702
Progress of Theoretical Physics, Vol. VI, No.5, September-October, 1951
On the Zero Point Entropy of Methane Crystal*
Takeo
NAGAMIYA
Department of Physics, Osaka U1tiversity, Osaka
(Received July 21, 1951)
Possible crystalline structures are suggested for the low temperature modifications of CH4, CD4 , and
CHaD crystals from the consideration of interatomic distances, assuming that the molecules are vibrating
rather than rotating in the crystals. The degrees of degeneracy of several lowest eigenstates of the molecular
motion are investigated by group theoretical considerations on the simplifying assumption that each molecule
is subjected to a field of a certain symmetry, namely, of octahedral, tetrahedral, trigonal, or of no symmetry. It results that the lowest rotaional state of the para-CH4 molecule,j=2, splits into several components
in the octahedral and tetrahedral field, of which the lowest state has a degree of spatial degeneracy 4 and
2, respectively. An explanation of Clusius's measurement of the zero-point entropy of the CH4 crystal can be
given by assuming a tetrahedral field and assuming that the conversion between different symmetry types of the
molecule does not occur. The alternative and more probable explanation can be given by the assumption
that 12 lowest levels resulting from the symmetry operations of the molecule are quasi-degenerate and the
symmetry of the field is tetrahedral or lower. The occurrence of the conversion does not matter in the
latter case. The zero-point entropy of CH3D can also be accounted for by the second type of assumption.
§ 1. Introduction
The thermal properties of methane crystal at low temperatures (above lOOK)
have been investigated mainly by Clusius and his coworkers.1l CH4 crystal undergoes a phase transition at 20.42°K accompanied by a sharp peak of the specific
heat curve. CHaD and CD 4 crystals have two transition points, with two corresponding sharp peaks in their specific heat curves, and the crystals are optically
anisotropic in their lowest modifications. A similar behaviour is found for CH2D 2,
but not in detail. The transition in CH 4 corresponds to the upper transition in
CHsD and CD 4 , as shown clearly by Eucken's2) study of the mixed crystals of
CH 4 and CD 4 • With increasing concentration of CD 4 the transition point increases
monotonically, and a straight line is obtained which connects the transition point
of the pure CH 4 with the upper transition point of the pure CD4 • The lower
transition appears for the concentrations of CD 4 higher than 20%, and the corresponding curve runs nearly parallel to the former. The transition points of the
crystal of CHaD coincide nearly with the points on these curves with 25 % CD 4 •
Eucken 2) derived the rotational part of the specific heat for CH4 crystal and
found that it is about 3.0cal/Mol above the transition point, showing that the
molecules are in the state of free rotation~ He also found the corresponding value
'" The main part of this work was read at the Meeting of the Phys. Soc. of Japan in Osaka,
November, 1947 (Symposium on Phase Transition).
On the Zero Point Entropy of Methane Crystal
703
of 3.8 -3.3 ca1jMol for CD 4 above its upper transition point, and inferred that
there is some hindrance for the molecular rotation. However, these results cannot be considered as conclusive, for the theoretical rotational specific heat obtained by Maue3) shows a maximum value of 3.7ca1jMol at about 25°K for CH 4
and that the classical value of 3 ca1jMol can only be attained above 50 oK.
Alpert's4) proton magnetic resonance experiments seemed at first to show that
the transition in CH4 and the upper transition in CHsD are the transition from
the state of oscillation to the state of nearly free rotation, but a more recent
research by himself 5) indicates the result which favours rather the oscillational state
nearly throughout the solid state.
The most recent X-ray structure analysis of CH4 is that carried out by
Schallamach. 6l Although some anomalous patterns were observed in the transition
range, no definte conclusion was drawn from them, and the structure (f. c. c.)
seems to remain unchanged.
It is the purpose of the present paper to make some inferrence to the crystal
structures of light and heavy methanes and to compute their zero-point entropies
on the basis of the model that each molecule in the crystal is subjected to a
field with definite symmetry. We assume that the interaction between neighbouring molecules can be replaced by a mean potential with a definite symmetry.
