Lattice dynamics and crystal stability of bromine

Lattice dynamics and crystal stability of bromine Z. Gamba, E. Halac, and H. Bonadeo Citation: The Journal of Chemical Physics 80, 2756 (1984); doi: 10.1063/1.447021 View online: http://dx.doi.org/10.1063/1.447021 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/80/6?ver=pdfcov Published by the AIP Publishing

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Lattice dynamics and crystal stability of bromine8 ) Z. Gamba, E. Halac, and H. Bonadeo b) Division Fisica del Solido, Cornision Nacional de Energ[a A tornica, Avenida del Libertador 8250, Buenos Aires (1429), Argentina

(Received 29 April 1983; accepted 22 September 1983) The la~tice d~n~ics and crystal stability of bromine have been studied using an interaction potential which mcludes atom-atom and Coulombic terms. It was found that in addition to high order elec~ros~a~ic te~s, a spe~ific interaction of the form - D /,-8 between nearest neighbor atoms, which is identified as an mduced dipole-induced quadrupole term, is needed to stabilize the observed orthorhombic structure and to give a good fit with other static and dynamical observables.

INTRODUCTION 2

The three halogen crystals chlorine, I bromine, and iodine 3 are isostructural, and belong to the orthorhombic space group Cmca (D i~ )with two molecules in the primitive unit cell located at sites of C2h symmetry. The simplicity of these structures has made them a much explored, although quite controversial, field for probing and testing models of intermolecular interaction. The crystal structure itself is not accounted for by single atom-atom or quadrupole-quadrupole interactions: English and Venables4 have shown that regardless of the form of the atom-atom potential or parameters used, a hypothetical Pa3 cubic structure is calculated to be more stable than the orthorhombic observed. Hsu and Williams5 stabilize the observed structure by introducing molecular dipoles, not easy to justify on symmetry grounds. Nyburg and Wong-Ng6 find that the electron cloud around the nuclei is very anisotropic, and so are the resulting atom-atom forces; this anisotropy leads to stabilization of the observed orthorhombic structure. The fact that the non bonded nearest neighbor atoms lie at distances which are significantly shorter than twice the corresponding van der Waals radii seems to be the cause-or the result, depending of the point of view-of the particular form of the intermolecular interactions in these crystals. Most dynamical calculations have not been concerned with the stability of the crystal structure. English and Leech 7 use a Lennard-Jones potential plus charges distributed on the molecules for the crystals of Cl 2 and Br2 ; they find that the Au mode is always calculated imaginary. Vovelle and Dumas8 use a Buckingham potential plus a Morse potential for the nearest neighbor atoms only in their calculation, but our calculations with their model showed that acoustical branches in some directions of k space are imaginary. Other calculations use a large number of parameters: the frequency fits of Suzuki et al.,9 who use seven force constants, or of Toukan and Chen,1O who use an interaction potential with ten parameters are excellent, but these results do not explain the physical nature of the interaction. Pasternak et al. II use a Lennard-Jones potential plus a simple bond charge model for C1 2 , Br2 , and 12 , They obtain a good frequency fit, and alSupported in part by Conicet Grant N. 8382 d/82. bl Fellow of Conicet. 2756

J. Chern. Phys. 80 (6),15 March 1984

include residual force and stress components for the crystals; however the cohesive energy is too small by a factor of about 6, and they do not compare with the hypothetical cubic structure. Very recently, Burgos et al. 12 have performed a thorough study of C12 ; their model consists of anisotropic atom-atom and electrostatic interactions. Their results regarding both the statics and dynamics of the crystal are in general in good agreement with experiment. Upon relaxation of the unit cell parameters to minimize the crystal packing energy, the orthorhombic structure is calculated to be more stable than the hypothetical cubic one; its volume, however, is about 20% larger than that of the observed structure. Since, as will be discussed later, the static equilibrium structure corresponds to a temperature of 0 K, a unit cell contraction is expected. For 12 , a coherent inelastic neutron scattering experiment 13 has provided a complete assignment of the optical frequencies. For the other crystals, theoretical studies have had to rely on tentative assignments, since no polarized measurements had been performed; this has led to considerable ambiguities because the optical modes could be divided only into translational (I.R. active) or librational (Raman active) modes. Since different existing models give rise to a different assignment of the unpolarized bands, the evaluation of the models themselves under these circumstances is rather doubtful. Very recently we have measured the polarized Raman spectrum of crystalline Br2 , 14 obtaining an unambiguous assignment of the observed bands to their proper factor group species; the availability of this experimental evidence has prompted us to undertake the study on the intermolecular forces responsible for the static and dynamic properties of this crystal. In a recent work we have studied the lattice dynamics and intermolecular forces of crystalline acetylene 15; the low temperature phase is isomorphic with bromine, while the high temperature phase is cubic, space group Pa3, the same space group that stabilizes the halogen structures under the assumption of atom-atom and quadrupoIe-quadrupoIe interactions. 4 We found that if high order (up to 26 pole) electrostatic interactions were taken into account, the lattice dynamics of both phases could be described correctly, and also the orthorhombic low temperature crystal structure was cal-

