NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF LINEAR CHAIN SYSTEMS

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NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF LINEAR CHAIN SYSTEMS

C. Wijers

To cite this version:

C. Wijers. NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF LIN- EAR CHAIN SYSTEMS. Journal de Physique Colloques, 1983, 44 (C10), pp.C10-397-C10-402.

�10.1051/jphyscol:19831082�. �jpa-00223540�

NUMERICAL CALCULATION OF THE OPTICAL PROPERTIES OF LINEAR CHAIN SYSTEMS

C . Wijers

Department of Applied Physics, Tuente University of TechnoZogy, P.O. Box 217, 7500 AE mschede, The Netherlands

Resum@ : La r6ponse optique d'une chaine atomique l i n e a i r e a @ t 6 c a l c u l e e nu- m6riquement dans l ' a p p r o x i m a t i o n d i p o l a i r e . On montre q u ' i l n ' e s t pas

p o s s i b l e d ' e x p l i q u e r avec l e modPle d i p o l a i r e l e s p o l a r i s a b i l i t & des a1 kanes observees exp6rimentalement.

A b s t r a c t : The o p t i c a l response of a l i n e a r c h a i n o f atoms has been c a l c u l a t e d n u m e r i c a l l y i n t h e d i p o l e approximation. I t has been shown, t h a t t h e d i p o l e model c a n ' t e x p l a i n t h e e x p e r i m e n t a l l y observed p o l a r i z a b i 1 i t i e s f o r a1 kanes.

I . INTRODUCTION

The a p p l i c a t i o n o f synchroton r a d i a t i o n t o spectroscopy has renewed the i n t e r e s t f o r t h e microscopic d e r i v a t i o n o f t h e o p t i c a l p r o p e r t i e s o f s o l i d s / I / . The r i g o r o u s s o l u t i o n o f t h i s problem r e q u i r e s t h e simultaneous s o l u t i o n o f Maxwell's and Schrodinger's equation f o r a surface r e s t r i c t e d many-particle system, which i s beyond t h e present s t a t e o f t h e a r t . The approaches used u n t i l now, s o l v e these equations separately, making the i n t e r c o n n e c t i o n by means o f t h e induced d i p o l e concept. I n p r a c t i c e t h e emphasis i s on electromagnetism and t h e quantummechanical behaviour e n t e r s o n l y throuo,h the p o l a r i z a b i l i t y a. The o l d Clausi us-Mossotti equa- t i o n represents such a s o l u t i o n under assumption o f t h r e e dimensional t r a n s l a t i o n a l symmetry, which was l a t e r confirmed by Oswald and Oseen f o r the t i m e dependent case / 2 , 3 j . S t a r t i n g w i t h Ewald, t h r e e dimensional s u r f a c e systems were s t u d i e d using two dimensional t r a n s l a t i o n a l symmetry /4/. Recent rereach s t i l l concentrates upon t h e same problem /5,6,7,1/. Apart from t h e p r a c t i c a l problems encountered by those i n v e s t i g a t o r s t h e r e i s s t i l l a b a s i c problem. A s p e c i f i c shape f o r the s t u d i e d vo- lume i s r e q u i r e d , t o a r r i v e a t a usable expression f o r the l o c a l f i e l d t h a t works upon a c e r t a i n d i p o l e . The same problem a r i s e s i f one wants t o s o l v e t h e average o f the e l e c t r i c f i e l d , as r e q u i r e d f o r t h e macroscopic form o f Maxwell's equations.

This leaves the separate s o l u t i o n o f t h e electromagnetic equations i n t h e t h r e e d i - mensional case w i t h a degree o f freedom, t h a t makes any conclusion on t h e quantum- mechanical i n f l u e n c e i n t h e model o r about the v a l i d i t y o f the d i p o l e concept i t s e l f questionable. E s p e c i a l l y i f t h e o p t i c a l response o f an adsorbed monolayer i s studied, as i s t h e c u r r e n t f i e l d o f i n t e r e s t , these l a t t e r questions a r e expected t o be v i t a l . The problem of c o n d i t i o n a l convergence /5,1/ does n o t a f f e c t f i n i t e systems. This i s the m o t i v a t i o n f o r t h e present work. The paper a p p l i e s the d i p o l e approximation t o f i n i t e l i n e a r chain systems, t h a t a r e s t u d i e d under t h e assumption o f constant a p p l i e d e l e c t r i c f i e l d s .