Clusius and Frankl) took a point of view that the molecular interaction is so
strong that it is meaningless to speak of any motion of a molecu)e independent of
the others, and they assumed therefore that the proton or deuteron spins are
distributed at random in the crystal. They showed that their measurements of
the zero-point entropy can be satisfactorily explained in this way. The point of
view here presented may be considered as the opposite approximation to the
problem, but it can be shown that the results of the measurements are equally
well explained.
Let us inquire somewhat closely into the comparison between the picture
presented by Clusius and Frank and that presented here. Consider an eigenstate
of the system of molecules in a CH 4 crystal and suppose that the molecules are
moving like gears coupled tightly together, or rather better that each molecule
is tightly bound to a definite direction with respect to the crystal axes, oscillating with a small amplitude. We may take another eigenstate with the same
energy eigenvalue which differs from the former by a symmetry operation of any
one molecule. If the coupling between molecules, or the coupling between each
molecule and the crystalline field, is not infinitely strong, then there will be a
resonance between these two states. Clusius and Frank's hypothesis is equivalent
to assuming that the energy associated with this resonance is not large compared
with the nuclear spin-spin coupling energy between neighbouring protons. The
latter is of the order of 10-22 erg per proton pair, so that the resonance frequency
must not be greater than 10-22/ h ___ 104 sec-I, in order that the assumption of
Clusius and Frank is correct. The nuclear magnetic resonance data show, however,
T.
704
NAGAMIYA
that this frequency (1/7:"0) is7) of the order of 106_10 7 sec-I, which is surely greater. We are thus lead to conclude that the eigenstates of the molecular motions are
independent of the spin configuration. Every eigenfunction of the space coordinates alone, describing the molecular motions, must have a certain symmetry character of the tetrahedral group with respect to the symmetry operations of anyone
molecule in the crystal, and, by the Pauli principle for the protons, the spin function of anyone molecule must have the corresponding symmetry character. This
is the point of view we are adopting here.
§ 2. On the crystal structure of methane crystal at low temperatures
The X -ray analysis of the CH4 crystal shows that it has a f. c. c. lattice both
above and below the transition point. In the lower modification, it may be supposed that the molecules are arranged in a most regular way. There are two
possible space groups for it. One is F43m which corresponds to the parallel arrangement of all the molecules with all their C-H arms directed towards the body
diagonals. The other is P213 where each of the four molecules in the unit cell
has one of its C-H arms directed to one of the four diagonals [111], [111.], [Ill]
[HI] (different for the four molecules), and its remaining three C-H arms directed as nearly as possible to the three cubic axes. The distances between atoms
in the neighbouring molecules and the numbers of the corresponding pairs of
atoms in the unit cell are listed in Table 1.
Table 1. The atomic distances between neighbouring molecules. The figures in parentheses
give the numbers of the corresponding pairs in the unit cell. The lattice constant is taken
to be 5.84 A. for the cubic lattices. For the hexagonal lattices the nearest carbon-carbon distance in the basal plane is taken to be equal to that in the cubic lattices, and the axial ratio
cIa
is taken to be 2V2/3 (1-tt).
Structure
H, H distance
F43m
2.35 (24)
3.58 (96)
hexagonal
(u-O)
-
hexagonal
(u-005)
-
.
2.56
2.69
3.06
3.28
3.58
3.63
(12)
(12)
(12)
(12)
(12)
(12)
2.35 (6)
2.82 (12)
2.35 (6)
2.71 (12)
3.30
3.65
3.98
4.27
4.55
(36)
(24)
(12)
(24)
(12)
3.30 (12)
3.65 (6)
3.24 (12)
3.41 (6)
4.13
4.13
4.09
4.13
...........
C, H distance
3.30 (48)
4.27 (96)
...........