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© 1984 American Institute of Physics

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Gamba, Halac, and Bonadeo: Lattice dynamics and crystal stability of bromine

culated to be more stable than the high temperature cubic structure as it should be; this has not been accomplished by any other model, In view of this we decided to apply the same model to crystalline bromine. CALCULATION METHOD

Full details of the method of calculation are given in Ref. 15. The intermolecular potential is written as a sum of an atom-atom potential of the Buckingham form Va-- in degrees. Ag and B 3, are librations in the plane of the molecular layer (symmetric and antisymrnetric, respectively), while the out ofplane librations areB 2, (symmetric) and B., (antisymmetric); we have assumed the C2 axis parallel to b as the element of symmetry relating the two molecules of the primitive cell. Expt.

JJH.ub Epack Lattice freq. A.

B 2• B,. A, B 3g B 2g B,g

Calc

11.66" - 11.95b - 12.44 26 49< 58 74< 65 79"71 92 d 98 54 53d 69 d 63

Equilibrium condition q:>(")

-2.4

Atom-atom

Electrostatic

First neighbor

0.22 - 1.82 2.71 2.82 0.98 0.91 -0.\0 -0.37

0.12 0.23 0.03 -0.06 0.16 0.40 0.61 0.58

0.66 2.59 -1.74 - 1.76 -0.14 -0.31 0.49 0.79

0.32

- 2.85

3.53

• Reference 29. b Estimated as in Ref. 27, using the Oebye-Einstein approximation for the calculation of the entropic contribution, Ref. 17. C Infrared frequencies at 77 K, Ref. 18. d Raman frequencies at 130 K, Ref. 19; assignments from Ref. 14.

small effect on the properties of this structure. In conclusion the effect of the induced dipole-induced quadrupole interaction is to single out the first neighbor contacts of the observed orthorhombic unit cell and to stabilize it with respect to the cubic one; its nonnegligible magnitude may come from the extreme closeness of the first neighbor atoms. RESULTS AND DISCUSSION

A new calculation of the static and dynamical properties of bromine was undertaken, adding a specific interaction of the form - D /r 8 forthe first neighbor nonbonded atomatom contacts; the refinement was now performed on the basis of five parameters, three for the atom-atom part, one for the electrostatic interaction as a whole, and D, the nearest neighbor interaction parameter. Unlike previous refinements, this one gives good agreement with the observables, TABLE III. Experimental lattice parameters and calculated orthorhombic and hypothetical cubic structures of minimum packing energy. Units: lattice parameters in A; tilt angle between the molecular axis and the crystal axis b, in the be plane, in degrees; E rack in kcal/mol.

and a crystal structure that contracts slightly upon relaxation of the cell axes towards minimum energy and is more stable than the cubic structure. The refined parameters are shown in Table I and the TABLE IV. Calculated frequencies (in cm - ') at some relevant points on the surface of the BZ. Symmetry species labeling according to Zaka • Subscripts ip and op refer to motions in and out of the be crystal plane, respectively.

Frequency

Symmetry species

R(C 2h l

80 66 58 44 37

R J •• R '.2 R '.2 R J• 4 R J. 4

S(C 2h 1

76 70 59 42 36

S,

Point

T(D 2h )

Observed

Minimum energy (0 Kl

Orthorhombic lattice at - 150 ·C· : 6.67 8.72 4.48 32.95

6.60 8.61 4.53 29.38

260.6 - 11.95

257.1 - 12.72

Hypothetical cubic lattice (space group Pa3):

a Unit cell Volume Epad

• Reference 2.

48

T, T2 T, T2 T,

87 61 46 32 17

Z, Z2 Z, Z, Z2

94 72 71 68 63 58 57 48 36 36

Y, Y8 Y7 Y2 Y, Y3 Y6 Y. Y7 Y8

51

Crystal structure

a b e Tilt angle Unit cell Volume E PBCk

88 62 55

Z(D2hl

Y(D2hl 6.78 311.8 -7.23

Type of motion

R'.2 =R R J •• =T

S,=T,R

T, = (TRl ip T 2 = (T, R)op

Z, = (T,R)ip Z2 =(T, R)op

• Reference 28.

