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Study of the phase transitions in ammonium bromide by Raman spectroscopy
J. Phys. Chem. Solids, 1973,Vol.34, pp. 787-799. PergamonPress. Printedin Great Britain
STUDY AMMONIUM
OF THE BROMIDE
PHASE TRANSITIONS IN BY RAMAN SPECTROSCOPY*
R. B. WRIGHT and C. H. W A N G t
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, U.S.A. ( R e c e i v e d 13 July 1972)
Abstract--A complete Raman scattering study of an oriented NH4Br single crystal has been carried out and new results are reported for temperature between 45 to 410~ The study includes measurements and interpretations of the Raman spectral bandwidths, intensities and frequencies of the lattice and internal modes as a function of temperature. Because of the ability of generating a single domain crystal, unambiguous assignments of the Raman active modes are made. From the present data together with those presented previously significant information on the behavior of the ordering and crystal structure associated with the phase transitions in NH4Br is obtained.
1. INTRODUCTION
THERE has been a recent renewal of interest in theoretical and experimental attempts to elucidate the mechanisms responsible for the order-disorder transitions which have been observed to occur in the ammonium halides (NH4Br, NH4CI and NH4I). Although great progress has been made toward qualitative explanations of this phenomenon, detailed quantitative predictions are still lacking. The infrared absorption and Raman scattering techniques have proven to be valuable tools for probing the dynamical nature of these phase transitions and the associated effects of ordering and disordering on the external and internal vibrations in single crystals of these compounds[I-15]. The usefulness of these techniques arises from the fact that the frequencies and lifetimes of individual phonon modes in the Brillouin zone can be studied directly in contrast to only observing the average effects of the many phonon modes as is the case in thermodynamic[16-18], elastic[19, 20] and dielectric [21 ] measurements. Due to the recent theoretical proposals and *Acknowledgement is made to the Research Corporation and the donor of PRF, administered by the American Chemical Society, for the support of this research. tAuthor to whom correspondence should be addressed.
experimental observations of 'soft' or unstable optical phonons associated with structural phase transitions[22, 23] in displacive ferroelectric crystals have raised the question that similar behavior may be responsible for the occurrence of phase transitions in general. It remains to be demonstrated, for example, that the order-disorder phase transition is also related to the softening of a particular phonon mode although recently such an interpretation has been used in NaNO2124]. Since the ammonium halides display a very general type of order-disorder or order-order phase transition, the search for such physical processes accompanying the transitions in these systems is highly justified. We have attempted to carry out a research program which consists of detailed Raman spectral studies of NH4CI and NH4Br oriented single crystals over a wide range of temperature. The investigations involve the measurement of the spectral intensity, bandwidth, frequency and polarization of all the Raman active modes appearing in these compounds. While the work on NH4C1 will be presented elsewhere [14], this paper reports the experimental results and discussions pertaining to NH4Br over the temperature of 45-410~ Because of the ability to produce single domain single crystals in the ordered tetragonal phase this paper presents 787
788
R.B.
WRIGHT
unambiguous symmetry assignments of the Raman modes in NH4Br for the first time. Also presented for the first time are new spectral anomalies associated with the order-disorder phase transition. Ammonium bromide has been found by X-ray [25] and neutron studies [26] to exist in four crystalline phases. Phase I (or-phase) is a disordered face-centered cubic structure similar to NaCl and possesses OhS(Fm3m) space group symmetry with four molecules per unit cell [25, 26]. The ammonium ions are not freely rotating but undergo restricted rotations hindered by a potential barrier corresponding to orientations in which one, two, or three hydrogen atoms make close approaches to the bromide ions[26]. The transformation from Phase I to Phase II (fl-phase) occurs at 41 I~ and is accompanied by a change of the crystal structure to the space-group oha(Pm3m) with one molecule per unit cell. In phases I and II, the ammonium ions are randomly distributed between two possible equivalent orientations giving rise to long-range orientational disorder of the ammonium ions. The site symmetry for both the NH4 + and Br- ions in these phases is Oh. Actually the ammonium ion transforms under the point group Ta, but due to the orientational disorder caused by the random NH4 + configuration in the single crystal in Phase II, the average X-ray determined structure appears to be Oh 1instead of Ta1. As X-ray measurements are time averaged and include scattering from the entire specimen, this average scattering is insensitive to the temporal and spatial disordering occurring in microdomains within the sample. It is this average scattering that is compatible with the Oh 1 symmetry group. Raman and infrared spectroscopy, however, sample across a disordered microdomain and because these processes interact at times faster than the expected ammonium ion disordering flipping rates (10 -13 sec compared to 10 -11 sec), they can instantaneously observe these disordered microdomains. The disordered microdomain
a n d C. H . W A N G
in Phase II has no real center of inversion (as it would if the On I group was strictly valid) and hence, the modes that are affected by the disordering can be both Raman and infrared active (consistent with the Ta1crystal group). Assuming that the crystal structure of the disordered phase is Tct1, the eighteen allowed vibrations are [27-29]: NH4Br = 1Al (x2-4- y2-4- z 2) + l E (2z2-- x 2-- y2, x 2 _ yZ) + I F I ( R ~ , R u , R~) + 4 F 2 ( x , y , z ; xy, xz, yz)
which can be further broken down into the internal vibrations, librations, and the lattice and acoustic phonon modes as follows: 'internal : NH4+ F llbrati~
NH4§
=
FNlattice ~ H4Br Facoustic = NH4Br
1A 1-q- 1E + 2F2
1F1 1 F2
1F2.
One of the F2 modes is an infrared active translational mode (labeled vs, a lattice restrahlen mode) and the other corresponds to the acoustic branch vT. The FI mode is an infrared and Raman inactive librational mode yr. The infrared optic mode is further split due to the long range electrostatic forces giving rise to Fz --~ F2(TO) + F2(LO) where TO and LO refer to transverse optic and longitudinal optic phonons respectively. Perry and Lowndes [9], have reported the position of these modes to be at 147 and 224 cm -1 respectively, for TO and LO at 300~ by use of reflectivity and transmission measurements for Phase II NH4Br. To our knowledge the external modes in Phase I have not as yet been studied in detail by optical methods. Phase III (),-phase) arises from an orderdisorder phase transition (h-transition) which occurs at Tx = 235~ accompanied by a
STUDY
OF
THE
PHASE
TRANSITIONS
slight distortion of one cubic axis. The resulting crystal structure is D4rh(P4/nmm) with two molecules per unit cell in which the ammonium ions are antiparallel ordered in the a-b (x-y) plane and parallel ordered along the c-axis (z-axis). The site symmetries are D2d and C4h for the ammonium ion and bromide ion, respectively, and give rise to the following factor group (D4h) vibrational symmetries [6, 29]. Finternal
2A~g(x2+y 2, z 2) + 1Blg(x2--y 2)
~__
NH4 +
+ 2B2g (xy) + 2Eg(xz, yz) + 1Alu(--) + 2 A e u ( z ) -k-2BlU(-- )
+ 2Eu (x, y) "nbration = 1A2g(Rz) + 1Eg(Rx, Ru; xz, yz) NH4+ + lBau(--) + 1Eu(x,y) FNH4Brlatice= 1Alg(x 2 + y2, z 2) + 1B2g(xy)
+ 2Eg(xz, yz) + IA2u(z) + 1Eu(x,y) ['acoustic_~ 1A2u (Z) + 1Eu (x, y). NH4Br
The nonvanishing components of the polarizability tensor for the lattice vibrations are shown for each Raman active mode [30].
Alg -----
Eg=
a 0
0 0
; B2g =
[! !] 0
=
0 e
.