2. THEORETICAL APPROACH

The treatment o f systems o f induced d i p o l e s , as can be found i n t h e textbooks, s t a r t s from o n l y two equations:

-f -?

p ( r ) = a

(F)

(1 )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831082

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Equation (1) presents the dipole

6

induced i n an atom a t

?

by the local e l e c t r i c f i e l d

z(F)

and a the p o l a r i z a b i l i t y expressed i n t h i s paper in Farad-meter 2

.

Eqauation ( 2 ) gives the microscopie e l e c t r i c f i e l d ):(; caused a t

?,

by a dipole

6

i n the o r i g i n . The l i n e a r chain, t o be investigated, consists of a number of atoms, placed a t a regular spacing a , along a s t r a i g h t l i n e , such t h a t - t h e axis o f t h e c h a i ~ coincides with the z-axis. E l e c t r i c f i e l d s i n the z-direction w ~ l l be called paral- l e l , i n the x-y direction perpendicular. For parallel f i e l d s of magnitude Eo, the dipole induced i n the atom with index i

,

p., has t o f u l f i l l , -as can be deduced from ( I ) , ( 2 ) and the principle of superpositioA,- the r e l a t i o n :

P; (3a)

o j f i I z .

-

2.1 J 1

where the index j runs over a l l the W atoms of the chain except f o r atom i . In a s i - milar way perpendicular f i e l d s y i e l d :

Collecting the r e s u l t s f o r a l l separate dipoles p . , one arrives from (3a,b) a t a l i n e a r system of equations, analogous t o , but lesslcomplicated as the equivalent threedimensional expressions. The system can be brought t o dimensionless form by defining:

These d i f i n i t i o n s transform (3a,b] i n t o the equations:

where i n addition has been assumed, t h a t :

The equations (5a,b) were solved numerically, f o r what purpose use was made of t h e CDC-Cyber computer of the university of Groni ngen. The programm was written i n FORTRAN-77 and the matrix-inversion, necessary t o solve (5a,b) was performed by means of the IMSL-routine LEQ2S. As a r e s u l t one obtains the normalized dipole s t r e n g t h ' s (a",

. . . . . . . . . ^, IT{)

^{or (nL,}

. . . . . . . . ,

ni) from these r e s u l t s already an

"observable" Ean be obtained, e.g. t i e t o t a l polar!zability, t h a t follows from:

= N

1

^ni//~L

t o t Clo i = l

Since the purpose of t h i s paper i s t o support the a p p l i c a b i l i t y of the dipole model f o r three-dimensional systems, i t i s i n t e r e s t i n g t o calculate something as t h e l o c a l d i e l e c t r i c constant and the local r e f r a c t i v e index, although those q u a n t i t i e s have not much meaning i n the one-dimensional case. For t h i s i t i s necessary t o calculate the average e l e c t r i c f i e l d . Each aton in the chain was surrounded by a cube of l i n e a r s i z e a , properly oriented. As such the f i e l d average follows from:

by means of a commonly known integral transform ( 8 ) can be transferred i n t o :

Where only surface i n t e g r a l s have t o be calculated. After some elementary algebra and use of an integral t a b l e , one can express the r e s u l t , in dimensionless form as:

For ( l l a , b ) use was made of the d e f i n i t i o n s

f ' ( ~ ) = 1/71 arctg

lh

^{:%(;:+ 11211}

f " ( ~ ) = ~ / I I arctg

From normalized dipole s t r e n g t h ' s and normalized e l e c t r i c f i e l d s , one obtains local d i e l e c t r i c constant and r e f r a c t i v e idex, a f t e r some algebra, as follows

1

(2%

+

1 ) h

+

(2R + 1)'

Also expressions ( l l a , b ) , (16) and (15) were evaluated i n the program, t h a t was men- tioned before and from which the r e s u l t s w i l l be discussed i n the followingsections.