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and nearest neighbor contributions to the dynamical matrix D, L is the eigenvector matrix, and Aij = 0oij. The numbers listed in Table II are then calculated as (LDaaL);;IA;. (LDeILl;;!A;. and (LDnnLl;;!A;. Table II shows that the nearest neighbor interaction contributes strongly to the packing energy, to the angular position of the molecule, and to the translational modes; as for other crystals 15.27, the rotational modes are more affected than the translational ones by the strongly anisotropic electrostatic interaction; furthermore, the atom-atom contribution is in general comparable in order of magnitude to that of the other two terms of the potential. The minimum energy structure has a slightly smaller volume than the observed one. The calculated cell parameter and volume, and the packing energy for the cubic cell are also given; it is clear that the orthorhombic one is definitely favored by our model. It is a common occurrence that models which fit optical frequencies show instabilities or imaginary frequencies at points with k #0. We have made a thorough scan of the Brillouin zone, and found everywhere real values for the lattice frequencies. In Table IV we give the frequencies and corresponding symmetry species of the vibrational modes at some relevant points on the surface of the BZ, and the dispersion curves in the r - Yand r - Z directions. We have followed Zak's notation;28 in order to be consistent we had to interchange here the labelings of the band c axes; therefore the r - Z line, for instance, is in the direction of the b axis of

results of the refinement in Tables II and III. It can be seen that the electrostatic parameter E is smaller than unity, as was found in calculations on other compounds. 15.27 It is possible that this effective reduction is related to polarization effects which have not been included explicity in the calculation, and to a reduction of the effective molecular volume upon crystallization. The values of the parameters C and D, on the other hand, are of the order of magnitude given by Hirshfelder et al. 26 for the corresponding averaged interactions of small molecules. With these values of parameters, we have verified numerically that the induced dipole-induced quadrupole interaction is nonnegligible only for the first neighbor atom-atom contacts of the orthorhombic structure, and that it is negligible for all contacts in the cubic structure, thus confirming our previous assumption. In Table II we have also included the fractional contribution of the different terms of the interaction potential (atom-atom, electrostatic, and nearest neighbor) to the calculated quantites; this is straightforward for the packing energy and the equilibrium conditions, since they are sums of the corresponding terms. For the frequencies VI' however, the diagonalization process has to be taken into account. The dynamical matrix may be split in three terms, and the secular equation written as

LDL = L(Daa

+ Del + DnnlL =

J...,

where Daa , Del' and Dnn are the atom-atom, electrostatic,

V(cm-')

fl., - -

fl.) -.-.-.-

A,--

A)-·-·-

fl.2 -----

fl.4 - .. - .. -

A2-"-"-

A

Y,--90 /

I

I

""

/

/

"

"...

4 -----

-- "-"- ____- -.'- -.'..

.. .. .. ..

..

..

..

90 Z,

. - - -......

FIG. 1. Calculated dispersion curves in the r - Yand r - Z directions. Symmetry species labeling according to Zak (Ref. 28).

Z,

"'.

...-----... ZI

30

"

..... rs

\.

'-'-'-'- '-'-.

\

\

\

\. Y

5 fl.

r

5

Z

./\.

J. Chem. Phys., Vol. 80, No.6, 15 March 1984 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 168.96.66.131 On: Tue, 07 Jan 2014 15:04:55


Table III. Modes ofthe.1 l , .1 2, A I' A2 species correspond to mixed rotations and translations in the be plane, while the other modes are perpendicular to this plane, It can be seen that in particular, the.1 1 and.1 2branches show a very strong dispersion. The dispersion curves in the r - Z direction are similar in shape to those observed for iodine l3 ; the assignment pattern at Z for Br2 and 12 is identical, as occurred for the observed frequencies at r. 14 CONCLUSIONS

In this work we present an interaction model which is capable of adequately reproducing the lattice dynamics and the crystal structure of crystalline bromine. Although this problem has been attacked before, the solutions found until now were partial, in the sense that only the dynamics or the statics of the crystal could be explained correctly by existing force fields. In our recent article on crystalline acetylene, we found that it was necessary to include high order multipole interactions in the potential, and found instabilities in acoustic and BZ boundary modes which could be identified as soft modes that lead to the phase transition. In the present case, no such instabilities appear, but the problem connected with the crystal structure led us to the identification of a specific interaction, of the form - D Ir 8 which may be ascribed to an induced dipole-induced quadrupole interaction between very close lying nonbonded atoms. Although we feel that not too much meaning should be attached, in general, to the specific parameters found in such particular refinements (a good example is the vast number of very different atomatom parameters found in the literature for hydrocarbon crystals) the form of the interaction potentials involved gives a good qualitative description of the main physical causes

2761

which are responsible for the behavior of the systems under study. 'R. L. Collin, Acta Crystallogr. 5,431 (1952); 9,537 (1956). 2B. Vonnegut and B. E. Warren, J. Am. Chern. Soc. 58, 2459 (1936). 3F. Vanbolhuis, P. B. Koster, and T. Migghe1sen, Acta Crystallogr. 23, 90 (1967).

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(1974).

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