The Optical Properties of Solids

The Optical Properties of SolidsAuthor(s): A. H. WilsonSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 151, No. 873 (Sep. 2, 1935), pp. 274-295Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/96547 .

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274 A. H. Wilson

excitation without electron exchange over the energy range of 20 to. 40 volts.

The excitation function for the 2 3P1 state reaches a maximum of 14 6 cm2/cm3 at 7 volts, that for the 2 'P1 state has a broad maximum of 15 cm2/cm3 at about 15 volts. The curve for the 3 1D2 state reaches a fairly sharp maximum at 15 volts.

The Optical Properties of Solids

By A. H. WILSON, Trinity College, Cambridge

(Communicated by R. H. Fowler, F.R.S.-Received March 27, 1935)

INTRODUCTION

I-The theory of the optical constants of metals remained in the state in which it was left by Drude until Kronigt applied the modern theory of metals to the problem. Since the appearance of Kronig's first paper many authors' have tried to extend the theory so as to bring the finer effects within its scope, and we may sum up the present position as follows. In the infra-red the optical constants of metals vary with temperature, and are therefore very much influenced by the thermal vibrations of the solid. For frequencies in this region the existing theory is effectively the Drude theory which relates the optical constants to the conductivity in static fields and to the inertia of the electrons. This theory is satisfactory in the far infra-red, but fails in the near infra-red. In the visible and ultra-violet the optical constants are approximately independent of temperature, which means that the effect of the thermal vibrations can be neglected, and the problem is thereby considerably simplified. In this region, therefore, we should expect the theory to be reasonably accurate, and we might hope to obtain useful information about the internal state of a metal. It must, however, be admitted that there is no really consistent theory extant, and, since our means of investigating the interior of a metal are very limited indeed, it seems desir-

t 'Proc. Roy. Soc.,' A, vol. 124, p. 409 (1929), and vol. 133, p. 255 (1931). - Fujioka, 'Z. Physik.,' vol. 76, p. 537 (1932); 'Sci. Pap. Imp. Univ., Tokyo,'

vol. 22, p. 202 (1933); Frbhlich, 'Z. Physik.,' vol. 81, p. 297 (1933); Mott and Zener, 'Proc. Camb. Phil. Soc.,' vol. 30, p. 249 (1934); Sergeiev and Tchernikovsky, 'Phys. Z. Sowjet.,' vol. 5, p. 106 (1934).



The Optical Properties of Solids 275

able that the theory should be put upon 'as sound a basis as possible. It is the main object of this paper to give a critical discussion of the phenomenon in the visible and ultra-violet, the effect of the lattice vibrations being entirely neglected.

In ? 2 the fundamental formula, which is a generalization of the Kramers-Heisenberg dispersion formula, is derived, and it is shown that it is quite unnecessary to make the usual assumption that the size of the crystal is small compared with the wave-length. In fact the problem becomes clarified if we do not make this assumption. It is also shown that there is no Lorentz-Lorenz correction to be introduced.

In ? 3 we discuss the dispersion formula for metals, with special reference to silver. It has usually been assumed that the theory gives quite wrong results for the threshold for internal photoelectric absorption. This is due to the fact that only simple cubic structures have been considered, and for these the theoretical threshold frequency is about one-twelfth the experimental. For face-centred cubic structures it is shown that the ratio is about two-thirds, so that there is no large discrepancy.

In ? 4 the dispersion formula for insulators is derived, and in ? 5 the relation between absorption and dispersion is discussed. In substances such as diamond, the internal photoelectric effect, as measured by the number of electrons liberated, does not depend in a simple way on the absorption coefficient K. The number of electrons liberated increases to a maximum and then falls off rapidly to zero in the interior of the absorption band. Now the number of photoelectrons must be equal to the number of photons absorbed, which is proportional to nK, where n is the refractive index. Therefore if n is abnormally low, the photoelectric response will be very different from the absorption coefficient. It is shown that throughout the whole of the absorption band K iS increasing more rapidly than n, and the dielectric constant may even become negative, so that K can reach very high values while nK is very small.

Finally the appendix is devoted to a proof of thef-sum rule for a metal.

