Polarized i.r spectra of monoclinic crystals

SpectrochimicsActa, Vol. 31A, pp. 1255to 1263. PergamonPress 1976. Printed in NorthernIreland

Polarized i.r spectra of monoclinic crystals J. HERRANZ and J. M. DELGADO Instituto de Q&nice F&ice “Rocaaolano” Consejo Superior de Investigeciones Cientfficas, Madrid, Spain (Received 24 November 1974) Al&a&-Expressions for the polarized absorption spectra of molecular monoclinic crystals are presented in 8 form which is useful in andysing the spectra in terms of crystal and molecular structure. Previous reported treatments 8re discussed. It is shown that the spectra with crossed polarizers c8n be inter&eted in an easier way. The polarized i.r. spectra of-in-plane modes of monoclinic a-glycine-d, have been an8lyzed.

INTRODUCTION Polarized

i.r. spectra

provide

useful

transition

is parallel

of molecular

information

moments.

single

on

the

can

be

from crystals with the directions

of

the

tensor

axes

spectra

with

running

the

parallel

to those

axes, [l]

other cases. Monoclinic tensor

axis

the other

parallel

information symmetry the

fixed,

electric

but may

but they

to the b crystallografic

are not symmetry

fixed,

on the frequency

of

monoclinic

plates

their

slow cooling

have

expressions

to

the

of

been

[2-41.

analyzed These

monocrystal.

that maximum the electric

made.

or minimum

vector

moments

are

imaginary

part,

absorption

is parallel

of the dielectric

associated they

with

the directions

This

result

the

axes

of maximum

of

has usually

of

the

crystals

crystallizes

system

[B] it appears

problem general

group of organic in

the

that

has not been made. treatment

formulation

a general

mum of absorption

study

It

is shown

[2] is valid

in molecular

crystals,

the maxi-

arises when the electric

vector

are

were

also

two

crossed

in front

-45’,

at the entrance

on a Perkinsix

To avoid

different

polarization

in front

respect

of the

to the slit

The light stroke in all the cases. The

shown

in

obtained plate

Figs.

with

polarizers

of the

coefficients

samples.

at

with

of by

(the

and the

1 and

the

polarizer analizer

of the monocromator).

spectra, at six different

2.

sample

orientations

at at The

of the

sample, are shown in Fig. 3. THEORY

larger than seems

+45’

to the plate

that the

ca8e, which

evaluated

obtained

was mounted

at

for the extreme

is appreciably

were

+45O

recorded

of the

In the present work a

but in the opposite

to be more likely

crystal

on a cloth

the absorption

of the sample.

spectra

Spectra

wa8

from polycrystalline

spectrophotometer.

observed

[3]. been

molecular

monoclinic

is given.

of WARD

case when birefringence dichroism,

the

by

A thin

to a KBr

was polished

thickness

the polarizer

between

the largest

was grown

uc plane was cut

spectrophotometer

perpendicular

supposed [ 51. Although

with

and water until a thickness

The

spectra

125

monocromator,

in general

absorption,

to what

i.r.

effects

plate

measurements,

orientations

arises when

will not coincide

is contrary

The Elmer

imply

Since the transition

with

10 pm.

was determined

to the axes of the real

tensor.

with alcohol

about

the validity

Such assumptions

The

wetted intensity

using

expressions

do not allow to check experimentally of the assumptions

of a-glycine-da

of a solution with a little seed.

out and stuck by its edges with par&e

Further-

UC plane

compared

plate parallel to the crystallograflc

directions in

results

to the

(monoclinic)

STALHBERQ [7] for a-glycine-d,,.

One monocrystal

axis;

part,

of a plate parallel

and

inter-

of crystals.

of a-glycine&

studied,

dielectric

simplified

EXPERIMRNTAL

UC plane

of the light.

parallel

crystals

approximate

been

the

i.r. spectra

spectra

in

will not coincide.

Spectra

part

have

with

of a crystal

those from

vector

more the axes of the real and imaginary general

The absorption

crystals have one dielectric

axes lie on the crystallografic

depending

agreement of polarized

uc plane

by

be difficult

to the axes of the imaginary

in

pretations

molecular

easily obtained dielectric

Such

tensor,

crystals

The problem of a plate

to be considered

parallel

is the transmission

to the crystallografic

ac plane,

when a plane wave strikes normal to the face of the crystal. 1255

1256

J. HERRANZ and J. M. DELGADO

cm-’ Fig. 1. Polarized ix. spectra of a-glycine-d, (ac plane). O” corresponds to incident electric vector parallel to c axis.

