68
M. Seto et al.
of its magnitude to obtain the intensity from the field and (2) the integrations on the
depth z and τ ; the τ integration corresponds to the time window at APD. In channel
C, NFS by the transmitter or scatterer is scattered by electrons at depth z. Thus, the
incident field at the depth satisfies
E C (w, w s , z) ∝ 1 − E t (w)E s (w, w s , z).
Considering cross section of electrons independent of w s , the detected intensity
I C (w s ) is obtained as
I C (w s ) = C C
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )E C (w, w s , z)
2
= C C
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )(1 − E t (w)E s (w, w s , z))
2
,
(2.4)
where C C denotes another proportionality factor. Now, we have the expression of the
detected intensity I (w s ) as
I (w s ) = I A (w s ) + I C (w s ) + I B ,
(2.5)
where I B denotes for other processes independent of w s , such as the process with
nuclear resonant scattering with recoil at the scatterer (called channel B).
The narrowing of the energy width and the wavy pattern in the background are
now discussed based on Eqs. (2.1)–(2.5). The narrowing effect depends on the time
window, as shown in Fig. 2.7. Here, a narrower energy width was obtained compared
to the ideal linewidth in conventional Mössbauer spectroscopy, which is twice the
natural linewidth. For example, although the ideal linewidth is 2.0 mm/s in conventional
174 Yb Mössbauer spectroscopy, a full-width at half maximum (FWHM) of
1.3 mm/s was achieved [8]. This property is advantageous when the nuclear hyperfine structure is small and precise analysis is required. In contrast, the wavy pattern
in the background is usually a drawback because it may conceal small components
in the spectra, at least in the initial guess for analysis. Proper analysis using the equations is required. Note that if the time window is [0, ∞], both the narrowing effects
and the wavy background vanish. In addition to the [0, ∞] time window, in the case
of thin transmitter and scatterer, the spectra show Lorentzian shape with the “ideal
linewidth,” similar to the conventional Mössbauer spectroscopy using RI. Therefore,
the Lorentzian approximation described at the beginning of this section corresponds
to the case. These situations are similar to the delayed coincidence Mössbauer spectroscopy using RI [27, 28]. Furthermore, the equations also show better conditions
for the measurement system, described previously. For example, the high-resolution
monochromator enhances the absorption depth of the spectra because it suppresses
the channel B process. Furthermore, the effective thickness of the transmitter and
that of scatterer (i.e., the sample and energy reference) should be similar.
M. Seto et al.
of its magnitude to obtain the intensity from the field and (2) the integrations on the
depth z and τ ; the τ integration corresponds to the time window at APD. In channel
C, NFS by the transmitter or scatterer is scattered by electrons at depth z. Thus, the
incident field at the depth satisfies
E C (w, w s , z) ∝ 1 − E t (w)E s (w, w s , z).
Considering cross section of electrons independent of w s , the detected intensity
I C (w s ) is obtained as
I C (w s ) = C C
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )E C (w, w s , z)
2
= C C
τ 2
∫
τ 1
dτ
z s
∫
0
dz
dw
2π
exp(−iwτ )(1 − E t (w)E s (w, w s , z))
2
,
(2.4)
where C C denotes another proportionality factor. Now, we have the expression of the
detected intensity I (w s ) as
I (w s ) = I A (w s ) + I C (w s ) + I B ,
(2.5)
where I B denotes for other processes independent of w s , such as the process with
nuclear resonant scattering with recoil at the scatterer (called channel B).
The narrowing of the energy width and the wavy pattern in the background are
now discussed based on Eqs. (2.1)–(2.5). The narrowing effect depends on the time
window, as shown in Fig. 2.7. Here, a narrower energy width was obtained compared
to the ideal linewidth in conventional Mössbauer spectroscopy, which is twice the
natural linewidth. For example, although the ideal linewidth is 2.0 mm/s in conventional
174 Yb Mössbauer spectroscopy, a full-width at half maximum (FWHM) of
1.3 mm/s was achieved [8]. This property is advantageous when the nuclear hyperfine structure is small and precise analysis is required. In contrast, the wavy pattern
in the background is usually a drawback because it may conceal small components
in the spectra, at least in the initial guess for analysis. Proper analysis using the equations is required. Note that if the time window is [0, ∞], both the narrowing effects
and the wavy background vanish. In addition to the [0, ∞] time window, in the case
of thin transmitter and scatterer, the spectra show Lorentzian shape with the “ideal
linewidth,” similar to the conventional Mössbauer spectroscopy using RI. Therefore,
the Lorentzian approximation described at the beginning of this section corresponds
to the case. These situations are similar to the delayed coincidence Mössbauer spectroscopy using RI [27, 28]. Furthermore, the equations also show better conditions
for the measurement system, described previously. For example, the high-resolution
monochromator enhances the absorption depth of the spectra because it suppresses
the channel B process. Furthermore, the effective thickness of the transmitter and
that of scatterer (i.e., the sample and energy reference) should be similar.