GENERAL METHOD

2.1-The propagation of light in a solid is a perfectly determinate problem in the sense that we can formulate the mathematical equations exactly, and there are no difficulties of principle involved. This is particularly true when we deal with the coherent scattered radiation, since most of the complications of the radiation theory are absent in this problem. The difficulties all arise in the practical solution of the equations,



276 A. H. Wilson

since we have to introduce simplifications which may well obscure important features.

The mathematical problem can be formulated as follows. Consider a solid in which there are N electrons moving in the field of the nuclei and of each other. Let E7 be the electric force acting on the electron y due to a light wave, and suppose we have solved the Schrodinger equation for the whole system, obtaining the wave function T which is a function of the 3N co-ordinates of the electrons and of the N electric forces Ey. The density p, and current jy associated with the electron y are then defined by

Pz =-e T*Tdr (1)

f * grad T P-T grad, T*? eAU * 1f )d (2)

where the integration is over all the co-ordinates ,except those of the electron y, and Al is the vector potential associated with Ey. Now the force E7 acting on an electron is the vector sum of the electric intensities of the incident light-wave and of the light-waves scattered by the other electrons, and can therefore be calculated from Maxwell's equations for free space

div El = 47 PT

curl H, = I (aEY + 47t3)

curl E _ ) c at

where PI/ E pz y' # Y

J,Y E j. Y' # Y

By eliminating Hy, we obtain integro-differential equations for E-, the solutions of which provide the answer to our problem. It is to be noted that the total electric intensity E of the light in the metal is different from E.. The total intensity is determined from the Maxwell's equations in which P, J take the place of Py, J,, where

P E - P= SP

JTen o r

The difference E. E Z is the local Lorentz-Lorenz force.




Having stated our problem, we must immediately proceed to simplify it. In the present state of the theory of metals we are forced to reduce the many-electron problem to a one-electron problem, so far as the solution of the Schrodinger equation is concerned. In the absence of a light- wave each electron moves in a field V, which is the field of the nuclei and the average field produced by the other electrons, while in the presence of a light-wave each electron moves in the.field V and in the field produced by the light scattered by the other electrons. The problem is therefore partly a one-electron problem and partly an N-electron problem, and it is a difficult question to decide whether this simplification is at all justified. Another profound simplification is an immediate consequence of this assumption. The use of one-electron wave functions implies that each electron is distributed over the solid and has no localized position. It then follows that the differences between the forces E and E are uniformly of the order 1 /N and never become large, as would occur if the electrons were localized, since I E - j can only become appreciable where the corresponding density py and current j, are appreciable. We may therefore use the same electric vector E for all the electrons, and this E is the electric vector of the light-wave in the solid. This does not mean there is necessarily no Lorentz-Lorenz correction term in the dispersion formula for a metal, but that such a correction is inconsistent with the approximations used in deriving the one-electron wave functions of present- day metal theory, and is automatically excluded by our assumptions. The use of single-electron wave functions is an unpleasant necessity in the theory of metals, and, at the moment, it is difficult to see the effect of simplifying the problem so much, except where the mutual interactions of the electrons play little or no part.

2.2-We have now to solve the problem of an electron moving in a given field of force, the potential energy of the electron being V, and being influenced by a light-wave, the vector potential of which is A, where

A = a e2vPt + a* e-27rivt (4) The wave equation is

h V2M _

eh h a( 8sm ~~27rmc 2nit

and we obtain a solution in terms of the unperturbed wave functions %, exp (- 27rivkt) by the method of the variation of constants. W, hVk is the energy of the state k. If we assume that at time t - 0 the electron is in the state k, then, to the first order in the amplitude of the light-wave, we have

Tk = kk e-27r Vkt + e 4 I21t (Ckl + ?ck) (6)



278 Al. H. Wilson

where

Ckl=_ 1 e (tk*(a . grad d) {e2dr( lk+) - l} (7A) - V + v 27timc

ckL I= 1 e2 i ',* (a* . grad 4k) dr . {e2-ilk-v) t-1}, (7B) 'Vak v 27rimc

"Vlbeig (vl- Vk.i

vf being (vd-v). It is conveiient to choose tk to be real, and we then find

J e2 t,2 e 2h0{ 1 1 cos2WT(Vk +V)t mc 87rM mC l VZk + V + Vik- Vtk+V

cos 2n (vlk - v) t} f l (A. grad +) dt

+ 1{ sin 27 (v, 1,+ v) t sin 2nt(vlk -v) t} J Az(A gr X - - t i +1t-. ~~~~~grad dr 2svt wa+ vl 1-v aJ t -

