96
M. Seto et al.
Here, we show that neither f Γ 0 (q) nor f E (q) cannot be determined by the conventional single-line TDI in the case of finite E width of approximately several meV
using a high-resolution monochromator. From eq. (2.29), we obtain
¯
I (q, t) ∝ |G(t)|
2
1 + 2S
(q, t)/[1 + f E (q)] cos(δ Et/)
.
(2.31)
In this equation, the cosine term has an additional factor 2/[1 + f E (q)]
compared to eq. (2.21). When this expression is used for the fitting, a function
F
(q, t) is assumed for 2S
(q, t)/[1 + f E (q)]. Using F(q, t), which represents
the form of S
(q, t) in the time window of the measurement, F
(q, t) can be
expressed as F
(q, t) = 2F(q, t)/[1 + f E (q)]. It follows that lim
t→0
F
(q, t) =
2 lim
t→0
F(q, t)/[1 + f E (q)] = 2 f Γ 0 (q)/[1 + f E (q)]. This equation suggests that
both free fitting parameters f Γ 0 (q) and f E (q) relate to lim
t→0
F
(q, t). Therefore, in
principle, neither f E (q) nor f Γ 0 (q) can be determined by the single-line TDI when
an identical pair of emitters are used in the incident SR condition with meV energy
width. The conventional TDI suffers this uncertainty of the physical meaning of
lim
t→0
F
(q, t).
In contrast, for the case of a multiline emitter case with |G 1 (t)|
2
= |G 2 (t)|
2 , both
f E (q) and f Γ 0 (q) can be determined based on the difference between |G 1 (t)|
2 and
|G 2 (t)|
2 as we discuss in Sect. 2.4.5 [121].
2.4.5 Time-Domain Interferometry Using Multiline
Mössbauer Gamma Rays
In the case where multiline γ-rays with |G 1 (t)|
2
= |G 2 (t)|
2 are used for TDI, the
intensity of the γ-rays from eq. (2.23) is written as
I (q, t s + t) ∝ |G 1 (t)|
2
|g(q, t s + t)|
2
+ |G 2 (t)|
2
|g c (q, t s )|
2
+ G
∗
1 (t)G 2 (t)g
∗
(q, t s + t)g c (q, t s )
+ G 1 (t)G
∗
2 (t)g(q, t s + t)g
∗
c (q, t s ).
(2.32)
Similarly, in Sect. 2.4.4, the observed time-averaging intensity ¯
I (q, t) can be
written as
¯
I (q, t) ∝ S(q, 0)|G 1 (t)|
2
+ S cc (q, 0)|G 2 (t)|
2
+ S(q, t)
G
∗
1 (t)G 2 (t) + G
∗
2 (t)G 1 (t)
(at t T ).
(2.33)
Using S
(q, t) and f E (q), ¯
I (q, t) can be rewritten as
¯
I (q, t) ∝
1 − S
(q, t)
|G 1 (t)|
2
+ |G 2 (t)|
2
+ S
(q, t)|G 1 (t) + G 2 (t)|
2
− [1 − f E (q)]|G 2 (t)|
2
(at t T ).
(2.34)
M. Seto et al.
Here, we show that neither f Γ 0 (q) nor f E (q) cannot be determined by the conventional single-line TDI in the case of finite E width of approximately several meV
using a high-resolution monochromator. From eq. (2.29), we obtain
¯
I (q, t) ∝ |G(t)|
2
1 + 2S
(q, t)/[1 + f E (q)] cos(δ Et/)
.
(2.31)
In this equation, the cosine term has an additional factor 2/[1 + f E (q)]
compared to eq. (2.21). When this expression is used for the fitting, a function
F
(q, t) is assumed for 2S
(q, t)/[1 + f E (q)]. Using F(q, t), which represents
the form of S
(q, t) in the time window of the measurement, F
(q, t) can be
expressed as F
(q, t) = 2F(q, t)/[1 + f E (q)]. It follows that lim
t→0
F
(q, t) =
2 lim
t→0
F(q, t)/[1 + f E (q)] = 2 f Γ 0 (q)/[1 + f E (q)]. This equation suggests that
both free fitting parameters f Γ 0 (q) and f E (q) relate to lim
t→0
F
(q, t). Therefore, in
principle, neither f E (q) nor f Γ 0 (q) can be determined by the single-line TDI when
an identical pair of emitters are used in the incident SR condition with meV energy
width. The conventional TDI suffers this uncertainty of the physical meaning of
lim
t→0
F
(q, t).
In contrast, for the case of a multiline emitter case with |G 1 (t)|
2
= |G 2 (t)|
2 , both
f E (q) and f Γ 0 (q) can be determined based on the difference between |G 1 (t)|
2 and
|G 2 (t)|
2 as we discuss in Sect. 2.4.5 [121].
2.4.5 Time-Domain Interferometry Using Multiline
Mössbauer Gamma Rays
In the case where multiline γ-rays with |G 1 (t)|
2
= |G 2 (t)|
2 are used for TDI, the
intensity of the γ-rays from eq. (2.23) is written as
I (q, t s + t) ∝ |G 1 (t)|
2
|g(q, t s + t)|
2
+ |G 2 (t)|
2
|g c (q, t s )|
2
+ G
∗
1 (t)G 2 (t)g
∗
(q, t s + t)g c (q, t s )
+ G 1 (t)G
∗
2 (t)g(q, t s + t)g
∗
c (q, t s ).
(2.32)
Similarly, in Sect. 2.4.4, the observed time-averaging intensity ¯
I (q, t) can be
written as
¯
I (q, t) ∝ S(q, 0)|G 1 (t)|
2
+ S cc (q, 0)|G 2 (t)|
2
+ S(q, t)
G
∗
1 (t)G 2 (t) + G
∗
2 (t)G 1 (t)
(at t T ).
(2.33)
Using S
(q, t) and f E (q), ¯
I (q, t) can be rewritten as
¯
I (q, t) ∝
1 − S
(q, t)
|G 1 (t)|
2
+ |G 2 (t)|
2
+ S
(q, t)|G 1 (t) + G 2 (t)|
2
− [1 − f E (q)]|G 2 (t)|
2
(at t T ).
(2.34)