10.2 NRVS Intensities for Discrete Normal Modes
How are vibrations and phonons with energies of meV coupled to nuclear transitions
at the tens of keV level? The occurrence of vibrational sidebands (analogous to those
in AM radio communication) is one way to understand the NRVS effect. A nucleus
vibrating at frequency Ω and illuminated by a monochromatic beam at ω N + Ω will
experience a negative sideband at (ω N + Ω) À Ω ¼ ω N and hence be in resonance for
a nuclear transition. In addition, a nucleus that is part of a vibrational excited state
will be in resonance via the positive sideband: (ω N À Ω) + Ω ¼ ω N . Finally, multiphonon resonances will also occur as higher-order sidebands, but their intensity will
be reduced.
The overall cross section σ(E) for nuclear resonant absorption of a photon with
energy E can be factored into two terms, one of which depends on the properties of
the nucleus and one that depends on the environment in which that nucleus is placed:
σ E
ð Þ ¼
π
2
σ E 0
ð ÞΓ 0 S E À E 0
ð
Þ
ð10:1Þ
and if one assumes a harmonic lattice or molecule:
S E
ð Þ ¼ f LM δ Γ E
ð Þ
|ffl ffl{zffl ffl}
M€ ossbauer
þ
X 1
n¼1
S n E
ð Þ
|fflfflfflfflffl ffl{zfflfflfflfflffl ffl}
NRVS
ð10:2Þ
where f LM is the Lamb-Mössbauer factor, δ Γ (E) is a Lorentzian of width Γ, and the
function S n (E) refers to events involving n phonons.
10.2.1 Stokes Fundamentals
Suppose we have a harmonic oscillator molecular system with a set of normal modes
labeled α that are described by the displacements r
!
kα for atom k and normal mode α.
We assume a lineshape function L to account for the experimental resolution and
lifetime broadening, and we convert the energy scale for our normal mode α to a
frequency v α in wave numbers. As a specific example, we use the FeCl 4
À ion, which
in T d symmetry will have normal modes with four types of symmetry species. The
particular frequencies are labeled ν 1 , corresponding to the totally symmetric A 1
Fe–Cl stretch, ν 2 , for the doubly degenerate E Fe–Cl bend; ν 3 , for the triply
degenerate T 2 Fe–Cl stretch; and ν 4 , for the triply degenerate T 2 Fe–Cl bend
(Fig. 10.5).
10.2 NRVS Intensities for Discrete Normal Modes
261
How are vibrations and phonons with energies of meV coupled to nuclear transitions
at the tens of keV level? The occurrence of vibrational sidebands (analogous to those
in AM radio communication) is one way to understand the NRVS effect. A nucleus
vibrating at frequency Ω and illuminated by a monochromatic beam at ω N + Ω will
experience a negative sideband at (ω N + Ω) À Ω ¼ ω N and hence be in resonance for
a nuclear transition. In addition, a nucleus that is part of a vibrational excited state
will be in resonance via the positive sideband: (ω N À Ω) + Ω ¼ ω N . Finally, multiphonon resonances will also occur as higher-order sidebands, but their intensity will
be reduced.
The overall cross section σ(E) for nuclear resonant absorption of a photon with
energy E can be factored into two terms, one of which depends on the properties of
the nucleus and one that depends on the environment in which that nucleus is placed:
σ E
ð Þ ¼
π
2
σ E 0
ð ÞΓ 0 S E À E 0
ð
Þ
ð10:1Þ
and if one assumes a harmonic lattice or molecule:
S E
ð Þ ¼ f LM δ Γ E
ð Þ
|ffl ffl{zffl ffl}
M€ ossbauer
þ
X 1
n¼1
S n E
ð Þ
|fflfflfflfflffl ffl{zfflfflfflfflffl ffl}
NRVS
ð10:2Þ
where f LM is the Lamb-Mössbauer factor, δ Γ (E) is a Lorentzian of width Γ, and the
function S n (E) refers to events involving n phonons.
10.2.1 Stokes Fundamentals
Suppose we have a harmonic oscillator molecular system with a set of normal modes
labeled α that are described by the displacements r
!
kα for atom k and normal mode α.
We assume a lineshape function L to account for the experimental resolution and
lifetime broadening, and we convert the energy scale for our normal mode α to a
frequency v α in wave numbers. As a specific example, we use the FeCl 4
À ion, which
in T d symmetry will have normal modes with four types of symmetry species. The
particular frequencies are labeled ν 1 , corresponding to the totally symmetric A 1
Fe–Cl stretch, ν 2 , for the doubly degenerate E Fe–Cl bend; ν 3 , for the triply
degenerate T 2 Fe–Cl stretch; and ν 4 , for the triply degenerate T 2 Fe–Cl bend
(Fig. 10.5).
10.2 NRVS Intensities for Discrete Normal Modes
261