0

!

I

1

I

I

I

I

I

I

lKlo

I 500

I

I

L

cm-’ Fig. 2. Polarized i.r. spectra of a-glycine-d, (QCplane). 00 corresponds to incident electric vector parallel to 0 axis. Inside

the crystal

waves are propagated

[4]

two

transverse

with wave vectors

have the direction of the incident light.

plane

K, which

wheneraS, lrr, are the components of the symmetric dielectric tensor, l, which can be written

Its values

r = r’ -

are given by the equation:

ic”

(2)

E’ being the real part and E” the imaginary (1)

part

(the notation of TURRELL [4] will be used);

E, and

Ey are the components

vector.

Each

wave

Equation

(1).

corresponds

of to

the an

electric

eigenvector

The ratio of the amplitudes

of

of the

Polarized i.r. spectra of monoclinic crystals

lo -

1267

67”

0 ax0

1500

cm-’ Fig. 3. Infrared spectra of a-glycine-d, with crossed polarizers (ac plane). O” corresponds to the polarizer parallel to c axis and the analizer at right angle. electric field components

for the waves associated

to the wave vectors Ki, Ks, are given by:

where q( - 1 < A linearly

(3) where

p, is in general a complex number. According

to Equation

(3), the waveS are elliptically

polar-

ized, with identical ellipticity and sense of rotation, and

their

major

axes K,

are

perpendicular.

wavevectors

K,,

components

of E from Equations

as a function

7 <

1) gives the ellipticity

The

incident

polarized

along the b (E

considered

as

the

plane

wave E”,

which is

y) axis, can be formally

superposition

of

two

waves

E,O, E,O with the same State of polarization those propagated inside the crystal. x axis, its electric vector components

of p and the

E,’

(1) and (3) are

are given by:

= ti cos L2 = Elao + E,,O (7)

E,' = @ sen $2 = E:$ + E,,O where, taking into account Equation

2 121 -

=

K22~2 = Co2 -

pczz

2

2

E$ -

2i?%,Ki

%Z

Czt --

=

K1

(4)

E,,O =

=

n2

-

K2

2

-

2in2K2

in)

the directions

of the major

P2 +

axes

of the ellipses

p reduces to a pure imaginary

number

SI -

1

=

ir]

&@

-

p

sin R)E”

For

the

crystal medium)

wave

plate

(-co8

Q+

(6)

psin hl)B?

1

with

(assuming

behaves

polarization

as an isotropic

state 1.

state

it is inside

refractive index ni and absorption polarization

(3) I9

P

the reflected and transmitted p

+

sin R COBn + -

E2; = P

(5)

the z and x axes are taken along

1 and 2 respectiveIy,

n

P

1 ( cos

2

Pa +

= z (n c

CO8

1

E,,,o = P

n, being the refractive index, and K the absorption When

p +

P2$.lP

where we have written K

(3),

Sin hz

P

e

P2

P 2

as

If the incident

light is polarized at an angle R with respect to the

given by:

constant.

of the

waves.

1, the

a vacuum

medium,

with

coefficient pi;

light are also in the

(As far as the authors know

1258

J. HERRANZ

this property it

can

has not been pointed

be

conditions

easily

verified

and J. M. DELGADO

out previously;

using

the

[S] at the plate surfaces).

boundary

Similarly,

for

the incident tions,

electric

Equation

interpretation

vector.

(13)

of the spectra.

the wave with polarization

state 2, the crystal plate

it is easier to analyze

behaves

media,

crossed polarizers.

index

as an ns,

isotropic

and

transmitted

absorption

electric

with

refractive

coefficient

vector,

Q.

The

Et, has components

for the wave t

1

= (1 1 -

-

VI2 exp [( -4&/c)(nI

T

EL =

(9)

of the plate

i+]

(10)

if+]

1 +n,

amplitude

being

the

expression. intensity

t,

is, p = iq).

using Equation

+

The

trans-

a

similar

by

and the transmission T1

=

Equation

r12J2

( )

equal

to

measured 2;

@+2*

to Equation

ir], which

means

-

for which

asterisk indicates

c = cos Cl; complex

1

(13)

that

the

R

angle

is

and

The

polarized

in terms

of single

crystals,

the

spectra

Since

of the direction

electric

is a minimum

=

with

Equation

of

with

(16)

(2rlll + q92

orientation

(17)

ellipses

and

the

of Y, can be found from

crossed

simplifications

coincides

(T,,,),

From

polarizers,

without

in the analysis

of the

ORIENTATION OF THE IMAQINARY DIELECTRIC TENSOR

conjugate.

i.r. spectra

analyzed

of the

data.

the of

SPECTRA WITH CROSSED POLARIZERS commonly,

lines.

7, as functions

introducing

axis of ellipse

I0 = Eo8/8n

1 (1’3)

values for Cl = m/2,

the transmission

transmission

bisecting

ellipticity the

cos2 2LI

coincides with the axes of the ellipses, while

Therefore,

(6), p has been taken

with respect to the major

a = sin a;

5*t21

by,

values for R = (2n + l)a/4, n = 0,

TminlTm*x

w,t,*+ t,*t,)

is given

r12)2

. Thus, the orientation

1,2,...

(15)

1

( = 1,/I,)

(16) has minimum

and maximum

their

where according

v2)

2 sin22h2 +

(T,,&,

rl

axis (14),

ItI - t2,2L

for maximum

1 + 2isc - rl

that the R

to the major

cos2 a)

0 -

(1 -

vector

-

(14)

Now Equation

t2) L (1 -

TL

(8) and

‘I2 (1 +

respect

i(sin’

of Cl, the transmitted

lEtI2 = 10

cos R

(6) becomes,

of indices n1

I’, taking into account Equations

&

tsE2,0)

we assume, as before,

(9), is given by,

It(a) =

sin R

between

(12)

given

As a function

at (9),

X [-(~+~)sin&2cosQ

coefficient. is

+

of ellipse 2. (That

-iq medium,

reflection

coefficient

+(QE,,~

EL = ,!#‘(G -

Rl = Id2 R,,

with

by Equation

+ tgzzo)

with

and K~, that is,

mission

obtained

(11)

at the boundary

and a semi-infinite

a simple

by an analizer

-(tlE,,O

angle is measured

of the plate, and

-1 -“1$_%

1-

is the reflection

-

the spectra

EL transmitted

For convenience

r12) exp [( -277iv/c)(nI

to

As it will be shown,

90’ with respect to the polarizer,

1, given by, [S, 91,

where d is the thickness

vacuum

coefficient

The amplitude

approxima.

lead

is given by,

Ezt = Elzt + E2at = tlElzo + t,E,,' E,t = El; + E,; = tlElzo + t2E2Zo where t, is the transmission

Without

does not

the

the direction crystal

principal

are

axes of the imaginary

is

one is mainly

of

tion.

The

of the transition

vibrations

interested

elements

related

moments with

dielectric

in determining

of the imaginary

the

tensor,

its orientapart

of the

1259

Polarized i.r. spectra of monoclinic arystals dielectric

tensor

referred

to

the

ellipses 1 and 2 from Equations

major

axes

In general,

of

(2), (4) and (6) are,

it can be expected

be easily verified with

crossed

that 71<

experimentally

polarizers).

In

1 (it can

from the spectra

such case Equation

(13) becomes I’/10

= 2-. [?s2K2c2Zn 1 - r12 h” =These

the

l_111(

elements

equation

mum values

n12 -n,")

2

This

T2W,l (20)

(IC,” - f~,“)]

-

of the eigenvalues

with

the

Furthermore

ellipses

Equation

dielectric

axes

of the imaginary

(nr2 - n,2) - ( Kis 1 II

%Kl

(24) allows

-

K,“)

n2K2

to determine

1

needed.

such evaluations

A possible method

may be the following:

Q are obtained

from Equations plate

thickness

< 0.2,

transmissivity

It I2 is approximately

]tl2 W (1 -

from

from

Equation

is a function evaluate

n.

expected

that

of n. and K

K

thickness. crystals

it

11 is to

r212 r212 to be

in such case, from Equation

n,

11 -

r2] W 4n/(l

+ n)2

(26)

ANALYBIS OF TEE BPECTRA FOR R < 1 method

above

indicated

due to the difficulty

of obtaining

the thickness plate.

However,

the spectra can be analyzed way.

is not accurate

axes

moment

minimum

absorption

electric

vector

transition

M.

arises

that

larger

molecular

the

spectra

of crystals.

of Equation

than

the

unity.

n1 -

n, <

to the

with the simpli-

5 N 0, is valid

crystals,

and

incident

and perpendicular

that the term in parenthesis

of the

maximum

when

the

tensor

to the direction

of polarized

conclusion

much

dielectric

in agreement

fied interpretation The

of the transition

Therefore

is parallel

moment,

(28)

At this frequency,

of the imaginary

are along and perpendicular

(25)

and 11 As

the

(11)

The

dielectric

In K,

assuming (24) is not

general,

for

therefore

by,

it is also possible

K,

molecular <

so that

r212exp (-2,WKd/C)

at two

For

Equation (lo),

given

(25) allows to determine

by measurements

of all

for doing

First q and

is chosen

exp[-&Kd/c]

Equation

the

imaginary

[4],

transition

(16) and (17), and

then ]t,12 and ]t212 are determined the

the

the

and C is a constant.

principal

5, once nr, n2,

tions (13) and (16)), allow for the evaluation

If

At

W’f, MS moment

of

of the principal tensor.

where M,, M, are the components (24)

(24) shows that and minima

with

(23)

In principle, transmission Kr’ KZV are known. measurements made at several values of n (Equathe constants

M

(and (16)).

associated

( 18-23)

rl

=-

zsK) sin 2E

moment

the with

Equation

dielectric

isolated-band

tensor can be written

r12+

(13).

imaginary

(22)

Equations

Equation

an

Ea2” = 9” sir? t + -

of

(27) give us the direction

of

which

coincides

inside the crystal

values

the maxima

transition

= 4 .c;’

for

value,

when 7 Q 1, Equation

(21)

B2” cosz e

vector

of polarization

Qr” = Q” cos2 E + F2” sins #$

li2”

tan2E

of electric

has an extreme

of the

peak

From

and mini-

. . . . Thus

E N 0 and therefore axes

tensor are given by

maximum

minimum

E; and c2” of the tensor cn and the angle 6 (v/4 > E > -7714) between the major axes of the and the principal

has alternate

(27)

at R = m/2, n = 0, 1,2,

orientation

transmission the direction

as a function

= It,]2 sins rR + It212CO82R

At

the

K1 -

peaks

the

main

thus for Equation

K2

absorption

to have

than 0.1 (it can be easily verified

\

I

bands

larger

values

experimentally)

(29) to have values appreciably

larger than the unity,

the birefringence

has to be

about 0.6, which is very unlikely. In

molecular

crystals

can usually be neglected, Equations

losses

[tl2 = exp [ therefore relative

due

to

reflection

that is R Q 1. Thus from

(10) and (12)

values of

in a much more simple

of

Kz, can be expected

practical

in most of the cases,

K1 -

?trK1 - n2K2

from

measurements

values of

the spectral

K1

range.

(30)

-6h’Kd/c]

of

]t,12 and

and Kg can be obtained Assuming

lt212, for all

7 N 0, and t N 0,

J. HERRANZ and J. M. DELGADO

1260 t,he eigenvalues Equations

of

(18-23)

Since the variation is small [lo]), to

the

imaginary

tensor,

of n, aa a function

(anomalous

reflection

the

2

eigenvalues

of

Equation

the

ratio

1-4

D

‘(Principal

to be written

It is interesting

to note that for an isolated

this ratio has been usually found to be nearly this

Equation

result

is

to

be

expected

part of Equation

band

cos 22 tan 2E -

zero

(COB2Xltan 2[)

because

(28) shows that in such cases one eigen-

value of .f” is zero.

The elimination

POLAIU!ZATIOl'i ON THE DIELECTRICTENSOR

The conclusion

that

q N 0 implies

from the expressions and

P, where

for not to weak t N 0, may

also be reached r] and E with x,

x is the angle between

P P2 + 1

tan 4l

axes of E” (see Fig. 4)

Equations P = (Ei -

<2’)/(C1” -

c;)

(34)

with Q’, Ed’, Ed”, Q,” being the eigenvalues I, E .

the case that to the major

Equation

%IZ)P/(1 -

P’)

(35)

the z and r axes are chosen axes of the ellipses

(6), Equation

1 and 2,

(35) becomes Q)H

(36)

where H = 2rl/(7?2+ 1)

23)

Equation

and

Equation

(2) together

similar

x) i[P

where

6 -

with for

of

E (-a/4

solutions.

2%

(41)

(42)

P2cos4x tan2 2[

(43)

1

to determine

of P and x. Within

< 8 Q ?r/4) Equation

Equations cI1’,

(21-

paa’, eIO’,

has

solution

from Equations

that

values

when

x) -

and the principal

i CO82X]H the major

(38) axes

axes of the real

tensor (see Fig. 4). Equating

the real and

E/x < 0,

> 1 or <0,

(44)

(43) and (33) taking

into account that from Equation Note

(37). 0 < H2 <

Equation

and when

negligible

P sin

2~ N 0,or P > 1, (Equations

At,

the

ellipticity

peak

of

an

is unlikely

(7 N 0) implies isolated

P >

their 1.

Therefore

band

the

that

of 6’ is zero. from

later

Equations

The (44)

E is negligible. assumed

of

absorbing

crystals

from

Equations

(42) and

5 N x, which means that maximum absorption

either

l

as follows

treatment

is

(37) and (41)).

WARD [2] and TURREL [4] have actually in

1.

take

to be fulfilled (see Equation ,’ usually is very small and by

(34)) because Ed’ Equation (28) one eigenvalue condition,

(43),

E/x > 1 its value

to. A

condition

two

which

using the condition

and (42) implies

i sin 2f = CO82(E -

which follows

H

the range

(42)

One of them is an extra

can be eliminated

former

x, is the angle between

of the ellipses dielectric

expressions

(37)

(36) yields

P sin 2(E -

(40)

0 < E/x < 1

2612 = i(czn -

Using

1+

(41) and (42) allow

(4)

%Z = (%Z -

from

of Q’ and

sin

tan 2% tan 25 -

(33) and 6 as a function

parallel

---I/HP

P2 sin 41

=

H2 = tan 2~ tan 2E -

and P,

For

(39)

of tan 2E, Hand P, from Equations

the principal

- ““14 < x Q ““14

Equation

sin 2~ = H/P

+ sin 2% =

H -=-H2 + 1

absorbing

which relate

axes of B’ and the principal

From

(38),

(39) and (40). yields respectively,

DEPENDFJ?CE OF WAVE

bands,

of c’)

axis

Fig. 4. imaginary

[3, 71;

>

X

dielectric

the dichroic

axis of 6”)

e:

as proportional

imaginary

(31) allow

(Principal

of frequency

can be neglected

g1 and IQ can be considered

tensor.

from

are,

arise when the electric

that (44)

and minimum vector

is along

Polarized i.r. spectra of monoohnia crystals the principal

axes of the real dielectric

the transition

moment

to the direction

tensor and

The full line given in Fig. 5, represents

is at an angle 2 with regard

of maximum

Y,

absorption.

Fig.

3.

Spectral

extinction seen

that

in

extinction electric

regions

are indicated the

vector.

corresponding

whole

occurs

for

observed

some

Therefore,

(17), the ellipticity Only

in

the

region

values,

this region,

to

total

region

orientation

seconding

the

to Equation

are linearly

715-770 cm-l,

but, since there

it has no interest

polarized

in our case) and the

determined

has

been (16).

and

is valid

minimum

when

are represented

by

agreement

the line

and

have

with

spectra.

the

isolated ratios,

in our analysis.

the

minimum

of

dots

in Fig. drawn

the incident

crystallogra6c

c axis)

in these

plots

are in

the crossed (Equation

electric vector

corresponding are very

small

7

-,BS

l

0 60’

-

0

a.

00 0 -60

-

aMinima 0 Maxima

I

I

I

3/”

of of absorption obsorption

I

I

I

81

1500

II

”

1000

500

cm-’

Fig. 5. Angle Y,

so

I-

between polarization direction of wave 1 and the o crystallogra& function of wavenumber.

axis, as a

792 cm-’

933 cm-’

so

40

lIlL_Ll 0

0

20

60

100

140

IS0

20

60

-

I

loo

140

to

6. The dichroic

1200 -

(27)

obtained

they

Some plots of 1,/I,

bands, are shown in Fig. observed

5; from

of from

Equation

q << 1. The values

27) vs. 0 (angle between

are no bands in

of the frequency.

from

Y has been also determined

maximum

polarizers

polarized.

q may

of one of the waves

It

which

total of

are linearly

the angle

axis

c axis, as a function

the

It may be

the major

crystallograflc Equation

7 is zero, this means that the

two waves inside the crystal non-zero

are shown in

by a thick line.

between

(which

BPECYBA OF a-l+LYCXNE-& The spectra with crossed polarizers

1261

I.30

FIG. 6. Transmission, as a function of the direction of incident electric vector. O0 corresponds to incident electric vector parallel to c axis. The numbers in the Figures denote the wavenumber of the bands.

in

1262

J. HJGRRANZand J. M. DELWDO

2000

I600

900

ICC0

1600

I400

1200

1000

E

600

700

6cO

5c0

4co

cm-’ Fig. 7. Transmissivity

Jtl’ N_ exp [-&~d/o].

Full line, wave 1; dotted line, wave 2. d _

Table 1. Angle between the transition moments (specie Bu) and the c crystsllografic axis, for glycine d, and d,. 6, [ref. I 1620 ?

73 Y

76

4’

1640

05

orientation

90

1548

II

1540

0

1530

72

1500

1445 ?

72

1040

15

1421

44

1416

0

1330

42

1217

140

1311

72

1176

120 (60)

1107

30

1119

_L

933

90

1042

27

909

0

918

66

792

90

902,

25

867

701

45

666

ItsI

(40)

according

544

167

167 (13)

(13)

518 ?

L

467

157 (23)

JK~ -

to the discussion

Fig.

~~1 >

7, 0.1.

previously

been

at the maxima

of the

for a-glycine-d,

The

bands

and the values are described vibration

It may be observed between

as should

directions

which

the same molecular

general agreement bands,

tensor

by STAHLBERG [7] for a-glycine-d,,

given.

approximately

the

dielectric

axis.

1, the angle Y bands

been correlated.

496

be seen from

bands

an axis of the imaginary

reported

46

field

made, t N 0; that is Y (Fig. 5) represent,s the angle

In Table

17

is given.

absorption

absorption

20

(180- q)

10 pm, it may

between

157 (23)

with

Since the sample thick-

that

Therefore

field, coincides 7,

to the spectra with electric

and the c crystallografic

533

,

In Fig.

ness is about the

1 and 2 for which the

electric

+ a/2, is also given. at

R

to exp ( -4nv~rd/c).

N exp (--4xv~~cZ/c),

which corresponds at Y

621

a When 9 > 90

5).

frequency

using in each spectral

of Figs.

of incident

the angle Y (Fig.

105 (75)

of

that the reflectivity

lt,12 is equivalent

range the spectrum

1600

on the meaning

(t,( 2, as a function

The curve has been drawn

II ?

II

The transmissivity, is negligible,

1590 ?

1138

with the discussion (32).

is shown in Fig. 7. Assuming

d2

*a

agreement Equation

10 pm.

of

thevalues

be expected the

by have

that there is for correlated

for they

corresponding

have

represent transition

moments. Acknowledgement-The authors gratefully acknowledge support from the Spanish “Fond0 National para el Desarrollo de la Investigacibn Cientifica”.

Polarized ix. spectra of monoclinic crystals REFEREI’XES [l] R. NEWMAN and R. S. HALBORD,J. Chem. Phye. 17,1276 (1950). [2] J. G. WARD, Proc. Roy. Sot. 2288, 205 (1966). [3] H. SUSI, Spectrochim. Actu 17,1267 (1961). [4] G. TURRELL, Infrared and Raman Spedm of Cryatala. Academic Press, London and New York (1972). [a] L. W. DnasqJ. MoZ. Spectry. 8,86 (1962).

1263

[6] E. E. KOCH, A. OTTO, Chem. Phye. 8,362 (1974). [7] U. STAELBERCI,E. STEGIER, Spectrochim. Acta BA, 476 (1967). [a] L. D. LANDAU, Electrodynomkca of ContinMedia p. 278, Pergamon Press, London (1960). [9]M. BOEN end E. WOLF, Prineiplee of Optics, p. 61 Pergamon Press London (1970). [lo] J. C. CANTI, M. BI-ON, J. BADOZ, J. Phy8. 82, 691 (1971).