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00 o o 0^ «o O" < " r i l ' I i iyi.iiitiMiJ iiiiiiiitt leiiii iiii•o UI » -O) o BIBLIOTHEEK TU Delft P 1966 3184 660180
CLASSICAL CALCULATIONS CONCERNING
THE DOUBLE REFRACTION, OPTICAL
ROTATION AND ABSOLUTE CONFIGURATION
of Te, Se, Cinnabar (HgS), a- and /?-Quartz, ^-Cristobalite, N a N O o , N a C l O , and N a B r O j
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN D O C T O R IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE H O G E S C H O O L DELET, O P G E Z A G VAN DE RECTOR MAGNIFICUS DR. IR. C ]. D. M. VERHAGEN. HOOGLERAAR IN DE AFDELING DER TECHNISCHE N A T U U R K U N D E , VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP DINSDAG 7 lULI 1970 TE 14.00 U U R D O O R
RENÉ REIJNHART
natuurkundig ingenieur geboren te VelsenBil^LIOTHEEK
DER
TECHNISCHE HOGESCHOOL
DELFT
Dit proefschrift is goedgekeurd door de promotor PROF. DR. }. A, PRINS.
I
1
STELLINGEN
1. De verschuivingen in de reflectiespectra van Se die Tuomi vindt als functie van aangelegde mechanische spanningen, kunnen in verband gebracht worden met veranderingen in inwendige elec-trische velden. De gekozen richting van de aangelegde spanning loodrecht op de c-as is onverstandig; de afgeleide coëfficiënt voor hydrostatische druk is onjuist.
T. O. Tuomi, (1970) Phys. Stat. Sol. 38, 623.
2. De informatie die Peterson vindt uit nucleaire quadrupool-kop-pelingsconstanten over de ladingstoestand in kristallen zoals NalnOa, NalnSg en NalnSeg zou vergeleken dienen te worden met effectieve atomaire polariseerbaarheden die uit een berekening van de dubbele breking in deze kristallen volgen.
G.E.Peterson, P.M. Bridenbauch, (1969) J. Chem. Phys. 51, 2610. 3. De grote droge plekken die onder de bladeren van kamerplanten
op een overigens metwaterdamp beslagen ruit te zien zijn, geven een duidelijk inzicht in de fysische processen die in het spel zijn bij het beslaan van ramen.
4. Het woord "motivatie" dat de laatste tijd nogal eens door fysici in spe en anderen gebruikt wordt die geen duidelijke motivering zien van, noch aandrang voelen tot de bestuderir^ van bepaalde onderwerpen uit het huidige studieprogramma aan de T. H. te Delft, komt terecht niet in van Dale's groot nederlands woorden-boek voor.
5. Door meting van de draaiing van het polarisatievlak van een lineair gepolariseerde golf in een roterend medium zou men op elegante manier de lichtsnelheid kunnen bepalen.
6. Er lijkt een verband te bestaan tussen de orde van grootte van optische draaiing, piëzo-electriciteit en optische boventonen in verschillende kristallen. Dit verband is theoretisch nog onvol-doende duidelijk en verdient nadere bestudering.
7. Onder bepaalde omstandigheden kan men vanuit een vliegtuig een ronde mistboog waarnemen, en in het midden daarvan niet de schaduw van het vliegtuig, doch het spiegelbeeld van de zon op een ijs- of sneeuwoppervlak.
8. Er is een eenvoudige verklaring te geven voor het "bellen blazen" dat het droge strand lijkt te doen, wanneer hierover bij opko-mende vloed voor het eerst weer zeewater begint te vloeien.
9. Brildragende mensen zien vaak een helderblauw waas onder een s t r a a t l a n t a a r n met kwiklamp. Dit is toe te schrijven aan een prismawerking van het b r i l l e g l a s .
10. In biologieboekjes op middelbare-schoolniveau wordt e r te weinig aandacht aan besteed dat ook het "staafjes-zien" van de golf-lengte afhankelijk i s .
11. De invoering van een college " r e l a t i v i t e i t " , of een college m e t soortgelijke filosofische aspecten voor de propaedeuse aan de afdeling d e r Technische Natuurkunde aan de T. H. te Delft, v e r -dient de voorrang boven de invoering van een " s o c i a a l " college, en dit niet alleen vanuit fysisch oogpunt.
12. Hermann e. a. vinden dat verschillende e l e c t r i s c h e eigenschap-pen van Te op analoge wijze beïnvloed worden door uniaxiale r e k en door hydrostatische d r u k ; de analoge veranderingen in de g e o m e t r i e van het k r i s t a l r o o s t e r die beide mechanische s t o r i r ^ e n teweegbrei^en, geven een verklaring voor deze w a a r -neming.
K . H . H e r m a n n . M. Ochel, J . Puhlmann, (1969) Phys. Stat. Sol. 36, 665.
F . T u i n s t r a , d i s s e r t a t i e (1967), Delft, pag. 63.
':.jüv..
13. Elk element v met Tr(v) = O van een gesplitste Gay ley-algebra C m e t k a r a k t e r i s t i e k ongelijk aan 2, kan g e s c h r e v e n worden in de v o r m V = (xy) z - x ( y z ) , waarbij x, y en z eveneens tot C behoren, 14. Het verdient aanbeveling in een wetenschap, die veel gebruikt
wordt als hulpwetenschap, b . v . wiskunde, de terminologie zo langzaam mogelijk te v e r a n d e r e n , ook al maakt de ontwikkeling d e r opvattingen veranderingen wenselijk.
Hetzelfde geldt a fortiori voor een taal. In beide gevallen zijn lijsten van betekenissen, die geregeld bijgehouden worden, w e l -haast onmisbaar,
ill
"You'll make me giddy soon, if you go on turning round like that". . .
Lewis Garroll
V
C O N T E N T S
Introduction. vii CHAPTER I. Enantiomorphism in c r y s t a l s of Selenium
and T e l l u r i u m . 1
I - l . Introduction. 1 1-2. C l a s s i c a l enantiomorphous c r y s t a l p r o p e r t i e s . 3
1-3. Enantiomorphous p r o p e r t i e s of Se or Te described
by t e n s o r s . 8 CHAPTER II. Birefringence and optical activity in the
c l a s s i c a l o s c ü l a t o r model. 16
I I - l . Introduction. 16 n - 2 . The point-dipole approximation, polarizabilities and
local field c o r r e c t i o n s . 17 n - 3 . Phenomenological d e s c r i p t i o n of the optical rotation
in c r y s t a l s . 22 II-4. Double refraction and optical rotation in the Ewald
theory. 28 CHAPTER HI. Application of the c l a s s i c a l point-dipole
theory to the calculatiqnof the birefringence
and the optical rotation in v a r i o u s c r y s t a l s . 37
m - l . Introduction. 37 n i - 2 . Calculations on the birefringence and optical rotation
of c r y s t a l s with a helix s t r u c t u r e . 39 i n - 3 . Discussion about the limitations and some
modifica-tions of the c l a s s i c a l point-dipole approximation. 53 n i - 4 . Galculations on c r y s t a l s of NaNOj, NaClOg and NaBrOg. 65 III-5. Galculations on ;8-cristobalite and s u m m a r y of the
r e s u l t s for all c r y s t a l s . 72 APPENDIX A. Derivation and discussion of some of the
formulas in section II-4. 76 APPENDIX B. Quantummechanical a s p e c t s of the d i e l e c
t r i c t e n s o r and the optical rotation in c r y s
-t a l s . " "' ' " 79
L i t e r a t u r e . 85 List of symbols and some specific t e r m s . 88
Summary. 92 Samenvatting. 94
v i l
I N T R O D U C T I O N
Macroscopic c r y s t a l p r o p e r t i e s may s o m e t i m e s give valuable information about the s y m m e t r y of c r y s t a l s t r u c t u r e s and, occasion-ally, even about atomic c o - o r d i n a t e s . In almost all c a s e s , however, the s a m e information i s obtained from x - r a y diffraction, the method
par excellence tor determining c r y s t a l s t r u c t u r e s . The p r e s e n c e of
threefold screw axes in Se and Te, for example, follows very easily from macroscopic p r o p e r t i e s of these c r y s t a l s such a s the m o r p h o -logy, etch figures or the optical rotation. F u r t h e r m o r e , some of these p r o p e r t i e s d i s t i n g u i s h between c r y s t a l s with different absolute configurations, i. e. between the handedness of the s c r e w axes o r the chirality. An x - r a y diffraction pattern also shows the threefold s c r e w axes, but, in Se and Te, never the handedness. Even the special method of a n o m a l o u s x r a y diffraction, wellknown for d e t e r -mining absolute s t r u c t u r e s of c r y s t a l s containing m o r e than one kind of atom (cf. quartz), is not powerful enough to determine the s c r e w s e n s e in c r y s t a l s of Se and Te. In other words, the macroscopic p r o p e r t i e s which a r e not generally used for an absolute s t r u c t u r e determination of the majority of c r y s t a l s , a r e very e s s e n t i a l for a determination of the absolute s t r u c t u r e s of Se and T e .