C, C distance
4.13
If we take 1.2A. for the van der Waals radius of H, the nearest H,H dis-
On the Zero Point Entropy of Methalle Crystal
705
tance in F43m, 2.35 (24), is nearly equal to twice this radius. The nearest R,R
distance in P"2 13, 2.56 (12), is somewhat larger. but there are abundant nextnearest distances in this structure. As to the C,R distances. if we take for the
carbon radius 2.0A. (the van der Waals radius of the CR g group), we find that
the nearest C,R distance in F43m. 3.30 (48). is nearly equal to 2.0 + 1.2= 3.2.
The same distance is found in P213 with 36 pairs. We are here assuming that
the molecules are at rest, but actually they are oscillating. and the atomic distances are distributed continuously. It is not possible to draw a definite conclusion
from the consideration above of the atomic distances alone which of the two structures
is more stable, but it appears to the writer that the structure F43m is more stable.
A possible structure of the lowest modification of heavy methanes. CRsD
and CD 4 • is a hexagonal close packing. The best packing condition is obtained
by arranging the molecules as shown in Fig. 1. where the molecules in each of
the basal planes are parallel to one another. but the orientations in the successive
two layers are different through a rotation of 180 0 about the hexagonal axis. A
small contraction along the hexagonal axis may be possible compared with the
closest-packing structure.
Each molecule in the structure F43m moves in a field of tetrahedral symmetry. and in other structures here considered. it moves in a field of trigonal
symmetry. In the highest modification. the field must be of octahedral symmetry
in the statistical average.
Fig. 1.
Projection of two successive net
planes upon one of them.
Fig. 2.
§ 3. The lowest eigenstates of CH4 and CD4 in a field of various symmetries
We assume in the following that each molecule in the crystal is subjected to
a field with a definite symmetry and moves each independent of others. This
approximation may be valid near absolute zero where the amplitude of the molecular oscillation is small.
The eigenfunction describing the motion of anyone molecule can be classified
according to the irreducible representations of the tetrahedral group for CH4 and
706
T.
NAGAMIYA
CD4 • As the starting approximation, let us take an eigenfunction which describes
the lowest oscillational state of a molecule about one of its equilibrium positions
and 11 other eigenfunctions which are obtained from the first by the symmetry
operations of the molecule. We shall denote the first by tP(E), and others by
tP(C",) , tP(Cy )
,
tP(Cn,
tP(Cz ) ,
tP(C22 ) ,
tP(CI )
tP(C2 ) ,
,
tP(C32 )
,
tP(Cs)
,
tP(C4)
tP(Cl) ,
,
(1)
where C'" means the rotation through 11: about the x-axis (see Fig. 2), CI the
rotation through 211:/3 about the line joining the carbon atom and the first hydrogen atom, C I 2 the double this operation, and so on. By linear combinations of
these functions we have the following irreducible eigenfunctions:
(2)
Total symmetric: tPl = '2JtP (C),
o
where C takes E and those appearing in (1).
Doubly degenerate:
(3)
tP2*=the conjugate complex of tP2'
Three sets of triply degenerate functions: tPij='2JUij(C-1) tP(C),
o
C»
(4)
where (Uti
is the three-dimensional unitary irreducible representation of the
tetrahedral group and will be given in Appendix. It can be shown that the
functions with the same i but with different j's mix with each other by the
symmetry operations of the group (Appendix).
If the symmetry of the field is tetrahedral and the minimum of the potential
energy corresponds to the coincidence of the molecular symmetry axes with the
symmetry axes of the field, then it can be shown that the functions with the same
i but with different i's mix with each other by the symmetry operations of the
field, so that these functions form a nine-fold degenerate state. In a trigonal
field, this level splits into a three-fold le"t'el and a six-fold level, if the minimum
position corresponds to the coincidence of the trigonal axis of the field with one
of the trigonal axes of the tetrahedron, and in an asymmetric field it splits into
three three-fold levels.
In a field of octahedral symmetry, again assuming that the minimum position
corresponds to the coincidence pf the symmetry axes of the tetrahedron with
those of the field, we have two sets of singly and nine-fold degenerate levels and
one four-fold degenerated level. All the mathematical details will be postponed
to Appendix. In Fig. 3 is shown an example of the level scheme for various
symmetries of the field.