3 . NUMERICAL RESULTS

A short look a t (5a,b) reveals t h a t the model contains only one f r e e parameter: the value f o r the diagonal element:

or

^,'c hence i t i s obvious t o use one of them as a c l a s s i f i c a t i o n parameter. In t h i s paper three special cases will be shown, each r ~ f r e s e n t i n g a d i f f e r e n t class of solutions. A t f i r s t a solution will be shownwhere C > 10. The special value, concerned here i s C" = 15.021, chosen t o correspond t o a case, where@= 1.0 x 10-40~tn2 and a = 3.08. Figure 1 shov,s the normalized dipole s t r e n g t h ' s f o r parallel and perpendicular f i e l d s as they occur i n a chain consisting of 7 atoms. Figure 2 shows the normalized e l e c t r i c f i e l d f o r the same case, i n p a r a l -

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Fig. 1. Fig. 2 .

Normalized dipoles, n = 7 ,

c

^{II =} 15.021 Normalized e l e c t r i c f i e l d , n = 7 ,

c#=

15.021 l e l orientation. The overall shape of the e l e c t r i c f i e l d resembles the behaviour of the dipoles i n the corresponding orientation.

As a r e s u l t the r e l a t i v e d i e l e c t r i c constant i s completely f l a t , as a function of atomic index, differences occuring only in the fourth d i g i t behind the comma. The reaction of the chain f o r t h i s value of

c",

on the applied e l e c t r i c f i e l d i s r a t h e r modest. The best i l l u s t r a t i o n f o r the behaviour of chains with C d > 10, i s repre- sented by the r e s u l t s f o r the t o t a l p o l a r i z a b i l i t y a;ltot f o r a number of chains with increasing chainlength n . If the chain i s allowed t o have any orientation with respect t o the f i e l d with equal probability

,

one observes an average En, tot t h a t follows from the expectation value f o r p Z , as:

- -

Pz = 'n, t o t Eo =

4

I

%n, t o t = 1/3[0n, t o t ⁺ 2 an, tot]

0 i s the angle between chain and e l e c t r i c f i e l d , 4 represents the azimuth. Figure 3 shows the r e s u l t s , f o r n = 2 t o n = 9 . In order t o emphasize the d e t a i l s , a l l t o t a l p o l a r i z a b i l i t i e s got a subtraction from na. The excess or defect in parallel and perpendicular cases i s due t o enrichment or decrement of the applied f i e l d by t h e neighbouring dipoles. The difference behaves l i n e a r l y , but has a small o f f s e t in the origin.

The next case studied, representative f o r the interval C " E (1.5,10) had a parallel diagonal matrix element

c N =

3.004, based on a = 5.0 x and a = 3.08.

Figure 4 shows the normalized dipole values f o r both orientations. The excessive growth of the parallel dipoles has no counterpart in the perpendicular ones. Except f o r the outer dipoles, the perpendicular dipoles become close t o constant in t h e central region. This behaviour i s independent from the chain length n. The

congruency between dipoles and the average of the e l e c t r i c f i e l d , as mentioned i n the previous, case, gradually s t a r t s t o disappear, as becomes c l e a r from the behaviour of local d i e l e c t r i c constant and r e f r a c t i v e index, shown i n f i g . 5. The most remarkable aspects of t h i s type of chain however a r e exhibited again by the t o t a l p o l a r i z a b i l i t i e s . Fig. 6 . gives these r e s u l t s . The para1 l e l p o l a r i z a b i l i t i e s , i n t h i s case, have become such pronounced, t h a t i t dominates as well the average t o t a l p o l a r i z a b i l i t y . Hence t h i s i s no longer a s t r a i g h t l i n e i n the close neigh- bourhoud of a curve i a . In stead the perpendicular t o t a l p o l a r i z a b i l i t i e s have become a s t r a i g h t l i n e as function of the atomic index i

,

but s t i l l with a c l e a r o f f s e t i n the o r i g i n . This i s e n t i r e l y compatible with t h e behaviour of the perpendicular dipoles.

The l a s t case studied represents the neighbourhoud of

c N

= 1.

As an exam~le

c'/=

^.702332 was taken, representing the case with a = 1.54

8

^and

a = 2.893 x

.

In t h i s case the r e s u l t s a r e completely odd. Parallel orienta- tions produce negative t o t a l polarizabi l i t i e s . Perpendicular orientations behave b e t t e r and produce positive r e s u l t s . B u t f o r both orientations individual dipoles produce positive and negative r e s u l t s i n an irregular way. In t h i s case t h e

electromagnetic interaction i s strongest, but i t i s useless t o show f i g u r e s , because there a r e no c h a r a c t e r i s t i c examples.

