88
M. Seto et al.
We write the electric field amplitude of the incident SR in the time t domain
as E 0 (t), which is related to ˆ
E 0 (ω) by Fourier transformation. The time response
functions of emitters 1 and 2 are defined to be as follows [117, 118]:
R 1 (t) = δ(t) + G(t)e
iδ Et/
and
R 2 (t) = δ(t) + G(t),
(2.13)
where δ(t) represents a transmission part without nuclear excitation process and
the second term including G(t) represents the nuclear excitation and de-excitation
processes with the Mössbauer effect. The energy shift of γ-rays from the upstream
emitter is considered by a term that includes the corresponding angular frequency
δ E/. Here, we ignored a transmittance factor because it does not affect the shape of
the time spectrum. When the energy shift is sufficiently large (δ E Γ 0 ), unfavorable
radiative coupling (RC; photons experience nuclear excitation processes in both
emitters) can be neglected [119]. We consider the δ E Γ 0 case in the following
discussion.
The electric field amplitude after emitter 1 is written as R 1 (t) ⊗ E 0 (t), where ⊗
denotes the convolution integral [117, 119]. After transmitting emitter 2, the electric
field amplitude is written as R 2 (t) ⊗ {R 1 (t) ⊗ E 0 (t)}. When
ˆ
E 0 (ω)
2
shows a finite
bandwidth E/, |E 0 (t)|
2 also shows a finite width T ∼ h//E. In the timescale
of T , the incident SR partly shows time coherence. T is much smaller than the
nuclear response as shown below. Therefore, we assume E 0 (t) ∝ δ(t). Here, the
electric field amplitude E tot (t) at the detector position can be written as
E tot (t) ∝
∞
∫
0
R 1
t
R 2
t − t
dt
≈ δ(t) + G(t)
1 + e
iδ Et/
.
(2.14)
Here, the origin of the time is the detection time of the SR pulse. The electric
field amplitude of γ-rays E(t) is written as E(t) ∝ G(t)
1 + e
iδ Et/
. The intensity
is given by
I (t) = |E(t)|
2
= 2|G(t)|
2 [1 + cos(δ Et/)].
(2.15)
Here, the factor |G(t)|
2 represents the NFS time spectrum from one emitter. In a
thin limit of the emitter thickness, it follows that |G(t)|
2
∝ e
−t/τ 0 , where τ 0 is the
lifetime of the nuclear excited state. Otherwise, |G(t)|
2 shows more complex time
dependence known as a dynamical beat [120]. In the
57 Fe case, τ 0 is ~141 ns. A
factor 1 + cos(δ Et/) represents a quantum beat modifying |G(t)|
2 .
Here, the velocity transducer used to drive emitter 1 brings a constant velocity with
positive and negative sign, alternatively. Note that the sign of the velocity of the driven
emitter 1 does not affect the obtained time spectrum I (t), because only an absolute
M. Seto et al.
We write the electric field amplitude of the incident SR in the time t domain
as E 0 (t), which is related to ˆ
E 0 (ω) by Fourier transformation. The time response
functions of emitters 1 and 2 are defined to be as follows [117, 118]:
R 1 (t) = δ(t) + G(t)e
iδ Et/
and
R 2 (t) = δ(t) + G(t),
(2.13)
where δ(t) represents a transmission part without nuclear excitation process and
the second term including G(t) represents the nuclear excitation and de-excitation
processes with the Mössbauer effect. The energy shift of γ-rays from the upstream
emitter is considered by a term that includes the corresponding angular frequency
δ E/. Here, we ignored a transmittance factor because it does not affect the shape of
the time spectrum. When the energy shift is sufficiently large (δ E Γ 0 ), unfavorable
radiative coupling (RC; photons experience nuclear excitation processes in both
emitters) can be neglected [119]. We consider the δ E Γ 0 case in the following
discussion.
The electric field amplitude after emitter 1 is written as R 1 (t) ⊗ E 0 (t), where ⊗
denotes the convolution integral [117, 119]. After transmitting emitter 2, the electric
field amplitude is written as R 2 (t) ⊗ {R 1 (t) ⊗ E 0 (t)}. When
ˆ
E 0 (ω)
2
shows a finite
bandwidth E/, |E 0 (t)|
2 also shows a finite width T ∼ h//E. In the timescale
of T , the incident SR partly shows time coherence. T is much smaller than the
nuclear response as shown below. Therefore, we assume E 0 (t) ∝ δ(t). Here, the
electric field amplitude E tot (t) at the detector position can be written as
E tot (t) ∝
∞
∫
0
R 1
t
R 2
t − t
dt
≈ δ(t) + G(t)
1 + e
iδ Et/
.
(2.14)
Here, the origin of the time is the detection time of the SR pulse. The electric
field amplitude of γ-rays E(t) is written as E(t) ∝ G(t)
1 + e
iδ Et/
. The intensity
is given by
I (t) = |E(t)|
2
= 2|G(t)|
2 [1 + cos(δ Et/)].
(2.15)
Here, the factor |G(t)|
2 represents the NFS time spectrum from one emitter. In a
thin limit of the emitter thickness, it follows that |G(t)|
2
∝ e
−t/τ 0 , where τ 0 is the
lifetime of the nuclear excited state. Otherwise, |G(t)|
2 shows more complex time
dependence known as a dynamical beat [120]. In the
57 Fe case, τ 0 is ~141 ns. A
factor 1 + cos(δ Et/) represents a quantum beat modifying |G(t)|
2 .
Here, the velocity transducer used to drive emitter 1 brings a constant velocity with
positive and negative sign, alternatively. Note that the sign of the velocity of the driven
emitter 1 does not affect the obtained time spectrum I (t), because only an absolute