On the Zero Point Entropy of Metham Crystal
707
§ 4. The zero-point entropy of CH4 and CD 4
We can now comptlte the zero-point entropies of CH 4 and CD 4 • We distinguish between two cases: 1. All the energy levels computed above are very
close to each other compared with the lowest kT of the measurements, and 2.
their separations are very large compared with it. As cited before, the nuclear
resonance data show a flipping frequency (1/,0) of the order of 106~1O7 sec- 1
which corresponds to level splittings of the order of 10- 20 erg, or the temperature
of the order of 10-4 deg. abs. If this is truly the case, the case 1 is valid, and
the degeneracy of the levels is almost perfect.
There are three types of CH4 molecule, namely, those with resultant spin 2,
1, O. Their spin functions belong to the three irreducible representations of
the tetrahedral group of dimensions I, 3, 2. The spin weights are thus 5 x 1 = 5,
3 x 3=9, 1 x 2=2, respectively. In normal methane they are contained with the
-~~I4,S
j=2
Fig. 3. CH4, CD4 •
The figures attached to the right of the levels
indicate their degrees of degeneracy.
,-_r"::'%_1
~j_=_'_~
f,.
,
~
4-~---______ 1--_______ 2 _________ 2
~
~9
'~3-------!
__J~'=_O_1--_...;r.:.:.'_1--_ _ _ f
Free rot.
Octah.
3
6___
Tetrah.
i
Trig.
of
Asymm.
ratio 5: 9 : 2, the most abundant being called ortho, the next meta, and the least
abundant para. As Maue 3) has shown, each type couples only with the molecular
motion of the same irreducible representation, and the total weight of a given
level is equal to its spatial degeneracy for ortho and para, and multiplied by 5 for
meta.
vVe shall at first consider the first case cited above.
a) Octahedral field. Weights for meta, ortho, para are 2 x 5= 10, 18, 4,
respectively. The abundance ratio is thus unchanged whether the conversion between different types occurs or it does not occur, but the zero-point entropy is
greater by Rln 2 compared with that measured.
b) Tetrahedral or lower field. vVeights for meta, ortho, para are 1 x 5=5,
9, 2 respectively. Therefore the zero-point entropy is just equal to the measured
value whether or not the conversion occurs.
vVe next go to the second case, and assume that conversion does not occur.
708
T.
NAGAMIYA
For, if it occurs, the molecule will all drop to the non-degenerate lowest level
of total symmetry and becomes meta at sufficiently low temperatures.
a) Octahedral field. The spatial degeneracies of the lowest states for meta,
ortho, and para molecules are 1, 9, and 4, respectively, so that the total weights
are 5, 9, and 4. The zero-point entropy is again greater than the measured value
by (2/16) ·Rln 2.
b) Tetrahedral field. The spatial degeneracies are 1, 9, 2, so that just the
required value is obtained.
c) Trigonal field. The spatial degeneracies are 1, 3 or 6, and 2, so that
the zero-point entropy is smaller than the required value by (9/16) ·Rln (9/3)
or (9/16) ·Rln (9/6). The former value applies also to the case of an asymmetric field.
We are thus lead to conclude that the symmetry of the field is tetrahedral
or lower and the ground levels are spaced very closely, or, alternatively, that they
are largely spaced, the field is of tetrahedral symmetry, and the conversion between different types of the spin symmetry does not occur. The former is more
probable in view of nuclear magnetic resonance experiments.
Similar considerations can be carried out for CD 4 , and the same conclusions
can be drawn, with minute differences that the excess in entropy in the case of
second a) is (12/81) ·Rln 2, and the defficiency of it in the case of second c)
is (54/81) ·Rln 3 or (54/81) ·Rln (3/2). The quasi-degeneracy of the lowest
levels for CD 4 may be more valid because of its larger moment of inertia. However Clusius does not seem to have given explicitly the value of the pero-point
entropy for this methane.
§ 5. The zero-point entropy of CHsD
The lowest levels for this molecule in a field of definite symmetry can be
treated in a similar way as for CH4 , the mathematical detail of which will be
given in Appendix.