This is the p r i m e r e a s o n for my study of the optical activity and s e v e r a l other p r o p e r t i e s of Se and T e . Another r e a s o n i s the r e -m a r k a b l e observation that a satisfactory calculation of the optical rotation in c r y s t a l s has never been given before, in spite of the fact that the phenomenon i s about one and a half c e n t u r i e s old and i s m e n -tioned in practically any elementary textbook on basic physics.
I have chosen a c l a s s i c a l model for the description of the optical rotation, which i s essentially the point-dipole model used already by e . g . Lorentz around the t u r n of the century. Although a c l a s s i c a l approach would, after all, not be expected to yield very a c c u r a t e r e s u l t s for the semiconducting c r y s t a l s of Se and Te, I find s u r -p r i s i r ^ l y good quantitative r e s u l t s for the o-ptical rotation in these c r y s t a l s . At the s a m e t i m e , however, I find an infinite r e s u l t for the refractive index for polarization in the direction of the s c r e w axes where strong bonding o c c u r s .
This r e s u l t led me to the study of other c r y s t a l s with different electronic p r o p e r t i e s and, m o r e generally, to a study of the nature of the c l a s s i c a l approximation. F r o m this study I come to the g e n e r a l conclusion that the c l a s s i c a l point-dipole approximation, applied in the special manner outlined in this d i s s e r t a t i o n , is considerably m o r e powerful than i s commonly assumed. In p a r t i c u l a r a s r e g a r d s the calculation of the optical rotation, the c l a s s i c a l method should most probably be p r e f e r r e d to a quantummechanical approach for the m a jority of c r y s t a l s . I have, n e v e r t h e l e s s , worked out a q u a n t u m m e chanical formulation of the optical rotation, which i s presented s e p a r -ately in an appendix.
s e c . I - l 1 C H A P T E R I
Enantiomorphism in c r y s t a l s of Selenium and T e l l u r i u m .
R é s u m é .
After an introductory discussion of some " c l a s s i c a l " enantio-morphous c r y s t a l p r o p e r t i e s and a special analysis of the etch fig-u r e s which appear on the p r i s m faces of Te, a g e n e r a l t r e a t m e n t of the enantiomorphism in c r y s t a l s of Se and Te (class 32) i s given in t e r m s of tensor s y m m e t r i e s . A n e c e s s a r y and sufficient s y m m e t r y condition is introduced, determining which (tensor) p r o p e r t i e s of these c r y s t a l s a r e enantiomorphous. Several phenomena a r e d i s -cussed which may d e t e r m i n e the a b s o l u t e s t r u c t u r e of Se and T e .
I - l . Introduction.
The t e r m enantiomorphism i s generally used in r e f e r e n c e to any two physical phenomena which show "opposite p r o p e r t i e s " . Originally, the name only r e f e r r e d to the opposite e x t e r n a l shapes of c r y s t a l s *). More p r e c i s e l y , one now defines two c r y s t a l s t r u c t u r e s A and B to be enantiomorphous if and only if a c e r t a i n r o t a t i o n -reflection, but no simple rotation in space will c a r r y A(B) over into B(A).
In general, the enantiomorphism of the microscopic s t r u c t u r e s of two c r y s t a l s will show up in s e v e r a l , though not all, macroscopic p r o p e r t i e s of these c r y s t a l s . This i s only a specific example of the g e n e r a l r u l e that the s y m m e t r y of m a c r o s c o p i c p r o p e r t i e s of c r y s -t a l s canno-t be lower, bu-t may be higher -than -the symme-try of -the c r y s t a l s t r u c t u r e . Considering how long the topic has been known in c r y s t a l physics, it i s r e m a r k a b l e to notice how little u n d e r s t a n d -ing one has nowadays about the relation between m a c r o s c o p i c and m i c r o s c o p i c enantiomorphism.
It i s our aim in t h i s chapter to find enantiomorphous c r y s t a l p r o p e r t i e s in a s y s t e m a t i c m a n n e r , and to give a phenomenological desciption of some of them.
Only c r y s t a l s t r u c t u r e s of the 11 c r y s t a l c l a s s e s listed in Table I - l can show enantiomorphism. In agreement with our definition of two enantiomorphous s t r u c t u r e s , the s y m m e t r y elements belongir^ to the point groups of these c l a s s e s contain neither reflections nor i n v e r s i o n s . F u r t h e r m o r e it i s seen that two enantiomorphous c r y s -t a l s do no-t necessarily belong -to enan-tiomorphous space groups.
*) iuavT L oa = opposite, fj jLtopcprj = the form.
s e c . I - l c r y s t a l system t r i c l i n i c monoclinic orthorhombic tetragonal rhombohedral hexagonal cubic c l a s s 1 2 222 4 4 2 2 3 32 6 622 23 4 3 2 s y m m e t r y e l e m e n t s E E Ca
E c; cy
E 2C^ Ca E 2G4 C2 2 c ; 2 c ; ' E C3 E 2C3 3C2 E 2 G , 2G3 G2 E 2Cg 2C3 Cg «JC2 "^^2 E 8C3 3C2 E 8G3 3G2 6C2 6G^ s p a c e g r o u p s P 4 i P 4 i 2 2 P 4 3 22 P 3 , P 3 i l 2 P 3 2 I 2 P61 P 6 s P 6 i 2 2 P 6 s 2 2 P 4 3 3 2 P 4 i 3 2 P 4 i 2^2 P 4 3 2 , 2 P 3 i 2 1 P 3 2 2 1 P62 P 6 , P 6 2 2 2 P 6 4 2 2 Table I-lEnantiomorphous crystal classes and space groups.
Enantiomorphous crystals of NaG103 in the cubic class 23, for ex-ample, belong to the same space group P2i3; the symmetry elements belonging to P2i 3 carry the enantiomorphous (mirror image) struc-tures over into themselves but not into each other. We shall here con-sider specifically the class 32 and the space groups P3i21 and P3221. As special examples we shall take crystals of Se and Te,bothof which consist of chains of atoms in the parallel arrangement of Fig. I - l . Two crystal structures of these elements with right- (P3i21)and
left-s e c . 1 - 2
a 4.35517 B. 4 . 4 4 6 9 3 & '^ 4.94945 & 5.91492 &
b | 1 . 8 9 0 1 4 & 2.39244 l
F i g . I - l
Trigonal trapezohedral crystal structure of Selenium or Tellurium: left: threefold lefthanded screw axes (P3221),
right: projection of the atoms of one helix onto a plane perpendicular to the c-axis.
Parameters from Wyckoff (1965).
(P3221) handed s c r e w axes probably provide the simplest example of two enantiomorphous atomic a r r a n g e m e n t s in c r y s t a l physics.
T h e r e f o r e , the discussion of the v a r i o u s symmetry p r o p e r t i e s of t h e s e c r y s t a l s will be relatively s i m p l e , but will n e v e r t h e l e s s show
all e s s e n t i a l enantiomorphous f e a t u r e s . A straightforward extension of our t r e a t m e n t to m o r e complicated s t r u c t u r e s should therefore be p o s s i b l e .
The atomic m e c h a n i s m s which e s t a b l i s h the relation between the m i c r o s c o p i c and m a c r o s c o p i c p r o p e r t i e s of c r y s t a l s in g e n e r a l will not be d i s c u s s e d in t h i s chapter. One of the enantiomorphous phenom-ena, the optical rotation, however, will be examined in detail from a m i c r o s c o p i c point of view in the following c h a p t e r s .
1-2. C l a s s i c a l enantiomorphous c r y s t a l p r o p e r t i e s .
T h e r e a r e t h r e e " c l a s s i c a l " phenomena which can show enantio-m o r p h i s enantio-m : enantio-morphology, optical activity and etch figures.
Quartz is the text book example of enantiomorphous m o r p h o -l o g y ; c e r t a i n t r a p e z o h e d r a -l c r y s t a -l faces distinguish -laevo- from d e x t r o - q u a r t z ( s e e e . g . J a e g e r , 1924). Although Se and Te both belong to the same space group(s) a s quartz (the l o w - t e m p e r a t u r e modification), morphological differences between laevo and d e x t r o -Se or Te have never been observed. This might be the r e a s o n that, in e a r l i e r l i t e r a t u r e , both Se and Te were thought to belong to the ditrigonal scalenohedral c l a s s 3 m ( D s d ) instead of the trigonal t r a p e z o h e d r a l c l a s s 32 (D3) (Groth, 1905). P e r h a p s morphological differences will at some time be found in properly vapour-grown c r y s t a l s .
