Optical constants of an isolated hexa-gonal crystal of silenium

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1 University of Iowa Iowa Research Online Theses and Dissertations 1916 Optical constants of an isolated hexa-gonal crystal of silenium Charles Henry Skinner State University of Iowa This work has been identified with a Creative Commons Public Domain Mark 1.0. Material in the public domain. No restrictions on use. This thesis is available at Iowa Research Online: Recommended Citation Skinner, Charles Henry. "Optical constants of an isolated hexa-gonal crystal of silenium." MS (Master of Science) thesis, State University of Iowa, Follow this and additional works at:

2 The O p tic a l Constants o f an is o la te d hexagonal c r y s ta l o f selenium by C harles Henry S kinner A T hesis Subm itted to the F a c u lty o f the Graduate o f C ollege the S ta te U n iv e rs ity o f Iowa in p a r t ia l f u lf ilm e n t o f the requirem ents f o r th e Degree o f M aster o f Science J u ly, 1916

3 Hexagonal C r y s ta l. Faces ABCD, BCEF, e t c., are known as "p ris m fa c e s, as d is tin g u is h e d from the "p in a c o id " fa c e s, DCE and ABF. The plane a1a2a3 is drawn p e rp e n d ic u la r to the prism fa c e s, and is known as the p r in c ip a l plane o f symmetry. The a x is CC', drawn th ro u g h the c e n te r o f the c r y s ta l and p e rp e n d ic u la r to the p r in c ip a l plane o f symmetry, w i l l be d e sig nate d as the g e o m e trical a x is, o r sim p ly th e a x is, o f the c r y s t a l.

4 THE OPTICAL PROPERTIES OF AN ISOLATED CRYSTAL OF SELENIUM. V a rio u s experim enters have used the p o la r im e tr ic method fo r in v e s tig a tin g the o p tic a l p ro p e rtie s o f selenium, b u t none o f them agree very c lo s e ly in t h e ir r e s u lts. The v a r ia tio n is due, w ith o u t doubt, to the d i f f i c u l t y in p re p a rin g a r e f le c t in g s u r face th a t re p re se n ts the a c tu a l c o n d itio n o f the m a te ria l w ith in. M e ie r* o b ta in e d a r e f le c t in g surfa ce o f amorphus selenium by f i r s t p o u rin g the m elted m a te ria l over a brass p la te. A fte r s o lid if y in g, i t was removed, and then ground and p o lis h e d by the Win k e l F irm, in G o ttin g e n. F o e rs te rlin g and F rée d e ric k s z * *, in t h e ir in v e s tig a tio n s, used m irro rs o b ta in e d by spreading m elted selenium over a w e ll cleaned g la ss p la te. W ith o u t fu r th e r m odif ic a t io n, m irro rs o f the amorphus type were o b ta in e d by removing the g la s s. When heated in a sand-bath a t 180 C., b e fo re removin g the g la s s, m irro rs o f c r y s t a llin e selenium were o b ta in e d. G rip e n b e rg *** o b ta in e d a r e f le c t in g su rfa ce by p re s s in g m elted selenium between p la te -g la s s. Small c ir c u la r p o rtio n s, le s s than two te n th s o f a m illim e te r in d ia m e te r, became c r y s t a llin e. - *M e ie r, Ann. d. Phys., 31. p.1029, ^ F o e r s t e r lin g and F r^ e d e ric k s z, Ann. d. P h ys., p.1227, ^ ""G ripenberg, Phys. Z e its c h r. 14, pp , F e b.l, 1913.

5 2 Wood* used an e n t ir e ly d if f e r e n t method f o r o b ta in in g the r e fr a c tiv e index o f selenium. He f i r s t o b ta in e d u n ifo rm film s by d e p o s itin g the selenium on p la te s o f plane p a r a lle l g la ss from a f l a t selenium cathode, in a h ig h vacuum. He n e xt obta in e d wedge-shaped film s by s h ie ld in g the g la s s p la te d u rin g d e p o s itio n w ith a s t r ip o f m ica mounted a c e n tim e te r o r so above i t s s u rfa c e. Wedges, o r p rism s, were o b ta in e d in th is manner w hich showed s tr a ig h t in te rfe re n c e bands when viewed by r e fle c te d l i g h t. He was a b le, by means o f the p rism s, to o b ta in the r e f r a c t iv e in d ic e s o f the red p a rt o f the spectrum from measurements o f the angle o f the p rism used and the angle o f minimum d e v ia tio n. W ith the values o f the r e fr a c t iv e index thus o b ta in e d as a b a sis f o r c a lc u la tio n, he o b tain ed the r e f r a c t iv e in d ic e s f o r the r e s t o f the spectrum from in te rfe ro m e te r measurements, u sin g the u n ifo rm film s f i r s t o b ta in e d. ( I f nr = the in dex o f r e fr a c tio n f o r wavele n g th λ r, and dr = the d ispla ce m e n t, measured in terms o f frin g e w id th ; th e n, f o r any o th e r wave-length, λ g, and d is placem ent d g, the index o f r e f r a c t io n is g iven by the exp re s s io n : ng = 1 + [n r - 1](dgλg/drλr). ) Wood, R.W., P h il. Mag., S.6, V o l.3, N o.18, June, p p

6 3 The p re sent in v e s tig a tio n is a study o f the o p tic a l p ro p e r t ie s o f an is o la te d c r y s ta l o f sele n iu m *. C ry s ta ls o f the type used are hexagonal in form. Owing to t h e ir p u r it y, such c ry s ta ls have the advantage o f p re s e n tin g r e f le c t in g surfa ces which may be taken as p o s s ib ly re p re s e n tin g the s ta te o f m a tte r w ith in the c r y s ta l. The one e s p e c ia lly in v e s tig a te d presented a r e f le c t in g su rfa ce a p rism face o f the c r y s ta l a p p ro xim a te ly one m illim e te r wide and two m illim e te r s lo n g. This is n e a rly seven tim es as la rg e as the m irro rs used by G ripenberg. There is no suggest io n, in any o f the works above m entioned, th a t the o p tic a l p ro p e rtie s are d if f e r e n t in d if f e r e n t d ir e c tio n s ; th a t is, th a t c r y s t a llin e selenium is a doubly r e fr a c tin g medium. W ith the type o f c r y s ta l used in the p re sent s tu d y, the consta n ts appear to be d if f e r e n t, depending on the o r ie n ta tio n o f the a x is o f the c r y s ta l w ith re s p e c t to the plane o f in c id e n c e. A p p a re n tly, there is one d is t in c t s e t when the a x is o f the c r y s ta l lie s in the plane o f in c id e n c e ; and another when the a x is is p e rp e n d ic u la r to the plane o f in c id e n c e. The o p tic a l p ro p e rtie s, as determ ined p o la r im e t r ic a lly, va ry f o r in te rm e d ia te p o s itio n s o f the c r y s ta l. 'These c ry s ta ls were made a t the P h y s ic a l L a b o ra to ry o f the S tate U n iv e rs ity o f Iowa, by P ro f. F. C. Brown (See Phys. R ev., N. S., V o l.4, p.8 5, 1914; and V o l.5, p.2 36, 1915). Through h is c o u rte s y, they were a v a ila b le f o r t h is in v e s tig a tio n.

7 4 G eneral Theory. I f p la n e -p o la riz e d lig h t is r e fle c te d from a m e ta llic su rfa ce the r e fle c te d l i g h t is e l l i p t i c a l l y p o la riz e d, unle ss the plane o f p o la r iz a tio n o f the in c id e n t l i g h t is e ith e r p a r a lle l o r p e r- p e n d ic u la r to the plane o f in c id e n c e. L e t the plane o f in cid e n ce be h o r iz o n ta l, and the plane o f p o la r iz a tio n o f the in c id e n t lig h t make an angle o f 45 degrees to the plane o f in c id e n c e. The azim uth o f the in c id e n t l i g h t is then s a id to be 45 degrees. Resolve the in c id e n t v ib r a tio n in to two components, one p a r a lle l and the o th e r p e rp e n d ic u la r to the plane o f in c id e n c e. These components w i l l be eq u a l in m agnitude and in th e same phase befo re r e f le c t io n. They w i l l n o t be changed a t norm al in c id e n c e. As the angle o f in cid e n ce in c re a s e s, the component p a r a lle l to the plane o f in cid e n ce undergoes a g re a te r change than the o th e r The angle o f in cid e n ce a t which the p h a s e -d iffe re n c e o f the two component v ib r a tio n s has the value ^ is termed the angle o f p r in c ip a l in c id e n c e. A t t h is in c id e n c e, the r e fle c te d l i g h t has i t s -gp&erbe&k e l l i p t i c i t y. I f now one o f the components be re ta rd e d so th a t the e l l i p t i c a l v ib r a t io n becomes again p la n e -p o la riz e d, the plane o f p o la r iz a tio n w i l l be d if f e r e n t from th a t o f the in c id e n t l i g h t, sin ce the am p litu des o f the component v ib r a tio n s are changed u n e q u a lly by r e f le c t io n. The azim uth o f the re s to re d plane v ib r a tio n is always le s s than the azim uth o f the in c id e n t v ib r a tio n. I f the in c id e n t l i g h t is a t the angle o f p r in c ip a l in c id e n c e, the corre spondin g azim uth o f the r e fle c te d l i g h t is c a lle d the p r in c ip a l azim uth.