Since the factor group has an inversion center the mutual exclusion rule is valid and the infrared and Raman spectra will not coinside. Most of the internal vibrations have been assigned, and detailed temperature studies have shown that most of the internal modes are quite insensitive to the h-transition and the associated ordering processes which occur as the temperature is lowered through this
IN AMMONIUM
BROMIDE
789
transformation. An exception to this statement appears to be a v4 triply degenerate bending vibration which exhibits anomalous behavior arising from the ordering processes which accompany the Phase II to III transition[5,6, 10]. Although the ordering effects on a 1435cm -1 infrared active mode (a component of v4) have been studied in detail by Garland and Schumaker[10] in NH4Br, the influence of ordering on the Raman active v4 modes has not been explored in detail. This mode was carefully studied in the present research and will be discussed in Section 3A of this paper. Wang and Fleury have reported the temperature dependent Raman scattering for the v6 (Eg) librational mode (326 cm -1 at 200~ and observed an exponential behavior of the linewidth which they attributed as arising due to a rotational diffusion thermal activation process[6]. We have carried out the analysis of the angular correlation function of v6 as a function of temperature. Because the motion of the NH4 + ions in Phase l l l is not part of the theme of this paper, to avoid confusing the present discussions, we will defer publication of these results elsewhere. The Raman active lattice modes were all studied in great detail in Phases II and III and wilt comprise the major presentation of this publication (Section 3(b)). Phase" IV (y-phase) is an ordered cubic phase with symmetry Td~ (isomorphous with Phase IV of NH4CI) and occurs at 108.5~ K on warming and - 77~ on cooling, via an order-order transition in which the bromide ions occupy Oh sites and the parallel ordered ammonium ions are situated on Td sites. This phase has recently been studied by Garland and Schumaker[10] and by Harvey and McQuacker [ 12] but was not studied in detail by the present authors. 2. E X P E R I M E N T A L
ARRANGEMENT
The Raman spectra were obtained using a constant intensity (140 mW) argon ion laser oscillating at either 4880 or 5145 ,~, a double
790
R.B. WRIGHT and C. H. WANG
monochromator for dispersion, and a photoncounter-descriminator with a cooled FW-130 photomultiplier detection system. The polarization measurements were achieved using an analyzer and polarization scrambler for the scattered light. A half-wave plate to rotate the E-vector of the incident laser light was also used to obtain the desired laser polarization. G o o d optical quality NH4Br crystals ( - 1 cm 3) were grown by slow evaporation from an aqueous solution with urea added as a habit modifier to prevent dendridic growth and to promote the formation of large cubic crystals with (001) faces exposed. The crystals were then cut and polished before use and were free of any visible imperfections. The low temperature Raman scattering studies were achieved using a Joule-Thomson refrigerator operating with compressed N2 and H2 gases. The sample crystal was mounted by cementing a (001) crystal face to a copper block mounted in the cryotip and the temperature was monitored simultaneously at the base and side of the crystal by two standardized chromel-constantan thermocouples. Temperature stability was maintained to _ 0.5 ~ K. It was found that the tetragonal (c- or z-) axis in Phase III appeared to be predominantly formed along the direction perpendicular to the (001) face cemented to the cryotip sample holder. Great care was taken to align the crystal with the spectrometer optics to avoid 'leakage' of one polarization into another, and with slow cooling at the phase transition ( ~ 1~ K per 20 min) the crystal was single domain in Phase III. Single domain crystals in Phase III which were not formed from pre-annealed samples, which can easily bo verified by checking the polarization dependence of the low frequency lattice modes with a half-wave plate and polarization analyzer. The ability of the present investigation to produce single domain crystals in Phase III enables the correct elucidation of the polarization assignments for the various Raman active modes as a distinct c- or z- axis can now be defined and as such the crystallographic axes can be used
to orient the single crystal for the polarization studies. The high temperature studies (297-410 ~K) were conducted using a home-made doublewalled pyrex dewar containing pure mineral oil as the heat bath into which the NH4Br crystal was immersed after glueing a (001) face to a copper block which was then held rigidly in the bath. The mineral oil was found not to interfere with the Raman spectra and was inert towards the NH4Br crystal. Temperature variation was controlled by a proportional temperature controller coupled to a 125W quartz heater. An electric stirrer was used to gently agitate the mineral oil to facilitate a uniform temperature in the bath. The temperature was determined by using two standardized independent chromel-alumel thermocouples mounted on the periphery of the crystal. Temperature stability w a s - - 0 . 5 ~ Because of the occurrence of fracturing of the crystal in going through the Phase II to I transition at 411 ~ K, the Raman study of the crystal in this phase was not undertaken. 3. RESULTS AND DISCUSSION
(a) Internal vibrations Shown in Fig. l(a) are representative Raman spectra of the u2(E) doubly degenerate NH4 + bending mode in Phase II observed in the a (cc) b [31 ] scattering configuration. The u2 vibration corresponds to an atomic motion in which the hydrogen atoms vibrate perpendicular to the N-H-bonds. As shown in Fig. l(a) the line shifts to lower frequency as temperature is raised (1688 cm -1 at 295~ to 1681 cm -1 at 406 ~K) and the line also broadens from 11 to 20 cm -a (measured as full-width at half-intensity) and decreases in intensity. Presented in Fig. l(b) are the frequency and spectral bandwidth of this mode as functions of temperature from 295 to 406 ~K. The frequency has a linear temperature dependence (frequency decreases as temperature increases), while the bandwidth exhibits a non-linear dependence, decreasing as temperature is lowered. Note the extreme increase in spec-
S T U D Y O F T H E PHASE T R A N S I T I O N S IN A M M O N I U M BROMIDE
791
1,688
x(zz)y
I
b. top = Peak -
1,681
(cb.mpl}/
frequency
F' = Spectral band width
r'
(cm-~)
/
20
- '"~ - -
t,6Sol I
1,725
311~ I I 1,700 1,675 Wevenumber, cm -I
I 1,650
1,6751/I 3
I t I I I 0 320 8 3 64 0 0 360 OK
I/
I / 30
40
Fig. l(a). Representative Raman spectra for Phase II of the doubly degenerate bending mode v.2(E) in the a (cc) b or (x (zz) y) scattering configuration. Fig. l(b). Temperature dependence of the peak frequency and spectral bandwidth for the vz(E) internal mode in Phase I1.
tral bandwidth as the Phase II to I transition is approached, indicating significant anharmonic influence at the transition region. Figure 2 shows representative depolarized a ( c a ) b = b ( c b ) a spectra of the internal triply degenerate bending mode v4 as a function of temperature. This mode corresponds to an internal NH4 + motion in which the nitrogen atom vibrates with respect to the four hydrogen atoms as a whole. In Phase II the mode has been assigned as having F2 symmetry. In Fig. 2 the line appears as a single, fairly symmetrical, broad line at 1400cm -1 (see the 406~ spectrum) whose spectral shape narrows and increases in intensity with progressive asymmetry developing on the high frequency side of the spectra. This continues until at 236~ two definite shoulders appear at approximately 1430 and 1420 cm -1. The weak 1430 cm -1 line disappears on further cooling but the 1420 cm -1 (vj) line continues to develop as temperature is lowered. The appearance of two bands for v4 in the Raman spectra in Phase III arises from the fact that
the symmetry of the crystal field about any ammonium ion is no longer strictly tetrahedral. As the phase transition temperature is approached, from above, the long-range ordering of the NH~ + ions begins to occur which slightly displace the Br ions and in turn gives rise to the major NH4 + ion perturbation by modifying the tetrahedral crystal field. The lifting of the degeneracy for the triply degenerate vibrations is thus expected, and is observed to manifest itself in the shoulder which develops on the high frequency side of v4. It is also reasonable to expect that, as in disordered alloy systems, the nearneighbor structures are predominantly those of the most stable crystalline forms although over any extended crystal volume there is still complete randomness. That is, there exists residual short-range order above Tx in the disordered phase. This statement has been quantitatively verified for NH4Br[11, 15] and NH4C1 [ 13] where an anomalous, totally polarized low frequency mode was found to correlate with the evolution of long-range order
792
R. B. W R I G H T and C. H. W A N G
1,4PO0 1,440 1,420 1,400 1,380 1,360 1,340 ~M]venumber, crn-I
Fig. 2. Raman spectra for the internal bending mode, ~4, region at various temperatures in Phases II and III obtained in the a(ca)b (= b(cb)a) scattering configuration.
i.e. the decay of short-range order at the phase transition. A similar interpretation was used by Garland and Schumaker [5, 10] in that the intensity decrease as Tx is approached from above of an anomolous component at 1444 cm -1 in the infrared spectra of NH4CI indicates the presence of disorder in the ordered (cubic) phase of NH4CI and the resulting intensity variation could be correlated with a long-range order parameter. As the crystal temperature is lowered through the Phase II to III transition (235~ the intensity of the 1420 cm -~ component increases and its frequency remains constant within the experimental resolution (§ l cm -1) but the 1400 cm -1 component decreases in inte'nsRy
and tends to shift slightly to higher frequency (1402 cm -1 at 201 ~K.) The scattering arrangement applicable to Phase III (tetragonal) which was used for the spectra shown in Fig. 2 below 235~ is a(ca)b=b(cb)a. This polarization should therefore select only the Eg(xz, yz) depolarized modes in a single domain (i.e. z or c crystal axis defined) NH4Br crystal in Phase III. Since the experimental results clearly indicate that the 1420 cm -1 mode is present and continues to increase in intensity in this polarization, the observed 1420cm -~ line should be assigned as an Eg, Raman active only mode. This is in contradiction to previous [1-7, 10] assignments of this vibration which was assigned to B2g symmetry. The previous investigators either could not discriminate between B2g and Eg symmetry because the experimental conditions would not allow the formation of single domain crystals[I-7,10,12] or the region under discussion was not examined [9]. Harvey and McQuaker[19, 32] have also recently questioned the assignment for the 1420 cm -1 line on the basis of transverse-longitudinal splittings but their results were obtained from pressed polycrystaUine samples which would not enable unambiguous polarization assignments. We therefore assign the 1420cm -~ line as a factor group allowed Eg, not a B2g mode, and assign the 1402 cm -~ line as B2g. This latter assignment is based on prior results [6, 10] which indicate that the 1402 cm -1 line (as measured in a multi-domain crystal which allows both B2gand Eg modes to appear in the depolarized spectra simultaneously) does not disappear (in contrast to that shown in Fig. 2) but is a symmetry allowed mode possessing the polarizability, crab= ctba, i.e. having B2g symmetry. In the a(ca)c configuration, as temperature is lowered, the 1402 cm -1 component, observed to decay (Fig. 2), is therefore an anomalous mode arising from interactions which allow coupling of the v4 phonon (B~gcomponent) to fluctuations of the order parameter in the vicinity of the phase
STUDY OF THE PHASE TRANSITIONS IN AMMONIUM BROMIDE transition. This phenomenon has been recently observed and interpreted in a similar manner for an anomalous low frequency (56 cm -~) mode which appears in the b(aa)c (or a (bb) c) scattering configuration [11], an extraneous mode not predicted by group theory. The observed intensity decay of this line is attributed to the final evolution of longrange order in the tetragonal phase. In Ref. [11] it was found that the temperature reduced integrated Raman intensity (i.e. the integrated intensity divided by In(co) + 1] where n(to) ---[ e x p ( h t o / k T ) - l ] -~) correlated with the residual specific heat which arises from the h-phase transition in NH4Br. Figure 3 shows the results of the intensity versus temperature behavior of the anomalous 1402 cm -~ component, and also presents the earlier results for the 56 cm -~ mode from Ref. [11] for comparison. The 8Cp was obtained from the specific-heat data, measured by Sorai et al. [18], by subtracting the measured specific heat of NH4Br from a smooth curve baseline representing all the non-configurational contributions to the heat capacity (such as lattice vibrations and the librational and internal ammonium ion motions) [l l]. The
793
1402 cm -~ intensity data was not temperature reduced as this factor had negligible effect for this mode over the temperature region of interest. The correspondence reflects a general correlation, although not as satisfactory as the low frequency mode, indicating that the above proposed explanation has definite merit. These results support previous conclusions that the ~'4 internal mode is a very sensitive 'probe' which reflects slight changes in site symmetry and local crystalline processes that occur as the crystal undergoes transformation from one phase to another.