4 s e c . 1 - 2 The second c l a s s i c a l phenomenon which may distinguish between enantiomorphous c r y s t a l s , i s the o p t i c a l r o t a t i o n * ) . Although the effect itself had been known since the beginning of the last century
(Arago, 1811), an atomistic interpretation was not given before the b e g i n n i i ^ of t h i s century (Oseen (1915), Born (1915), Ewald (1916), Born (1933)). Several attempts have since been made to perform a quantitative calculation of the optical rotation in c r y s t a l s , none of which seem to have been completely successful (Hylleraas (1927), Endeman (1965) etc.; see chapter II). We shall, in the following c h a p -t e r s , be concerned wi-th a calcula-tion of -the op-tical ro-ta-tion in some relatively simple inorganic c r y s t a l s such as Se, Te, or quartz, and shall then d i s c u s s the nature of the phenomenon in m o r e detail.
E t c h f i g u r e s have also long been known to distinguish between enantiomorphous c r y s t a l s (quartz, NaC103 e t c . , Groth (1905)). Well defined enantiomorphous etch figures have been observed on the hexa-gonal p r i s m faces of T e - c r y s t a l s , which have been etched with sulphur-ic acid a t ± 1 5 0 O G (Blakemore, Nomura, 1960, 1961). Etch figures have also been observed on c r y s t a l s of Se, but the detailed shape of t h e s e figures has never been examined (Eckart(1963), Harrison(1965)). We shall d i s c u s s the etch figures on Te h e r e in some more detail.
Fig. 1-2 shows the etch figures which appear on the p r i s m faces
(1102) 1102
IB ne ic nc
Fig.1-2
Projection of the etch p y r a m i d s onto the hexagonal p r i s m faces (10Ï0) and (1100) of T e . See the text for a discussion of the different o r i e n -tations of the figures.
*) C r y s t a l s , showing optical activity, on the other hand, do not n e c e s -sarily_have t o b e enantiomorphous; cf.the c r y s t a l s in the c l a s s e s m m 2 , 4 and 4 2 m on page 67.
sec.1-2
Model I-A Model II-A
Model I-C Model II-C
Fig.1-3
Four models of etch p y r a m i d s which may a p p e a r on the hexagonal
p r i s m faces of Te, c o r r e s p o n d i n g with the p r o j e c t i o n s I-A. II-A. I-C and II-C In Fig. 1-2. All four m o d e l s have r i g h t - h a n d e d s c r e w a x e s .
6 sec.1-2 of Te in the form of pyramids with a top T, and faces as indicated*). We distinguish between pyramids of the t y p e s I and II (mirror im-ages) and the o r i e n t a t i o n s A, B and C. Only pyramids of one type appear on one particular crystal. The pyramids I-A ane I-B or II-A and II-B alternate on subsequent prism faces, in agreement with the presence of two-fold axes in the crystal structure, which carry the orientation A over into the orientation B. _We have taken the example that orientation A appears on face (1010) and orientation B on face (1100), but this is an arbitrary choice. We should also con-sider an orientation A of the pyramids on face (1100). We call these pyramids I-C and II-C, in order to distinguish them from the pyra-mids I-A and II-A on face (1010) (Fig. 1-2).
It is generally assumed that a unique relation exists between the t y p e of pyramid that appears on a crystal of Te and the absolute structure (screw sense, handedness, chirality) of this crystal. We have tried to determine this relation by building models of the pyra-mids and by studying the location of the pyramid faces in the crystal structure. Fig. 1-3 shows four models, each of which contains right-handed screw axes. The models I-A and II-A have the (1010) face as base plane, and correspond to the projections I-A and II-A in Fig. 1-2. The models I-G and II-C, having the (1100) face as base plane, correspond to the projections I-G and II-C in Fig. 1-2. We would have expected to observe essential differences in the construc-tion of the pyramid faces of the models of type I and II (of the same orientation), because different types are supposed to appear on enan-tiomorphous crystals. However, inspection of the photographs shows no essential differences whatsoever.
Assuming that the area of the pyramid faces is inversely proportional to the surface energy (i. e. the energy per surface to r e -move one layer of atoms; cf. Grosse, 1969), it can be shown that the pyramids of the orientation C will ^ p e a r instead of A. This, how-ever, does not solve the problem. Looking at the opposite sides of the models I-C and II-C in Fig. 1-3, we see the pyramid faces of Fig. 1-4. The construction of the edges in these two models is seen to be different. We presume that one edge better " r e s i s t s " the etchant than the other. Because very little is known about the actual nature of the etchii^ mechanism, however, it is difficult to decide which edge is m o r e resistant. In a nai've picture, one could consider the working of the etchant on each atom separately. This leads to the conclusion that the edge in Model II-C, with the loosely bound shaded atoms (chain-ends, Fig. 1-4) would be more easily dissolved by the etchant than the corresponding edge in Model I-C. Pyramids of type I should then appear on crystals with right-handed screw axes.
*) Blakemore et.al. (1960) give a last index of 3 to the faces we have given the indices (10Ï2)etc. The index 2 seems to be in better agree-ment with the observations of the etch figures we made (in projection) (Reijnhart (1967), van Aken (1968)).
sec.1-2
Model I-C Model II-C
Fig.1-4
Opposite side of the models I-C and II-C in Fig. 1-3. If the bond between nearest neighbour atoms in one chain is called bi and the bond between nearest neighbour atoms in adjacent chains b2, the shaded atoms in Model II-C are held to the pyramid by one bond bi and one bond b2. whereas the corresponding atoms in Model I-C (shaded) are held by one bond bi and three bonds
hz-Crosse (1969) comes to the same conclusion from an analogous discussion of the different etching and evaporation rates forjhe pos-itive and negative sides (arbitrary choice) of the faces (1210) and equivalents (Fig. 1-5). *) Roughly, his explanation amounts to the
polar a x i s
Fig.1-5
Different orientations of the helices in Te with respect to the positive or negative side (arbitrary choice) of a crystal surface perpendicular to a polar axis.
*) The positive and negative sides of crystal surfaces perpendicular to polar axes differ most probably in several other properties (e.g. mechanical properties, hardness etc.)
8 sec.1-3 following: the atom 1 of chain A is more strongly attached to the (+) crystal surface than one of the atoms 2 or 3 of chain B to the (-) crystal surface. Therefore chain A will be more difficult to "detach" than chain B.
Unfortunately, our calculation of the optical rotation in Te (chap-t e r s n and in) shows (chap-tha(chap-t pyramids of (chap-t y p e II appear on crys(chap-tals with r i g h t - h a n d e d s c r e w a x e s . Considerir^ the reasonable agreement we find between the calculated and experimental values of the rotation for Se and Te and the correct s i g n of the rotations we find for quartz, we tend to believe that the etching and growing mech-anisms in Te are controlled by more complicated processes. It is, for example, highly questionable whether the individual atoms may be considered separately in explaining these phenomena.
Although the existence of a relation between the shape of the etch pyramids and the absolute structure seems obvious, it would be de-sirable to check this relationship again on several crystals, grown in different ways, from different seeds, using different etchants. The influence of impurities on the etch phenomena or the optical activity of Te (Se) has never been examined, but may be important*). 1-3. Enantiomorphous properties of Se or Te, described by tensors.
The absolute structure of the elemental crystals of Seor Te can-not be determined by means of an (anomalous) x-ray diffraction experiment. Neither the screw sense, nor the orientation of the chains, i.e. the directions of the polar axes 1, 2 or 3 in Fig. 1-6, can be found. If one of these data is known from a d i f f e r e n t exper-iment, however, the other is readily found from an additional x-ray analysis (cf. Reijnhart (1967), Arlt (1969), Grosse (1969)).
2
3
Fig. 1-6
Three polar directions in the Se- or Te-structure.
) In some organic materials the influence of certain polluting mole-cules or even bacteria of one handedness, is known to be very im-portant.
s e c . 1 - 3 9 In the previous section we d i s c u s s e d some familiar phenomena which may give information about the s c r e w s e n s e and/or the d i r e c -tions of the polar a x e s . In this section we shall d i s c u s s , in a m o r e s y s t e m a t i c manner, enantiomorphous p r o p e r t i e s of the Se ( T e ) s t r u c -t u r e , described by -t e n s o r s . *)
We shall always consider orthogonal co-ordinate s y s t e m s (x,y,z), s o m e t i m e s denoted by (1, 2, 3). An orthogonal t r a n s f o r m a t i o n from a co-ordinate system ( x , y , z ) to another co-ordinate system ( x ' , y ' , z ' ) will be d e s c r i b e d by a t r a n s f o r m a t i o n m a t r i x aim.- defined such that
x ' = a,., X + ai2 y + a.,3 z , y ' = 321 x + a22y+ a23z , z ' = a,! x + a32y +a33Z .