8 5 The r e la tio n s e x is tin g between the o p tic a l consta n ts and the angles o f p r in c ip a l in c id e n c e and p r in c ip a l azim uth are as fo llo w s :* L e t ϕ = the angle o f p r in c ip a l in c id e n c e, ψ = the p r in c ip a l azim uth, n = the in dex o f r e f r a c t io n, k = the c o e f fic ie n t o f a b s o rp tio n, and R = the c o e f fic ie n t o f r e f le c t io n ; *These r e la tio n s as here g iven appear in Winklem ann' s Handbuch d. P h ys., p.822 sq. See a ls o, Wood: P h y s ic a l O p tic s, p p , second e d itio n ; Drude: Theory o f O p tic s, p p

9 6 D e riv a tio n o f the E q u a tions. The d e riv a tio n o f the form ulas w i l l here be g iv e n, accordin g to the E le ctro m a g n e tic Theory. We must f i r s t understand the fundam ental e q uations o f M axw ell. Suppose a c u rre n t i ' (measured e le c tro m a g n e tic a lly ) to flo w p e rp e n d ic u la rly across a re c ta n g le ABCD, o f dim ensions dx and dy ( F ig. l ). I f the c u rre n t flo w s up from the plane o f the f ig u r e, and the p o s itiv e d ir e c t i o n o f the c o o rd in a te s is as shown in the diagram, a p o s itiv e m agnetic pole w i l l be c a rrie d around the re c ta n g le in the d i r e c tio n in d ic a te d by the arrow s. I f α, β, α', and β' denote the components o f the m agnetic fo rc e a c tin g along AB, AD, DC, and BC, the work done by a u n it p o s itiv e pole in moving once around the re c ta n g le w i l l be W = αdx + β'dy - α'dx - β dy (1) When dy and dx are s u f f ic ie n t ly s m a ll, (α'-α)/dy becomes the d i f f e r e n t ia l c o e ffic ie n t α/ y, so th a t α' = α + ( α/ y ).dy. S im ila r ly, β ' = β + ( β/ x).d x. S u b s titu tin g these values in (1 ), W = (( β/ x)-( α/ y)) dx dy. (1 ) But now, in the e le c tro m a g n e tic system, the work done in moving a u n it p o s itiv e p o le once around a c u rre n t i ', re g a rd le ss o f the path o r the n a tu re o f the medium surro u n d in g the c u rre n t, is, W = 4πi '. L e t j ' be the c u rre n t d e n s ity ( v a l ue o f the c u rre n t when the re c ta n g le has u n it a re a ), and the components i f the d e n s ity along the x -, y -, z-axes be j x', j y',

10 7 and j z '. Then i = j z' dx d y. T herefore W = 4 π jz ' dx dy = (( β/ x)-( α/ y ) dx dy, from which 4πjz' = ( β/ x)-( α/ y). S im ila r ly, 4 π j x ' = ( γ/ y) - ( β/ z), (2) 4πjy' = ( α / z ) - ( γ / x ). These equations may be w r itte n withthecurentin the e le c t r o s t a tic system by in tro d u c in g a consta n t c, d e fin e d as the r a t io o f the e le c t r o s t a t ic to the e le c tro m a g n e tic u n it, th u s: c = i/i' = jx/jx', e tc. (3) These equations h o ld f o r a l l m edia, sin ce the work done by a m agnetic p o le in moving around a c u rre n t is independent o f the natu re o f the medium. By s im ila r re a sonin g, we may fin d the r e la t io n between the m agnetic c u rre n t and the accompanying e le c t r ic f i e l d. I f sx, sy, and sz denote the d e n s ity o f the m agnetic c u rre n t along the th re e axes, ( 3 ') These equations are g e n e ra l, and h o ld f o r a l l media. In the case o f displace m e nt c u rre n ts, the d e n s ity o f the lin e s o f fo rc e change w ith the tim e. The p ro d u ct o f the c u r re n t d e n s ity and 4rr w i l l be the change in the number o f lin e s o f fo rc e in u n it tim e, sin ce a charge e sends o u t 4rfe lin e s o f fo rc e. A ls o, in the fre e e th e r, the e le c t r ic fo rc e is num e ric a lly equal to the d e n s ity o f the lin e s o f fo rc e. But in

11 8 ponderable m edia, the e le c t r ic fo r c e - is s m a lle r than in fre e e th e r, e being the d ie le c t r ic consta n t o f the medium. In lik e manner, the m agnetic fo rc e is p s m a lle r, p being the m agnetic p e rm e a b ility. So th a t, (4) Since the m agnetic changes are so ra p id in the case o f lig h t v ib r a tio n s, we may w r ite p = 1 in a l l o p tic a l problem s. S u b s titu tin g the above values f o r the e le c t r ic and m agnetic c u rre n ts in (5) and ( 3 '), we o b ta in the fo llo w in g e xp re ssio n s, which com p le te ly determ ine a l l p ro p e rtie s o f the e le c t r ic and m agnetic f ie ld s in an is o tr o p ic d ie le c t r ic : (5) These are the M axwell e q u a tio n s. We can now determ ine the e q uations o f m otion o f a p a r t ic le a c tin g under the in flu e n c e o f an e le c tro m a g n e tic wave. D if f e r e n tia tin g the f i r s t e q u a tio n o f (5) w ith re s p e c t to t, S u b s titu tin g the values o f ^ and as g iven by the la s t two equations o f (5 ), D iffe r e n t ia t in g the f i r s t th re e e q uations o f (5) w ith re s p e c t to x, y, and z, and adding, we g e t 4~ (4^ + 4^- + 4 ^ ) = For o t ox oy dz p e rio d ic changes in e le c t r ic and m agnetic fo rc e s, the d if f e r e n t i a l c o e ffic ie n t w ith re s p e c t to tim e is p ro p o rtio n a l to the (6)

12 9 changes them selves (when the phase ^ has been added), so th a t in the above e x p re s s io n, = o. E quation (6) now becomes (7) S im ila r equations h o ld f o r Y and Z; also f o r d, js and /. D if f e r e n t ia l equations o f the above form re p re s e n t waves which are propagated w ith a v e lo c ity V = When a wave passes from one medium whose d ie le c t r ic co n sta n t is e, to another whose d ie le c t r ic consta n t is e, we have the r e la t io n, (8) where n is the in dex o f r e f r a c t io n. We w i l l n e xt deduce the laws o f r e f le c t io n and r e fr a c tio n f o r plane waves in c id e n t upon the boundary s e p a ra tin g two media o f d if f e r e n t o p tic a l d e n s itie s. A plane wave is re p re se n te d by the e q u a tio n s, (9) in which X, Y, and Z are the va lu e s o f the components o f the e le c t r ic fo rc e along the th re e axes a t any tim e and p o in t in space; Ax, k tj, kz, the am p litu des o f the -&#- the- re s p e c tiv e components; T, the p e rio d o f the d is tu rb a n c e ; V, i t s v e lo c ity o f p ro p a g a tio n ; and m, n, p, the d ir e c tio n cosines o f the norm al to the wave f r o n t. Suppose a ra y o f lig h t I to he in c id e n t a t an angle in the x -z plane ( F ig.2 ). The d ir e c tio n cosines may now be w r itte n : m = s in 0,' n = 0, p = cos $. Resolve the in c id e n t e le c t r ic fo rc e in to two components, E^ p e rp e n d ic u la r to the plane o f in cid e n ce ( p a r a lle l to the y - a x is ), and E^, p a r a lle l to the plane o f in c i-