Fig. 5. Fig. 6 .

Dielectric constant/refractive index Theoretical t o t a l polariza i l i t i e s n = 7 ,

c

= 3 . 0 0 4

c

^{= 3.004, a = 5.0 x 10- 4B ~m 2}

4. COMPARISON WITH EXPERIMENTAL RESULTS

Linear chain systems on an atomic s c a l e , as calculated i n t h i s paper are the domain of organic chemistry. The compound t h a t comes c l o s e s t t o our model a r e alkanes.

From the 1 i t t e r a t u r e /9/ tne polarizabi l i t i e s of a1 kane moleculas were found from n = 1 t o n = 10, where n represents the number of C-atoms. As can be seen in f i g . 7.

the p o l a r i z a b i l i t i e s form a s t r a i g h t l i n e as function of the number of C-atoms. Now l e t us look f o r a purely electromagnetic explanation of these r e s u l t s . Suppose t h e r e i s very weak coupl i n g , so C" > 10, then indeed the calculated t o t a l average polarizabil i t y builds a s t r a i g h t l i n e . However: t h i s s t r a i g h t l i n e goes through the o r i g i n , whereas the experimental curve has an o f f s e t . For weak coupl ing, hence C"Q 3, the perpendicular t o t a l p o l a r i z a b i l i t y produces a good s t r a i g h t l i n e , with a very acceptable o f f s e t . The correspondence of t h a t curve with experiments, as f a r as shape i s concerned, i s almost perfect. Correspondence of the experimental r e s u l t s w i t h strong coupling cases, C" < 1, can be forgotten, f o r reasons mentioned before.

However, the l i t t e r a t u r e values f o r the C

-

C distance in alkanes i s 1.54

8

^{and the}

measured p o l a r i z a b i l i t y f o r CH4 i s 2.893 x 1 0 - ~ ~ F t n ~ . As shown i n the previous

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paragraph, these values produce a

c"=

-702332, c l e a r l y defining alkanes a strong coupl ing case.

5. CONCLUSION

To conclude l e t us look f o r t h e physical inplications of the three p o s s i b i l i t i e s , discussed in the previous paragraph. What makes very weak coupl ing, C~ t 10, an a t t r a c t i v e solution i s t h a t the average p o l a r i z a b i l i t y behaves l i n e a r l y . Since the experiments were performed in gaseous o r liquid phase, one expects an averaged r e s u l t . This solution however lacks the o f f s e t . The good agreement w i t h the perpendicular response for c f l c 3 has already been mentioned. The conclusion from t h i s however i s unphysical: every time the alkane catches a photon, i t will o r i e n t i t s e l f perpendicular t o i t s e l e c t r i c f i e l d . Such conclusion has t o be rejected. B u t both these cases have values f o r C

,

f a r d i f f e r e n t from the one calculated from experimental data. That CN= .7032 produces with t h i s theory i r r e g u l a r r e s u l t s . The overall conclusion has t o be t h a t t h i s model c a n ' t describe the optical response of alkanes. A few arguments can be given i n addition: alkanes a r e not exactly a l i n e a r chain, but r a t h e r a zig-zag s t r u c t u r e . The assumption of constant p o l a r i z a b i l i t y f o r each segment of the chain can also be questioned. If t h i s will change the overall conclusion can be doubted.

Fig. 7.

Experimental polarizabil i t e s CnH2,, ⁺

a0

10.

6. REFERENCES

1. LITZI4AN 0. and DUP P., Optica Acta 29 (1982) 1317.

2. MOSSOTTI O.F., Mem. Soc. Sci Elodenaf4 (1859) 49.

3. OSEEN C.W., Ann. der Physik, 42 (191- 1257.

4. EWALD P.P., Ann. der Physik, (1916) 117.

5. NIJBOER B.R.A. and DE WETTE F x . , Physica 23 (1957) 309.

6. PHILPOTT M.R., J . of Chem. Phys., 60 (197471410.

7. MAHAM G.D. and OBERIIAIER G . , Phys .-Rev.

183

(1969) 834.

,

i lo-" Fm'

~ " . w , e % p

155: ,/

A,';

^{1 2 3 4 5 6 7 n e+ 9}