4.8 Application: The Vibrations of UF 6
83
An alternative option would be to construct a pure bending mode, based on the
tangent motions. Let us denote this by |π 2 . The remainder is then denoted by |σ 2 .
|σ 2 =
−
√
2mMQ U + (M + 4m)Q σ − m
√
8Q π
√
(M + 4m)(M + 6m)
|π 2 =
−
√
4mQ U +
√
MQ π
√
M + 4m
(4.106)
In Fig. 4.6 we present both choices of bases. The angle, α, between both basis sets
is defined by
cos α =
M(M + 6m)
(M + 2m)(M + 4m)
(4.107)
In the case of UF 6 (m = 18.998, M = 238.050) this angle is −10.5 ◦ . The actual
eigenmodes are found by setting up the Hessian in one of these coordinate sets and
diagonalizing it. This Hessian matrix is symmetric and thus contains three independent parameters: the two diagonal elements and the single off-diagonal element.
The sum of the resulting eigenvalues is equal to the trace of the matrix, and the
product is equal to its determinant; this leaves still one degree of freedom, which
can be associated with the composition of the normal mode, viz. the angle of rotation in the diagram. It is important to realize that this composition also gives rise to
observables, albeit not the eigenfrequencies, but a variety of other properties, such
as the intensities of the vibrational transition, isotope shifts and isotope splittings,
or electron diffraction amplitudes. For most octahedral complexes, as in the case of
UF 6 , the rotation angle for the actual T 1u eigenmodes lies in the interval [0,α].This
means that the modes may approximately be assigned as a stretching and a bending
mode. In the spectrum their frequencies are denoted as ν 3 and ν 4 , respectively. The
isotope effect of the radioactive nucleus U 235 , as distinct from U 238 , is absent for all
modes, except for the T 1u modes, since these involve the displacement of uranium.
Of the latter two, the strongest effect is expected for the stretching vibration, since
this involves the largest displacement of the central atom. The pure stretching mode,
|σ 1 , can be expressed in terms of the displacements along the z-direction as
|σ 1 =
2mM
M + 2m
−Z 0 +
Z 1 + Z 6
2
(4.108)
This is precisely the antisymmetric mode for a triatomic F–U–F oscillator. The
square root preceding the modes corresponds to a mass weighting by the reduced
mass, μ, for such an oscillator:
μ =
1
M
+
1
2m
−1
=
2mM
M + 2m
(4.109)
Substitution of U 238 by the U 235 isotope will reduce this effective mass by a factor
0.9982. The frequency is accordingly increased by the square root of this factor.
83
An alternative option would be to construct a pure bending mode, based on the
tangent motions. Let us denote this by |π 2 . The remainder is then denoted by |σ 2 .
|σ 2 =
−
√
2mMQ U + (M + 4m)Q σ − m
√
8Q π
√
(M + 4m)(M + 6m)
|π 2 =
−
√
4mQ U +
√
MQ π
√
M + 4m
(4.106)
In Fig. 4.6 we present both choices of bases. The angle, α, between both basis sets
is defined by
cos α =
M(M + 6m)
(M + 2m)(M + 4m)
(4.107)
In the case of UF 6 (m = 18.998, M = 238.050) this angle is −10.5 ◦ . The actual
eigenmodes are found by setting up the Hessian in one of these coordinate sets and
diagonalizing it. This Hessian matrix is symmetric and thus contains three independent parameters: the two diagonal elements and the single off-diagonal element.
The sum of the resulting eigenvalues is equal to the trace of the matrix, and the
product is equal to its determinant; this leaves still one degree of freedom, which
can be associated with the composition of the normal mode, viz. the angle of rotation in the diagram. It is important to realize that this composition also gives rise to
observables, albeit not the eigenfrequencies, but a variety of other properties, such
as the intensities of the vibrational transition, isotope shifts and isotope splittings,
or electron diffraction amplitudes. For most octahedral complexes, as in the case of
UF 6 , the rotation angle for the actual T 1u eigenmodes lies in the interval [0,α].This
means that the modes may approximately be assigned as a stretching and a bending
mode. In the spectrum their frequencies are denoted as ν 3 and ν 4 , respectively. The
isotope effect of the radioactive nucleus U 235 , as distinct from U 238 , is absent for all
modes, except for the T 1u modes, since these involve the displacement of uranium.
Of the latter two, the strongest effect is expected for the stretching vibration, since
this involves the largest displacement of the central atom. The pure stretching mode,
|σ 1 , can be expressed in terms of the displacements along the z-direction as
|σ 1 =
2mM
M + 2m
−Z 0 +
Z 1 + Z 6
2
(4.108)
This is precisely the antisymmetric mode for a triatomic F–U–F oscillator. The
square root preceding the modes corresponds to a mass weighting by the reduced
mass, μ, for such an oscillator:
μ =
1
M
+
1
2m
−1
=
2mM
M + 2m
(4.109)
Substitution of U 238 by the U 235 isotope will reduce this effective mass by a factor
0.9982. The frequency is accordingly increased by the square root of this factor.