13 10 dence and p e rp e n d ic u la r to the d ir e c tio n o f p ro p a g a tio n. E η may now be re s o lv e d in to two components, Eχ p a r a lle l to the x - a x is, and -E z p a r a lle l to the z -a x is (the n e g a tiv e s ig n is used because the p o s itiv e d ir e c tio n o f the z -a x is is taken downwards). Hence Ex= Eη cos ϕ, Ez= -Eη s in ϕ. I f now we l e t Xl, Yl, and Zl stand f o r the components o f the in c id e n t e le c t r ic fo rc e along the th re e axes, and V1the v e lo c i t y o f p ro p a g a tio n in the f i r s t medium, we have from (9 ), Xl = Eη cos ϕ cos ((2π/T)( t - (x s in ϕ + z cos ϕ)/v1))), Yl = Ey cos ((2π/T)(t-((x sin ϕ + z cos ϕ)/v1))), (10) Zl =-Eηs in ϕ cos ((2π/T)( t - ((x sin ϕ + z cos ϕ)/v1))). The m agnetic fo rc e s a sso cia te d w ith the above e le c t r ic f o r ces in the in c id e n t wave are o b ta in e d by d if f e r e n t ia t in g the above and s u b s titu tin g in e q uations (5 ). For example, the x - component αl o f the m agnetic fo rc e is, by e q u a tio n (5 ), (1/c)( αl/ t) = ( Yl/ z) -( Zl/ y). From e q u a tio n (1 0 ), the p a r t ia l d i f f e r e n t i a l o f Yl w ith re s p e c t to z is Yl/ z = Ey sin ((2π/T)(t-((x sin ϕ + z cos ϕ)/v1)))((2π/t)(cos ϕ/v1)), and the p a r t ia l d i f f e r e n t i a l o f Zl w ith re s p e c t to y is = Zl/ y 0. In te g ra tin g t h is e x p re s s io n, αl/ t =+cey((2π/t)(cos ϕ/v1))(sin((2π/t)(t-(x sin ϕ + z cos ϕ)/v1))). αl = +c Ey(cos ϕ/v1)(cos((2π/t)(t-((x sin ϕ + z cos ϕ)/v1)))).

14 11 S im ila r eχp re ssio n s are o b ta in e d f o r β l, and γl. Remembering th a t V1= c/ (ε1), the th re e components o f the m agnetic fo rc e become: (1 0 ' ) When the in c id e n t wave reaches the boundary s e p a ra tin g the two m edia, i t is d iv id e d in to a r e fle c te d and re fra c te d wave. The e le c t r ic and m agnetic fo rc e s in both the r e fle c te d and r e fra c te d waves can be re p re se n te d by expressions analogous to those in equations (10) and ( 1 0 '). L e t the r e fle c te d wave make an angle ϕ' w ith the p o s itiv e d ir e c tio n o f the z -a x is (See F ig. 2 ). The d ir e c tio n cosines are now m = s in ϕ ', n = 0, and p = cos ϕ '. Resolve the r e fle c te d e le c t r ic fo rc e in to two components, Ry p e rp e n d ic u la r to the plane o f in c id e n c e, o r r e f le c tio n (and p a r a lle l to the y - a x is ), and Rη p a r a lle l to the plane o f in cid e n ce and p e rp e n d ic u la r to the d ir e c tio n o f p ro p a g a tio n o f the r e fle c te d wave. The components o f Rη in the d ir e c tio n o f the x - and z-axes are r e s p e c tiv e ly, Rx = Rη cos ϕ', and Rz = -Rη s in ϕ '. The components o f the r e fle c te d e le c t r ic and m agnetic fo rc e s, along the th re e axes, may now be w r itte n : (11) (1 1 ') L e t % be the angle o f r e f r a c t io n (see F ig. 2 ), the a m p litude o f the re fra c te d wave, and X15 Y^, Zt, the components o f

15 12 the e le c t r ic fo rceaalong the th re e axes. A s im ila r n o ta tio n w i l l h o ld fo r the m agnetic fo rc e. Using V4 as the v e lo c ity in the second medium, and e, as the d ie le c t r ic c o n s ta n t, we may w rite f o r the re fra c te d wave: (12) (12' ) B efore e s ta b lis h in g a r e la t io n between the angles o f in cidence, r e f le c t io n, and r e f r a c t io n, we must understand the boundary c o n d itio n s o f the e le c t r ic and m agnetic fo rc e s. The e le c t r ic fo rc e p a r a lle l to the boundary must be the same on both s id e s. O therw ise, an u n lim ite d amount o f work could be d e riv e d by c a rry in g a charged p a r t ic le along the boundary in one medium a g a in s t the e le c t r ic fo rc e, and th e n, c a rry in g i t across the boundary, a llo w i t to move back in th e second medium through the same d is ta n c e. Hence X, = X2, and Y; = Yz. The z-component on the two sid e s o f the boundary is determ ined as fo llo w s : L e t the e le c t r ic fo rc e R ; in the f i r s t medium be in c id e n t a t an angle 0, and l e t R^, the re fra c te d e le c t r ic fo rc e, make an angle χ to the norm al. The norm al component o f the e le c t r ic fo rc e in the upper medium is th e re fo re Z; = R ^ o s $ j and in the low er medium, Z%~ R ^ o s χ. The same number o f lin e s pass through a plane o f u n it area which is p e rp e n d ic u la r to the z - a x is, in w hichever medium we consid e r the p la n e. That is, the d e n s ity o f the lin e s o f fo rc e p a r a lle l

16 13 to the z -a x is is the same in the two m edia. Or, p u ttin g i t in the form o f an e q u a tio n, ε1r1 cos ϕ/4π= e2r2 cos χ/4π. Hence ε1z1= ε2z2. S im ila r ly, the boundary c o n d itio n s f o r the m agnetic fo rc e a re : α1= α 2, β 1 = β 2, and μ1γ1 = μ 2 γ 2. But since μ1 = μ2 f o r r a p id ly o s c illa t in g m agnetic fo rc e s such as accompany lig h t waves,γ 1 = γ 2. These boundary c o n d itio n s can be f u l f i l l e d o n ly i f, f o r z = 0, a l l fo rc e s become p ro p o rtio n a l to the same fu n c tio n o f t, x, and y. That is, the fo llo w in g r e la tio n s between the angles o f in c id e n c e, r e f le c t io n, and r e f r a c t io n, must h o ld : (13) E xpressions f o r the am p litu des o f the r e fle c te d e le c t r ic fo rc e, re s o lv e d in to components p a r a lle l to and p e rp e n d ic u la r to the plane o f in c id e n c e, w i l l now be deduced. The fo rc e X f on the upper sid e o f the boundary is equal to the sum o f the fo rc e s o f the in c id e n t and r e fle c te d waves, Xx + X/ l; and th is is equal to the fo rc e X^ on the low er sid e o f the boundary. X ^is o p p o site in d ir e c tio n to X^. W ith s im ila r expressions f o r the o th e r components, we have, (14) The exp ressions to the r ig h t in (14) are o b ta in e d by s u b s titu tin g in the expressions to the l e f t the values o f the seve r a l components as g iv e n by e q uations (1 0 )-(1 2 ), to g e th e r w ith

17 14 the r e la tio n s expressed in (1 3 ). Adding (b) and ( c ), and e lim in a tin g D y, we g e t Ey ((( ε1cos ϕ)/( ε2cos χ))- 1 ) = Ry ( (( ε1 cos ϕ)/( ε2 cos χ))+1). Adding (a) and ( d ), and e lim in a tin g Dη, we g e t Eη(( cos ϕ/cosχ)-( ε1/ ε2)= R η ( ( c o s ϕ / c o s χ ) + ( ε 1 / ε 2 ) ). ( 1 4 ' ) Remembering th a t (ε2)/ (ε1) = n = sinϕ/sinχ, the expressions f o r the r e fle c te d am p litu des o f the e le c t r ic fo rc e become: Ry = - Ey(si n(ϕ-χ))/(sin(ϕ+χ))), Rη = Eη((tan(ϕ-χ))/(tan(ϕ+χ))); (15) and Rη/Ry= -((Eηcos(ϕ+χ))/(Eycos(ϕ-χ))). (16) The p re cedin g d is c u s s io n a p p lie s o n ly to the p ro p a g a tio n o f waves in media which are p e rfe c t in s u la to r s. The c u rre n t may c o n s is t o f a displacem ent c u rre n t in the e th e r, the component p a r a lle l to the x -a x is being re p re se n te d by (ε/4π)( x/ t); and a convectio n c u rre n t due to the m otio n o f the e le c tro n s in s id e the atoms. The e le c tro n s are bound to p o s itio n s o f e q u ilib riu m by fo rc e s o f r e s t it u t io n, and experience o n ly s lig h t displacem ents under the in flu e n c e o f a steady e le c t r i c fo rc e. When the fo rc e is removed, the e le c tro n s re tu rn to t h e ir o r ig in a l p o s itio n s. In the case o f n o n -in s u la tin g media, a conduction c u rre n t, σx, w i l l be set up under the a c tio n o f a steady e le c t r ic fo rc e. The c u rre n t is due to the m otion o f e le c tro n s w hich are n o t bound to p o s itio n s o f e q u i lib r iu m, as in the case o f in s u la to r s, b u t are fre e to move along under the in flu e n c e o f the fo rc e. I f the c u rre n t dens it y and the e le c t r ic fo rc e are measured in e le c t r o s t a tic u n its, σ re p re s e n ts the a b solu te c o n d u c tiv ity in the e le c tr o s t a t ic system. We may th e re fo re w r ite f o r the c u rre n t p a r a lle l

18 15 to the x -a x is, jx = ((c/4π)( X/ t)) + σx (17) No fu r th e r m o d ific a tio n s are needed f o r absorbing media than the one in tro d u c e d in (1 7 ), and the fundam ental M axwell Equatio n s, 4πjx/c = ( γ/ y) - ( β/ z), e t c., and (1/c)( α/ t) = ( Y/ z) - ( Z/ y), e t c., s t i l l h o ld. To o b ta in the d i f f e r e n t i a l e q u a tio n o f the d is tu rb a n c e, d if f e r e n t ia t e (17) w ith re s p e c t to t, jx/ t =(ε/4π)( 2X/ t2)+(σ( X/ t). (18) D iffe r e n t ia t in g the f i r s t e q u a tio n o f (5) w ith re s p e c t to t, and s u b s titu tin g the va lu e s o f γ/ t and β/ t as g iv e n by the la s t two equations o f (5 ), we g e t (ε/c2)( 2X/ t2) = 2X/ z2, (19) since f o r p la n e -p o la riz e d plane waves, p a r a lle l to the z -a x is, 2X/ x2 = 2X/ y2 = 0 (see e q u a tio n 7 ). From (4 ), 2X/ t2 = (4π/ε)( jx/ t). Hence, (4π/c2)( jx/ t) = 2X/ z2. Substitu tin g the va lu e o f jx/ t in (1 8 ), we g e t f o r the d i f f e r e n t i a l e q u a tio n o f m o tio n : 2X/ z2=((ε/c2)( 2X/ t2)) + ((4πσ/c2)( X/ t)). (20) E quation (20) has f o r i t s s o lu tio n, X = A e((i(2π/t))(t-m z )), (21) in w hich m is complex. D iffe r e n t ia t in g (2 1 ), X/ t = i (2π/T)X, (22) and s u b s titu tin g in (1 7 ), jx = ((ε/4π)( X/ t)) + (σ(( X/ t)(t/2πi))) = ((ε/4π) + (σt/2πi))( X/ t). M u ltip ly in g the num erator and denom inator o f the second term in the pare n th e s is by 2 i, the above e xp re ssio n becomes: j x = ((ε - 2iσT)/4π)( X/ t). (23) By comparison w ith (4 ), we see th a t f o r conductors the com

19 16 p le x d ie le c t r ic c o n s ta n t, ε - 2 i σt, re p la c e s the r e a l c o n s ta n t, ε, i n the exp re ssio n f o r p e rfe c t in s u la to r s. W ritin g ε ' f o r ε - 2 i σt, and s u b s titu tin g in e q u a tio n (1 9 ), we get (ε'/c2)( 2X/ t2)= 2 x / z 2. D iffe r e n t ia t in g (21) two tim es w ith re s p e c t to both t and z, 2X/ t2 = -((4π2/T2)X); 2x/ z2 = -((4π2m2/T2)X); ( and s u b s titu tin g these va lu e s in (2 4 ), (ε'/c2)((4π2/t2)x) = (4π2m2/T2)X. Hence m2= ε'/c2. m has the dim ensions o f a re c ip ro c a l v e lo c ity. Since ε ' is complex, m is a lso complex, and we may w r ite m = (1-ik)/V in w hich k and V are r e a l, V being the v e lo c ity o f p ro p a g a tio n o f the wave in the absorbing medium. S u b s titu tin g t h is va lu e in (2 1 ), X = A e((2πi/t)(t-((1-ik)/v)z)) = A e(2πit/t) -(2πiz/TV) -(2πkz/TV)) = A e-2πk(z/λ) e2πi((t/t)-(z/λ)), (25) since TV = λ. In t h is e x p re s s io n, Ae-2πk(z/λ) re p re s e n ts the amplit u d e, which decreases as z in c re a s e s. When z = λ, th a t is, when the wave has tra v e rs e d a th ic k n e s s equal to the wave- le n g th, the a m p litu d e has decreased by the amount e-2πk. The co n sta n t k measures the a b s o rp tio n, and is c a lle d the coe f f ic ie n t o f a b s o rp tio n. Since m2= ε'/c2, we have ε'/c2 = ((1-ik)/V)2, ε ' = (c2/v2)( 1 - k 2-2 i k ) ; o r, c a llin g c/v the r e f r a c t iv e index o f the medium, ε ' = n2 (1 - k 2-2 i k ). (26) ε ' may be s u b s titu te d f o r ε in e q u a tio n (8 ), so th a t

20 17 (27) in w hich χ is complex. W ith t h is u n d e rs ta n d in g, e q u a tio n (16) s t i l l h o ld s. Now, i f the in c id e n t lig h t is p o la riz e d in a plane making an angle o f 45 degrees to the plane o f in c id e n c e, Eη = Ey, so th a t (16) becomes (28) But now, Rη and Ry are com plex. L e t Rη = Rη ' e iδ η, and Ry = Ry' eiδ y, in w hich Rη ', Ry', δη and δy are re a l q u a n tit ie s. δη and δy re p re s e n t the advances in phase o f the components Rη' and Ry' w ith re s p e c t to the in c id e n t wave. Hence, Rη/Ry = (Rη' eiδη)/(rη' eiδy) = (Rη'/Ry')(ei(δη-δy)) = ρ eiδ, (29) where Rη'/Ry' = ρ and δη - δy = Δ. From (28) and (2 9 ), Since the rig h t-h a n d member o f (30) is complex, the o th e r member must a lso be complex; th a t is, Δ cannot be zero. This means th a t when p la n e -p o la riz e d lig h t is r e fle c te d from the surfa ce o f a co n d u ctin g medium, the r e fle c te d l i g h t i s e l l i p t i c a l l y p o la riz e d. Expanding the cosine terms in the above e xp re ssio n, we have ρei Δ = -((cos ϕ co s χ - s in ϕ sinχ)/(cosϕcosχ+sinϕsinχ). (31) (30) In (2 7 ), ε1 may be taken as u n it y, so th a t s in χ = (sin ϕ/ε'). From t h is, cos χ = (1 - s in 2χ) = (1 - (sin2ϕ)/ε'2). S u b s titu tin g these values in (3 1 ), (32)

21 18 A t norm al in c id e n c e, ρeiδ = -1 ; and a t g ra z in g in c id e n c e, ρei Δ = 1. That is, f o r these two p o s itio n s, Δ = 0, and the r e fle c te d lig h t is p la n e -p o la riz e d. When Δ = π/2, the r e f le c ted lig h t has i t s l e a s t e l l i p t i c i t y i.e.,nearestcircular. A t t h is angle, eiδ = i, since by expansion eiδ = cos Δ + i s in Δ. The corresponding angle o f in cid e n ce is c a lle d the angle o f p r in c ip a l in c id e n c e, ϕ. E quation (32) may now be w r itte n : (1 + i ρ)/(1- i ρ) =(s in ϕ tan ϕ)/ (ε' -sin2ϕ).(3) M u ltip ly t h is e q u a tio n by i t s complex c o n ju g a te, (1 - i ρ)/(1+i ρ) =(s in ϕ tan ϕ)/ (ε"-sin2ϕ), in w hich ε " = n 2 (1 - k ik ), the complex conju gate o f ε '. The le ft-h a n d member reduces to 1. S quaring the rig h t-h a n d member and c le a rin g, s in 4ϕ tan4ϕ = (ε ' - s in 2ϕ) (ε" - s in 2ϕ). S u b s titu tin g the above va lu e s o f ε ' and ε ", we g e t the exp re s s io n : s in 4ϕ ta n 4ϕ = n 4 (1+k2)2-2n2(1- k 2) s in 2ϕ + s in 4ϕ. (34) This e q u a tio n, w hich in v o lv e s o n ly n, k, and ϕ, may be p u t in to a more u s e fu l form, as fo llo w s : F ir s t, e x tra c t the square r o o t, and we g e t s in 2ϕ tan2ϕ = n2 (1+k2) - s in 2ϕ + 2k2s in 2ϕ - 2k 4s in 2ϕ and again e x tra c tin g the square ro o t o f each member, s in ϕ tan ϕ = n + (nk2/2) - (nk4/8) (sin2ϕ/2n) + (5k 2s i n2ϕ/4n) - (27k4s in 2ϕ/16n) +. F a c to rin g o u t (n + (nk2/2) - (nk4/8) +.) in the second member, s in ϕ tan ϕ = (n + (nk2/2) - (nk4/8) +... ) (1 -(sin2ϕ/2n2)+ ( 3 k 2 s in 2 ϕ /2 n 2 ) - (5k 4s in 2ϕ/2n2) )

22 19 The la s t exp re ssio n may now be w r itte n in the form : s in ϕ ta n ϕ = n( 1+k 2 ) [ 1 - ((sin2ϕ/2n2)( 1-3k2 +5k4-... )]. M u ltip ly in g the term in the p a re n th e s is by (1+k2 )2, and d i v id in g the r e s u lt by the same q u a n tity,- we have s in ϕ tan ϕ = n ( 1+k 2)[ 1 - ((sin2ϕ/2n2)(((1-3k2+5k4-...)((1+k2)2))/((1+k2)2))], which reduces, a f t e r p e rfo rm in g the o p e ra tio n in d ic a te d, to the d e s ire d form : s in ϕ ta n ϕ = n ( 1 + k 2 )[ 1 -((1/2n2)sin2ϕ)((1-k2)/((1+k2)2))]. (35) I f the p h a s e -d iffe re n c e, w hich is in tro d u c e d by r e f le c t io n from the surfa ce o f a co n d u cto r, be a n n u lle d, so th a t the e l l i p t i c a l v ib r a tio n is converted in to a p la n e -p o la riz e d one, the plane o f p o la r iz a tio n w i l l make an angle ^ w ith the plane o f in c id e n c e. V«h.en the in c id e n t lig h t is a t the angle o f p r in c ip a l in c id e n c e, the corre spondin g a n g le d is c a lle d the p r in c ip a l azim uth, \p. In the f ig u r e, the x z-p la n e is the plane o f in c id e n c e, the r e f le c t in g surfa ce being in the xyp la n e. The l i g h t is i n c i dent a t the angle ]?, the plane o f p o la r iz a tio n makin g an angle o f 45 degrees F ig. 3. to the plane o f in c id e n c e. L e t the d ir e c tio n and m agnitude o f the in c id e n t v ib r a tio n be re p re se n te d by the arrow aa1. The component E^ o f the in c id e n t am plitude s u ffe rs a g re a te r change in m agnitude a t r e f le c t io n than the E^ component. Remembering th a t the plane o f p o la r iz a tio n is always p e rp e n d ic u la r to the plane o f v ib r a t io n, we

23 20 see th a t Rγ/Ry= ρ = tan ψ. T h e re fo re, (1 + iρ)/(1 - iρ) = (1 + it a n ψ)/(1- itanψ) =(cosψ+isinψ)/(cosψ- isinψ). M u ltip ly in g both num erator and denom inator o f the la s t member by the denom inator, we g e t (1 + iρ)/(1 - iρ) = (cos2ψ + sin2ψ)/((cos2ψ-sin2ψ) - 2icosψsinψ) = 1/(cos2ψ - isin2ψ). S u b s titu tin g t h is value in (3 3 ), 1/(cos2ψ- isin2ψ) = (sinϕtanϕ)/ (ε'-sin2ϕ) R eplacing ε ' w ith i t s va lu e as g iven in (2 6 ), s in ϕ ta n ϕ (cos2ψ - is in 2 ψ ) = (n 2( 1- k 2-2 i k ) -s in 2 ϕ ). (36) E x tra c tin g the square ro o t o f the rig h t-h a n d member, s in ϕ ta n ϕ (cos2ψ - is in 2 ψ ) = n - n ik - (s in 2ϕ/2n) - (ik sin2ϕ/2n) + (k 2s in 2ϕ/2n) + (ik 3 s in 2ϕ/2n) E quating r e a l and im a ginary p a rts, s in ϕ ta n ϕ cos2ψ = n - ( s in 2 ϕ /2 n ) + ( k 2 s in 2 ϕ / 2 n ) -... (a) - s i nϕ ta nϕ is in 2 ψ = - n ik - (iksin2ϕ/2n) + (ik3sin2ϕ/2n) (b) D iv id in g (b) f i r s t by - i and then by e q u a tio n (a ), tan 2ψ = (2n2k + k s i n2ϕ- k3sin2ϕ+...)/(2n2 - sin2ϕ+k2sin2ϕ-...) = k ( ( 2n2 + s in 2ϕ - k 2s in 2ϕ +... )/(2n2-sin2ϕ+k2sin2ϕ-.) = k (1 + (s in 2ϕ/n2) - (k2 s i n2ϕ/n2) ) = k [1 + (s in 2ϕ/n2)(1 - k2 + k )). Hence, ta n 2ψ = k [1 + (sin2ϕ/(n2(1+k2)))]. (37) E quations (35) and (37) may be used s im u lta n e o u s ly f o r f in d in g n and k, when ϕ and ψ are known. The c o e f fic ie n t o f r e f le c t io n o f any substance is d e fin e d as the r a t io o f the in te n s ity o f the r e fle c te d l i g h t to th a t o f the in c id e n t lig h t when the angle o f in cid e n ce is ze ro. In t h is case, cosϕ = cosχ = 1; and sin ce ε ' may re p la c e (ε2/ε1), equatio n ( 1 4 '), when a p p lie d to conducting m edia, becomes:

24 21 S u b s titu tin g Rη'ei δη f o r Rη, and n 2( 1- ik ) 2 fo r ε ', the above exp ression becomes, (Rη'ei δη/eη) = (n ( 1- ik ) - 1 )/(n(1-ik)+1). (38) M u ltip ly in g (38) by i t s complex c o n ju g a te, (Rη'eiδη/Eη) = (n(1+ik)-1)/(n(1+ik)+1), and remembering th a t the in te n s ity o f a v ib r a tio n is p ro p o rtio n a l to the square o f i t s a m p litu d e, we g e t f o r the c o e f fic ie n t o f r e f le c t io n, R = (n 2 ( 1+k 2)+ 1-2n)/(n2(1+k2)+1+2n). (39)

25 22 Apparatus and Method o f Procedure. I t is thus p o s s ib le, from the fo re g o in g, to co m p le te ly determ ine the o p tic a l consta n ts o f a m e ta l, when the angles o f p r in c ip a l in cid e n ce and p r in c ip a l azim uth are known. The method o f procedure f o r o b ta in in g these a n g le s, in the case o f the -Xc r y s ta l o f selenium p re v io u s ly re fe rre d to, was as fo llo w s : A N ic o l, to be used as a p o la r iz e r, was mounted on the c o l lim a to r o f a sp e ctro m e te r. A second N ic o l and a B a b in e t compen- s a to r were a tta ch e d to the te le s c o p e. T his la s t N ic o l,' to be used as an a n a ly z e r, was mounted between the e ye -piece and the B a b in e t. In the diagram, L is the c o llim a tin g le n s, P the pola r iz in g N ic o l, C the p o s itio n o f the c r y s ta l, B the B a b in e t, A the a n a ly z in g N ic o l, and E the e y e -p ie c e. An e le c t r ic arc and a monochromator (n o t shown in the diagram ) fu rn is h e d the d e s ire d illim in a t io n. The c r y s t a l, mounted on the p ris m -ta b le o f the sp e ctro m e te r, was a d ju s te d to be p e rp e n d ic u la r to the plane o f the te le scope and c o llim a to r. L ig h t o f a known w a v e -le n g th, p o la riz e d in a plane making an angle o f 45 degrees to the plane o f in c id e n c e, was allo w ed to f a l l on the face o f the c r y s t a l, and the r e fle c te d ra y caught by the t e le scope. The angle o f in c id e n c e was then v a rie d u n t i l the c e n tra l fr in g e o f the B a b in e t, p re v io u s ly d is p la c e d thro ugh a ^Wood's O p tic s, p.jl.64, second e d itio n ; Mann's Manual o f Advanced O p tic s, Chap, on M e ta llic R e fle c tio n. **Wood, i b i d., p.3 33; Drude, Theory o f O p tic s, p.2 57.