(b) Lattice phonons The lattice phonon region (< 300cm -1) in Phase II ammonium bromide, as well as in Phase II ammonium chloride and iodide, should consist of only an infrared active zone center translational mode and an inactive acoustic mode. But here again the disordering inherent in this phase generates spectral anomalies. Within the crystal dimension corresponding to the wavelength of the exciting light, there is a loss of translational symmetry and consequently, the sampling of phonons with only very small wave vectors, k = 0, is not mandatory upon the Raman scattering process; k----0 wave vector seleclntee/alad i~lmsity of t ~ 1,402r ~ . tion rule is not obeyed. The spectra are there~ Raau cad inleO~ed i~lemi~ " ~Cp ~IQ ~tQin~l from: ~ fore weighted by selection rules, coupling constants and the corresponding density of states [33]. Thus, the usual sharp and symmee; trical lines should not be expected and the resulting Raman spectrum would then mirror d the one-phonon density of states. However, the appearance of a sharp and symmetrically 4 /~/ shaped polarized resonance at 56cm -1 in NH4Br has been observed. This discrepancy 0 L t i I L h I has been interpreted as arising from a largewave-vector phonon at the zone b.oundary, "K possibly a transverse acoustic mode at the Fig. 3. The integrated intensity of the 1402cm-~ band (solid circles) plotted vs the absolute temperaturein the X (or M) point[6, 11, 15]. The existence of vicinity of the h-transition. The temperature-reduced short-range order in Phase II, as discussed integrated intensity of the 56 cm-~ band (open circles, previously, was thought to allow the coupling from Ref. [11]. and the residual specific heat data 8Cp (in units of 1.67eallmoledeg) obtained from Ref. [18] of the zone-boundary phonon to the incident are includedfor comparison. light wave [6]. Since the detailed presentation of l~
" "
56era-' t~nd.
,
794
R.B. WRIGHT and C. H. WANG
o f the scattering at the 56 cm I region has b e e n given elsewhere[13, 15], there is no need to amplify the discussion o f this m o d e further, except to say that it is the progenitor o f the (Section I) totally s y m m e t r i c A lg lattice m o d e expected in Phase I I I NH4Br. Details of the t e m p e r a t u r e dependent depolarized R a m a n scattering o f the low frequency lattice m o d e s a b o v e r o o m t e m p e r a ture are shown in Fig. 4(a), w h e r e a broad, slightly asymmetrical band centered at 134 cm -~ (406~ K) is o b s e r v e d to shift slightly to higher frequency ( 1 3 8 c m -~ at 311~ and ,
,
,
,
,
,
r
,
134
X(ZX) y
increase in intensity as t e m p e r a t u r e is decreased. T h e spectral linewidth decreases in a non-linear m a n n e r as t e m p e r a t u r e is lowered (Fig. 4(b)) to 310~ the high frequency shoulder which develops below 300~ (Fig. 5(a) and 5(b)) p r e v e n t e d the measuring of reliable bandwidths below this temperature. T h e spectral bandwidth of the 134 cm -1 m o d e a s s u m e s a rapid growth as the 411~ phase transition is approached. This large increase in spectral width prior to the 41 I~ transition indicates that large anharmonic interactions are associated with the us m o d e and m a y play a significant role in the ensuing phase transition. Shown in Figs. 5(a) and 5(b) are representa-
13%6oK --
~k~
I
I
I
I
(].
122
I
I
Resolution
*K
c >.. o D
O.
I
h
200
I i i ,~o Wovenum~r,
,do
i
do
c
~-i
Fig. 4(a). Phase II Raman scattering showing the temperature dependence for the depolarized a(ca)b lattice modes over the 50-200 cm -~ region. 120 I00
b,
"g sc 6C
I
250
2(3 i 410
NO ~
Fig. 4(b). Bandwidth for the depolarized mode shown in Fig. 4(a) at temperatures between 310-410~ --
I
210
I
I
IT0 Wovenurnber,
I
I
130 crn-I
I
90
Fig. 5(a). Representative Raman spectra for the a(ca)b scattering arrangement which selects the depolarized lattice translational mode v'5 (Eg) (high frequency mode). Appearance of v~ (B.2g) (low frequency mode below Tx = 235~ is a spectral anomaly discussed in text.
S T U D Y O F T H E P H A S E T R A N S I T I O N S IN A M M O N I U M B R O M I D E
795
ing, the low frequency band shifts gradually to lower frequency, while the high frequency shoulder remains constant in frequency with both lines increasing marginally in intensity. In Phase III, below Tx, the spectra obtained with the two different scattering configurations show pronounced differences. The a(ca)b = b(cb) a configuration (Fig. 5(a)) indicates the high frequency mode increases in intensity, shifts to higher frequency (183 cm -1 at 122~K) and narrows in bandwidth. This 183 cm -1 line has been previously assigned as an Eg (Raman active only), v~, translational optic mode[9], consistent with our findings. The overlap existing between the two modes in Phase II prevented reliable bandwidth and intensity measurements to be obtained. The trends of the spectral features we observed in Phase III were the same as those reported in Ref. [9] and, thus, will not be presented. The intensity of the low frequency (134cm -1) component shown in Fig. 5(a) displays unusual behavior. It decreases rapidly to a finite, though small, constant value (see Fig. 6(a)) as t
250
210
170
Wovenumber,
1:30
90
u
i
c m -I
a(co)b
Fig. 5(b). Raman spectra at various temperatures. The v~ (B2g) translational lattice mode is selected by the b (ab)c arrangement. Leakage of the polarizer allows observation of v'5 (Eg) mode. Relative gain settings for the spectra "rare: gain X 1 for 273 and 236~ K; gain X 89 220~ K, and gain X~for 122~
tive Raman spectra for the v5 lattice modes obtained respectively in the a(ca)b and b(ab)c configurations in Phase II and Phase III below 300~K. It is seen that in Phase II (above 235 ~K) the gross features of the spectra are identical in both configurations. Any apparent differences arose because the specimen under study, as well as the cryoptip refrigerator, had to be rotated 90 ~ and the optics realigned to observe the different polarizations. In Phase II the spectra consist of a broad band, the major peak of which is centered at approximately 140 cm -1 with a high frequency shoulder developing at roughly 175 cm-L On cool-
T X = 2 3 5 *t
.=.it -=-
Z I J --m-
o
9
-ul. - m . . . .
'
I00
140
9 ""m
9 no
-= ......
"
180 T (~
am'- _ _=,=, m..ml
'
220
260
Fig. 6(a). Peak intensity for the anomalous low frequency component shown in Fig. 5(a) as a function of the absolute temperature.
796
R . B . W R I G H T and C. H. W A N G
the temperature of the crystal is lowered below T~. The constant intensity value is a leakage component of the other depolarized scattering configuration (b(ab)c) due to imperfections in optical alignment. The rapid decrease of the intensity is associated with the ordering of the NH4 + ions in the vicinity of the order-disorder phase transition in an analogous manner as that described for the anomalous 1402 and 56 cm -1 components. While the spectral peak frequency of the Eg mode observed at 183 cm -~ was not unusually affected by the phase transition, the ordering processes have a peculiar effect on the peak frequency of the n 2g~t/jtv\ 5/ mode observed in the b(ab)c configuration (Fig. 5(b)). In Fig. 6(b) the intensity and frequency of the B2g translational optic mode are shown as a function of temperature. The data clearly indicate that in the disordered phase the mode frequency increases as temperature is lowered to 260 ~K, below which an anomalous decrease of the frequency occurs. The decrease in 7
i o(balc
6
-i----
4
.---
2
F1 ~Z
I
I
I
frequency continues even through the phase transition (141 cm -1 at 260 ~K and 132 cm -1 at 220 ~ At 220~ the mode frequency assumes again a normal temperature behavior. The anomalous softening behavior of the Bzg mode observed between 260 and 220~ reflects that the precursor of the orderdisorder phase transition appears at 260~K, which is 25~ higher than Tx. This result is consistent with the anomalous intensity versus temperature behavior that was observed in the 56 cm -1 Raman line reported earlier [ 15]. Our normal coordinate analysis has shown that the B2g(v~) mode is connected with the translational vibration along the c-axis of the two sets of equivalent NH4 + ions in the unit cell. From the intensity and frequency data (Fig. 6(b)), the mode reveals that the interaction between the neighboring NH4 + tetrahedra in NH4Br is greater than that in NH4CI [34]. The stronger interaction may also be associated with the precipitation of the unusual ordering behavior of NH4 + ions in the 3,-phase.
i
I
c y ~
I
I
1414
Tk = 235 * i
qun
/
~
Peok Intensity
J
-...>_
al "'~'''-
i z
... .
mm--l~an'-"nL'--,nL---.-~
ul
I00
o.
I
I 140
I
I 180
I
, ~l 220
I 260
I
30 300
T(~ Fig. 6(b). Peak frequency and relative intensity values for the v~ (B2g) translational mode (shown in Fig. 5(b)) over the temperature interval 120-300~ Error bars are shown for the frequency data representing the experimental spectral resolution of • 1.
S T U D Y OF T H E P H A S E T R A N S I T I O N S IN A M M O N I U M BROMIDE
Recently Yamada et al. have published a microscopic theory on the phase transitions in NH4Br [35]. In their paper they adopted the phonon-order parameter coupling picture first proposed by Wang and Fleury for NH4Br[6] and have demonstrated mathematically that the direct interaction between N H , + ions stabilizes parallel ordering whereas the indirect interaction through the Br- ions stabilizes the antiparallel ordering. The indirect interaction is closely associated with the apparent softening of the B2g lattice mode in the 260-222~ region (Fig. 6(b)). This is because the decrease in the B2g mode frequency arises from the weakening of the NH4+-Br - interaction as the crystalline parameter change (c-axis elongates). In addition, since the N H , § ions can also undergo 180~ rotations about the tetragonal c-axis (which may be accomplished via a rotation over an energy barrier or tunneling), the possibility of a translational-librational mode coupling may also exist. However, because the NH4 + libration about the c-axis (belonging to A2g) is neither Raman nor i.r. active, it will require other techniques, such as the neutron scattering, for its elucidation. Nevertheless, it should be pointed out that Hartwig, WiernerAvnear and Porto have recently proposed a similar interpretation for the observed spectral anomalies observed in the temperature dependent Raman scattering in NaNO2 which undergoes a ferroelectric orderdisorder phase transition [24]. In addition to the A~g(v~) mode mentioned above, there is another factor group allowed Raman active mode of Eg symmetry. The Eg optic mode appears in the b ( c a ) c scattering configuration and exhibits a dramatic intensity increase at constant frequency as the temperature is lowered below Tx (the intensity enhancement was observed only as T is lowered from 236 to 210~ remaining constant below 210~K, see Fig. 7). If the cartesian components of the set of the two non-equivalent Br- ions and the set of the two non-equivalent NH4 + ions are designated
J P C S Vol. 34 No. 5 - C
-7
797
l Retolutlon
b(co)c
9----13 3 *K
==
"7,
~161
*K
o
A *K
I00
90
80
70
60
50
Wovenumber, cm-*
Fig. 7. Temperature dependence of the external v~ (Eg) optic mode for the b (ca)c = a (cb)c configuration. The low frequency sideband is due to polarization leakage of the v; (A~g) optic mode.
by {(XlylZ,), (xzY2Z2)} and {(x'ly'lr (x~y'2z~)}, respectively, our mode analysis has shown that the normal coordinates for A ~g is S (A~g) = 89 -- z2) for B~g is s (B
g) =
1
!
-
and for Eg is S ( E g ) = 89 (x, + y , ) -- (x2 + Y2) + (x'l+y'1) -- (x~+y~) ]. One sees that A lg, B2g and Eg are symmetrical with respect to the inversion operation of
798
R . B . W R I G H T and C. H. W A N G
is necessary for them to be Raman active. Since the progenitor of the A ~g and Eg modes is the zone boundary TA mode (which appears in the 56 cm -1 region observed in the Raman spectrum of phase II), the behavior of the Eg mode is also associated with the ordering of the NH4 + ions as the k-transition occurs. Since the presence of the 56 cm -~ scattering in Phase II depends on the existence of order with the periodicity of a reciprocal lattice vector[6, 35, 36], there is probably no peak in the density of states at this position. It should also be pointed out that while the 76 cm -1 mode appears and grows in intensity between 236-210~ the total scattering intensity is considerably less than that of the Alg mode reported in Ref. [15]. This probably has to do with the fact that the original population associated with the zone-edge T A transfers more to the A lg than to the Eg mode due to the presence of the mode coupling mechanism in the vicinity of Tx[15]. We notice from the normal coordinate analysis that the displacements of the Br-ions in Phase III correspond to the polarization of the zone boundary normal coordinate of the lattice vibration in the CsC1 type crystal at (2-q0), the A lg and Eg Raman active phonons thus come from the regions near the point M. It is hoped that in the future neutron scattering will reveal the exact location of the anomalous mode observed at 56 cm -~ in Phase II. D4h a s
4. SUMMARY AND CONCLUSIONS
Raman scattering experiments from NH4Br single crystals in the disordered phase reveal selection rule violations and line-shape distortions that may be explained by the instantaneous disordering in the crystal and by the residual short-range ordering of the ammonium ions. By using oriented, single domain crystals, polarization assignments have been made unambiguously for the first time. Anomalous scattering has been studied in Phase II and III. Those Raman active modes which appear to be most significantly influenced by the o n s e t o f
ordering processes (as the NH4Br crystal approaches, from above, the Phase II to III transition) are the Alg(v'~), Eg, (v~) and B2g (v's) external optic modes and the B2g(v4) internal mode. These modes exhibited peculiar line shape and intensity variations as the k-transition was approached. Raman scattering from extraneous polarization components for the v4 and v~ modes were observed and found to correlate with the residual specific heat of the crystal at the order-disorder phase transition. The v5 mode was also observed to have an extraneous component and displayed a softening in frequency at the htransition. We have associated these peculiar spectral features with (1) the presence of residual short-range order in the disordered (Phase II) phase which permits the Raman inactive vibrations to give rise to Raman scattering via the coupling of the order parameter to these vibrations; (2) the rapid replacement of the short-range order by long-range order at the phase transition causes the intensity variations and the extraneous polarization components; (3) the evolution of tetragonal distortions of the Br- ions along the crystalline c-axis starting at 260~ K induce anharmonic perturbations in the motions of the Br- and NH4 + ions parallel to the tetragonal c-axis. This perturbation produces angular distortions in the NH4 + ion tetrahedron caused by this fluctuating potential field. We have shown that our experimental results on the intensities, frequencies and the linewidths of various Raman lines together with those presented previously[l l, 15] can provide significant information on the phase transitions in NH4Br. By careful examination of the polarization properties of various scattering spectral components in an oriented NH4Br single crystal, we have also found that well-defined Raman selection rules are operative for the K ~ 0 phonons and the internal vibrational modes. Further theoretical work is therefore needed on the spectra of disordered tt
S T U D Y O F T H E P H A S E T R A N S I T I O N S IN A M M O N I U M B R O M I D E
crystals and on the selection rules appropriate for various spectral bands affected by the disorder. REFERENCES
1. K R I S H M A N R. S., Proc. Ind. Acad. Sci. 26A, 432 (1947); 27A, 32 (1948). 2. W A G N E R E. L. and H O R N I G D. F., J. chem. Phys. 18, 296,305 (1950). 3. BOVEY L. F. H., J. Opt. Soc. Am. 41, 836 (1951). 4. C O U T U R E - M A T H I E U L. and M A T H I E U J. P., J. chim. Phys. 49, 266 (1952). 5. G A R L A N D C. W. and S C H U M A K E R N. F., J. Phys. Chem. Solids 28, 799 (1967). 6. W A N G C. H. and F L E U R Y P. A., Proceedings of the Conference on Light Scattering Spectra of Solids (Edited by G. B. Wright), p. 665, Springer, New York (1969). 7. R I M A I L., C O L E T. and PARSONS J. L., ibid., p. 651. 8. D U R I G J. R. and A N T I O N D. J., J. chem. Phys. 51, 3639 (1969). 9. PERRY C. H. and L O W N D E S R., J. chem. Phys. 51, 3648 (1969). 10. S C H U M A K E R N. F. and G A R L A N D C. W., J. chem. Phys. 53, 392 (1970). 11. W A N G C. H., Phys. Rev. Lett. 26, 1226 (1971). 12. H A R V E Y K. B. and M c Q U A K E R N. R., J. chem. Phys. 55, 4390 (1971). 13. W A N G C. H. and W R I G H T R. B., J. chem. Phys. $6, 2124 (1972). 14. W A N G C. H. and W R I G H T R. B., J. chem. Phys., in press. 15. W A N G C. H. and W R I G H T R. B.,J. chem. Phys. 57, 4401 (1972). 16. S I M O N F., Ann. Phys. 68, 241 (1922).
799
17. SIMON F. and B E R G M A N N R., Z. Phys. Chem. Leipzig 8B, 225 (1930). 18. SORAI M., S U G A H. and SEIKI S., Bull. Chem. Soc. Japan 38, 1125 (1965). 19. G A R L A N D C. W. and Y A R N E L L C. F., J. chem. Phys. 44, 1112 (1966). 20. G A R L A N D C. W. and R E N A R D R.,J. chem. Phys. 44, 1130 (1966). 21. K A M I Y O S H I K., Sci. Res. Inst, Tohoku Univ. 8A, 252 (1956). 22. S C H N E I D E R T., S R I N I V A S A N G. and E N Z C. P., Phys. Rev. 5A, 1528 (1972). 23. SCOTT J. F., Light Scattering in Solids (Edited by M. Balkanski), p. 387 and references therein. Flammarion Sciences, Paris (1971). 24. H A R T W l G C. M., W l E N E R - A V N E A R E. and PORTO S. P. S., Phys. Rev. 5B, 79 (1972). 25. K E T E L A A R , J. A. A., Nature, Lond. 134, 250 (1934). 26. L E V Y H. A. and P E T E R S O N S. W.,J. Am. Chem. Soc. 75, 1536 (1953). 27. H A L F O R D R. S.,J. chem. Phys. 14, 8 (1946). 28. H O R N I G D. E.,J. chem. Phys. 16, 1025 (1948). 29. F A T E L Y W. G., M c D E V I T T N. T. and BENTLEY F. F., Appl. Spectry, 2,5, 155 (1971). 30. L O U D O N R.,Adv. Phys. 13,423 (1964). 31. D A M E N T. C., PORTO S. P. S. and T E L L B., Phys. Rev. 142, 570 (1966). 32. H A R V E Y K. B. and M c Q U A K E R N. R., J. chem. Phys. 55, 4396 (1971). 33. W H A L L E Y E. and BERTIE J. E., J. chem. Phys. 46, 1264 (1967); 46, 1271 (1967). 34. T E H H. C. and B R O C K H O U S E B. N., Phys. Rev. 3B, 2733 (1971). 35. Y A M A D A Y., M O R I M. and N O D A Y., J. phys. Soc. Japan 32, 1565 (1972). 36. S O K O L O F F J. B., Phys. Rev. 5B, 4962 (1972).
STUDY AMMONIUM
OF THE BROMIDE
PHASE TRANSITIONS IN BY RAMAN SPECTROSCOPY*
R. B. WRIGHT and C. H. W A N G t
Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, U.S.A. ( R e c e i v e d 13 July 1972)
Abstract--A complete Raman scattering study of an oriented NH4Br single crystal has been carried out and new results are reported for temperature between 45 to 410~ The study includes measurements and interpretations of the Raman spectral bandwidths, intensities and frequencies of the lattice and internal modes as a function of temperature. Because of the ability of generating a single domain crystal, unambiguous assignments of the Raman active modes are made. From the present data together with those presented previously significant information on the behavior of the ordering and crystal structure associated with the phase transitions in NH4Br is obtained.
1. INTRODUCTION
THERE has been a recent renewal of interest in theoretical and experimental attempts to elucidate the mechanisms responsible for the order-disorder transitions which have been observed to occur in the ammonium halides (NH4Br, NH4CI and NH4I). Although great progress has been made toward qualitative explanations of this phenomenon, detailed quantitative predictions are still lacking. The infrared absorption and Raman scattering techniques have proven to be valuable tools for probing the dynamical nature of these phase transitions and the associated effects of ordering and disordering on the external and internal vibrations in single crystals of these compounds[I-15]. The usefulness of these techniques arises from the fact that the frequencies and lifetimes of individual phonon modes in the Brillouin zone can be studied directly in contrast to only observing the average effects of the many phonon modes as is the case in thermodynamic[16-18], elastic[19, 20] and dielectric [21 ] measurements. Due to the recent theoretical proposals and *Acknowledgement is made to the Research Corporation and the donor of PRF, administered by the American Chemical Society, for the support of this research. tAuthor to whom correspondence should be addressed.
experimental observations of 'soft' or unstable optical phonons associated with structural phase transitions[22, 23] in displacive ferroelectric crystals have raised the question that similar behavior may be responsible for the occurrence of phase transitions in general. It remains to be demonstrated, for example, that the order-disorder phase transition is also related to the softening of a particular phonon mode although recently such an interpretation has been used in NaNO2124]. Since the ammonium halides display a very general type of order-disorder or order-order phase transition, the search for such physical processes accompanying the transitions in these systems is highly justified. We have attempted to carry out a research program which consists of detailed Raman spectral studies of NH4CI and NH4Br oriented single crystals over a wide range of temperature. The investigations involve the measurement of the spectral intensity, bandwidth, frequency and polarization of all the Raman active modes appearing in these compounds. While the work on NH4C1 will be presented elsewhere [14], this paper reports the experimental results and discussions pertaining to NH4Br over the temperature of 45-410~ Because of the ability to produce single domain single crystals in the ordered tetragonal phase this paper presents 787
788
R.B.
WRIGHT
unambiguous symmetry assignments of the Raman modes in NH4Br for the first time. Also presented for the first time are new spectral anomalies associated with the order-disorder phase transition. Ammonium bromide has been found by X-ray [25] and neutron studies [26] to exist in four crystalline phases. Phase I (or-phase) is a disordered face-centered cubic structure similar to NaCl and possesses OhS(Fm3m) space group symmetry with four molecules per unit cell [25, 26]. The ammonium ions are not freely rotating but undergo restricted rotations hindered by a potential barrier corresponding to orientations in which one, two, or three hydrogen atoms make close approaches to the bromide ions[26]. The transformation from Phase I to Phase II (fl-phase) occurs at 41 I~ and is accompanied by a change of the crystal structure to the space-group oha(Pm3m) with one molecule per unit cell. In phases I and II, the ammonium ions are randomly distributed between two possible equivalent orientations giving rise to long-range orientational disorder of the ammonium ions. The site symmetry for both the NH4 + and Br- ions in these phases is Oh. Actually the ammonium ion transforms under the point group Ta, but due to the orientational disorder caused by the random NH4 + configuration in the single crystal in Phase II, the average X-ray determined structure appears to be Oh 1instead of Ta1. As X-ray measurements are time averaged and include scattering from the entire specimen, this average scattering is insensitive to the temporal and spatial disordering occurring in microdomains within the sample. It is this average scattering that is compatible with the Oh 1 symmetry group. Raman and infrared spectroscopy, however, sample across a disordered microdomain and because these processes interact at times faster than the expected ammonium ion disordering flipping rates (10 -13 sec compared to 10 -11 sec), they can instantaneously observe these disordered microdomains. The disordered microdomain
a n d C. H . W A N G
in Phase II has no real center of inversion (as it would if the On I group was strictly valid) and hence, the modes that are affected by the disordering can be both Raman and infrared active (consistent with the Ta1crystal group). Assuming that the crystal structure of the disordered phase is Tct1, the eighteen allowed vibrations are [27-29]: NH4Br = 1Al (x2-4- y2-4- z 2) + l E (2z2-- x 2-- y2, x 2 _ yZ) + I F I ( R ~ , R u , R~) + 4 F 2 ( x , y , z ; xy, xz, yz)
which can be further broken down into the internal vibrations, librations, and the lattice and acoustic phonon modes as follows: 'internal : NH4+ F llbrati~
NH4§
=
FNlattice ~ H4Br Facoustic = NH4Br
1A 1-q- 1E + 2F2
1F1 1 F2
1F2.
One of the F2 modes is an infrared active translational mode (labeled vs, a lattice restrahlen mode) and the other corresponds to the acoustic branch vT. The FI mode is an infrared and Raman inactive librational mode yr. The infrared optic mode is further split due to the long range electrostatic forces giving rise to Fz --~ F2(TO) + F2(LO) where TO and LO refer to transverse optic and longitudinal optic phonons respectively. Perry and Lowndes [9], have reported the position of these modes to be at 147 and 224 cm -1 respectively, for TO and LO at 300~ by use of reflectivity and transmission measurements for Phase II NH4Br. To our knowledge the external modes in Phase I have not as yet been studied in detail by optical methods. Phase III (),-phase) arises from an orderdisorder phase transition (h-transition) which occurs at Tx = 235~ accompanied by a
STUDY
OF
THE
PHASE
TRANSITIONS
slight distortion of one cubic axis. The resulting crystal structure is D4rh(P4/nmm) with two molecules per unit cell in which the ammonium ions are antiparallel ordered in the a-b (x-y) plane and parallel ordered along the c-axis (z-axis). The site symmetries are D2d and C4h for the ammonium ion and bromide ion, respectively, and give rise to the following factor group (D4h) vibrational symmetries [6, 29]. Finternal
2A~g(x2+y 2, z 2) + 1Blg(x2--y 2)
~__
NH4 +
+ 2B2g (xy) + 2Eg(xz, yz) + 1Alu(--) + 2 A e u ( z ) -k-2BlU(-- )
+ 2Eu (x, y) "nbration = 1A2g(Rz) + 1Eg(Rx, Ru; xz, yz) NH4+ + lBau(--) + 1Eu(x,y) FNH4Brlatice= 1Alg(x 2 + y2, z 2) + 1B2g(xy)
+ 2Eg(xz, yz) + IA2u(z) + 1Eu(x,y) ['acoustic_~ 1A2u (Z) + 1Eu (x, y). NH4Br
The nonvanishing components of the polarizability tensor for the lattice vibrations are shown for each Raman active mode [30].
Alg -----
Eg=
a 0
0 0
; B2g =
[! !] 0
=
0 e
.
Since the factor group has an inversion center the mutual exclusion rule is valid and the infrared and Raman spectra will not coinside. Most of the internal vibrations have been assigned, and detailed temperature studies have shown that most of the internal modes are quite insensitive to the h-transition and the associated ordering processes which occur as the temperature is lowered through this
IN AMMONIUM
BROMIDE
789
transformation. An exception to this statement appears to be a v4 triply degenerate bending vibration which exhibits anomalous behavior arising from the ordering processes which accompany the Phase II to III transition[5,6, 10]. Although the ordering effects on a 1435cm -1 infrared active mode (a component of v4) have been studied in detail by Garland and Schumaker[10] in NH4Br, the influence of ordering on the Raman active v4 modes has not been explored in detail. This mode was carefully studied in the present research and will be discussed in Section 3A of this paper. Wang and Fleury have reported the temperature dependent Raman scattering for the v6 (Eg) librational mode (326 cm -1 at 200~ and observed an exponential behavior of the linewidth which they attributed as arising due to a rotational diffusion thermal activation process[6]. We have carried out the analysis of the angular correlation function of v6 as a function of temperature. Because the motion of the NH4 + ions in Phase l l l is not part of the theme of this paper, to avoid confusing the present discussions, we will defer publication of these results elsewhere. The Raman active lattice modes were all studied in great detail in Phases II and III and wilt comprise the major presentation of this publication (Section 3(b)). Phase" IV (y-phase) is an ordered cubic phase with symmetry Td~ (isomorphous with Phase IV of NH4CI) and occurs at 108.5~ K on warming and - 77~ on cooling, via an order-order transition in which the bromide ions occupy Oh sites and the parallel ordered ammonium ions are situated on Td sites. This phase has recently been studied by Garland and Schumaker[10] and by Harvey and McQuacker [ 12] but was not studied in detail by the present authors. 2. E X P E R I M E N T A L
ARRANGEMENT
The Raman spectra were obtained using a constant intensity (140 mW) argon ion laser oscillating at either 4880 or 5145 ,~, a double
790
R.B. WRIGHT and C. H. WANG
monochromator for dispersion, and a photoncounter-descriminator with a cooled FW-130 photomultiplier detection system. The polarization measurements were achieved using an analyzer and polarization scrambler for the scattered light. A half-wave plate to rotate the E-vector of the incident laser light was also used to obtain the desired laser polarization. G o o d optical quality NH4Br crystals ( - 1 cm 3) were grown by slow evaporation from an aqueous solution with urea added as a habit modifier to prevent dendridic growth and to promote the formation of large cubic crystals with (001) faces exposed. The crystals were then cut and polished before use and were free of any visible imperfections. The low temperature Raman scattering studies were achieved using a Joule-Thomson refrigerator operating with compressed N2 and H2 gases. The sample crystal was mounted by cementing a (001) crystal face to a copper block mounted in the cryotip and the temperature was monitored simultaneously at the base and side of the crystal by two standardized chromel-constantan thermocouples. Temperature stability was maintained to _ 0.5 ~ K. It was found that the tetragonal (c- or z-) axis in Phase III appeared to be predominantly formed along the direction perpendicular to the (001) face cemented to the cryotip sample holder. Great care was taken to align the crystal with the spectrometer optics to avoid 'leakage' of one polarization into another, and with slow cooling at the phase transition ( ~ 1~ K per 20 min) the crystal was single domain in Phase III. Single domain crystals in Phase III which were not formed from pre-annealed samples, which can easily bo verified by checking the polarization dependence of the low frequency lattice modes with a half-wave plate and polarization analyzer. The ability of the present investigation to produce single domain crystals in Phase III enables the correct elucidation of the polarization assignments for the various Raman active modes as a distinct c- or z- axis can now be defined and as such the crystallographic axes can be used
to orient the single crystal for the polarization studies. The high temperature studies (297-410 ~K) were conducted using a home-made doublewalled pyrex dewar containing pure mineral oil as the heat bath into which the NH4Br crystal was immersed after glueing a (001) face to a copper block which was then held rigidly in the bath. The mineral oil was found not to interfere with the Raman spectra and was inert towards the NH4Br crystal. Temperature variation was controlled by a proportional temperature controller coupled to a 125W quartz heater. An electric stirrer was used to gently agitate the mineral oil to facilitate a uniform temperature in the bath. The temperature was determined by using two standardized independent chromel-alumel thermocouples mounted on the periphery of the crystal. Temperature stability w a s - - 0 . 5 ~ Because of the occurrence of fracturing of the crystal in going through the Phase II to I transition at 411 ~ K, the Raman study of the crystal in this phase was not undertaken. 3. RESULTS AND DISCUSSION
(a) Internal vibrations Shown in Fig. l(a) are representative Raman spectra of the u2(E) doubly degenerate NH4 + bending mode in Phase II observed in the a (cc) b [31 ] scattering configuration. The u2 vibration corresponds to an atomic motion in which the hydrogen atoms vibrate perpendicular to the N-H-bonds. As shown in Fig. l(a) the line shifts to lower frequency as temperature is raised (1688 cm -1 at 295~ to 1681 cm -1 at 406 ~K) and the line also broadens from 11 to 20 cm -a (measured as full-width at half-intensity) and decreases in intensity. Presented in Fig. l(b) are the frequency and spectral bandwidth of this mode as functions of temperature from 295 to 406 ~K. The frequency has a linear temperature dependence (frequency decreases as temperature increases), while the bandwidth exhibits a non-linear dependence, decreasing as temperature is lowered. Note the extreme increase in spec-
S T U D Y O F T H E PHASE T R A N S I T I O N S IN A M M O N I U M BROMIDE
791
1,688
x(zz)y
I
b. top = Peak -
1,681
(cb.mpl}/
frequency
F' = Spectral band width
r'
(cm-~)
/
20
- '"~ - -
t,6Sol I
1,725
311~ I I 1,700 1,675 Wevenumber, cm -I
I 1,650
1,6751/I 3
I t I I I 0 320 8 3 64 0 0 360 OK
I/
I / 30
40
Fig. l(a). Representative Raman spectra for Phase II of the doubly degenerate bending mode v.2(E) in the a (cc) b or (x (zz) y) scattering configuration. Fig. l(b). Temperature dependence of the peak frequency and spectral bandwidth for the vz(E) internal mode in Phase I1.
tral bandwidth as the Phase II to I transition is approached, indicating significant anharmonic influence at the transition region. Figure 2 shows representative depolarized a ( c a ) b = b ( c b ) a spectra of the internal triply degenerate bending mode v4 as a function of temperature. This mode corresponds to an internal NH4 + motion in which the nitrogen atom vibrates with respect to the four hydrogen atoms as a whole. In Phase II the mode has been assigned as having F2 symmetry. In Fig. 2 the line appears as a single, fairly symmetrical, broad line at 1400cm -1 (see the 406~ spectrum) whose spectral shape narrows and increases in intensity with progressive asymmetry developing on the high frequency side of the spectra. This continues until at 236~ two definite shoulders appear at approximately 1430 and 1420 cm -1. The weak 1430 cm -1 line disappears on further cooling but the 1420 cm -1 (vj) line continues to develop as temperature is lowered. The appearance of two bands for v4 in the Raman spectra in Phase III arises from the fact that
the symmetry of the crystal field about any ammonium ion is no longer strictly tetrahedral. As the phase transition temperature is approached, from above, the long-range ordering of the NH~ + ions begins to occur which slightly displace the Br ions and in turn gives rise to the major NH4 + ion perturbation by modifying the tetrahedral crystal field. The lifting of the degeneracy for the triply degenerate vibrations is thus expected, and is observed to manifest itself in the shoulder which develops on the high frequency side of v4. It is also reasonable to expect that, as in disordered alloy systems, the nearneighbor structures are predominantly those of the most stable crystalline forms although over any extended crystal volume there is still complete randomness. That is, there exists residual short-range order above Tx in the disordered phase. This statement has been quantitatively verified for NH4Br[11, 15] and NH4C1 [ 13] where an anomalous, totally polarized low frequency mode was found to correlate with the evolution of long-range order
792
R. B. W R I G H T and C. H. W A N G
1,4PO0 1,440 1,420 1,400 1,380 1,360 1,340 ~M]venumber, crn-I
Fig. 2. Raman spectra for the internal bending mode, ~4, region at various temperatures in Phases II and III obtained in the a(ca)b (= b(cb)a) scattering configuration.
i.e. the decay of short-range order at the phase transition. A similar interpretation was used by Garland and Schumaker [5, 10] in that the intensity decrease as Tx is approached from above of an anomolous component at 1444 cm -1 in the infrared spectra of NH4CI indicates the presence of disorder in the ordered (cubic) phase of NH4CI and the resulting intensity variation could be correlated with a long-range order parameter. As the crystal temperature is lowered through the Phase II to III transition (235~ the intensity of the 1420 cm -~ component increases and its frequency remains constant within the experimental resolution (§ l cm -1) but the 1400 cm -1 component decreases in inte'nsRy
and tends to shift slightly to higher frequency (1402 cm -1 at 201 ~K.) The scattering arrangement applicable to Phase III (tetragonal) which was used for the spectra shown in Fig. 2 below 235~ is a(ca)b=b(cb)a. This polarization should therefore select only the Eg(xz, yz) depolarized modes in a single domain (i.e. z or c crystal axis defined) NH4Br crystal in Phase III. Since the experimental results clearly indicate that the 1420 cm -1 mode is present and continues to increase in intensity in this polarization, the observed 1420cm -~ line should be assigned as an Eg, Raman active only mode. This is in contradiction to previous [1-7, 10] assignments of this vibration which was assigned to B2g symmetry. The previous investigators either could not discriminate between B2g and Eg symmetry because the experimental conditions would not allow the formation of single domain crystals[I-7,10,12] or the region under discussion was not examined [9]. Harvey and McQuaker[19, 32] have also recently questioned the assignment for the 1420 cm -1 line on the basis of transverse-longitudinal splittings but their results were obtained from pressed polycrystaUine samples which would not enable unambiguous polarization assignments. We therefore assign the 1420cm -~ line as a factor group allowed Eg, not a B2g mode, and assign the 1402 cm -~ line as B2g. This latter assignment is based on prior results [6, 10] which indicate that the 1402 cm -1 line (as measured in a multi-domain crystal which allows both B2gand Eg modes to appear in the depolarized spectra simultaneously) does not disappear (in contrast to that shown in Fig. 2) but is a symmetry allowed mode possessing the polarizability, crab= ctba, i.e. having B2g symmetry. In the a(ca)c configuration, as temperature is lowered, the 1402 cm -1 component, observed to decay (Fig. 2), is therefore an anomalous mode arising from interactions which allow coupling of the v4 phonon (B~gcomponent) to fluctuations of the order parameter in the vicinity of the phase
STUDY OF THE PHASE TRANSITIONS IN AMMONIUM BROMIDE transition. This phenomenon has been recently observed and interpreted in a similar manner for an anomalous low frequency (56 cm -~) mode which appears in the b(aa)c (or a (bb) c) scattering configuration [11], an extraneous mode not predicted by group theory. The observed intensity decay of this line is attributed to the final evolution of longrange order in the tetragonal phase. In Ref. [11] it was found that the temperature reduced integrated Raman intensity (i.e. the integrated intensity divided by In(co) + 1] where n(to) ---[ e x p ( h t o / k T ) - l ] -~) correlated with the residual specific heat which arises from the h-phase transition in NH4Br. Figure 3 shows the results of the intensity versus temperature behavior of the anomalous 1402 cm -~ component, and also presents the earlier results for the 56 cm -~ mode from Ref. [11] for comparison. The 8Cp was obtained from the specific-heat data, measured by Sorai et al. [18], by subtracting the measured specific heat of NH4Br from a smooth curve baseline representing all the non-configurational contributions to the heat capacity (such as lattice vibrations and the librational and internal ammonium ion motions) [l l]. The
793
1402 cm -~ intensity data was not temperature reduced as this factor had negligible effect for this mode over the temperature region of interest. The correspondence reflects a general correlation, although not as satisfactory as the low frequency mode, indicating that the above proposed explanation has definite merit. These results support previous conclusions that the ~'4 internal mode is a very sensitive 'probe' which reflects slight changes in site symmetry and local crystalline processes that occur as the crystal undergoes transformation from one phase to another.
(b) Lattice phonons The lattice phonon region (< 300cm -1) in Phase II ammonium bromide, as well as in Phase II ammonium chloride and iodide, should consist of only an infrared active zone center translational mode and an inactive acoustic mode. But here again the disordering inherent in this phase generates spectral anomalies. Within the crystal dimension corresponding to the wavelength of the exciting light, there is a loss of translational symmetry and consequently, the sampling of phonons with only very small wave vectors, k = 0, is not mandatory upon the Raman scattering process; k----0 wave vector seleclntee/alad i~lmsity of t ~ 1,402r ~ . tion rule is not obeyed. The spectra are there~ Raau cad inleO~ed i~lemi~ " ~Cp ~IQ ~tQin~l from: ~ fore weighted by selection rules, coupling constants and the corresponding density of states [33]. Thus, the usual sharp and symmee; trical lines should not be expected and the resulting Raman spectrum would then mirror d the one-phonon density of states. However, the appearance of a sharp and symmetrically 4 /~/ shaped polarized resonance at 56cm -1 in NH4Br has been observed. This discrepancy 0 L t i I L h I has been interpreted as arising from a largewave-vector phonon at the zone b.oundary, "K possibly a transverse acoustic mode at the Fig. 3. The integrated intensity of the 1402cm-~ band (solid circles) plotted vs the absolute temperaturein the X (or M) point[6, 11, 15]. The existence of vicinity of the h-transition. The temperature-reduced short-range order in Phase II, as discussed integrated intensity of the 56 cm-~ band (open circles, previously, was thought to allow the coupling from Ref. [11]. and the residual specific heat data 8Cp (in units of 1.67eallmoledeg) obtained from Ref. [18] of the zone-boundary phonon to the incident are includedfor comparison. light wave [6]. Since the detailed presentation of l~
" "
56era-' t~nd.
,
794
R.B. WRIGHT and C. H. WANG
o f the scattering at the 56 cm I region has b e e n given elsewhere[13, 15], there is no need to amplify the discussion o f this m o d e further, except to say that it is the progenitor o f the (Section I) totally s y m m e t r i c A lg lattice m o d e expected in Phase I I I NH4Br. Details of the t e m p e r a t u r e dependent depolarized R a m a n scattering o f the low frequency lattice m o d e s a b o v e r o o m t e m p e r a ture are shown in Fig. 4(a), w h e r e a broad, slightly asymmetrical band centered at 134 cm -~ (406~ K) is o b s e r v e d to shift slightly to higher frequency ( 1 3 8 c m -~ at 311~ and ,
,
,
,
,
,
r
,
134
X(ZX) y
increase in intensity as t e m p e r a t u r e is decreased. T h e spectral linewidth decreases in a non-linear m a n n e r as t e m p e r a t u r e is lowered (Fig. 4(b)) to 310~ the high frequency shoulder which develops below 300~ (Fig. 5(a) and 5(b)) p r e v e n t e d the measuring of reliable bandwidths below this temperature. T h e spectral bandwidth of the 134 cm -1 m o d e a s s u m e s a rapid growth as the 411~ phase transition is approached. This large increase in spectral width prior to the 41 I~ transition indicates that large anharmonic interactions are associated with the us m o d e and m a y play a significant role in the ensuing phase transition. Shown in Figs. 5(a) and 5(b) are representa-
13%6oK --
~k~
I
I
I
I
(].
122
I
I
Resolution
*K
c >.. o D
O.
I
h
200
I i i ,~o Wovenum~r,
,do
i
do
c
~-i
Fig. 4(a). Phase II Raman scattering showing the temperature dependence for the depolarized a(ca)b lattice modes over the 50-200 cm -~ region. 120 I00
b,
"g sc 6C
I
250
2(3 i 410
NO ~
Fig. 4(b). Bandwidth for the depolarized mode shown in Fig. 4(a) at temperatures between 310-410~ --
I
210
I
I
IT0 Wovenurnber,
I
I
130 crn-I
I
90
Fig. 5(a). Representative Raman spectra for the a(ca)b scattering arrangement which selects the depolarized lattice translational mode v'5 (Eg) (high frequency mode). Appearance of v~ (B.2g) (low frequency mode below Tx = 235~ is a spectral anomaly discussed in text.
S T U D Y O F T H E P H A S E T R A N S I T I O N S IN A M M O N I U M B R O M I D E
795
ing, the low frequency band shifts gradually to lower frequency, while the high frequency shoulder remains constant in frequency with both lines increasing marginally in intensity. In Phase III, below Tx, the spectra obtained with the two different scattering configurations show pronounced differences. The a(ca)b = b(cb) a configuration (Fig. 5(a)) indicates the high frequency mode increases in intensity, shifts to higher frequency (183 cm -1 at 122~K) and narrows in bandwidth. This 183 cm -1 line has been previously assigned as an Eg (Raman active only), v~, translational optic mode[9], consistent with our findings. The overlap existing between the two modes in Phase II prevented reliable bandwidth and intensity measurements to be obtained. The trends of the spectral features we observed in Phase III were the same as those reported in Ref. [9] and, thus, will not be presented. The intensity of the low frequency (134cm -1) component shown in Fig. 5(a) displays unusual behavior. It decreases rapidly to a finite, though small, constant value (see Fig. 6(a)) as t
250
210
170
Wovenumber,
1:30
90
u
i
c m -I
a(co)b
Fig. 5(b). Raman spectra at various temperatures. The v~ (B2g) translational lattice mode is selected by the b (ab)c arrangement. Leakage of the polarizer allows observation of v'5 (Eg) mode. Relative gain settings for the spectra "rare: gain X 1 for 273 and 236~ K; gain X 89 220~ K, and gain X~for 122~
tive Raman spectra for the v5 lattice modes obtained respectively in the a(ca)b and b(ab)c configurations in Phase II and Phase III below 300~K. It is seen that in Phase II (above 235 ~K) the gross features of the spectra are identical in both configurations. Any apparent differences arose because the specimen under study, as well as the cryoptip refrigerator, had to be rotated 90 ~ and the optics realigned to observe the different polarizations. In Phase II the spectra consist of a broad band, the major peak of which is centered at approximately 140 cm -1 with a high frequency shoulder developing at roughly 175 cm-L On cool-
T X = 2 3 5 *t
.=.it -=-
Z I J --m-
o
9
-ul. - m . . . .
'
I00
140
9 ""m
9 no
-= ......
"
180 T (~
am'- _ _=,=, m..ml
'
220
260
Fig. 6(a). Peak intensity for the anomalous low frequency component shown in Fig. 5(a) as a function of the absolute temperature.
796
R . B . W R I G H T and C. H. W A N G
the temperature of the crystal is lowered below T~. The constant intensity value is a leakage component of the other depolarized scattering configuration (b(ab)c) due to imperfections in optical alignment. The rapid decrease of the intensity is associated with the ordering of the NH4 + ions in the vicinity of the order-disorder phase transition in an analogous manner as that described for the anomalous 1402 and 56 cm -1 components. While the spectral peak frequency of the Eg mode observed at 183 cm -~ was not unusually affected by the phase transition, the ordering processes have a peculiar effect on the peak frequency of the n 2g~t/jtv\ 5/ mode observed in the b(ab)c configuration (Fig. 5(b)). In Fig. 6(b) the intensity and frequency of the B2g translational optic mode are shown as a function of temperature. The data clearly indicate that in the disordered phase the mode frequency increases as temperature is lowered to 260 ~K, below which an anomalous decrease of the frequency occurs. The decrease in 7
i o(balc
6
-i----
4
.---
2
F1 ~Z
I
I
I
frequency continues even through the phase transition (141 cm -1 at 260 ~K and 132 cm -1 at 220 ~ At 220~ the mode frequency assumes again a normal temperature behavior. The anomalous softening behavior of the Bzg mode observed between 260 and 220~ reflects that the precursor of the orderdisorder phase transition appears at 260~K, which is 25~ higher than Tx. This result is consistent with the anomalous intensity versus temperature behavior that was observed in the 56 cm -1 Raman line reported earlier [ 15]. Our normal coordinate analysis has shown that the B2g(v~) mode is connected with the translational vibration along the c-axis of the two sets of equivalent NH4 + ions in the unit cell. From the intensity and frequency data (Fig. 6(b)), the mode reveals that the interaction between the neighboring NH4 + tetrahedra in NH4Br is greater than that in NH4CI [34]. The stronger interaction may also be associated with the precipitation of the unusual ordering behavior of NH4 + ions in the 3,-phase.
i
I
c y ~
I
I
1414
Tk = 235 * i
qun
/
~
Peok Intensity
J
-...>_
al "'~'''-
i z
... .
mm--l~an'-"nL'--,nL---.-~
ul
I00
o.
I
I 140
I
I 180
I
, ~l 220
I 260
I
30 300
T(~ Fig. 6(b). Peak frequency and relative intensity values for the v~ (B2g) translational mode (shown in Fig. 5(b)) over the temperature interval 120-300~ Error bars are shown for the frequency data representing the experimental spectral resolution of • 1.