Clusius obtained for this methane an excess zero-point entropy of Rln 4 over
that one would expect from the random distribution of spins. He explained this
by pointing out that D occupies every corner of the molecular tetrahedron in
random way. Similar explanation can be given from our point of view.
There are two types of spin configuration for this molecule; one is invariant
against the rotation about the trigonal axis of the molecule, and the other has
the degenerate character 1, e, e2 or 1 e2 , e. Normal gas contains them with a
ratio of 1: 1, because the proton spin weight of the former (quartet) is 4 and
that of the latter (doublet) is also 2 x 2 = 4. The former couples with the total
symmetric modes of molecular motion, and the latter with the degenerate modes.
The total weight of a level is equal to its spatial degeneracy multiplied by 4 for
the former type, and equal to that multiplied by 2 for the latter type, as was
shown by Maue.
On the Zero Point Entropy of Methane Crystal
709
In a field of tetrahedral symmetry or of lower symmetry, we have 12 lowest
levels, and if these are considered as quasi-degenerate, we have 4 levels for the
first type of methane and 8 levels for the second. The zero-point entropy is
therefore
I
I
1
I
1
1] =Rln32
R [ -In(4x4)
+-In(8
x2)--ln---In2
2
2
2
2
2
(the last two terms in the bracket is the mixing entropy), namely, greater by Rln 4
than that required from the random distribution of proton spins. Here again is
immaterial whether the conversion between different types occurs or it does not.
H the level distances are large and the non-occurrence of the conversion is
assumed, we have too small a value of the zero-point entropy even for the case
of free rotation, as one would see from the level scheme shown in Fig. 4.
t t £a
j .. 2., "l::±2
1O
f (. f.'l
~ ,
't"=:tt
• • "1::0
f 1 1
Fig. 4.
~
1 E £a
' " £2
f
t
t f....
j 'G1, 't" ... :to,
.. , "1:=0
'11
1 1 f
h
Free rot.
'1--3
111
I
111
Octah.
1 Ii i&
1
~~6
, 1 1
j=o
11
[,
The level scheme of CH3D. (111) and (lee 2 )
indicate the characters of the levels with respect to the rotations about the molecular trigonal axis.
3
1
f ",,-
f f 1
f 1f-
2.
2.
2
6--------2.--1-
.... I.
2
.2
.2
3-::--- ~::::::-
1
t
1
Tetrah.
Trig.
f
1
1
1
Asymm.
§ 6. Discussion
The present consideration would appear as not very much different from that
given by Clusius and Frank. But the following point must be pointed out. In
their point of view the spins are distributed at random in the crystal, and as its
consequence the vapours of methanes in equilibrium with their corresponding crystals at low temperatures would contain the greatest amount of the most stable
type of the molecule (meta type in the case of CH4 ), because the conversion from
one type of molecule to another can take place through the crystals. In our
point of view of the quasi-degenerate levels the occurrence of the conversion is
not necessary, so that the vapour can contain different types in the normal ratio.
The problem of the conversion in the crystals of methanes requires another analysis.
710
T.
NAGAMIYA
Appendix
1.
The irreducible representations of the tetrahedral gronp
The Table 2 gives the simple characters and the following is the three-dimensional unitary irreducible representation:
Cz
E
ClI
2
( 010
1 0 0 )
001
~.
(-1 -2e -2e)
-2e -1
2e 2
-2e2 2e-1
C1
-2e -2e)
2e
-2e
2e2 -1
(-1 2 -2e -2e)
-2e -e
2
,-2e
2 _S2
C22
Cl
( o
1 082 00 )
o0 e
!
-2
(-1 -2 e -2e)
-2e -e"
2
-2e 2 2-e
0
-2e -2e)
2e 2
2e - e2
-~
-1
)
(-1 -2
l -2e -8 -228 )
1(-1
-2 - e
-2
2
~
Ca
2
?!
-2
l (-1
-2 -1
(-1
11 -2e2 -1
(;2
( o
1 0e 00 )
o 0 e2
C.
(-1
Ca2
,-2£
2e-e
_282 282 _e2
i -2e2 -2
_e 2 -2)
2e2 l
Cl
-2e2
-2 _82
-2
282
(-1
-28)
2::
-!
Table 2.
2.
The transformation properties of the
eigenfunctions
Operation of a group element A on (4),
§ 3 gives
Class;
Simple
1
1
1
character;
1
1
e
e2
1
1
e
3
-1
e2
0
A1'fj= 2JuilC- 1) CP(AC)
1
0
a
= 2Jui/B-1A) l' (B)
(B=AC)
= 2J2J UUC(B-l) uk/A) 1'(B) = 2J1'ik uk/A),
R
R
k
k
and successive operations of two elements give the corresponding product of the
matrices.
The energy eigevalues (CH 4, CD4 )
We first construct the energy matrix elements from 12 functions, 1'(E) and
11 functions given by (1), § 3 :
3.
J1'(E)H1'(E)dT=a,
J1'( C,,) H1'(E) dT=j9",
J1'(C )H1'(E)dT=j9, J1'CC.) H1'CE)dT=j9.,
J1'C
f1'(
Ci )H1'CE) dr:=ri=
lI
Cl)H1'CE)dT,
i= 1,2,3,4.
Here we assume that all 1"s are real. The symmetry of the field can be described by certain relations among these parameters. For a tetrahedral field, one has
On the Zero Point Entropy of Methane Crystal
711
and for a trigonal field, one has, taking the trigonal axis to coincide with the
axis when the molecule takes the original position E,
c,-
The case of an octahedral field will be treated afterwards.
The Hamiltonian matrix elements with respect to the eigenfunctions given by
(4), § 3 can be calculated as follows:
J1'~
H1'IcI
dr=~~uij(A-l)Ukl(B-l)
f
1'(A)H1'(B)dr
=~~ut(C-l B-1 )UIcI (B- l ) f1'(BC)H1'(B)dr
=>JIjuZ (C- 1B- 1 )UIcI(B- 1 )Ha,
a
JJ
where
Ha= J1'(C)H1'(E)dr.
Since we are taking a unitary representation, we have
Iju~(C-l B-1) Ukl (B- l )
JJ
= IjUji) BC) Ukl (B-1) = IjIj 7l,~, (B) u",.( C) Nlcl( B- 1)
B
B
m
The sum in the parenthesis vanishes unless m=k, j=l, and when these relations
hold it is equal to hi! (h=order of the group= 12, f--dimension of the irred-u
cible representation=l or 3) .8) We thus have
J1'~ H1'1c1dr=O,
U4=I)
J1'~ H1'lcjdr= ~IjuH(C)Ha.
fa
Thus the Hamiltonian matrix elements exist only between those 1'.;s which
have a common f. F or the three-dimensional irreducible representation we have
thus to solve a three-dimensional secular equation.
The results for the energy eigenvalues are given in Table if we neglect the
overlap integral between any two different 1'(C)'s.
Table 3.
f/1:
a+3,g+8r
a+3,g+2rl+6r2
a+3,g--4r
a+3,g--rl--3r2
(9) ... tetrahedral
(3)
.
1
(6)···tng ona
T.
712
NAGAMIYA
The second row refers to a tetrahedral field and the third and the fourth rows refer
to a trigonal field. The parenthesized figures in the third column give the degrees of degeneracies.
4.
Tetrahedron in an octahedral field
We may here rather change the roles of the symmetry operations of the
molecule and those of the field. Thus C now implies a symmetry operation of the
octahedral field. The reduction of the regular representation of the octahedral
group yields the following irreducible representations:
where Bethe's notation is used. 9l ['1 and ['2 are one-dimensional, and should be
both total-symmetric with respect to the symmetry operations of the molecule.
