72
M. Seto et al.
approximately several mill electron volts or less. Usually, SR produced by an undulator installed in a recent hard X-ray storage ring is approximately several electron
volts, and it is possible to generate X-rays with a bandwidth of approximately several
milli electron volts (in special cases, approximately 0.1 meV can be achieved [43])
using Si monochromators. Sapphire (α-Al 2 O 3 ), a material for the monochromator, in
addition to Si, has been used particularly for high-energy nuclides [44, 45]. The incident energy can be scanned at approximately the nuclear resonant excitation energy
by changing the Bragg angle of the monochromator. Note that the respective nuclear
resonant energies of the available isotopes are much more than the usual scan range
(100 meV). Therefore, pure element (isotope)-specific measurement is assured.
To observe the phonon energy spectrum, we irradiate a sample containing the resonance isotope as a function of the energy of the generated X-rays with a bandwidth
of approximately several milli electron volts. From the irradiated sample, a strong
scattering is emitted due to photoelectron effects and so on. Therefore, to discern
the relatively weak nuclear scattering from the strong scattering, we observe only
delayed photons (e.g., fluorescence X-rays and γ-rays) and electrons (e.g., conversion electrons) emitted at the de-excitation in the time domain because scattering
due to electronic processes is promptly emitted at the irradiation. The schematics
of the measurement of nuclear resonant excitation accompanied by phonon creation
or annihilation are shown in Fig. 2.8. As shown in the figure, if the energy of the
incident photon is equal to the sum of the energy of the nuclear resonant excited state
and a phonon, one phonon is created. Alternatively, one-phonon annihilation occurs
if the sum of the energy of the incident photon and that of a phonon is equal to the
sum of the energy of the nuclear resonant excited state. The detection system with
a closed-cycle refrigerator cryostat for NRIS measurement using an eight-element
APD detector is shown in Fig. 2.9.
We can obtain the phonon density of states (PDOS) weighted by the projection
of the polarization vectors on the direction of the incident X-ray radiation from the
NRIS measurement. Based on [46], the following expression is relevant:
g(E, κ) =
V
(2π )
3
j
dq
κ · e j (q)
2 δ
E − ω j (q)
,
(2.6)
where V is the volume of the unit cell, q is the phonon wave vector, ω j (q) is the
phonon dispersion relation for the branch j, κ is the normalized wave vector of the
incident X-ray (κ = k/|k|, k: wave vector of incident X-ray), and e j (q) is the
polarization vector of the vibrations of the resonant atom. For single crystals with a
cubic Bravais lattice and polycrystalline materials composed of resonant atoms only,
g(E, κ) is the exact PDOS. In the general case of a polycrystalline material, that is,
a material composed of not only resonant atoms but also other nonresonant atoms,
averaging over all directions of the incident radiation results in g(E), representing a
PDOS weighted by the square amplitude of the resonant atoms. Therefore, we can
obtain a partial PDOS for a specific element (isotope). The nuclear resonant inelastic
absorption cross section can be expressed using the following weighted PDOS:
M. Seto et al.
approximately several mill electron volts or less. Usually, SR produced by an undulator installed in a recent hard X-ray storage ring is approximately several electron
volts, and it is possible to generate X-rays with a bandwidth of approximately several
milli electron volts (in special cases, approximately 0.1 meV can be achieved [43])
using Si monochromators. Sapphire (α-Al 2 O 3 ), a material for the monochromator, in
addition to Si, has been used particularly for high-energy nuclides [44, 45]. The incident energy can be scanned at approximately the nuclear resonant excitation energy
by changing the Bragg angle of the monochromator. Note that the respective nuclear
resonant energies of the available isotopes are much more than the usual scan range
(100 meV). Therefore, pure element (isotope)-specific measurement is assured.
To observe the phonon energy spectrum, we irradiate a sample containing the resonance isotope as a function of the energy of the generated X-rays with a bandwidth
of approximately several milli electron volts. From the irradiated sample, a strong
scattering is emitted due to photoelectron effects and so on. Therefore, to discern
the relatively weak nuclear scattering from the strong scattering, we observe only
delayed photons (e.g., fluorescence X-rays and γ-rays) and electrons (e.g., conversion electrons) emitted at the de-excitation in the time domain because scattering
due to electronic processes is promptly emitted at the irradiation. The schematics
of the measurement of nuclear resonant excitation accompanied by phonon creation
or annihilation are shown in Fig. 2.8. As shown in the figure, if the energy of the
incident photon is equal to the sum of the energy of the nuclear resonant excited state
and a phonon, one phonon is created. Alternatively, one-phonon annihilation occurs
if the sum of the energy of the incident photon and that of a phonon is equal to the
sum of the energy of the nuclear resonant excited state. The detection system with
a closed-cycle refrigerator cryostat for NRIS measurement using an eight-element
APD detector is shown in Fig. 2.9.
We can obtain the phonon density of states (PDOS) weighted by the projection
of the polarization vectors on the direction of the incident X-ray radiation from the
NRIS measurement. Based on [46], the following expression is relevant:
g(E, κ) =
V
(2π )
3
j
dq
κ · e j (q)
2 δ
E − ω j (q)
,
(2.6)
where V is the volume of the unit cell, q is the phonon wave vector, ω j (q) is the
phonon dispersion relation for the branch j, κ is the normalized wave vector of the
incident X-ray (κ = k/|k|, k: wave vector of incident X-ray), and e j (q) is the
polarization vector of the vibrations of the resonant atom. For single crystals with a
cubic Bravais lattice and polycrystalline materials composed of resonant atoms only,
g(E, κ) is the exact PDOS. In the general case of a polycrystalline material, that is,
a material composed of not only resonant atoms but also other nonresonant atoms,
averaging over all directions of the incident radiation results in g(E), representing a
PDOS weighted by the square amplitude of the resonant atoms. Therefore, we can
obtain a partial PDOS for a specific element (isotope). The nuclear resonant inelastic
absorption cross section can be expressed using the following weighted PDOS: