VOL. 21 NO. I, 2
CHINESE JOURNAL OF PHYSICS
SPRING/SUMMER, 1983
Lattice Dynamics of the Hydrogen Halide Crystals
CHIA-NAN CH A N G ( f&&. 3 )
Department of Electronic Engineering and Technology National
Taiwan Institute of Technology, Taipei, R. 0. C.
and
W
A N-
SU N T
Institute
SE
($#p&*)
a n d C H A N G LI (& 3~)
of Physics, Academia Sinica Nankang.
Taipei, Taiwan, R. 0. C.
(Received 4 December, 1982)
A lattice dynamical model for three of the four hydrogen halide molecular
crystals in their low temperature phase Ill has been investigated by using the
Born-Van Karman model with different interatomic force constants and the results
are compared to observed zone centre (q=O) frequencies (Raman). This simple
model is also used to estimate those hydrogen bond strengths in different crystals.
I. INTRODUCTION
s
O L I D HCl, and HBr form an interesting group of isomorphic molecular crystals which has
been the subject of several experimental and theoretical studies. The phase temperature
HCl and HBr have 3 solid
diagram for the hydrogen and deuterium halide is shown in Fig. 1.
phases. Phase I and II show orientational disorder, while phase III is order. In HF the intermolecular binding is stronger than in the other hydrohalides. Molecular aggregates form in the
gas phase”‘ . In contrast to the other two halides, there is no phase transition in the solid phase,
the molecules condense directly into a structure, which corresponds to the low temperature
structure of HCl and HBr. In this low temperature phase, the crystals have ordered orthorhombic
structures’*-‘).
The molecules form planar zig-zag hydrogen-bonded chains, with two molecules in the
primitive unit cell on sites of C, symmetry forming space group Ci: (Bb21m). T h e s t r u c t u r a l
properties of those zig-zag chains are listed in Table 1.
Standard group theoretical techniques shows that there are three translational (A,+&+&)
and four librational modes (A,+&+&+&) corresponding to the external motions of rigid
molecules. There are also two stretching modes (A,+&). All of the nine optically active modes
are active in Raman scattering and, except for two A , modes, are also active in infrared
absorption. It is the aim of this work to attempt to interpret these normal modes of vibrations
obtained from spectroscopic techniques ’+” I. n terms of such a simple force constant model and
to estimate those hydrogen bond strengths in those three crystals.
( 1 ) Gmelin. Handbuch der anorg. Chem., Eng-Bd., 5 (1959).
( 2) E. Sandor and R.F.C. Farrow, Nature, 213, 171 (1967).
( 3 ) E. Sandor and M. W. Johnson, Nature, 217, 541 (1968).
( 4 ) M. W. Johnson, E. Sandor and E. Arzi, Acta Cryst. B31, 1998 (1975).
( 5 ) A. Anderson, B. H. Torrie and W. S. Tse, Chem Phys. Lett. 70, 300 (1980).
(6 ) J. E. Vessel and B. H. Torrie, Can. J. Phys. 55, 592 (1977).
( 7 ) T. S. Sun and A. Anderson, Chem. Phys. Lett. 17, 104 (1972).
11
LATTICE DYNAMICS OF THE HYDROGEN HALIDE CRYSTALS
12
HF
M.P.
DF
HCI
H Eir
D Cl
186.3
18 9.6
D Br
185.6
120
111
III
80
Fig. I. Phase-temperature diagram for the crystalline HF, HCl and HBr.
Table 1. Properties of the phase III crystals
~_~._____
_______..__
~~~ ~
Molecular Bond Length a (A)
Hydrogen Bond Length b (A)
Angle (#) (0=#-90’)
HF (4.2K)
0.97
1.53
120.1’
HCI (77.4K)
HBr (84K)
1.275
1.414
2.413
93”31’
2.513
91”48’
II. MODEL CALCULATION
Assuming that the vibrational motions of each atom in the zig-zag chain are harmonic, the
equations of motion of the Eth one of the k- kind atom are
MA(:)=-
22 #CQ(:; ::)a@(::)
11, A’. 1
(1)
where A,8(:; Fe) is the negative of the force exerted in the a-direction on the atom (:) when the
atom (t:) is displaced a unit distance in the B-direction, all other atoms being kept at their
equilibrium position.
If we choose, as a solution to equation (I), a function of the travelling-wave from
u,(i) = (M‘) -I/2U,0(:, q)elcVr(:)--r~l
(2)
We find that
~*a,,(:, q) = x (M,M,t)-1/*4.,(:; :;)e-‘q. (r(i)-r(G) )uaO(c,, q)
(3)
In the long wave length limit, i.e. q*(r(:)--r(g)) < 1, and for nontrivial solution of (3),
i. e. uPO(& q) #O, we have the following secular equation
C.N. CHANG,
w.S.TsE
AND t. CHANG
Fig. 2. Geometry and force constants of phase III solid HF. HCl and HBr adopted
in the calculations described in this paper. Empty circles are atoms in the
plane z=O, the circles with crosses inside are atoms in different plane parallel
to z=o.
‘In this planar chain model, interaction between adjacent zig-zag chains were ignored if
compared with intramolecular forces. The model is outlined in Fig. 2 where all the force constants are shown. We limit the range of interaction of each hydrogen atom -and halogen atom
to their fourth-nearest neighbors in each planar zig-zag chain structure. Assuming the axiallysymmetric force constant”‘, CY, a, 6 between each atom to its first and second nearest neighbors.
and the central force constant”’ 7, E between each atom to its third and fourth nearest neighbors.
we have the following matrix of force constant between various pairs of atoms.
Let
P=oz+bz+2ab sin 0
Q= (a+b)*+2u(a+b) sin 8-l-u’
U=(a+b)+asinO
W=2ubcos0+(u*+bZ) sinBcos0
-q-&-)
d( ;; :)=f$(;; ;)=
0
.._ - ____._ ~__ --(8) N. Wakabayashi and R. hl. Nicklow.
(9)
X=(aZ+6*)(1+sin*8)+40bsinB
V=(u)+(a+b) sin0
N=F, Cl, or Br
0
0
-02
0
~
Neutron scattering and lattice dynamics of material with
layered structures in “Material Science”, Vol. 2, Addison-Wesly Publishing Company, 413 (1979).
A. A. Maradudin, E. W. Montroll, G. II. Weiss and I. P. Ipatova, “Theory of lattice dynamics in the
harmonic approximation”, Academic Press, New York, 14 (1971).
---
14
LATTICE DYNAMICS OF THE HYDROGEN HALIDE CRYSTALS
I
(-8(~)--~-_(=+b)‘cosl8
-+(a+b)(V) cost9
d( ii; :)=a; ii)= /
I
I
1
-$(W)-$(Y)(o+b)
cos~6+6
coso
-8 (+-
- ; (W)-$-(V)(a+b)cosB
$_a+B+ -; (X)f$(V
0
-a!cos’0 -8(--&-)
sin*0 +(cY-&&~)-)
0
0
[&(6-cr-B)sin20-s(U)(ocosB)-(($)
4( N,N
0. 0 1 _ -+ (B-8-&-) sin20-$( acos8)
(W)] 0
0
0
--i- (LX--~-~- ) sin20+!-(a+b) (Y)cosO
a+b
Q
0 /
(6!+/3sin28)+~(U)2+6(~b-)cos*B 0 ~( 5 )
~(a+b)z)coszB
I -(a+bsinV)+(cr+
6
1 (a+b)
I
/
0
6
-+-(~--cT~&J sin2B-+-(aces 6)
0
/
I
/
I
0
0
4 b+nsinV
) + (P+$ a* co9 8)
a+b
j
0
(cr+~)sin’t9+cos20+$o’+$(x)
0
I
i
0
sin20
--a: sin’ 0-6 (TFb-) co? 0
4( **I3
1. 1 1 _ +(6-l)-a)sin2B-$(U) (acosB)-$(W)
)
0
-%S%)i
1
2 ( n--6 ( -(h8b,) sin20
‘~(~+B)+~(~z+b’)+~-az]coszQ+Bsin’B
,.I_
NSN -
/
I
__Bsi*Zo--(<+6)
6a co? &-$I)
0
4(
0
sin20 +(/3--6-~~b-)sin28+-~(acos0)(U)
-;- (a-s-&-) sin20+ -l- (a cod) ( U)
I
)
0
I -(B+$) co~~B--6(-~~b~~)
.
0
0
0
4( :; :)=a:; A)=
0
--~--(v)2-~
0
1 (-~(n’+b’)+-~-_(o+6)2)
9( i; iI=
-g(V)(a+b)cosB
61
--a (CY-~~-$~) sit&?+;-(nfb) (V)cosB
B+Sk$)
0
cos’B+--( V)*+crsinV
;
0
0
0
8
All other matrixes that are not listed are zero matrixes.
In the long wavelenth limit, q-+0, we
have the following 12X12 symmetric secular equation with Aij=A,i
=o
where
A,=&$(:, ;A) is an 3X3 matrix with Iji, jS2
.~_.