26 23 d is ta n c e corresponding to a p h a s e -d iffe re n c e o f a q u a rte r o f p e rio d, re tu rn e d to the zero p o s itio n. T his gave the angle o f p r in c ip a l in c id e n c e. A t t h is a n g le, the B a bin et a n n u lle d the p h a s e -d iffe re n c e in tro d u c e d by the m e ta llic r e f le c t io n, so th a t the e l l i p t i c a l v ib r a t io n was converted in to a p la n e - p o la riz e d one. The plane o f p o la r iz a tio n o f t h is v ib r a tio n was determ ined b y a d ju s tin g the a n a ly z in g N ic o l u n t i l the frin g e s were b la c k e s t. The p r in c ip a l azim uth is thus the ang le between t h is plane and the plane o f in c id e n c e. On r o t a t in g the a n a lyzer through an angle equal to tw ic e the p r i n c i p a l azim uth, the o r ig in a l frin g e s d isappear, and another s e t appears, the la t t e r frin g e s occupying p o s itio n s midway between the p o s itio n s occupied by the form er s e t. Owing to a wide range in w hich the apparatus may be adju s te d, any o f which adjustm ents appear e q u a lly w e ll to the o b serve r, s e v e ra l re a d in g s o f the same q u a n tity must be take n. In a l l o b se rva tio n s o f the p re se n t s tu d y, the fo llo w in g p la n was used: W ith a g ive n w a ve -le n g th, te n re a d in g s o f ^ were f i r s t taken when the in c id e n t v ib r a tio n was p o la riz e d in a plan e making an angle o f +45 degrees to the pla n e o f i n c i dence, and another s e t o f te n re a d in g s when the in c id e n t v i b ra tio n was p o la riz e d in a plane making an angle o f -45 degrees to the plane o f in c id e n c e. W ith the angle o f in cid e n ce thus determ ined, ten re a d in g s were taken o f the a n a lyzer when a d ju ste d f o r the c e n tra l fr in g e s, and te n when a d ju s te d f o r the a lte rn a te fr in g e s, w ith the in c id e n t lig h t p o la riz e d in both the +45-degree plane and the -45-degree p la n e. The re a d ings o f the a n a ly z e r when s e t f o r the c e n tra l fr in g e s, the

27 24 in c id e n t lig h t being p o la riz e d in the +45-degree p la n e, should check w ith the re a d in g s when s e t f o r the a lte rn a te fr in g e s, the in c id e n t l i g h t t h is tim e being p o la riz e d in the -45-degree p la n e. The a rith m e tic a l means o f the sets o f lik e re a d in g s, to g e th e r w ith t h e ir probable e r r o r s, are ta b u la te d on the sane page w ith the o b s e rv a tio n s them selves. The probable e rro rs appear as p lu s or minus q u a n titie s. They have been determ ined from the approxim ate fo rm u la, where r 0 is the probable e r r o r, n the number o f o b s e rv a tio n s, and S v the sum o f the r e s id u a ls, a l l taken w ith the p o s itiv e s ig n. (A re s id u a l is the d iffe re n c e between any s in g le observ a tio n and the a r ith m e tic a l mean o f the n o b s e rv a tio n s.) In d e te rm in in g the probable e rro rs o f the f in a l r e s u lts, as obta in e d from equations (3 5 ), (3 7 ), and (3 9 ), the fo llo w in g ru le s were found to be s u f f ic ie n t ly c o rre c t: When adding, subt r a c t in g, m u ltip ly in g, and d iv id in g numbers, the e r r o r o f the f in a l r e s u lt is equal to the sum o f the e rro rs in each number used; and when a number is ra is e d to any power, the e rro r o f the r e s u lt is equal to the e rro r o f the number m u ltip lie d by the number re p re s e n tin g the power. M errim an: M e th o d 'o f L e ast Squares, p.9 3, fo rm u la (3 6 ).

28 25 Data. The fo llo w in g d e f in itio n s o f the symbols used in ta b u la tin g the data w i l l make the ta b le s c le a r. Readings were take n, n o t o f ϕ d ir e c t ly, b u t o f the supplement o f 2ϕ ; so th a t, i f A = 180-2ϕ, ϕ = (180 -A)/2. A' = re a d in g o f spectrom eter when the in c id e n t lig h t is p o la r ize d in a plane making an angle o f +45 to the plane o f in cidence. A" = re a d in g o f spectrom eter when the in c id e n t l i g h t is p o la r ize d in a plane making an angle o f -45 to the plane o f in cidence. A = the mean o f A' and A ". When the l i g h t is in c id e n t a t the angle ϕ and p o la riz e d in the +45 p la n e, a = re a d in g o f a n a lyze r when a d ju s te d f o r the c e n tra l fr in g e, a' = " " " " " " " a lte rn a te fr in g e s. When the lig h t is in c id e n t a t the angle ϕ, and p o la riz e d in the -45 p la n e, b = re a d in g o f a n a lyze r when a d ju s te d f o r the a lte rn a te fr in g e s, b ' = " " " " " " " c e n tra l fr in g e. 2 ψ = a '- a = b ' - b. C ry s ta l H o riz o n ta l = a x is o f c r y s ta l p a r a lle l to plane o f in c id e n c e. C ry s ta l V e r tic a l = a x is o f c r y s ta l p e rp e n d ic u la r to plane o f in c id e n c e.

29 26 Table I. C ry s ta l H o riz o n ta l. λ = 406μ μ. A' A" A Ø ' ' ' ± 2.57' ' ± 1.3' a a ' b ' b ψ ' ' ' ' ' ± 5.94' ψ = ' ± 11.88'

30 27 Table I I. C ry s ta l V e r t ic a l. λ = 406μ μ. A' A" A ϕ 36 15' ' ' ± 4. 6' ' ± 2.3 ' a a ' b b ψ ' ' ' ' '± 6. 06' ψ = ' ± 12. l2 '

31 28 Table I I I. C ry s ta l H o riz o n ta l. λ = 441μ μ. A' A" A ϕ 23 50' ' ' ± 3.49' ' ± 1. 75' a a b ' b ψ ' ' ' ' ' ± 4.8 7' ψ = ' ± 9.74'

32 29 Table IV. C ry s ta l V e r t ic a l. λ = 441μ μ A' A" A ϕ ' ' ' ± 2. 25' ' ± 1.12' a a' b ' b ψ ' ' ' ' ' ± 4.73' ψ = ' ± 8. 47'

33 30 Table V. C ry s ta l H o riz o n ta l. λ = 480μ μ. A A" A Ø ' ' ' ± l.8 4 ' ' ± 0. 92' a a b b ψ ' ' ' ' ' ± 1.84' ψ = ' ± 7.56'

34 31 Table V I. C ry s ta l V e r t ic a l. λ = 480μ μ. A' A" A ϕ ' 37 45' ' ± 1.06' ' ± 0. 53' a a ' b ' b ψ ' ' ' ' ' ± 2.71' ψ = ' ± 5.42'

35 32 Table V I I. C ry s ta l H o riz o n ta l. λ = 505μ μ. A' A" A Ø 25 25' 25 00' ' ± 1.6' ' ± 0.8' a a ' b ' b ψ ' ' ' ' ' ± 3.4' ψ = ' ± 6.8'

36 33 Table V I I I. C ry s ta l V e r t ic a l. λ = 505μ μ. A' A" A ϕ ' 37 55' ' ± 1. 94' 71 0 l. 2 5 ' ± 0. 97' a a ' b ' b ψ ' ' ' ' ' ± 4.84' ψ = ' ± 9. 68'

37 34 Table IX. C ry s ta l H o riz o n ta l. λ = 550μ μ. A' A" A Ø ' 25 50' ' ± 1.26' ' ± 0. 63' a a ' b ' b ψ ' ' ' ' ' ± 3. 09' ψ = ' ± 6.18'

38 35 Table X. C ry s ta l V e r t ic a l. λ = 550μ μ. A A" A Ø 36 30' ' ' ± 2.08' ± 1.04' a a* b ' b ψ ' ' ' ' 7 4 l.7 5 ' ± 3.43' ψ = ' ± 6.86'

39 36 Table X I. C ry s ta l H o riz o n ta l. A = 589μ μ. A' A" A ϕ 26 05' ' 26 10' ± 1. l ' ' ± 0.5' a a ' b b ψ ' ' ' ' 6 18' ± 2.7' = ' ±5.39'

