2 Synchrotron-Radiation-Based Energy-Domain Mössbauer …
93
E(q, t s + t) =
∞
−∞
dt
g(q, t s + t)G 1
t − t
E 0
t
+ G 2
t − t
g
q, t s + t
E 0
t
.
(2.23)
The first and second terms of eq. (2.23) describe the electric field amplitudes of
the γ-rays passing paths I and II in Fig. 2.19, respectively.
In this subsection, we consider the experimental setup with single-line γ-ray
emitters shown in Fig. 2.18a. Here, we assume G 1 (t) = G 2 (t)e
iδ Et/ and G 2 (t) =
G(t). The electric field amplitude of the γ-rays is as follows:
E(q, t s + t) =
∞
∫
−∞
dt
g(q, t s + t)G
t − t
e
iδ E(t−t
) E 0
t
+ G
t − t
g
q, t s + t
E 0
t
.
(2.24)
The time variation of |G(t)| is much slower than that of |E 0 (t)|
from the relation τ 0 δT 12 T . Therefore, the first term
in eq. (2.24) follows that g(q, t s + t)
∞
∫
−∞
dt
G
t − t
e
iδ E(t−t
) E 0
t
∼ =
g(q, t s + t)G(t)e
iδ Et/
∞
∫
−∞
dt
E 0
t
. Similarly, the second term follows that
∞
∫
−∞
dt
G
t − t
g
q, t s + t
E 0
t
∼ = G(t)
∞
∫
−∞
dt
g
q, t s + t
E 0
t
. Here, we
define g c (q, t s ) as g c (q, t s ) ≡
∞
∫
−∞
dt
g
q, t s + t
E 0
t
/
∞
∫
−∞
dt
E 0
t
. In case of
E 0 (t) = δ(t), it follows that g c (q, t s ) = g(q, t s ). Using g c , eq. (2.24) can be written
as
E(q, t s + t) ∼ = ˆ
E 0 (0)G(t)
g(q, t s + t)e
iδ Et/
+ g c (q, t s )
,
(2.25)
where we used the general relation
∞
∫
−∞
dt
E 0
t
= ˆ
E 0 (0) ≡ ˆ
E 0 (ω = 0). Here,
ˆ
E 0 (ω) is the angular frequency-domain representation of E 0 (t).
Here, we consider the meaning of g c (q, t s ), which is an integration of the product
of g
q, t s + t
and E 0
t
by t
. We show examples of paths I and II of the γ-rays
detected at t with an incident time t
= 0 (long dashed line) and t
= 0 (short dashed
line) in the time–space diagrams of Figs. 2.20a and b, respectively. Gamma rays
with different incident times t
interfere at the detector owing to the finite coherent
width of E 0 (t) in both γ-ray paths I and II. The integration in g c (q, t s ) originates
from a characteristic of path II: γ-rays scattered by the sample at various times t s + t
(
t
T ) interfere with each other at the detector position at t. Alternatively, in path
I, the γ-rays scattered by the sample at unique time t s + t interfere with each other
at the detector position at t. Therefore, the γ-rays passing path I are not affected by
the time width T of the incident radiation. This is the interpretation of eq. (2.25).
From eq. (2.16), we obtain the detected γ-ray intensity:
93
E(q, t s + t) =
∞
−∞
dt
g(q, t s + t)G 1
t − t
E 0
t
+ G 2
t − t
g
q, t s + t
E 0
t
.
(2.23)
The first and second terms of eq. (2.23) describe the electric field amplitudes of
the γ-rays passing paths I and II in Fig. 2.19, respectively.
In this subsection, we consider the experimental setup with single-line γ-ray
emitters shown in Fig. 2.18a. Here, we assume G 1 (t) = G 2 (t)e
iδ Et/ and G 2 (t) =
G(t). The electric field amplitude of the γ-rays is as follows:
E(q, t s + t) =
∞
∫
−∞
dt
g(q, t s + t)G
t − t
e
iδ E(t−t
) E 0
t
+ G
t − t
g
q, t s + t
E 0
t
.
(2.24)
The time variation of |G(t)| is much slower than that of |E 0 (t)|
from the relation τ 0 δT 12 T . Therefore, the first term
in eq. (2.24) follows that g(q, t s + t)
∞
∫
−∞
dt
G
t − t
e
iδ E(t−t
) E 0
t
∼ =
g(q, t s + t)G(t)e
iδ Et/
∞
∫
−∞
dt
E 0
t
. Similarly, the second term follows that
∞
∫
−∞
dt
G
t − t
g
q, t s + t
E 0
t
∼ = G(t)
∞
∫
−∞
dt
g
q, t s + t
E 0
t
. Here, we
define g c (q, t s ) as g c (q, t s ) ≡
∞
∫
−∞
dt
g
q, t s + t
E 0
t
/
∞
∫
−∞
dt
E 0
t
. In case of
E 0 (t) = δ(t), it follows that g c (q, t s ) = g(q, t s ). Using g c , eq. (2.24) can be written
as
E(q, t s + t) ∼ = ˆ
E 0 (0)G(t)
g(q, t s + t)e
iδ Et/
+ g c (q, t s )
,
(2.25)
where we used the general relation
∞
∫
−∞
dt
E 0
t
= ˆ
E 0 (0) ≡ ˆ
E 0 (ω = 0). Here,
ˆ
E 0 (ω) is the angular frequency-domain representation of E 0 (t).
Here, we consider the meaning of g c (q, t s ), which is an integration of the product
of g
q, t s + t
and E 0
t
by t
. We show examples of paths I and II of the γ-rays
detected at t with an incident time t
= 0 (long dashed line) and t
= 0 (short dashed
line) in the time–space diagrams of Figs. 2.20a and b, respectively. Gamma rays
with different incident times t
interfere at the detector owing to the finite coherent
width of E 0 (t) in both γ-ray paths I and II. The integration in g c (q, t s ) originates
from a characteristic of path II: γ-rays scattered by the sample at various times t s + t
(
t
T ) interfere with each other at the detector position at t. Alternatively, in path
I, the γ-rays scattered by the sample at unique time t s + t interfere with each other
at the detector position at t. Therefore, the γ-rays passing path I are not affected by
the time width T of the incident radiation. This is the interpretation of eq. (2.25).
From eq. (2.16), we obtain the detected γ-ray intensity: