OH- DIPOLE CENTERS IN ALKALI HALIDES

(1)

HAL Id: jpa-00213307

https://hal.archives-ouvertes.fr/jpa-00213307

Submitted on 1 Jan 1967

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

OH- DIPOLE CENTERS IN ALKALI HALIDES

F. Lüty

To cite this version:

(2)

JOURNAL DE PHYSIQUE Colloque C 4, Supplkment au no 8-9, tome 28, Aozit-Septembre 1967, pages C 4-120

OH- DIPOLE CENTERS IN ALKALI HALIDES

(*)

by F. LUTY

University of Utah, Department of Physics

RBsume. - Les halogknures alcalins qui contiennent des ions OH - de substitution sont devenus les exemples principaux et les mieux ktudiCs, et les systkmes modkles de substances cristallines paraklectriques et paraklastiques.

Le dopage contr8lk entre 10 - 6 et 10 -2 permet l'ktude des systkmes de dipoles sans interaction ou rkagissants, par diffkrentes techniques expkrimentales. On examine ici les techniques optiques et calorimktriques.

Une connaissance approfondie de l'absorption optique est B la base de l'ktude Clectro-optique et klasto-optique. On passe ici en revue les absorptions Clectronique ( - 6 eV) et vibrationnelle (- 0,45 eV) du centre OH

-,

ainsi que son absorption vibrationnelle directe ( - 0,04 eV), rCcem- ment dkcouverte. Les effets klectro- et Blasto-optique (changements d'absorption optique sous l'action de champs klectriques ou Clastiques) montrent que, dans la plupart des cas ktudiks, les dipoles sont maintenus dans les directions < 100 > par le potentiel cristallin octakdrique. Ces expkriences donnent les valeurs des moments dipolaires klectriques et Clastiques efficaces, et l'ani- sotropie optique des centres.

A partir de l'effet klectrocalorique, et en faisant varier les temps de croissance et de dkcroissance du champ klectrique appliquk, on mesure directement le temps de relaxation dipole-rkseau. Des expkriences rkcentes sur six halogknures alcalins diffkrents ont montrk que ce temps de relaxation dkpend considkrablement du rkseau hbte, variant entre 2 X 10 -5 et 2 X 10 -9 s a 2 OK. On peut dCduire, des relations de ce temps avec le champ et la tempkrature, des conclusions relatives au mkcanisme de couplage entre dipole et phonons.

On a Ctudik quantitativement, aussi bien par les techniques Clectro- et klasto-optique qu'klectrocalorique, les interactions dans les systkmes constituCs par des dipoles OH - concentrks ou diluks et des dkfauts ponctuels chargks (paires de Frenkel).

Abstract. - Alkali halides containing substitutional OH- ions have become prominent and well investigated examples and model systems for paraelectric and paraelastic crystal substances. Controlled doping between 10 - 6 and 10 - 2 allows the study of both interaction-free and interacting

dipole systems with various experimental techniques. Here the optical and caloric techniques are reviewed.

The basis for electro-optical and elasto-optical work is a detailed knowledge of the optical absorption itself. This is reviewed first, covering the electronic ( - 6eV), vibrational ( - 0.45 eV) and the recently discovered direct librational absorption ( - 0.04 eV) of the O H - center. The electro- and elasto-optical effect (change of optical absorption under applied electric and elastic fields) reveals for most investigated cases that the dipoles are confined to the c 100 > directions by the octahedral crystalline potential. Values for the effective electric and elastic dipole moment and the optical anisotropy of the center are obtained from these experiments.

Using the electro-caloric effect and varying the rise- and decaytime of the applied electric field, the dipole-lattice relaxation time can be directly measured. Recent experiments on six different alkali halides show a considerable dependence of this relaxation time on the host lattice, varying between 2 X 10 - 5 and 2 X 10 - 9 sec at 2 OK. From the field- and temperature-dependence of the dipole-lattice relaxation time, conclusion can be drawn about the coupling mechanism between dipole and phonons.

Interaction effects among concentrated OH - dipole systems and among systems of diluted OH - dipoles and charged point defects (Frenkel pairs) were quantitatively studied, using the electro- and elasto-optical as well as the electro-caloric technique.

The earlier interest in OH- impurities developed effects o n many physical properties of these crystals, from investigations which showed that the presence of like optical absorption [l, 21, thermal conductivity ,[3], these defects in alkali halide crystals has pronounced photochemical reactions [4, 5, 61 and ionic conducti-

vity [7]. These findings became especially important in (*) Supported in part by AFOSR Grant no 1141-66. view of the fact that all crystals grown from the

(3)

OH- DLPOLE CENI'ERS IX ALKALI HALIDES C 4 - 121 mclt at the open air (and until recently all crystals wcrc

grown in this way !) containOH- impurities in concentrations of several t 0" cm- 3 . Higher concentrations of OH- ions in the crystal (up to about 10'' can be obtained by doping the melt with the appropriate alkali hydroxide (e, g., KOH in KCl). General bin- ding considerations and the observation, that OH- ions can be converted by light [1,4] and X-irradiation [6] into reaction products containing aniun v:tcancies ( U and Fcenters), made a substitutiona1 plam for the

OH

-

ion very probable. Combined measurements of the Iattice pararnelcr and ~nacroscopic density on KC1 : KOH crystals [R] have finally confirmed this substitutional model.

More recciltly a slroilg intcrcst in the OH- dcfect has developed from the fact that thc OH- Ion has a permanent clcctric and - due to its non- spherical shape

-

an u elastic )) dipolc moment.

Thus the application of electric and elastic fields can give rise to an alignment of the dipolc centers, i, e., paraelectric arid paraelastic behavior of the crystal. These phenomena hwe becn demonstrated and investigated using different types of experimental

techniques : 1) Electro-optical measurements [9, 10, 1 l]. 2) Elasto-optical measurements [12, 131. 3 ) Electro-caloric measurements 114, 151. 4) Dielectric measurements [16, 171. 5) Paraelectric resonance 118, 191. Abrorpfmi or 6'K I cm-'

-

₅ n c S -0 F B -X

.

-. 0 -- 6.6 6 1 5 8 5 4 ,Y Photon cnergy

These investigations (and othcrs at different dipole centers) have opened a field, which tends to parallcl the fieId of pararnagnctisrn in its diflerent phenomena and techniques by electric and elastic counterparts. It is planned here to review briefly the first three types of investigations and to develop from them the main physical aspects and properties of the O H -

dipole center. For many details howevcr it will be referred to the literature and to publications which are in progress.

Before discussing the first two techniques it is appropriate to review briefly thc optical properties of the OH- centers. Similar to a free diatomic molccule the

OII- ion in alkali halides gives rise to three types of optical excitations, a n electronic, vibrational and r( r o t ~ t i o ~ l a l absorption.

cij The electronic absorptionconsists of a broad and structureless band in the UV-region (5-7 eV). For the systems so far investigated the observed band maxima follow an Tvey relation of the form

A,,,,

= 752. do." [l 11.

Figure 1, shows two examples of this band in KBr and

RbCI. Substitution o f OD for OH produces only minor changes in the absorption shapc [20]. Compa- rison of the absorption with chemical OH analysis yields vaIues for the oscillator strength varying between

0.1 and 0.2 for 6 difTcrent alkali halides. lrradiatiot~ into the electronic band produces a characteristic emission [2 1, 221 and photochemical decomposition of the center 14, 51.

(4)

Ktrr 0 7-

/o'*

KOH Infrared Absorptron

1

FIG. 2 . - Infrared absorpti011 o f O H - ions in KBr. (A) librational (B) strctching and (C) stretching .-l- librational excitations as described in text.

b) The vibrational absorption consists mainly of a band i n the near infrared region (about 2.7 g) produ- ced by the optical excitation of a stretching vibration of the OH- molecule [2, 71. The absorption strength of this band varies considerably for different host lattices and is in general very small, as expressed by an effec- tive oscillating chargc c* much smaller than P (c, g., in

KC1

e* 0.1 e). Thc band sharpens considerably to low temperatures 1231. A prominent side band appears on the high energy side of the strctching band, sepa- rated by about 300 cm-'. Figure 2 shows on the left side the stretching absorption ( B ) and this side band ( C ) in KBr. A substitution of On for OH shifts the position of rile stretching band to 11jghcr energies

(e.

g., in KBr from 3 620 to 2 670 cm-'). The factor between these frequencies 1.35 is close to J2 indica- ting that mainly the H and D ion is vibrating. More- over the splitting to the side band (B - C in figure. 2) is reduced by about the same factor (1.3) when going

from OH to OD [23]. 'Ihis has led to the interpretation of this side band as an excitation of the stretching mode plus a (( librational (torsional) excitation

of the hydroxyl rnolecule [23].

c) The direct excitation of this librationnl motion

OF

the OH- has been recently found in the far infrared region at an energy corresponding to the splitting energies between the bands B and C 1241. This new band

(c< A 1) in figure 2) which is located close to the edge of

the fundamental vibrational absorption, increases and sharpens strongly towards Low temperatures, where it becomes much more prominent than the weak near infrared absorption. Substitution of On for OH shifts the band from 313 to 236 cm- %in KBr which

corresponds to the shift in the splitting energy between bands C and B. This band A can be visualized as being developed o u t of the rotational excitation of the free OH- moleculc : When bringing the OH- ion into the crystal, the Free rotor is placed into a octahedral crystalline potential which hindcrs its rotational motion and modifies the free rotor lcvcls considerably. The calculation of this problem ((( Devonshire-model n

[25, 261) shows that this poteiitial can have energy minima eithcr along the < 100 > or c 11 1 > dircc- lions, so that the dipoIes at low temperatures should be confined to these crystalliue directions. Absorption in the band A would then correspond to the excitation

or a torsional ((c librational motion of the dipole around this equilibria.

(5)

O H - DIPOLE CENI'ERS 1N ALKALI HALIDES

Chpole molecules

l%. 3. - Principle and gcometry of clwtro-optical expcrimcnts. Dipoles are assuined to be confined by crystalline potential to thc six < 100 > dircctions. Anisotropic absorption of molecules indicated by direrent transition voftots in moleculc axis and in plane perpendicular to it. Invcl diagram, distribution of ccnlers and optical transition vectors are shown

a) without ficld, b) field in c 100 > and c) ficld in < I1 1 >

.

to the < 100 > oricntations, so that wc havc

-

in classical description -a six-fold dcgeneratc orientntio- nal grou~ld state. Application of an electric ficld liks this degeneracy, the split levels become occupied by a Boltzmann distribution, i. e., the dipoles bccome aligned. For cc seeing )) this alignment optically, one can in principle makc use of eitl~cr of the discussed optical transitions : Fach of thcsc cxcitations should have different oscillator strength whcn being excitcd parallel or perpendicular to the dipolc axis (U or n

polarization). Redistribution of thc dipolc oricntations by the applied field thus will cause changcs in the optical absorption when measured parallel and perpendicular to the field. Figure [ shows a mcasurcment of the absorption change (right ordinate) introduced

into the U V absorption by an electric ficld applied in a < 100 > direction parallel to thc k vector of the light. The field causes an incrcasc in absorption which has the same spcctral distribution as the band itself. Figure 4 shows this electric field induced extra- absorption in dependence of applied f cld and inversc temperature for two crystal systems. The electro- optical effect can quantitatively be understood by

the assumption that the

OH-

dipolcs absorb perpendicular to the dipole axis much strongcr than parallel to it. (For the frcc

OH-

ion one expccls a full n-pola- ri7ation for thc lowest energy clcctronic transition.)

Figure 3 explains schenlatically how the longitudinal

<

100 > field redistributes thc ccnter axes and transi-

tion vectors, thus producing the absorption increase.

It further illustrates (part C) that the application of a ficld i n the

<

11 1 > direction should produce no absorption change. This was indeed observed experimentally [9] and established the model of the

<

100 > oricntcd dipoles. Thc measured electro- optical effect and its field and tcrnpcrature dependence (points in figurc 4) can quantitatively be explained by the described mode1 (the lines i n figurc 4 arc calculated) after fitting two open parameters : The effective electric dipole moment p and the ratio of absorption strengths parallel and perpendicular to the dipolc

axis

fi,lfi.

For

OH-'

in several alkali halides (KCI,

KBr, KaCI, RbCl) one finds

/I,,/fl

closc to 0.3, while the dipofe moment p (whcn using a Lorentz correc-

(6)

I I I I I I r I

I NaCI: OH' (44 . rolecm-JI I

FIG. 4.

-

Electro-optical cffcct of OH- centers in NaCl and RbCI. Absorption under field K ( E ) coniparcd to absorption without ficld K(0) in dependence on ficld and inverse tenipera- lure.

for OH- in CsBr ; this prevents the determination of the equilibriu~n orientations of the dipoles in the cesium halide structure.

Besides electric effects from the dipole moment the substituted OH- ion introduces elastic effects - due to its misfitting size

-

into the lattice. Density and lattice parameter measurements [8] reveal a general inward displacement of the lattice around the OH- in KCI. Due to the non-spherical shape of the defect this diplacement should be different in the direction of its axis and perpendicular to it. Following a concept first introduccd by Kroner [27]

the defect can be described a s . an (( elastic dipole D,

having in our case tetragonal < 100 > oriented symmetry and being charactcrizcd by the tensor components of the strain which the defect introduces into the lattice parallel ( I , , ) and perpendicular

(A,)

to its axis. As described in detail in [28], an elastic dipole interacts with an applied stress field similar to the way in which an electric dipole interacts with an electric ficld. These considerations have led to elasto-optical experiments, the principle and gcometry of which are schematically dcscribed in

Porarlastic Alignment of OH- Dipoles

FIG. 5.

-

Principle and gcomctry of c!asto-optical effect (similar to figure 3). For strcss equal zero, stress in < 100 > , < 110 > and < 111 > direction it is sllown thc experimental geometry, thc levcl splitting, and the expcctcd change

(7)

OH - DIPOLE CENTBRS IN ALKALI HALIDES C 4

-

125 figure 5. Tlie six-fold degenerate orientational ground

state of the < 100 > oriented dipoles is split by

the application of external strcss fields of different directions in the way explained i n figure 5. Again a

redistribution of centcr orientations

-

and therefore of optical transtion vectors

-

takesplace. When measuring with light polarized parallel

(0)

and perpendicular (-I.)

to the stress, one cxpccts characteristic absorption changes which are given for stress in < 100

>,

110 >

and 11 1 > direction in the lower parl of figure 5. The experimental resuIts [12, 131 are in full agreement with these expecrations, confirming f u r h e r thc model of the < 100 > oricntcd dipole, derived from the electro-optical experiments. For our tetragonaI < 100 >

oriented elastic dipole (characterized by the components

1,, and;-,) the initial alignment under a < 100 > applied

uniaxial pressure P should be proportional to

( R ,

-

A2)

P

, .-..

k T

Figure 6 shows a plot of the ~ncasured stress dichroism i n dependence of the inverse temperature ; from the slope of the straight lilies the elastic dipole moment or shape factor (1,

-

2,) of the defect can be obtained.

As seen in figure 6 different values result for OH- in different host lattices. Together with results from

F ~ G . 6.

-

Elas~o-uprical t f f c c t in dependence on the invcrsc tenlperaturc for KBr, RbCl and KC1 (strcss applied in < 100 > direction ; for KCI a mcasurcment of < 110 > strcss is addcd).

lattice parameter measurements (which yicld the sum of the strain components i,

+

2 R, around the defect) the tensor components of the elastic dipole 1, and 1, can be determined separately.

Preliminary experiments on the stress-induced

dichroism of the near infrared absorption (band

B and C in figure 2) yielded results which are not

consistent with the simple picture developed here [13]. Detailed measurements on the optical absorption as well as the field and strcss dichroism, applying much higher resolution than used in [23], are under way to clarify the situation.

M u c h

progress in un- derstanding is hoped to be obtained from stress and electric field effects on the far infrared band of 300 cm-' corresponding to the direct librational excitation of the dipole in the crystalline potentiat. Experiments on this question are under way too.

Similar to the well-known magnetic counterpart, the ordering and disordering of the clectric dipole system by a field causcs changes in entropy and thus

-

i11 an adiabaticexperiment - changes in the temperature of the crystal. In the region where the dipole polariza-

d

tion P is given by

P

= N,u

.

, one expects tempera-

3 kT

ture changes AT = $.

"'

6 c , k T

-

(viiliid for AT 6 T ) under application and re~noval of an electric field E. Figure 7 shows quantitative results of this eIectroca-

Iorl

to-'

-

10' V / C ~ to5 I 3 L 6

ongelegles ekklr. Feld krnperotur

FP Id

on aus

o T =4,0 OK

a T = Z ? ' K

FIG. 7.

-

Para-electric heating and cooling, in dcpcndence on field and temperature.

Feld

a n uus

0 E = 2.3,~' V/cm

(8)

ReverslbIe

md

l t r e w ~ s ~ ~ M e Paraelectrk Heating and Cooling 4 & E t Cm * . C, t r, t

mm-

*

ubsorprbn A d , . --

-

1 --

E-t 4 - - - - , k m m e m A m dSWm I .-

FIG. 8.

-

Schematic representation of phonon. emission and absorption when rise time of tield TE

is long against dipole lattice relaxation timc (above) and for Z E < .E.,,, (below).

loric effect, demonstrating the reversibility of heating and cooling and its

E 2

and dependence (c, is proportional to T 3 ) . This reversibie temperature change is obtained only when the electric field is changed slowly compared to the dipole-lattice relaxation time. This is explained in figure 8 schematically for a model dipole with two possible orientations parallel and antiparalld to the field : When changing the ficld slowly compared to the dipole-lattice relaxation time (upper half, z, > z,,,) rt thermal equilibrium

between lattice and dipole system exists a t every moment. Heating and cooling can readily be understood as the reversible energy emission or absorption respectively which is necessary t o establish the Bolt- zmann distribution at each moment. For r , < z,,, however (lower part figure 8), all the relaxation takes place after the full splitting of the energy levels, thus dissipating a higher (double) amount of energy ; the cooling effect on the other hand shouId disappear

in this .case because the aligned dipoles relax only

after the splitting has been removed and thus do

not necd the help of energy. Exactly this,behavior is found cxpcrimentally (Fig. 9) when the rise and decay-time .c, of the field is varied from large to small

values : Starting with the reversible behavior, the heating effect rises to about the double value while the cooling eRect disappears. The lines in figure 9 are

calculated under the assumption of a single field-

independent relaxation time T,,, which i s chosen for a best fit to the experi~nental measurements. As seen in figure 9 the dipole lattice relaxation time varies considerably (between about 2 X 10- and 2 X 10-g

sec at 2 OK) for different host lattices. Preliminary measurements on the temperature depcndence of T,,, suggest direct phonon processes as the predominant relaxation mechanism at low temperatures (in agreement with results from dielectric loss measurements [17Jj. The fieid independence of T,,, indicates a cou- pling of the dipole to the stress field of the phonons

(9)

OH- DIPOLE CENTERS IN ALKALI HALWES C 4

-

127

Rise

-

and Gecaytime of Field

FIG. 9.

-

Variation of the relative electro-caloric effect on the rise and decay time of the electric field for OH- in different alkali halides.

All described experiments and their interpretation were so far restricted to

a) high electric and elastic fields, for which the classical description of < 100 > localized dipoles can be applied ;

b) diluted dipole systems (N(0H-)

< 1019 cm-3),

for which interaction effects can be neglected.

The eigenstates of the OH- in small or zero fields are no longer the classical six

< 100 >

oriented states, but (due to the overlap between them) proper linear combinations of these local states, split by a tunneling energy A . For the simplest model of a dipole in an octahedral crystalline field [25,26, 301 a ground state singlet A,,, an intermediate triplet T,, and an upper doublet state Eg are expected, with splitting energies 2 A and A between them. Contrary to our high field experiments, measurements at low or zero fields can yield information about the tunneling energy A [16, 17, 18, 191. From paraelectric resonance measurements a value of A = 2.6 X 10- eV is

derived. In the Devonshire model both tunneling energy and libration energy are dependent on one parameter only, the strength (barrier height) of the

octahedral potential. The attempt to fit thz twa mzasu red energy values A and v,,,, into such a model fails completely. Two modifications have been proposed to explain this failure : Lawless [31] shows that assu- ming an off-center position for the dipole in the vacancy and a rotation around a point which is not the center of gravity, the changed moment of inertia I and rota- tional constant A2/I allows the fitting of both energy values into a Devonshire model. Shore [32] on the other hand notes that the measured large value of the elastic dipole moment (= tetragonal lattice distortion) suggests that the instantaneous crystal field as seen by the dipole is lowered from octahedral to tetragonal symmetry. A calculation of the tunneling matrix elements between states of different orientations now has to include terms representing the change in lattice distortion. This (( rotational polaron model ))

allows a consistent explanation for the measured values for tunneling and librational energy and elastic dipole moment.

When the OH- concentration in the crystal exceeds 1019 cm-3 interaction effects among the dipoles become important, as first observed in dielectric measurements [16]. These effects can be studied too with the experimental techniques described here :

KC1 9 KOH

T = 4,7'K

ongelegles elrktr. Feld

FIG. 10. - Dependence of the

(10)

C 4 - 128 F. LUTY

figure 10 shows the reduction of the relative electro- i —i \

optical effect with rising O H- concentration, figure 11 4r KCl * KOH

. Electrocaloric Effect I / Concentration Dependence KBr + K2„H / ° 3 nnr.wK P-20 IOe-yjr / \ cm / 2 F. 2fll0' V/cm <(2)T*2fi°K / \j3) T. 2.0'K no , -I , ^ 10° , , D-D sw' / ^-T F. ifl-10 V/cm (4) T« 2jO'K K(P) +-+»•«« y ^ ^ 1 ' i-oUW-' / ^ Ell P [1001 ^ i ^ V . ^ /^ ° 2 . / ^ ^ \

j^^^~~

s

/ /i v

S^ 3 / / _ _ \ L

w

|^_

1 L 1— , / y*

\ \ p

;

+~ - - . ° 1 5 / / A \l) a

4 I 1

*--^

i 1

3 /

/Jt

0 0.2 O.i 0.6 I'KJ-' 2 / ** [Temperature]'"'

FIG. 11. — Elasto-optical effect (measured parallel and 1(f 2 3 5 J0" 2 3 5 )0*> -3

perpendicular to applied < 100 > stress) in dependence on OH'-Dipole Concentration inverse temperature for different O H- concentrations. The

fitted curves are calculated with a Curie Weiss law. FIG. 12. — Electro-caloric effect in dependence on the OH _ concentration.

illustrates the same behavior for the elasto-optical

effect. The electro-caloric effect, which varies for energy AE (similar to the case of antiferromagnetism). interaction-free systems linearly with the dipole concen- The curves in figure 11 are calculated with this law tration (Fig. 12) drops drastically above O H- concen- and titled to the measurements by a choice of Aj&for

trations of about 101 9cm~3. All these measurement each O H- concentration. In this way a dependence

demonstrate that with increasing concentration of AE on the dipole concentration is obtained which interaction effects among the dipoles prevent more is roughly a linear one.

and more the alignment ofthedipoles in external fields. Finally it shall be mentioned here that a dilute The interesting question of the nature of this interaction interaction free dipole system can be used to study the and possible cooperative phenomena of the random interaction with other defects in the crystal. Success-dilute dipole system at low temperatures has stimulated ful experiments have been done measuring the interac-several theoretical treatments [33, 34, 35, 36]. All of tion of charged Frenkel-pairs (introduced into the these papers consider only electric dipole interaction crystal by x-irradiation at 4 °K) with the paraelectric and arrive at partly controversial conclusions about dipole system [10]. These types of experiments open possible parallel and antiparallel ordering effects. the way to many investigations in which the dipoles Experiments and model calculation by Hartel [13] can be used as a very sensitive probe to measure however show that the elastic interaction effects internal electric and elastic fields from other defects between the dipoles are at least comparable if not in the crystal.

(11)

Of1

-

DIPOLE CENTERS IN ALKALI HALDES C 4 - 129

References

[ I ] ROLFE ( J . ) , Phys. Rev. Letrer.~, 1958, 1, 56.

121 ETZEL (H. W.) and PATTERSON (D. A.), Phys. Rev.,

1958, 112, 1112.

131 K L E ~ (M. V.), Phys. Rev., 1961, 122, 1393.

[4] RERCKHOFF (F.), 2. Physik, 1960, 158, 595.

[S] CAPE (J. A.), Phys. Rev., 1961, 122, 18.

[6] LGTY (F.), J. Phys. Chern. Solids, 1962, 23, 677.

171 F R ~ T Z (B.), LUTY (F.) and ANGER (J.), 2. Physik, 1963, 174,

2f?0.

[8] PAUS (H.) and EUTY (F.), Phys. Stat. Sol., 1965, 12, 341.

[9] KUHN (U.) and LUTY (F.), Solid Stare Comm., 1964,

2 , 281.

[l01 KAPPHAN (S.) and LUTY (F.), BuII. Am. Phys. Soc.,

1967, I t , 82, and to be published.

[ I I ] WEINMANN (K. F.) and Liim (F.), 31tll. Am. Phys. Soc., 1967, 12, 82, and to bc published.

[l21 HARTEL (H.) and L ~

(F.),

Y Phys. Slat. Sol., 1965,

12, 347.

[l 31 HARTEL (H.), Thesis Stuttgart 1966.

1141 KUHN (U.) and L ~ ~ T Y ( F . ) , Solid Stale Comm., 1965,

4, 3 1 .

[ l 5 1 SHEPHERD (I.) and FEHER (G.), Phys. Rev. LRilers, 1965, 15, 194.

[l61 KANZIG (W.), HART

(H.

R.) and ROB ER^ (S.), Phys. Rev. Letters, 1964, 13, 543.

[l71 BOSSHARD, DREYFUS (R. W.) and Kiinzrct (W.), Physs, Konder~sier re Ma~erie, 1965, 4, 254.

1181 BRON (W. E.) and DREYFUS (R. W.), Phys. Rev. Lerters, 1966, 16, 165.

[l91 FEHER (G.), SIIEPHERD (I.) and SHORE

(H.

B.), Phys. Rev. Letters, 1966. 16, 500.

[20] FISCHER ( F . ) , Solid Stare Conim., 1964, 2 , 51.

121 l KOSTL~N (H.), Solid Sraie Comm., 1965, 4, 81.

[22) PATTERSON (D. A.) and KABLER (M. N.), Solid S~ate

Comm., 1965, 4, 75.

[23] CHAU (C. K.), KLEIN (M. V.) and WEDDING (B.),

Phys. Rev. Letters, 1966, 17, 521.

[24] HARR~SON (D.) and L u w (F.), Bull. Am. Phys. Soc.,

1967, 12, 82.

[25] DEVONSHIRE (A.

F.),

Proc. Roy. SOC., 1936, A 153,

601.

[26] SAUER [P.), SCHIRMER (0.) and SCHNEIDER (J.), Plzys. Sfat. Sol., 1966, 16, 79.

[27] KRONER (E. Z.), Meral/kuttde, 1960, 51, 457.

[28] NOWICK (A. S.) and HELLER

(W.

R.), Adv. Physics,

1963, 12, 251.

[29] VREDEVOE (L. A.), Phys. Rev. (to be published). [30] SHORE (H. S.), Phys. Rev., 1966, 151, 570.

[31] LAWLESS (W. N.) (to be published).

I321 SHORE (H. B.), Phys. Rev. Leliers, 1966, 17, 1142.

1331 ZERNICK (W.), Phys. Rev., 1965, 139, A 1010.

[34] BROUT ( R . ) , Phys. Rev. Letters, 1965, 14, 175.

[35] KLEM (M.), Phys. Rev., 1966, 141, 489.

1361 LAWLESS (W. N.), Phys. Korrdcns. Maferie, 1 966, 5,