S T U D Y OF T H E P H A S E T R A N S I T I O N S IN A M M O N I U M BROMIDE
Recently Yamada et al. have published a microscopic theory on the phase transitions in NH4Br [35]. In their paper they adopted the phonon-order parameter coupling picture first proposed by Wang and Fleury for NH4Br[6] and have demonstrated mathematically that the direct interaction between N H , + ions stabilizes parallel ordering whereas the indirect interaction through the Br- ions stabilizes the antiparallel ordering. The indirect interaction is closely associated with the apparent softening of the B2g lattice mode in the 260-222~ region (Fig. 6(b)). This is because the decrease in the B2g mode frequency arises from the weakening of the NH4+-Br - interaction as the crystalline parameter change (c-axis elongates). In addition, since the N H , § ions can also undergo 180~ rotations about the tetragonal c-axis (which may be accomplished via a rotation over an energy barrier or tunneling), the possibility of a translational-librational mode coupling may also exist. However, because the NH4 + libration about the c-axis (belonging to A2g) is neither Raman nor i.r. active, it will require other techniques, such as the neutron scattering, for its elucidation. Nevertheless, it should be pointed out that Hartwig, WiernerAvnear and Porto have recently proposed a similar interpretation for the observed spectral anomalies observed in the temperature dependent Raman scattering in NaNO2 which undergoes a ferroelectric orderdisorder phase transition [24]. In addition to the A~g(v~) mode mentioned above, there is another factor group allowed Raman active mode of Eg symmetry. The Eg optic mode appears in the b ( c a ) c scattering configuration and exhibits a dramatic intensity increase at constant frequency as the temperature is lowered below Tx (the intensity enhancement was observed only as T is lowered from 236 to 210~ remaining constant below 210~K, see Fig. 7). If the cartesian components of the set of the two non-equivalent Br- ions and the set of the two non-equivalent NH4 + ions are designated
J P C S Vol. 34 No. 5 - C
-7
797
l Retolutlon
b(co)c
9----13 3 *K
==
"7,
~161
*K
o
A *K
I00
90
80
70
60
50
Wovenumber, cm-*
Fig. 7. Temperature dependence of the external v~ (Eg) optic mode for the b (ca)c = a (cb)c configuration. The low frequency sideband is due to polarization leakage of the v; (A~g) optic mode.
by {(XlylZ,), (xzY2Z2)} and {(x'ly'lr (x~y'2z~)}, respectively, our mode analysis has shown that the normal coordinates for A ~g is S (A~g) = 89 -- z2) for B~g is s (B
g) =
1
!
-
and for Eg is S ( E g ) = 89 (x, + y , ) -- (x2 + Y2) + (x'l+y'1) -- (x~+y~) ]. One sees that A lg, B2g and Eg are symmetrical with respect to the inversion operation of
798
R . B . W R I G H T and C. H. W A N G
is necessary for them to be Raman active. Since the progenitor of the A ~g and Eg modes is the zone boundary TA mode (which appears in the 56 cm -1 region observed in the Raman spectrum of phase II), the behavior of the Eg mode is also associated with the ordering of the NH4 + ions as the k-transition occurs. Since the presence of the 56 cm -~ scattering in Phase II depends on the existence of order with the periodicity of a reciprocal lattice vector[6, 35, 36], there is probably no peak in the density of states at this position. It should also be pointed out that while the 76 cm -1 mode appears and grows in intensity between 236-210~ the total scattering intensity is considerably less than that of the Alg mode reported in Ref. [15]. This probably has to do with the fact that the original population associated with the zone-edge T A transfers more to the A lg than to the Eg mode due to the presence of the mode coupling mechanism in the vicinity of Tx[15]. We notice from the normal coordinate analysis that the displacements of the Br-ions in Phase III correspond to the polarization of the zone boundary normal coordinate of the lattice vibration in the CsC1 type crystal at (2-q0), the A lg and Eg Raman active phonons thus come from the regions near the point M. It is hoped that in the future neutron scattering will reveal the exact location of the anomalous mode observed at 56 cm -~ in Phase II. D4h a s
4. SUMMARY AND CONCLUSIONS
Raman scattering experiments from NH4Br single crystals in the disordered phase reveal selection rule violations and line-shape distortions that may be explained by the instantaneous disordering in the crystal and by the residual short-range ordering of the ammonium ions. By using oriented, single domain crystals, polarization assignments have been made unambiguously for the first time. Anomalous scattering has been studied in Phase II and III. Those Raman active modes which appear to be most significantly influenced by the o n s e t o f
ordering processes (as the NH4Br crystal approaches, from above, the Phase II to III transition) are the Alg(v'~), Eg, (v~) and B2g (v's) external optic modes and the B2g(v4) internal mode. These modes exhibited peculiar line shape and intensity variations as the k-transition was approached. Raman scattering from extraneous polarization components for the v4 and v~ modes were observed and found to correlate with the residual specific heat of the crystal at the order-disorder phase transition. The v5 mode was also observed to have an extraneous component and displayed a softening in frequency at the htransition. We have associated these peculiar spectral features with (1) the presence of residual short-range order in the disordered (Phase II) phase which permits the Raman inactive vibrations to give rise to Raman scattering via the coupling of the order parameter to these vibrations; (2) the rapid replacement of the short-range order by long-range order at the phase transition causes the intensity variations and the extraneous polarization components; (3) the evolution of tetragonal distortions of the Br- ions along the crystalline c-axis starting at 260~ K induce anharmonic perturbations in the motions of the Br- and NH4 + ions parallel to the tetragonal c-axis. This perturbation produces angular distortions in the NH4 + ion tetrahedron caused by this fluctuating potential field. We have shown that our experimental results on the intensities, frequencies and the linewidths of various Raman lines together with those presented previously[l l, 15] can provide significant information on the phase transitions in NH4Br. By careful examination of the polarization properties of various scattering spectral components in an oriented NH4Br single crystal, we have also found that well-defined Raman selection rules are operative for the K ~ 0 phonons and the internal vibrational modes. Further theoretical work is therefore needed on the spectra of disordered tt
S T U D Y O F T H E P H A S E T R A N S I T I O N S IN A M M O N I U M B R O M I D E
crystals and on the selection rules appropriate for various spectral bands affected by the disorder. REFERENCES
1. K R I S H M A N R. S., Proc. Ind. Acad. Sci. 26A, 432 (1947); 27A, 32 (1948). 2. W A G N E R E. L. and H O R N I G D. F., J. chem. Phys. 18, 296,305 (1950). 3. BOVEY L. F. H., J. Opt. Soc. Am. 41, 836 (1951). 4. C O U T U R E - M A T H I E U L. and M A T H I E U J. P., J. chim. Phys. 49, 266 (1952). 5. G A R L A N D C. W. and S C H U M A K E R N. F., J. Phys. Chem. Solids 28, 799 (1967). 6. W A N G C. H. and F L E U R Y P. A., Proceedings of the Conference on Light Scattering Spectra of Solids (Edited by G. B. Wright), p. 665, Springer, New York (1969). 7. R I M A I L., C O L E T. and PARSONS J. L., ibid., p. 651. 8. D U R I G J. R. and A N T I O N D. J., J. chem. Phys. 51, 3639 (1969). 9. PERRY C. H. and L O W N D E S R., J. chem. Phys. 51, 3648 (1969). 10. S C H U M A K E R N. F. and G A R L A N D C. W., J. chem. Phys. 53, 392 (1970). 11. W A N G C. H., Phys. Rev. Lett. 26, 1226 (1971). 12. H A R V E Y K. B. and M c Q U A K E R N. R., J. chem. Phys. 55, 4390 (1971). 13. W A N G C. H. and W R I G H T R. B., J. chem. Phys. $6, 2124 (1972). 14. W A N G C. H. and W R I G H T R. B., J. chem. Phys., in press. 15. W A N G C. H. and W R I G H T R. B.,J. chem. Phys. 57, 4401 (1972). 16. S I M O N F., Ann. Phys. 68, 241 (1922).
799
17. SIMON F. and B E R G M A N N R., Z. Phys. Chem. Leipzig 8B, 225 (1930). 18. SORAI M., S U G A H. and SEIKI S., Bull. Chem. Soc. Japan 38, 1125 (1965). 19. G A R L A N D C. W. and Y A R N E L L C. F., J. chem. Phys. 44, 1112 (1966). 20. G A R L A N D C. W. and R E N A R D R.,J. chem. Phys. 44, 1130 (1966). 21. K A M I Y O S H I K., Sci. Res. Inst, Tohoku Univ. 8A, 252 (1956). 22. S C H N E I D E R T., S R I N I V A S A N G. and E N Z C. P., Phys. Rev. 5A, 1528 (1972). 23. SCOTT J. F., Light Scattering in Solids (Edited by M. Balkanski), p. 387 and references therein. Flammarion Sciences, Paris (1971). 24. H A R T W l G C. M., W l E N E R - A V N E A R E. and PORTO S. P. S., Phys. Rev. 5B, 79 (1972). 25. K E T E L A A R , J. A. A., Nature, Lond. 134, 250 (1934). 26. L E V Y H. A. and P E T E R S O N S. W.,J. Am. Chem. Soc. 75, 1536 (1953). 27. H A L F O R D R. S.,J. chem. Phys. 14, 8 (1946). 28. H O R N I G D. E.,J. chem. Phys. 16, 1025 (1948). 29. F A T E L Y W. G., M c D E V I T T N. T. and BENTLEY F. F., Appl. Spectry, 2,5, 155 (1971). 30. L O U D O N R.,Adv. Phys. 13,423 (1964). 31. D A M E N T. C., PORTO S. P. S. and T E L L B., Phys. Rev. 142, 570 (1966). 32. H A R V E Y K. B. and M c Q U A K E R N. R., J. chem. Phys. 55, 4396 (1971). 33. W H A L L E Y E. and BERTIE J. E., J. chem. Phys. 46, 1264 (1967); 46, 1271 (1967). 34. T E H H. C. and B R O C K H O U S E B. N., Phys. Rev. 3B, 2733 (1971). 35. Y A M A D A Y., M O R I M. and N O D A Y., J. phys. Soc. Japan 32, 1565 (1972). 36. S O K O L O F F J. B., Phys. Rev. 5B, 4962 (1972).