40 37 Table X I I. C ry s ta l V e r t ic a l. λ = 589μ μ. A' A" A ϕ ' 34 25' 34 40' ± 1.65' 72 40' ± 0.82' a a ' b ' b ψ ' ' ' ' 8 54' ± 4.08' ψ = ' ± 8. 16'

41 38 Table X I I I. C ry s ta l H o riz o n ta l. λ = 626μ μ. A' A" A ϕ 25 40' ' ' ± 1.46' ' ± 0.73' a a b ' b ψ ' ' ' ' ' ± 4.17' ψ = 10 39' ± 8. 34'

42 39 Table XIV. C ry s ta l V e r tic a l. λ = 626μ μ. A A" A Ø ' 33 35' ' ± 2. 38' l2 ' ± l. l9 ' a a' b ' b ψ ' ' ' ' ' ± 3.62' ' ψ = ' ± 7.24'

43 40 Table XV. ' C ry s ta l H o riz o n ta l. λ = 668 μ μ. A' A" A Ø ' 26 20' ' ± 2.0 4' ' ± 1. 02' a a b ' b ψ ' ' ' ' ' ± 4.97' ψ = ' ± 9.94'

44 41 Table XVI. C ry s ta l V e r t ic a l. λ = 668μ μ. A' A" A Ø ' ' ' ± 2.59' ± 1.3' , a a ' b ' b ψ ' ' ' ' ' ± 3.72' ψ = ' ± 7. 44'

45 42 F o llo w in g are the values o f ϕ and ψ, as found in Tables I. - X V I., w ith the c a lc u la te d c o n s ta n ts, n, k, and R. Table X V II. gives the r e s u lts when the c r y s ta l is in the h o riz o n ta l p o s itio n (a x is o f c r y s ta l p a r a lle l to the plane o f in c id e n c e ), and Table X V III. when the c r y s ta l is in th e v e r t ic a l p o s itio n (a x is o f c r y s ta l perpend ic u la r to th e plane o f in c id e n c e ).

46 43 Table X V II. C ry s ta l H o r iz o n ta l. ψ λ ϕ 406μ μ '± 1.3' 12 32'± 6 ' λ n k R 406μ μ ± ± ±

47 44 Table X V III. C ry s ta l V e r t ic a l. ψ λ ϕ 406μ μ '± 2.3' 8 40'± 6' λ n k R 406μ μ ± ± ±

48 45 Tables X V II. and X V III. show c o n c lu s iv e ly th a t the c r y s ta l is doubly r e f r a c tin g ; w h ile Tables I.- X V I. show th a t one o f the opt i c axes c o in c id e s w ith the g e o m e tric a l a x is o f the c r y s ta l, and th a t the o th e r o p tic a x is is a t r ig h t angles to the f i r s t. For i f one o f the o p tic axes d id n o t c o in c id e w ith the g e o m e tric a l a x is, the components o f the in c id e n t e le c t r ic fo rc e along the o p tic axes would be a lte re d when the angle between the plane o f p o la r iz a tio n o f the in c id e n t lig h t and the plane o f in cid e n ce changes from +45e to -4 5 ; and hence the observed va lu e s o f A' and A" would not check w ith each o th e r. A ls o, the fo llo w in g d a ta, taken w ith the g e o m e tric a l a x is o f the c r y s ta l in a plane making an angle o f a p p ro xim a te ly +45, and then a p p ro xim a te ly -4 5, to the plane o f in c id e n c e, in d ic a te th a t the o p tic axes o f the c r y s ta l in the two p o s itio n s were sym m etrica lly placed w ith re s p e c t to the plane o f in c id e n c e ; and hence th a t one o f the o p tic axes must c o in c id e w ith the g e o m e tric a l a x is. Angle between a x is o f c r y s ta l and plane o f in cid e n ce (=<* ) A p proxim ately +45 A pproxim ately -45 Table XIX. h = 589^/U- Angle between plane o f p o la r i z a tio n o f in c id e n t l i g h t and plane o f in cid e n ce (= 9) 1 V l' o / W ith in e xp e rim e n ta l e r r o r s, the observed values o f and y do not change i f, when the c r y s ta l is tu rn e d through an angle 2o(, the plane o f p o la r iz a tio n o f the in c id e n t lig h t is tu rn e d through an angle 20. This fo llo w s o n ly when one o f the o p tic axes o f the c r y s ta l c o in c id e s -w ith the g e o m e tric a l a x is.

49 46 The form ulas used in c a lc u la tin g n, k, and R, were developed on th e assu m ptio n o f a homogeneous medium. I t has been shown^beyond d o ubt, th a t the type o f c ry s t a l in v e s tig a te d is n o t homogenous, but doubly r e f r a c t in g. I t is th e re fo re c le a r th a t these form ulas are not s t r i c t l y a p p lic a b le to th e p re sent problem. Form ulas, w hich are a p p lic a b le to a non-homogeneous medium, w i l l now be g iv e n.

50 47 E quations f o r C r y s ta llin e Media. Let us remember th a t f o r th e type o f c r y s ta l under in v e s tig a tio n th e o p tic axes are p a r a lle l and perpend ic u la r, re s p e c tiv e ly, to the g e o m e tric a l a x is. For a c r y s ta l o f t h is ty p e, Drude* has shown th a t i f we le t r = (Rη/Ry)= ρ ei Δ, ( c. f. eq.29, p.1 7 ) where ζ is the angle in c lu d e d between the a x is o f the c r y s ta l and the plane o f in c id e n c e. Then, i f R = (1+r)/(1-r) = (1+ρeiΔ)/(1-ρeiΔ), ( c. f. eq.32, p. 17) Let R = R1 when ζ = 0, i. e., when the c r y s ta l is h o r iz o n ta l. Then, Now, le t us re tu rn to the e xp re ssio n, R = (1+ρ eiδ)/(1-ρ eiδ). R eplacing ρ by t a n ψ, and expanding eiδ, (=cosδ + is in Δ ) we can show th a t When Δ = π/2 the above e xp re ssio n f o r R reduces to : *P.Drude, " L ic h tr e fle x io n am A n tim o n g la n z", Ann. d. Phys.34, p. 529, 1888.

51 48 where the s u b s c rip ts, as b e fo re, r e fe r to the c r y s ta l in the h o riz o n ta l and v e r t ic a l p o s itio n s, re s p e c tiv e ly. E quating th e two expressions f o r R1 and R2, we g e t: Let the values o f α and β, as g iv e n by equations (A ), be α = a11+ i a 12, β = a21 + ia 22. Drude* has shown th a t: I f χ be d e fin e d as an a n g le, so th a t a 1 2 / a 1 1 = t a n χ, we can deduce from the above equations th e r e la t io n, (T his may be o b ta in e d from e q u a tio n (1) a lo n e, o r from (3)' and (4)' take n to g e th e r). *P. Drude, "R e fle x io n und B rechung", Ann. d. Phys. 32, p.61 6, "Ueber O b e rfläc h e n s c h ic h te n ", Ann. d, Phys. 36, p.544, "T h e o rie der M e ta llr e fle x io n ", A nn.d.p hys. 35, 1888, p.5 18, e q.(1 5 ). V o ig t has a rriv e d a t e q u iv a le n t expressions f o r n and k. See A nn.d.p hys. 31, 1887, p

52 49 Now, from e q u a tio n ( 4 ) ', w hich reduces to S im ila r ly f o r β= a 21 + ia 22, i f we d e fin e ε by the exp ression a2/a21=tanε, we get f o r k2 and n2 : Mü l l e r * has used th e e q u a tio n s, as g iven h e re, in d e te rm in in g the o p tic a l consta n ts o f antim ony c r y s ta ls. The e xp re ssio n f o r th e c o e ffic ie n t o f r e f le c t io n, given on page 21 (e q u a tio n 3 9 ), remains unchanged. * E. C. Mül l e r, "O ptisch e S tu d ie n am A n tim o n g la n z", Neues Jahrbuch f ürm in e ra lo g ie, e tc., B. B. X V II., 1903, p.215.

