−iδ Et/
. The experimentally observed
¯
I (q, t) is obtained by averaging I (q, t s + t) by t s over a long measurement time as
¯
I (q, t) ∝ |G(t)|
2
|g(q, t s + t)|
2
+
|g(q, t s )|
2
+
g(q, t s + t)g
∗
(q, t s )
e
iδ Et/
+
g
∗
(q, t s + t)g(q, t s )
e
−iδ Et/
(2.19)
where · · · indicates averaging by t s over a long measurement time. It follows that
S(q, 0) =
|g(q, t s + t)|
2
=
|g(q, t s )|
2
. In classic mechanical cases, it follows that
S(q, t) = g
∗
(q, t s + t)g(q, t s ) = g(q, t s + t)g
∗
(q, t s ) suggesting S(q, t) can be
treated as a real number [115, 119, 122, 123]. We define S
(q, t) as the intermediate
scattering function normalized by S(q, 0) as
S
(q, t) = g(q, t s + t)g(q, t s )/
|g(q, t s + t)|
2
,
(2.20)
we obtain
¯
I (q, t) ∝ |G(t)|
2 S(q)
1 + S
(q, t) cos(δ Et/)
.
(2.21)
When a single exponential relaxation is assumed for S
(q, t), we can write
S
(q, t) ∝ exp{−t/τ }, where τ is a relaxation time. Often, there is an intrinsic
relaxation in S
(q, t) even when standard samples with no detectable dynamics are
measured [115]. To express it, we introduce a relaxation function F
int
(t). By using
this factor, the time spectrum is given by
¯
I (q, t) ∝ |G(t)|
2
1 + F
int
(t)S
(q, t)cos(δ Et/)
.
(2.22)
The measured time spectrum ¯
I exp (q, t) is written as ¯
I exp (q, t) = ¯
I (q, t)⊗D(t)+B
as described in Sect. 2.4.3.1.
We calculated NFS and QEGS time spectra, as shown in Fig. 2.18c, where we
used the following conditions: the effective thickness of each emitter T e = 10, v
= 20 mm/s, and τ = 0.5τ 0 . The corresponding γ-ray energy spectra are shown
in Fig. 2.18b. The figure shows that the time-spectrum shape changes, reflecting
dynamics as the disappearance of the quantum beat.
In Fig. 2.18d, experimentally obtained time spectra by (I) NFS and QEGS studies
on o-terphenyl at (II) 270 K, (III) 280 K, and (IV) 290 K are shown. It can be
confirmed that the quantum beat disappears by heating the sample. Least squares
fittings using eq. (2.22) were successfully performed considering the time resolution
D(t) and background. The obtained relaxation time was confirmed to be consistent
with previous results [124].