(6)
C. N. CHANG, W. S. TSE AND L. CHANG
15
Table 2. Calculated frequencies and force constants for solid HF, HCl, HBr
in phase III (cm-‘)
HF
.4ssignment
Cal
A1
Acoustic { B I
(o/p)T
(i/PIT
(i/PIT
(o/p)L
(o/p)L
(i/p)L
(i/p)L
(i/p)S
(i/p)S
1:
Al
BI
AZ
Bz
AI
BI
Al
BI
Cal
0
0
0
57
187.5
363.5
548.5
569.5
742
943
3045.5
3386
0.07
0.71
0.75
28.5
94
103.5
178
180
300
444
2706.5
2728
_____~~ _~
0.08
0.22
0.37
121.5
199
334
543
556.5
674.5
950
3089
3313
HBr
____
HCl
Exp
EXP
0
0
0
61.0
88.5
114
141.5
223
336
409
2697
2741.5
Cal
EXP
0.09
0.95
0.97
18
65
69.5
165.5
166.5
272
393
2399
2417
0
0
0
45.0
61.0
75.5
147
209
297
376
2395
2431
Optimized force constants: (in units of lo5 dyne/cm)
HF
HCl
HBr
4.970
0.345
0.30
0.175
0.160
4.135
0.095
0.040
0.0188
0.032
3.273
0.102
0.029
0.0163
0.027
___ ~. ~__ ____
r
ii
.z
,
o/p. out of plane; i/p, in plane; L, libration; T, translation; S, stretch.
/Iii=
-&q L; A)
is an 3x3 matrix with 3<i, j<4. (N=F, Cl or Br)
Aij= /dLMi 4(;; ,j) is an 3x3 matrix with l<i<2, 3<&4
The elements in each diagonal submatrix of equation (6) are defined as follows
III. DISCUSSION AND CONCLUSIONS
The equations of motion for the four independent atoms in the x, y and z directions, relating
the five interacting force constants, are set up and a 12x12 dynamical matrix is estiblshed as discussed above. This is diagnoalized on an VAX 111780 computer using the Jacobi rotation method,
and the eigenvalues (giving normal mode frequencies) and eigenvectors (atomic amplitudes) are
calculated. Since the in-plane motions are less likely to be significally influenced by inter-chain
forces, the force constants are varied to give a best fit with these frequencies. The optimized
values of the force constants and a comparison of the observed and calculated frequencies are
listed in Table 2. In Fig. 3 the vibrational displacements for q=O in-plane and out-of-plane
motions of HF, HCl and HBr (same structure) are reported qualitatively.
Some physical arguments are used to assist in the determination of some of the force constants:
(1) The stretching frequencies mostly depend on combination of the force constant a: and
/3. The appropriate value of 13 is obtained by fitting the external inplane translational frequencies
due to the fact that the molecules (HBr, HCl or HF) translate in the plane is determined chiefly
by B as seen in Fig. 2.
--.._.‘-_.,*
16
LATTICE DYNAMICS OF THE HYDROGEN HALIDE CRYSTALS
G:
P_._____-
-_--___’
Ci/p)L
Bl
(+_I__--__
.
G/P > L
*1
1
p_t-_---_
WP)L
B2
1d
(o/p)T
A2
Acoustic
B1
+-
I
a
------c
Acoustic
B
i i& _ - _ : _ -
Fig. 3. Zone centre normal modes of a single chain of solid HF, HC1 and HBr
derived from a five-parameter force constant model.
.
.,
_,
C.N. CHANG, W.S. TSE AND L. CHANG
8
b
17
(2) The splitting of the internal stretching frequencies depends on 7.
(3) The lattice librational frequencies within the zig-zag plane depend on the combination
of r, 6 and E .
(4) The out-of-plane librations depend on 6 since only the 6 force constant is involved in
the z direction.
From Table 2, it is seen that the stretching, in-plane librational and in-plane translational
modes are quite well fitted. The agreement is not as good for the out-of-plane librations, which
is due to the fact that only a 6 bending force constant is introduced. Note that the pure translational modes are not identically equal to zero because of the round-off errors produced during
the diagonalization procedure. The optimized force constants show some interesting trends:
(1) (Y, representing the molecular bond, for HCl and HBr differs from its gas phase value
by -12 or -13% while the change is nearly -44% for HF.
(2) /3, representing the hydrogen bond, is much larger for HF than it is for HCl and HBr
which is consistest with the large changes mentioned above for LY, i.e. production of a strong
hydrogen bond weakens the molecular bond between the hydrogen and fluorine.
(3) r, 6 and E again have much larger values for HF than for the other hydrogen halides
reflecting the strong bonding between molecules.
The model adopted in these calculations for solid HCl HBr and HF shows generally good
agreement between the observed and calculated frequencies suggesting that the forces between
molecules in different chains are much smaller than those between nearest neighbours in the same
linear chain. A completely acceptable model, however, is unlikely until a satisfactory representation of the hydrogen bonds is found.