53 50 The r e la tio n s, th e re fo re, e x is tin g between the o p tic a l constants and the angles o f p r in c ip a l in cid e n ce and p r in c ip a l azim uth, in the case o f a c r y s t a llin e medium such as the hexagonal c r y s ta l o f selenium, are as f o llo w s : Let ϕ1 = the angle o f p r in c ip a l in c id e n c e, when the c r y s ta l is h o r iz o n ta l, ϕ2when the c r y a ta l is v e r t ic a l, ψthe p r in c ip a l azim1 uth, when h o r iz o n ta l, = ψ2 when v e r t ic a l, and n 1, n 2, k1, k2, and R1, R2, the corre spondin g in d ic e s o f r e f r a c t io n, a b s o rp tio n c o e ffic ie n ts, and c o e ffic ie n ts o f r e f le c t io n, re s p e c tiv e ly, th e n :

54 51 Using these form ulas f o r c a lc u la tin g n, k, and R, we get what we b e lie v e to be the tru e v a lu e s. The r e - i s u its are given in Table XX. We are in c lu d in g, f o r re fe re n c e, in Table X X I., the corre spondin g r e s u lts o f F o e rs te rlin g and F r^ e d e ric k s z. In Table XXII., Wood's r e s u lts f o r the index o f r e fr a c t io n are g iv e n. The values o c c u rin g in the ta b le are taken from the curve o f h is a r t ic le a lre a d y re fe rre d to (See W-W, F ig. 7 ). I t is to be observed th a t the r e s u lts f o r the re f le c t in g power, when the c r y s ta l is v e r t ic a l, are in good accord w ith r e s u lts o b ta in e d by P fu n d *, who used a d ir e c t method o f measurement (See Table X X I I I., also P-P, F ig. 9 ). The surfa ces used by Pfund were prepared in the same manner as those used by F o e rs te rlin g and F r^ e d e ric k s z. In Table X X IV., th e re is a summary o f th e r e s u lts f o r the index o f r e fr a c tio n o f selenium, f o r wavele n g th 589 / w. as given by v a rio u s o b serve rs. * H. A. Pfund, P h ysik. Z e its c h r. 10, p.3 4 0, 1909.

55 52 Table XX. (H o r iz o n ta l) ( V e r t ic a l) λ n 1 k 1 R1 n2 k2 R

56 53 Table XXI. From F o e rs te rlin g and F rée d e ric k s z. * λ ϕ ψ n k R 398μ μ 72 36' 8 32' 3.02 * * This r e s u lt f o r k is o b ta in e d from the a p p ro x i mate fo rm u la, k = tan 2ψ. A ll o th e r r e s u lts in above ta b le are o b ta in e d from the com plete fo rm u la. There is no apparent reason f o r the v a r ia t io n. The complete fo r mula g ives f o r k, T his is the value used when p lo t t in g the curve.

57 54 Table X X II. Wood s r e s u lts f o r the Index o f R e fra c tio n. (Taken from cu rve ) λ n Table X X III. P fund s r e s u lts f o r the C o e ffic ie n t o f R e fle c tio n. λ R

58 55 Table XXIV. Observer Form o f Se. Surface Method o f p re p a ra tio n G ripenberg C ry s ta llin e N a tu ra l M elted Se. pressed between g la ss p la te s Wood V itre o u s, tra n s p a re n t Cathode d e p o s it M eier Amorphous P o lish e d M elted Se. spread over p la te brass 71 10' F o e rs te r lin g and F réederic k s z S kinner Amorphous C ry s ta llin e ( in mass) Is o la te d hexagonal c r y s ta l Cast Cast N a tu ra l M elted Se. spread on g la ss p la te Above form c ry s t a lliz e d by h e a t in g By s u b lim a tio n 71 08' 73 06' * 72 40' ** ' *A xis o f c r y s ta l p e rp e n d ic u la r to plane o f in c id e n c e. * * A x is o f c r y s ta l p a r a lle l to plane o f in c id e n c e. 6 51' 3 15' 8 59' 8 54' 6 18' λ = 589μμ. n k R

59 56 D is c u s s io n o f Curves. The curves which fo llo w (P ig s. 5-9) show the r e la tio n s exis t in g between the consta n ts and w a ve-le n g th. In the process o f making the m irro rs used by F o e rs te rlin g and F r& e d e ricksz, the h o t selenium was b ro ught in to c o n ta c t w ith g la s s, which v e ry reasonably m ig h t a f fe c t the surfa ce in such a way th a t the s u rfa c e i t s e l f d id n o t t r u ly re p re s e n t the c o n d itio n o f the media w ith in. This would le a d to the c o n clu s io n th a t t h e ir r e s u lts would n o t agree w ith the r e s u lts o f the p re sent in v e s tig a tio n, so f a r as a b solu te valu e s are concerned, b u t th a t corresponding curves m ight agree in form. T his is found to be tr u e, in the case o f the curves w hich re p re s e n t the c r y s ta l in the v e r t ic a l p o s itio n. The a b solu te v a l ues o f are s lig h t ly low er than t h e ir v a lu e s, b u t the curves (F ig. 5) show an e x c e lle n t agreement in form. They b o th show a minimum a t about 500/^- and a maximum a t about 620^p..The values o f (fj are s lig h t ly h ig h e r than t h e ir v a lu e s ; b u t a g ain, the e xce lle n ce in agreement, as to form, is v e ry s t r ik in g (F ig. 6 ). Both show a d is t in c t maximum a t about ^. I t f o l low s, sin ce ^ and (// agree so c lo s e ly, th a t the values o f the c a lc u la te d consta n ts must a lso agree (F ig s. 7-9 ). Mien the c r y s ta l is place d in the h o riz o n ta l p o s itio n, i t acts lik e an a lto g e th e r d if f e r e n t substance. There is no lo n g er any resemblance to o th e r r e s u lts, even in form. The values o f J (F ig. 5) are v e ry much la r g e r. There is no maximum or minimum, the v a r ia tio n w ith w a ve-length being alm ost consta n t between the values 78,4 7 f o r w a ve-le n g th 406^U and fo r * *See an a r t ic le by T ate, Phys.Rev. 34, 1912, p.3 21.

60 57 668μ μ. The values o f ψ (F ig. 6) range from 12 32' to 4 58' w ith no maximum o r minimum. The curve in te rs e c ts the curve corre spondin g to the c r y s ta l in the v e r t ic a l p o s itio n a t two p o in ts, 545μ μ and 625μ μ. I t is in te r e s tin g to note th a t the maximum o f the l a t t e r curve occurs e x a c tly mid-way between these values o f λ. The values o f n (F ig. 7) are alm ost consta n t th ro u g h o u t the g re a te r p a rt o f the v is ib le spectrum. A t 589μ μ, n = 4.0 4, a r e s u lt more than 15% g re a te r than th a t found by any o th e r o b serve r. There is no o th e r known element w ith as h ig h a r e fr a c tiv e in d e x.* The values o f k (F ig. 8) l i e on alm ost a s t r a ig h t lin e between and , f o r w ave-lengths between 406μμ and 668μ μ. The carve in te rs e c ts the one corre spondin g to the c r y s ta l in the v e r t ic a l p o s itio n in very much the same manner as the curves in F ig. 6. The c o e f fic ie n t o f r e f le c t io n ( F ig. 9) is much h ig h e r than when the c r y s ta l is v e r t ic a l, the v a l ues dro ppin g g ra d u a lly from 0.47 to 0.3 7, between the w ave-lengths 406μ μ and 668μ μ. N o te.- Curves marked H-H are f o r the c r y s ta l when in the h o riz o n ta l p o s itio n ; those marked V-V are f o r the c r y s ta l in the v e r t ic a l p o s itio n ; and those marked F-F are from the r e s u lts o f F o e rs te rlin g and F rée d e ric k s z. Wood's r e s u lts f o r the in dex o f r e fr a c tio n are shown in curve W-W, F ig. 7 ; and P fu n d ' s r e s u lts f o r the c o e f fic ie n t o f re f le c t io n are shown in curve P-P, F ig. 9. * This does n o t a p p ly to compounds; f o r in s ta n c e, Sb2S3(S t ib n it e ). See Mül l e r, l. c., p. 240; Drude, A nn.d.p hys. 34, p.5 23, 1888.

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66 63 Summary. 1. An is o la te d hexagonal c r y s ta l o f selenium is doubly r e f r a c t in g. 2. The o p tic axes are p a r a lle l and p e rp e n d icu la r, re s p e c tiv e ly, to the geom etric a x is. 3. The o p tic a l c o n s ta n ts, as determ ined w ith the a x is o f the c r y s ta l p e rp e n d ic u la r to the plane o f in c id e n c e, are in e x c e lle n t agreement w ith r e s u lts o b tain ed by o th e r o b serve rs. 4. The o p tic a l c o n s ta n ts, as determ ined w ith the a x is o f the c r y s ta l p a r a lle l to the plane o f i n c i dence, are p e c u lia r to the is o la te d hexagonal c ry s t a l. P h ysica l L a b o ra to ry, S tate U n iv e rs ity o f Iowa.