['3 is two-dimensional, and each of the two should contain both the characters
(1, 1, e, e2 ) and (1, 1, £2, e) with respect to the symmetry operations of the
tetrahedral subgroup of the field. But these two ['3 are degenerate, as one can
verify directly, or as follows. The function 1'2' constructed previously, is an eigenfunction with respect to both molecular symmetry and field-symmetry, and we can
construct another function ¢/ by a rotation through 'IT:/2 about one of the cubic
axis of the field. Then 1'2 and ¢l are bases of ['3' This set and its conjugate
complex evidently belong to the same ,level. ['4 and ['5 are three-dimensional, and
both have the same character with respect to the tetrahedral subgroup, so that
both must contain the nine-fold degenerate functions constructed previously. Thus
we have two non-degenerate levels, one four-fold degenerate level and two ninefold degenerate levels.
5.
CHsD in a field of various symmetries
Tetrahedral field We have only to take the case of putting a tetrahedron
in a trigonal field and change the roles of the symmetries of the molecule and
those of the field. We have the following scheme (Table 4):
Table 4.
Degree of
degeneracy
Character with respect to the rotation about
a molecular trigonal axis through
00
1
6
3
2
1
1
120 0
1
1
1
1
e,
240 0
1
e, e2
S2,
s
1
S2
Energy
eigenvalue
S2,
e
a+ 3.9+2rl+ 6r2
a-.9-rl+r2
a-.9+2(rl-r2)
a+3.9-rl-3r2
713
On the Zero Point Entropy of Methane Crystal
Ottohedral field As before, all levels with the degrees of degeneracy 1, 6,
3 are doubled, and the level with degeneracy 2 becomes a level with degeneracy 4.
Trigonal field We distinguish two cases. a) The C-D arm oscillates
about the trigonal axis of the field. The level scheme is then as follows (Table 5):
Table 5.
Degree of
degeneracy
Character with respect to the rotation about
the trigonal axis
1
1
1
1
2
1
Energy
eigenvalue
a+2r
b) One of the C- H arms oscillates near the trigonal axis. Denoting by r'l the
Ho for the rotation about the C-D arm, by rl the lIo for the rotation about
the trigonal axis of the field, by ra' the I10 for the combination of these two rotations in the same sense, by ri that in the opposite sense, and putting HE=a',
we have the following scheme (Table 6):
Table 6.
Degree of
degeneracy
1
2
2
2
2
Character about
the molecular axis
1
1
1
1
1
1
1
1
e, e2
1
e, e2
Character about
the field axis
1
1
1
1
e
1
e, e2 e2, e
1
1
e, e2 e2, e
1
1
e'l., e
S2,
£2,
e
e, 82
£2,
e
e,
£2
Energy
eigenvalue
a'+2r/+6rl
a' +2rl'-rl -rs'-rl
a'-r/+ 2rl-rs'-rl
a'-r/ -rl-rs' +2rl
a'-r/-rl+ 2rs'-rl
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
K. Clusius u. L. Popp, Zeits. f. physik. Chern. B46 (1940), 63.
A. Kruis, L. Popp u. K. Clusius, Z. Elektrochem. 43 (1937), 664.
K. Clusius, L. Popp u. A. Frank, Physica 4 (1937), 1105.
A. Frank u. K. Clusius, ZS. f. physik. Chern. B36 (1937), 291.
Eucken, Zeits. f. Elektrochem. 45 (1939), 126.
W. Maue, Ann. der Phys. 30 (1937), 555.
L. Alpert, Phys. Rev. 75 (1949), 398.
J. T. Thomas, N. L. Alpert and H. C. Torrey, J. Chern. Phys. 18 (1950), 1511.
A. Schallamach, Proc. Roy. Soc. AI71 (1939), 569.
Calculated from the values of Tl and T2 given in Thomas, Alpert, and Torrey's paper with the
use of Fig. 14 of the paper by Bloembergen, Purcell and Pound, Phys. Rev. 73 (1948), 697.
I. Schur, Die algebl'aischen Gl'undlagen dey Darstellungstheoni? dey Gmppen (bearbeitet und herausgegeben von E. Stiefel) ZUrich, 1936, p. 41.
H. A. Bethe, Ann. der Phys. 3 (1929), 133.
a)
b)
c)
d)
A.
A.
N.