An inversion, for example, i s d e s c r i b e d by the t r a n s f o r m a t i o n m a t r i x / a i l a,2 ^3\ / - I 0 0 \
aim= ^^ ^ ^AA 0 "1 « •
We consider now two s e t s of quantities, T' and T, r e f e r r e d to two a r b i t r a r i l y chosen orthogonal c o - o r d i n a t e s y s t e m s ( x ' , y ' , z ' ) and (x, y, z). The set T ' c o n s i s t s of the e l e m e n t s t ' p q . . . r» with p = = ( x ' , y ' , z ' ) , q = ( x ' , y ' , z'), etc. and the set T of the e l e m e n t s t^j. _ _ k, with i = ( x , y , z ) , j = ( x , y , z ) , e t c . . If the e l e m e n t s of T' are obtained from the elements of T via a t r a n s f o r m a t i o n
*'pq. . .V ~ - ^pi ^ j - • • ^rk k j . . . k
with n transformation m a t r i c e s aj^j^^, T' and T a r e defined to be n-th rank p o l a r (for the + s i g n ) o r a x i a l (for the - sign) t e n s o r s and will b e denoted by I ' and 1 .
A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n for a t e n s o r Ï to be enantiomorphous i s that it changes sign on application of an i n -v e r s i o n . The proof of this statement, in which we r e f e r to Fig. 1-7, demonstrating the effects of an inversion and a reflection in the x-y plane (plane of Fig. 1-7), will be given below.
We shall first consider an a r b i t r a r y t e n s o r T that changes sign on application of an inversion. I can always be written as the sum of two t e n s o r s J ^ and J ^ , in which I ^ c o n s i s t s of the e l e m e n t s of I which have an uneven number of indices z and in which I ^ c o n s i s t s of all other e l e m e n t s of I . An i n v e r s i o n c a r r i e s 1 over into - I = = _ j u _ j e ^ and a reflection in the x-y plane (Fig. 1-7) into I " = *) This t r e a t m e n t applies m o r e generally to any c r y s t a l s belonging
10 s e c . 1 - 3 reflection in x-y plane indenticol : T" - T' I". - T " , T' Fig.1-7
The effect of an inversion and a reflection in the xy plane on a S e -chaln and on a t e n s o r T = ^ " + T^ (see text);
R = Right-lianded: L = Left-lianded.
= - T ^ + T^. A c o m p a r i s o n of the effects of t h e s e two s y m m e t r y operations on the s t r u c t u r e in Fig. 1-7 and the t e n s o r X shows that T^ gives direct information about the s c r e w s e n s e , whereas I ^ only gives information about the orientation of the chains, (cf. the change in sign of j e from a rotation of 60° around the c-ax;is in Fig. 1-7). T h e r e a r e three p o s s i b i l i t i e s :
1. I gives information only about the s c r e w s e n s e , I ^ = Q.;
2. 2 gives information only about the orientation of the chains, or the direction of polar a x e s , X^ = 0;
3. T gives full information about theTmicroscopic) absolute s t r u c t u r e , TU ^ 0, j e ^ g_
This shows that in all t h r e e c a s e s sufficient information i s provided to find the absolute s t r u c t u r e ( I = ^ i s a t r i v i a l c a s e ) ; t h i s , in other words, proves that the above condition i s s u f f i c i e n t .
We subsequently consider an a r b i t r a r y t e n s o r X that i s invariant with r e s p e c t to an inversion. If I i s again written a s the sum of two t e n s o r s , j u and I©, defined s i m i l a r l y , we notice from Fig. 1-7 that T^ may distinguish between the two figures on the left side. However, t h i s does not mean that X^ distinguishes between r i g h t - and left-handed s c r e w a x e s . The inverted s t r u c t u r e , for instance, (Fig. 1-7) with left-handed s c r e w axes i s a l s o described by a tensor + X^. F u r t h e r m o r e , it can be proven that I ^ i s equal to ^ b e c a u s e of the s y m m e t r y of the c r y s t a l s t r u c t u r e (i.e. uniaxial s y m m e t r y ) . This proves that the above condition i s n e c e s s a r y . Evidently, only p o l a r t e n s o r s of u n e v e n rank and a x i a l t e n s o r s of e v e n rank meet this condition.
A f i r s t r a n k p o l a r t e n s o r in the c l a s s 32, to which Se and Te belong, can easily be shown to contain only z e r o e l e m e n t s .
s e c . 1 - 3 11 form ' ' M I I ^22 *-i33 ^23 ^113 ^^2\ / ^ m ~^^^^ O ^211 ^222 ^233 1223 tj^j t2.|2 0 0 0 0 - t , „ -t.A . ( L I )
t,,
0 0 0 t11 0 0 0 t33 ^ S l l *322 ^333 *323 *313 ^312' ^ ^ ^ ^The splitting in T^ + T^ in this case r e a d s :
/O O O ti23 O O \ /t,.,! -t,i., 0 0 0
0 0 0 0 -t,23 0 + 0 0 0 0 0 -t,ii 1 . (1.2)
0 0 0 0 0 0 / \0 0 0 0 0
In o r d e r to find the absolute s t r u c t u r e of Se (Te) from such a t e n s o r , one can either determine the full t e n s o r ( j u + j e ) or one can d e t e r -mine X^ (t^^.,) and perform an additional x - r ^ analysis to find the s c r e w s e n s e .
A s e c o n d r a n k a x i a l t e n s o r in c l a s s 32 has the g e n e r a l form
(L3)
The t e n s o r X® i s seen to be z e r o in this c a s e , so that only the s c r e w sense and not the d i r e c t i o n s of polar a x e s can be found (cf. the optical rotation).
Although s e v e r a l physical phenomena a r e d e s c r i b e d by higher rank t e n s o r s , we shall not consider those h e r e , as they generally do not give any information about the s y m m e t r y of the microscopic s t r u c t u r e that is not already contained in the corresponding t h i r d r a n k polar or second rank axial t e n s o r s .
This simple analysis of tensor s y m m e t r i e s in the S e - s t r u c t u r e p r e s e n t s a possibility to find all those (tensor) p r o p e r t i e s which, in combination with a p r o p e r m i c r o s c o p i c theory, determine the m i c r o -scopic enantiomorphism. In Table 1-2 s e v e r a l physical quantities a r e introduced schematically, with some r e l a t i o n s between them, d e s c r i b e d by third rank polar or second rank axial t e n s o r s . Without analysing the details of the a t o m i c m e c h a n i s m s involved, 'we shall briefly d e s c r i b e some of the relations in Table 1-2, involving an e l e c t r i c field.
P i e z o - e l e c t r i c i t y (Ei = l i j k I j k ) i s , in c r y s t a l s of quartz or ZnS, due partly to a displacement of ions with r e s p e c t to each other and partly to changes in tonicity. (The latter effect, due to charge r e d i s t r i b u t i o n s , may be m o r e or l e s s important than the f o r m e r for different c r y s t a l s ; c f . Phillips, 1969.)In Se and Te, however, the p i e z o -electricity i s entirely due to charge r e d i s t r i b u t i o n s . In this case a prediction of the direction of the induced polarization (or the sign of
12 sec.1-3 RELATIONS BETWEEN P i e l e c t r i c a l polarization $ i heat c u r r e n t Mi magnetic polarization ei -j mechanical s t r a i n Ei
(©i
Hi l i j d e s c r i b e d by t h i r d rank polar t e n s o r s Ei —EjEk (non-linear ($.) optics) (aT\ /3T\ l a r J j U r ' k - - T i j (piezo-electricity) --HjHkMi --HjEk (modified chemical shift) H l'5T\ e i j —Ek (inverse p i e z o -electricity) 1 l a r ' k e l e c t r i c field t e m p e r a t u r e field (gradient) magnetic field mechanical field ( s t r e s s )
second rank axial t e n s o r s 2 i (4i) Mi Mi Hi - H j —Ej (magneto-electric effect) /9T\ - - E j (optical activity) Table 1-2
Relations between some v e c t o r s and t e n s o r s (components and e l e m e n t s Indicated by i, j . k) which a r e d e s c r i b e d by t h i r d r a n k polar or second rank axial t e n s o r s .
t.,.,., in (III. 1)) does not seem to be possible from simple arguments. Arlt and Quadflieg (1969) have experimentally determined the direc-tion of the stress-induced polarizadirec-tion with respect to the orientadirec-tion of the etch pyramids on the prism faces of Te (section 1-2). Our cal-culation of the optical rotation in Te (chapter III) combined with these measurements give a direction of the induced electrical moment in a deformed chain as indicated in Fig. 1-8.
sec.1-3 13
Fig.1-8
Direction of an e l e c t r i c a l moment In a deformed T e - c h a l n with a l t e r e d valence c h a r g e s (see text).
We assume that a macroscopic deformation of the crystal results in a microscopic deformation of the chains and that the induced macro-scopic polarization is due to a superposition of micromacro-scopic electri-cal moments in the chains. If, in a very simplistic approximation, the charge between nearest neighbour atoms in a chain (valence charge) is thought to be concentrated in a point charge -q, centred between the atoms, the electrical moment in a deformed chain could be brought about by charges (q' and q") in valence charges, as indicated in Fig. 1-8. The direction of the electrical moment indicates thatboth q' and q" are positive. This result implies that the change in charge distribution counteracts the deformation because of electrostatic r e -pulsion. Although the real mechanism which controls the piezo-electricity is presumably much more complicated, even this simple model appears to account correctly for a counteraction of the mechan-ical deformation. There may be a possibility to express the changes in valence charges in terms of charges in hybridization of atomic wave-functions or in terms of changes in valence angles a n d / o r distances (cf. Pauling, 1960). We have not worked out this concept further, but have only calculated the charges in the valence armies j3 and (3' in Fig. 1-9 as a function of a relative charge in width b' of a "Te-triangle". The angle p is seen to change faster than the angle |3' if a constant valence distance d is taken.
N o n - l i n e a r o p t i c a l e f f e c t s have been observed in Se and Te*), but the relation between the absolute structure and these effects has never been examined. The relations between the absolute configurations and the (absolute) non-linear permittivity tensors of NaClOg and NaBrOj (cubic) have been established by Simon (1968).
The l i n e a r e l e c t r o - o p t i c e f f e c t is essentially the static analogue of the non-linear optical effects. (Mason, 1966). This has been measured inSeby Turner(1968), without referrir^ to any structur-al aspect.
The screw sense could further show up in the relation between a magnetic moment M and a magnetic field H plus an electric field E. *) The measured non-linear coefficients in Se as well as Te belong
to the largest ever observed in any material (Patel (1965, 1966), Jerphagnon (1966, 1967), Landolt Börnstein (1969)).
14 sec.1-3 e - 5 9 U 9 2 X b . 1 79433 X >)> 2 e s X 0 4 O 0 4 0 8 Ab'/b' m % Fig.1-9
Valence angles j3 and (3' in "(degrees) in a T e - c h a i n as a function of a relative change A b ' / b ' in the width b ' of a " T e - t r i a n g l e " .
The valence distance d i s kept constant.
The relation between (a diamagnetic) M (or A H) and H is given by the second order tensor of the c h e m i c a l s h i f t and, accordingly, does not give information about the enantiomorphism of the crystal structure of Se or Te (cf. also Bensoussan, 1967). If, however, in addition to the magnetic field H, an electric field E is applied, the induced moment M is described by a third order polar tensor, which may give information about the absolute structure.
The o p t i c a l a c t i v i t y has been, for a long time, the only example of a second order axial tensor. The tensor (1-3) describir^ the relation between D and B (in Table 1-2 we have for simplicity taken E and H) is generally called the gyration tensor, (cf. chapter II for a discussion of the relation between D and B in uniaxial crys-tals).
The m a g n e t o - e l e c t r i c e f f e c t , which seems to have been observed in certain magnetic materials (Bhagavantam, 1966), could have a special significance in Te and Se. The symmetry of these crystals allows a magnetic field to be induced in a certain direction on application of an electric field in t h e s a m e direction. If the direction of the c-axis is taken (Fig. I-l), this effect could be caused by charge that is moving alor^ the helices under the influence of the
s e c . 1 - 3 15 applied electric field, thus inducing a magnetic field (moment) in the
direction p a r a l l e l to the h e l i c e s . The effect has never been ob-served, but could, in principle, provide a d i r e c t relation between the s c r e w s e n s e and the direction of the induced magnetic field * ) .
Table 1-2 gives only some examples of phenomena which may distinguish between enantiomorphous c r y s t a l s . Many other enantio-morphous phenomena may, of c o u r s e , exist which a r e not mentioned h e r e * * ) .
This general t r e a t m e n t of t e n s o r p r o p e r t i e s of Se or Te (class 32) shows on the one hand that, i n p r i n c i p l e , s e v e r a l methods a r e available for determining the absolute s t r u c t u r e of these c r y s t a l s . On the other hand it excludes many other phenomena (the Hall-effect, elasticity etc.) as possible methods for an absolute s t r u c t u r e d e t e r -mination. The phenomena we mentioned have all in common, however, that t h e i r theoretical interpretation is very involved and, in many c a s e s , still unknown. Practically no work s e e m s to have been done on these phenomena in Se and Te, whose relatively simple s t r u c t u r e would suggest an exceptional opportunity to improve understanding.
*) This effect has been suggested to me by F . T u i n s t r a .
See also Reijnhart (1966), w h e r e the effect has been studied in m o r e detail.
**) We have, for example, not included the m a c r o s c o p i c momentum or angular momentum of a c r y s t a l .
16 s e c . n - 1
C H A P T E R II
Birefringence and optical activity in the classical oscillator model.
Resume.
After an introductory discussion about the presuppositions and limitations of the classical point-dipole approximation, expressions will be derived for the double refraction and optical rotation in terms of the geometrical arrangement of the atoms in a generalized Lo-rentz-model of a crystal, i. e. in terms of local field corrections. This generalization is given a firm basis in an extension of Ewald's method of calculating the double refraction. The extension consists in taking into account the finite wave-length of the light to the first power, i.e. second order of approximation. Explicit expressions are given for the local electric fields in crystals of arbitrary symmetry. II-l. Introduction.
In this chapter a method for calculating the birefringence and optical rotation in single crystals will be discussed. The behaviour of the electrons in these crystals, which determines the above optical properties, can be described by either a classical or a quantum-mechanical model. Both models may still vary considerably in degree of sophistication.
Although many phenomena can only be understood by using quantum-mechanics, some effects may quite properly be described by using a classical model, a good example of what seems to be a general rule that oversimplified models eventually lead to good results (cf. the free-electron model in solids).
For the description of the double refraction and the optical activity we shall primarily use a very simple classical model, in which infinitely small electron oscillators ("point-dipoles") are situated at the centres of the atoms in a crystal. The co-ordinates of these centres are supposed to be known from an x-ray analysis. Ewald (1916, 1921) used this "dipole-model" in his theory of birefringence in crystals. He introduced the physical concept that a plane electromagnetic wave should exist self-consistently in an infinite, transparant medium. This means that all the spherical waves, emitted with a velocity c from the individual dipole oscilla-tors, should together sum up to a plane electromagnetic wave, travelling with velocity v, each oscillator beir^ in its turn excited by the total field due to all the other oscillators. The condition of self-consistency yields a ratio between c and v which represents the refractive index n = c/v. Birefringence in non-cubic crystals is then accounted for by the fact that the above dipole sums depend on the direction of linear polarization, giving rise to different refractive indices in different directions.
sec. II-2 17 Although the dipole approximation itself was already used much earlier, e.g. Lorentz (1909), Ewald first developed the proper tech-nique to calculate the double refraction in this model. As he suggested (1916), his method of calculation could be extended so as to explain the optical activity in crystals without the introduction of a new physical concept. In this case, circular polarization, of course, is a better method of decomposition than linear polarization. Optical activity is then understood as a difference between the refractive indices nj^ and UL for right- and left-handed polarization *).
In order to find different values for nj^ and nL, the first order theory of birefringence ((RA)i = 0 , 1 > 1) will have to be extended to the second order ((R/x)! = 0 , 1 > 2 ) . R represents a characteristic interatomic distance and X the wave-lei^th of the electromagnetic wave. Van Laar and Endeman (1965, 1968) worked out this extension for their calculations on the optical activity of NaC103 and NaBrOg. According to their work, the classical model would appear to be in-adequate for the calculation of the optical activity. The rotation was found to be strongly dependent on the atomic arrangement. The ex-perimental values could not be derived theoretically with any reason-able accuracy, even with the most accurate structure parameters (cf. also Bijvoet 1960). Application of their formulas to Se and Te also yielded unacceptable results.
Therefore, I have started again from Ewald's treatment and have extended it consistently to the second order. My final formulas, which differ slightly from those of Endeman and Van Laar, appear to give satisfactory quantitative results for Se and Te, as well as for NaC103, NaBrOg, and some other crystals. These results will be extensively discussed in the next chapter.
After having discussed some assumptions about the model and its limitations in section II-2, we shall describe our derivations in sections II-3 and II-4, with particular emphasis on some essential points regarding, for example, local and average fields.
II-2. The point-dipole approximation, polarizabilities and local field corrections.
We define the "classical model" of a crystal as the approxima-tion in which the individual atoms behave like free atoms in their reaction to an electromagnetic field. The electric vector in this field, being influenced by the atoms, will be discussed here. In particular, the "classical p o i n t - d i p o l e model" will be considered, in which the behaviour of an arbitrary atom h with respect to an electric field E is fully described by a p o i n t - dipole oscillator at its centre with the "reduced"polarizability ar- This quantity an having the dimension
*) This description of the optical activity has been known since the times of Fresnel (~ 1820).
18
sec.n-2
of a volume, i s defined a s the induced dipole moment ph divided by eg t i m e s the e l e c t r i c field E(rh) at the point-dipole h. * ).F o r the sake of simplicity we shall from h e r e on always w r i t e a instead of Ckir, and call this a the " a t o m i c p o l a r i z a b i l i t y " .
F o r each atom h in the c r y s t a l we may then w r i t e :
Ph = eo a E(rh). (H. 2) E(rh) i s composed of the dipole waves from all atoms except h, and i s consequently closely related to the c r y s t a l s t r u c t u r e .
It may be decomposed in an a v e r a g e field E = TT i E(r) d r , used in Maxwell's bulk equations, and a local field c o r r e c t i o n E L • (V = volume unit c e l l ) . The direction of E L is in g e n e r a l different from, and its value in g e n e r a l depends on the direction of E.
In i s o t r o p i c (cubic) c£ystals Ej^ is_given by the well-known L o r e n t z L o r e n z c o r r e c t i o n P / 3 e o , where P is the average p o l a r -ization in the c r y s t a l . This c o r r e s p o n d s with the local e l e c t r i c field
E(rh) = l + ^ . ( n . 3 ) The formulas (11-2) and (II-3) then yield the following equation for
the average polarization:
P = N p h = N So a ( I + ^ ) ( n . 4 ) (N= number of atoms p e r volume; only one atom p e r cell will be
con-sidered, so N= 1/V).
The d i e l e c t r i c t e n s o r £ r , which d e g e n e r a t e s into a s c a l a r e^ for an isotropic c r y s t a l , i s defined in the equation
Eo i r E = EQ E + P for the g e n e r a l c a s e , and SQ Er E = SO E + P for the isotropic c a s e . **)
*) The atomic dipole polarizability a in the S.I. i s equivalent to our eoOr* 1'he value of a-^ m a y b e derived from the older G.G.S.atomic polarizability, which i s often found in l i t e r a t u r e , by multiplying by 4ïï c m V m ^ = 47r-10-6. The An c o m e s from the fact that the S.I. i s " r a t i o n a l i z e d " and the G.G.S. not. Though i r r e l e v a n t for most of our work, we give the value of a r for the s i m p l e s t c a s e , a charge e with m a s s m and binding frequency v^, neglecting damping:
"'r " 47T2m(i.2 - v^2)s^ • (II-1) **) We shall u s e the t e r m d i e l e c t r i c constant (tensor) or permittivity
for what i s s t r i c t l y the " r e l a t i v e " dielectric constant er(= e/eo). The s u b s c r i p t r will be used throughout.
s e c . I I - 2 19 Substitution of P from (II. 4) into (II. 5) leads to the familiar C l a u s i u s
-Mossotti formula
1 . l + | a N
ej. - 1 1 3 4 Q ! N , or er= • (II-6)
^r + 2 J _^^^
Sj. b e c o m e s infinitely large f o r - ö ö N - l . This is usually called the Clausius-Mossotti " c a t a s t r o p h e " .
In a n i s o t r o p i c c r y s t a l s the local field c o r r e c t i o n E L can be calculated with Ewald's method (1916, 1921). It may, in uniaxial c r y s t a l s , lead to two different " C l a u s i u s - M o s s o t t i c a t a s t r o p h e s " (cf. F i g . I I - l a and c).
The c l a s s i c a l point-dipole approximation gives a satisfactory d e s c r i p t i o n of the double refraction in many c r y s t a l s . In some c a s e s , however, large d i s c r e p a n c i e s between the calculated and e x p e r i m e n -tal values of the birefringence a r e observed. Te and Se will be shown to b e examples of t h i s . Due to the considerable overlap of atomic charge distributions in t h e s e c r y s t a l s , the behaviour of the e l e c t r o n s cannot be d e s c r i b e d by a t o m i c wave-functions (or the corresponding c l a s s i c a l dipoles). C r y s t a l wave-functions ( e . g . Bloch-functions) have to b e considered instead. The c r y s t a l i s thus considered as one big atomic s y s t e m .
The calculation of the polarization in this system works in e s -sentially the s a m e way as in a one-atomic system and the formulas one obtains a r e very s i m i l a r . In a c o r r e c t t r e a t m e n t , the v a r i a t i o n s of the microscopic e l e c t r i c field over a unit cell will have to be taken into account by considering the l o c a l e l e c t r i c field in the p e r t u r -bation caused by the electromagnetic wave in the c r y s t a l . Usually, however, the variations of E(r) over a unit cell a r e neglected and the average field Ë i s considered instead * ) . It leads to the familiar a v e r a g e polarization
P = N So a 1 , and the d i e l e c t r i c t e n s o r
€ r = | + N g , ( n . 7) in which the polarizability t e n s o r a i s now given in t e r m s of c r y s t a l wave-functions:
20 s e c . n - 2 e^lk,k+K
o; = s , 2 / 2 " p — — T - ; II-8 k = wave v e c t o r .
K = r e c i p r o c a l lattice v e c t o r .
I k k+K = oscillator strength between the " c r y s t a l l e v e l s " k and k+K. 1/^ k+K ~ resonant frequency between the " c r y s t a l l e v e l s " k and k+K. The t e n s o r c h a r a c t e r of f d e t e r m i n e s the double refraction in this approximation.
The t e n s o r c h a r a c t e r of e j . is thus determined by two at f i r s t sight completely different m e c h a n i s m s in the c l a s s i c a l point-dipole model and the quantum-mechanical m o d e l - w i t h o u t - l o c a l - f i e l d - c o r r e c t i o n s . In the f o r m e r c a s e , the g e o m e t r i c a l a r r a n g e m e n t of the atoms c a u s e s the anisotropy via the local field c o r r e c t i o n s ; in the latter c a s e , the '.symmetry of) the c r y s t a l wave-functions cause(s) the anisotropy via the " c r y s t a l o s c i l l a t o r s t r e i ^ t h f". Both of these c a s e s (equations 'II. 6) and (II. 7)) a r e schematically illustrated in Fig. I I l , for an i s o -tropic a s well a s for an aniso-tropic c r y s t a l . The two curves in Fig. I I - l c correspond to two different local field c o r r e c t i o n s and the two c u r v e s in Fig. I l - l d correspond to the two s c a l a r equations which follow form (II. 7):
(sr)// = 1 + N a ,, and
(II. 9) (ei.)_L = 1 + N a _j_ .
Neither of t h e s e d e s c r i p t i o n s of the atomic s t r u c t u r e will be frenerally c o r r e c t , although they may each give valuable approxi-mations in different c r y s t a l s . A quantum-mechanical t r e a t m e n t with local field c o r r e c t i o n s , describing the r e a l physical situation, i s possible in principle, but difficult to apply in p r a c t i c e . Some a s p e c t s of it will be d i s c u s s e d in appendix B.
In the following p r e l i m i n a r y discussion and in the next chapter s o m e m o d i f i c a t i o n s of the c l a s s i c a l model will be considered, which may explain why the pointdipole approximation i s l e s s s a t i s -factory for some c r y s t a l s .
A f i r s t modification one might tentatively consider i s the i n t r o duction of the solid state polarizability g from (II. 8) into the c l a s s i -cal point-dipole t r e a t m e n t . The dipole polarizability will then have the form
sec.II-2 21 C M ( Q ) U N a (b) (c) C M„ C Mj_ Fig. II-l
Dielectric constant E^, as a function of the "reduced" polarizability
a in the c l a s s i c a l model with local field c o r r e c t i o n s and in the
quantum-mechanical model without local field c o r r e c t i o n s .
a. isotropic c r y s t a l ; c l a s s i c a l model; Clausius-Mossotti " c a t a s t r o p h e " at CM.
b. Isotropic c r y s t a l ; quantum-mechanical model.
c. uniaxial c r y s t a l : c l a s s i c a l model; Clausius-Mossotti " c a t a s t r o p h e s " at CM,, a n d C M j _ . * )
d. uniaxial c r y s t a l ; q u a n t u m - m e c h a n i c a l m o d e l . * * )
*) The symbols J. and // will, in this dissertation, indicate the directions perpendicular and parallel to the optic axis in uniaxi: crystals.
**) In uniaxial crystals which do not show optical activity, the tento; a in (II. 8) has the form
a j . 0 0 \ 0 a_L 0
0 0
aJ
or la, 0 0 ai \o 0 0 0 a22 s e c . I I - 3 for a uniaxial c r y s t a l showing optical activity. The f i r s t t e n s o r r e p r e s e n t s the original isotropic atomic polarizability, the second the c o r r e c t i o n following from (II. 8). The t e n s o r element ai2 d e t e r m i n e s the optical rotation, which i s found in a pure quantum-mechanical t r e a t m e n t without local field c o r r e c t i o n s . This modification, however, a s will be shown in the next chapter, yields entirely wrong r e s u l t s . The very introduction of this a implies that the pointdipole a p p r o x i -mation i s not valid anymore (cl. appendix B).
Another modification may be introduced by taking the finite size of the atoms into account, i. e. by considering dipoles with finite size instead of p o i n t d i p o l e s . This may lead to a c o r r e c t i o n in the c a l -culation of local fields. The fields at a c e r t a i n atom due specifically to n e a r e s t neighbour atoms may differ considerably from point-dipole fields. A polarizability t e n s o r analogous to (11.10) could again be introduced, in which the c o r r e c t i o n i s now due to a modification of the calculation of local fields ( A n y modification of the c l a s s i c a l point-dipole model will indeed effectively lead to a modified polarizability.) An analysis of this modification in Se and Te can be shown to indicate (see chapter i n ) that 0:3 should b e s m a l l e r than a^ in (II. 10). Quanti-tative information about the anisotropy of a, however, i s extremely difficult to obtain. Our calculations do indeed show that the e x p e r i -mental values of e J. and p a r e found if an anisotropic polarizability i s used with 03 < a^. F u r t h e r m o r e , our calculations show that a^j i s , for many c r y s t a l s , practically equal to z e r o .
Finally, it should be emphasized that only the ratio 0:3/01, and not the absolute magnitudes of 0:3 and a, a r e important in these con-s i d e r a t i o n con-s . Therefore, d i f f e r e n c e con-s in interatomic d i con-s t a n c e con-s (or bond c h a r a c t e r ) in different d i r e c t i o n s , r a t h e r than the very existence of short interatomic distances (or overlapping atomic charge d i s t r i -butions), seem to d e t e r m i n e the applicability of the c l a s s i c a l method.
(The applicability of the point-dipole approximation will be analysed in m o r e detail in the next chapter.) The study of Se and Te is p a r t i -cularly interesting in this r e s p e c t b e c a u s e the interatomic distances and the bond c h a r a c t e r in these c r y s t a l s a r e very different in the d i r e c t i o n s p a r a l l e l and perpendicular to the chains.
I I - 3 . Phenomenological description of the optical rotation in c r y s t a l s . Only cubic c r y s t a l s and uniaxial c r y s t a l s with propagation of the light in the direction of the axis will be considered in this section.
E x p r e s s i o n s will f i r s t be derived for the optical rotation p in t e r m s of either the refractive indices n^ and n-^ (sec. H - I ) , or the e l e m e n t s of a complex permittivity t e n s o r e'^' + i i p a n d a c o r r e s -ponding permeability t e n s o r y-j?' + i j i p ' . *)
*) cf. footnote on page 18. The t e r m permeability (tensor) will be used for what i s s t r i c t l y the " r e l a t i v e " permeability ytj- = M/MO-The s u b s c r i p t r will b e used throughout.
s e c . I I - 3 23 We introduce a right-handed orthogonal co-ordinate system (Fig. II-2) and consider the propagation of an electromagnetic wave along the z - a x i s f r o m z = - = o t o z = + =o.
Fig. II-2
Right-handed co-ordinate system with clockwise rotation of the dis-placement vector D.
Right and left c i r c u l a r l y polarized waves a r e then d e s c r i b e d by the two complex displacement v e c t o r s
(0) - i D ! ° ' , 0 ) e i ^ - ^ e - ^ ' " t y i k • r - i Lct D L = ( D - , i D y " ' , 0 ) e ^ ^ - ^ e and with (II. 11) ^ ' = i > ; ^ ' .
(Analogous formulas hold for the magnetic v e c t o r s , but these will not be explicitly written out h e r e ) . A right-handed c i r c u l a r l y polarized wave i s , by definition, a wave that t u r n s clockwise i n t i m e for an
o b s e r v e r looking f r o m z = = o t o z = - » .
For the description of the optical rotation only the wave c h a r a c t e r in s p a c e i s relevant; therefore we consider the situation at the time t = 0. If at z = 0 the p h a s e s $j^and <Ï>L of 2 R and D L a r e considered to b e z e r o , then at z = 1 the phases a r e
* R = J^R1 2ïï - Upjl and «•i k L l = 2 7 7 - n L l ,
if the wave-vector k i s taken in the direction of the positive z - a x i s . F o r the r e a l p a r t of the plane-polarized wave D = Dj^ + Dj^ at z = 0 we then find
and at z = 1
24
DS,°' (cos2ff^
iRl + cos2iT^nLl) + o'y' (sin2 c ^sec.n-- sin2ïïi^nLl). This indicates that D at z = 1 is rotated clockwise over an angle <^=-|27r-^(nL - nj^)l with respect to D at z = 0.
• 3
The optical rotation per length is consequently:
P = T (n^ - nj,) [rad/m} (clockwise). (II. 12)
If the exponential factor e"^-' I + i'^' is considered instead of g i k ' r - i uJt^ R should be replaced by L in(n. 11) and L by R. Equation (II. 12), on the other hand, is not changed. Therefore, the description of the optical rotation, unlike the double refraction, requires a c a r e -ful definition of the sign of the phase in the exponential factor.
For uniaxial crystals the complex tensor e^ + i Sr (with propagation of the light in the direction of the optic axis) has the general form
/ ^ 0 0,'°' / o ., - - ' • '
>ö Ö ' ; / / " r ? ö o I • »-'3)
A similar expression holds for jil^' + i i£^' . If the identities C = (ei2)r A x '
and C=(M,2)iVnx *) ^''-''^
are introduced, the following relations between D and E, and B and H are found for propagation of the light in the direction of the optic axis (k = kz):
D = e , ( ^ ' + i^i.' )E = eoi^r' I - i C ^ k x E = eo|'r°' E + C ^ ^ B, (n.i5) B = ^^o(M? +iHÏ' )H= ^oë? H - i ^ ^ ' k X H= M,^ï' H " 4 ^ 2 (to is positive in these formulas). The relation between the circular refractive indices nj^, nj_^ and Z, Ë, follows then from Maxwell's equa-tions and (II. 15):
VL
| l ( e J r ? + (f^i)r ^1 +-^l' **) ("• ^^^ *) nj_is the (first order) refractive index for polarizationperpendi-cular to the optic axis: n_L= '\/(eJ|^' (f^i)']?'.
(cf. the refractive index for isotropic crystals: n = N/e^+tj..) ^*) Quadratic terms in ^, g have been neglected,
sec.II-3 25
with the (+, -) sign corresponding with(nR, n-^). The optical rotation p is obtained from the equations (11.12) and (II. 16):
p = - f j ( f i J r ? + (si)r « H ^ ] (clockwise). (n.l7) The permeability tensor Mr is, in non-magnetic materials, generally approximated by the unit tensor ^. Furthermore, the factor 4 can be proven to be equal to zero in the classical point-dipole model (see appendix B). In this model we thus find:
P = - J C [ ^ ] (clockwise). (n.l8) In the next section it will be shown that e^^' (or ^) in the
classi-cal model is determined by the geometriclassi-cal arrangement of the atoms. This will here be illustrated in the classical Lorentz picture of an i s o t r o p i c crystal.
From the local field in this model (cf. (n. 3)) and the average electric field
the following equation in P is found:
P = N a ( ^ - ^ + 4 ) P . (11.19) This is in fact the kind of equation that is found in Ewald's theory
(sec.n-4). In isotropic crystals exhibiting optical activity, the local field at dipole h will have the form:
E h = Ë - . | - + i r ^ , (n-20) with - ^ ~ 3e, = e ,
With this kind of local field the equation in P becomes
or
^^Wa--3^-'Z}^-(l~-l)^-(N = number of atoms per volume; in an optically active crystal the unit cell has to contain more than one atom; this, however, is i r r e l -evant in the further discussion).
26
sec.
n-3
substituting g!^' + i i ' ^ ' for the d i e l e c t r i c constant e^. and solving for I r ' , or by substituting successively Uj^ and n £ and solving the equations for right and left c i r c u l a r l y polarized waves.
In the first c a s e we find:
XT/n, - Q - l y Na • 3 • L e ^ . l + icl;> e^ - 1 (e^ - 1 Na
^' , itl - ^r)
ï»Y» ~ J. T J. ^ y ^ T * leading to (Clausius-Mossotti) (IL 21) 1 - i N aand, with (11.14) and (11.18):
p = . ^ y ( f ^ j l l l ' . [ £^1 ] (clockwise) (II. 22)
AQ n m
(nj_= n for an isotropic c r y s t a l )
In the second c a s e we c o n s i d e r the right c i r c u l a r polarization (P, -i P, 0) with refractive index n^ and the left c i r c u l a r polarization (P, i P . ' 0) with n ^ (cf. (H. 11)).
If r ^ - -Ö is put equal to a, we find the equations: a - i y 0\ / P \ 1 / P \ i y a 0 - i P = 3 - i P
0 0 0 / \ 0 / " R - 1 \ 0 / and
from which Uj, and n^ and the optical rotation (II. 12) follow: 1
- = a - y j
Z ( e r - l ) ' n ^
p = ^ (nL - nR) = - ^ y ( i J U l I I [ £ M ] (clockwise).
AQ AQ n m
Both methods give, a s expected, the s a m e answer.
In an anisotropic c r y s t a l containing different kinds of atoms, the local electric fields will have a m o r e complicated form and will lead to an equation of self-consistency : ' ^ R - i /
1 - a , r i
„2 1 a 1 y ) " L - 1 " R -•"^»i^
sec. n - 3 27 0 F p U FiP 0 U 0 0 ü P = (F'°' + i F ' " ) ( ^ ^ ) P , (IL 23) — = n - 1 —
in which ten tensor F determines the double refraction and F^ the optical rotation, n is the refractive index for a certain direction of polarization. For uniaxial crystals with propagation of the light in the direction of the optic axis F has the form
( 0 ) , . _ , ( • > )
F = F ' ° ' + i / ' = 0^ F, o] +i
~ \o 0
FJApplying the same method of calculating URand UL as before, we find for the dielectric constants for polarizations perpendicular (1) and parallel (//) to the axis:
(sr)l=K)r = l + Fi°',
(^r)// = ( S ) r = 1 - F;°' and for the optical rotation:
. F J ^ ' rad
P=-X,-^ [ ^ ] (clockwise). (n.24)
Expressions for the tensors F ' ° ' and F'^' will be derived in the next section. ~
We shall be primarily concerned with the classical model. It should be mentioned, however, that both double refraction and optical rotation are also found in a quantum-mechanical model-without-local-field-correctijons. In such a model (cf. also section II-2) an average polar-ization P is found from a (crystal) polarizability tensor a'°' + i a'^' with uniaxial symmetry: ~ ~
P = NeJaf''' +ia'^) ) Ë , (11.25) From this equation we find a real dielectric tensor
ei.°'=I + N^'°l
and an optical rotation
P ^ . Z ^ ^ B L [ Ï ^ ] (clockwise). (n.26) The factor 2 appears in the last formula because z can be proven to
be equal to | (11.14) in this quantum-mechanical model (see appen-dix B).
28 s e c . I I - 4 II-4. Double refraction and optical rotation in the Ewald theory.
The central problem in the c l a s s i c a l point-dipole theory i s the summation of point-dipole fields. A special difficulty i s m e t with in the calculation of refractive indices in c r y s t a l s b e c a u s e the e l e c t r i c field at the site of one p a r t i c u l a r oscillator is sought; the field of this oscillator, having a singularity at its own origin, has to be subtracted from the total field.
In an isotropic dielectric between two condenser plates the local field Eh (= E(rh) at the oscillator h) can be shown to be equal to (II. 3):
_ P D £ P D 2 P / „ „„^
E + 7^ = f^- — + ^ = —-•^— . (11.27)
The t e r m . = i s the field due to the external c h a r g e s on the condenser
-£o
P
plates, - - = - i s the field due to the induced c h a r g e s on the edges of the P
dielectric and ^r= is the field due to the c h a r g e s on the " L o r e n t z -s p h e r e " in-side the d i e l e c t r i c . In thi-s model the contribution from all
2 P
dipoles to the local e l e c t r i c field is equal to - „ ~ . It has been shown by de Wette (1959) that a d i r e c t , planewise summation of the e l e c t r i c fields of the individual dipoles does indeed give this contribution. (The sum i s mathematically not uniquely determined; a " s p h e r i c a l " summation, for instance, can easily be shown to give no field at all). T h i s picture, in which the local field i s considered to b e partly due to the externally applied field and partly to the environment, i s commonly found in l i t e r a t u r e . In Ewald's theory, however, the problem i s considered from a different point of view. The c r y s t a l is thought to be infinitely l a r g e , and the existence of an electromagnetic wave, propagating in the direction of the wave vector k, i s accounted for by adding a phase factor e ^ - - - h to the dipoles p^. The local e l e c t r i c field E]^ at the dipole h is then given by the sum of the fields of all dipoles except h.
D 2 P 2 P This sum then equals -=- - -Q -=• instead of - ^ -=-.
So "5 So o Co
The refractive index is calculated from the condition of s e l f c o n s i s t -ency that one p a r t i c u l a r dipole should b e excited by the field of all other dipoles, which, a p a r t from phase differences, oscillate iden-tically. (In a composite lattice, corresponding dipoles in different unit c e l l s oscillate identically).
The Hertz vector Z i s particularly suited to the mathematical description of t h e s e dipole fields because only its magnitude is r e l e -vant in carrying out the s u m m a t i o n s . Different orientations have to be considered in the final determination of the electric fields from the equation:
sec.II-4 29 1 S^Z 1 ö^Z
E = curl curl Z + t^Z - ^^ ^^ = grad div Z - - ^ - ^ . (II. 28) The Hertz potential at the point r due to a dipole at r ^ is given by the formula
_ ^ - i a j ( t - | r - r i n l / c ) + i k . r m
7(T)-O'^°^ i i . (11.29)
2 ( 1 ) - P m 4 f f e j r - r m l
The total potential at r in an infinite medium is subsequently found by summation over all dipoles (m). We shall here consider a crystal with 3 atoms per unit cell. Each atom is located at the point of a vector R i + r t = Ijgi + l^a,^ + Igag + rt; the vector Ri(ai ,§3,^3 being primitive lattice vectors), determines the cell in which the atom is located, the vector rt the location of the atom t in the unit cell. The Hertz vector Z{ii(r) at r in the neighbourhood of atom h' (in cell 0, 0, 0) due to all dipoles except h', is then given by the formula:
1 "^ , , -1 i k n | r - r t - R i | i k - R i
h'^^ - 47reo t=l 2t ^ f | r - r t - R i l
Q i k o | r - r t | 4ïïCo t=l -* l l - I t l (the time factor e'i^* has been left out)
E ' ^ summation over all lattice points except (0, 0, 0). 1
E ' = summation over all dipoles in cell (0, 0, 0) except h'. t=l
k^ = 2ïï . ;^ wave-length in vacuo.
^ 0 - 'V > 0
Ao
|k| = — ; X = wave-length of the plane electromagnetic wave which ~ ^ ' propagates through the crystal.
Differentiations according to (H. 28) are subsequently carried out in order to obtain the electric field Eh» = E(r)r ^ rh' ^^ the atom h . The self-consistent equation of motion for the dipole h« is then finally given by
— i h ^ P h ' = ^ h ' - (^^-31)
^° " Ik|
This equation has solutions only for certain values of Y = ^> ^"^ thus provides a method for calculating the refractive index.
30 sec.II-4 succeeded in giving a rapidly converging expression for the badly converging summation over 1 in {11. 30):
TT' (r - rt) = E' , ^ = ^ 1 ll - I t - Bll 47r e' ^ISm +is|^ - k / } / 4 E " + Kg^^ +k) • (r - r^.) s i n k j r - r t i ^ " ^ m ISm + k l ' - k , ^ " ' l r - r t l - 2 f 7 ^ { e ^ ' ^ = ' - ^ - - ^ t l ^ ( | r - r t | E . i | p ) . e - i ^ o l r - r t l ^ ( | ^ . ^ ^ | E . i k , ) ^ ^ ^ ^ c o s k J r - r t - R i l ^ { e ^ ^ o l l - r f - R l l ^ d ^ , | ^ ^ i k 1^0 l l - I t - R l l 2 | r - r t - R i r ^ -^' 2 E ' . e - ^ ^ ° l ^ - - ^ t - ^ l U ( | r - R i - r t l E - % ) } ] e ^ ^ ' ^ ^ . (IL 32) 3m"27r(mjbi + mgbg + m3b3)with bj^,b2,b3 reciprocal lattice vectors, V = volume of a unit cell,
$(x) =—fr- / e dz (error function).
E is aparameter with the dimension of a reciprocal length which may be chosen in such a way that both series converge rapidly. TI' is
in-dependent of E.
In some problems the total potential is required which includes the term with Ri= 0. In that case the expressionfor the potential becomes:
4^ e-{|qm + i s l ' - k o ' } / 4 E % i ( q m + k ) . ( r - r t )
c o s k J r - r t - R l l 1 r i ^ o l l - I t - g l l ^ . i „ | „ i^o. ^f^ I r - I t - R i l - 2 | r - r t - R i l ^ " * ( | r - r t - R i l E + g^)^
- i k o l l - i t + Ril , , ik , i k . R i