X (4k grad 4 I grad k). (8) When the current is written in this form it is obvious that there is no singularity in jk when v = vk. If N (k) is the number of electrons in the state k, the total current is given by

i = E N (k) j (k). (9) k-

We should have obtained exactly the same equation for J if we had taken the wave functions to be complex instead of real, since if a complex wave function occurs its conjugate also occurs for some value of k, and the total current is independent of our choice of wave functions. So far we have not made use of the fact that we are dealing with a solid and the equations (8) and (9) are therefore perfectly general. In a solid, the electronic states lie very close together and, when the solid is infinite, form a continuum. The frequency vl is a continuous function of k, so that the summation over k can be replaced by an integral. Now, iff (x) is finite everywhere,

X2f() 1 cos-C dXt R f (x) I-e dx

f~~~~~ X dx R (X) x p X2fX dx

XI

for large values of t, where R denotes the real part and P denotes the principal value. Using the fact that

1 aA aE 472 V2 c at at c




we may rewrite J in the form

e2 rk 2 aE h a E (.r 4 =- mv2 N (k) k2- +i42m Vk J da

hv sin 2WT (vlk + v)t sin2 (v1k - V) t) ( * W I)IQ (;E. grad Qk) dl > ME I \ vlk + 'v Vlk -v

X (4kgrad 4I- I grad 4), (10)

where the principal value of the summation with respect to k and I is to be taken. Kronig and Groenewold,t using a different form of the per- turbation theory, obtained equation (10) with the following modifications. The terms involving E were absent, and the summation with respect to k and I was to be taken quite generally, and not restricted to the principal value. The disadvantage of this method is that we have then to take the principal values of the summations without any justification for doing so; and to obtain that part of the current containing E, the conduction current, we must use a relation, due to Kramers,t between dispersion and absorption. The method adopted here gives both the displacement and conduction currents without introducing any extraneous hypothesis, and shows that there are no infinities to be explained away. It seems to have been first pointed out by Waller,? using Dirac's radiation theory, that there is no difficulty in obtaining that part of the current vibrating in phase with the light-wave, provided the frequency is large enough to excite the electron into a state belonging to the continuous spectrum.

2.3-NVe have now to solve the Maxwell equations (3) inserting the value of J given by (10) and the corresponding p, which is most easily obtained from the relation ap / a8t + div J 0 O. In the classical theory -the electric vector is of the form (E., 0, 0) where

Ex = Fe.2wK/c cos 27v (t - [nz/c]), (11)

but we cannot expect to find such a simple exact solution here. The expression (11) is not, however, an exact solution of the equations of the Cclassical electron theory, but is only a solution when the equations have been suitably averaged over a physically small region. If we assume that (11) is an approximate solution of the quantum equations, it is a simple .matter to determine the constants n, K, but it is not so simple to prove that

t ' Physica,' vol. 1, p. 255 (1934). X ' Proceedings of the Volta Congress,' vol. 2, p. 545 (1927). ? 'Z. Physik,' vol. 58, p. 75 (1929).



280 A. H. Wilson

such an approximate solution exists. To do this we must use the symmetry properties of the wave functions, neglect the momentum of the light-wave in comparison with the momentum of the electrons, and average over a region whose dimensions are large compared with the lattice constant and small compared with the wave-length of the light. The averaging process is more difficult to carry out here than in the classical theory, since each electron is spread over the whole crystal instead of being localized.

If we consider a cubic lattice of lattice constant a, then

1kmk (xyz) = e27ri(k,x+k2,ykaz)IGa umk (xyz), (12)

where u is periodic in the lattice constant, and we are using the boundary condition that i is to be periodic in a large cube of side Ga. The energy spectrum consists of a series of nearly continuous bands, designated by the index m', the states forming a band being distinguished by the integers kj, k2, k3. All the wave functions (12) are orthogonal-those with the same values for kl, k2, k3 on account of the orthogonality of the functions ur', and those with different values of kl, k2, k3 on account of the orthogonality of the exponential functions. It follows immediately from these orthogonal and periodic properties that the matrix elements of the operator a/ ax are different from zero only for states with the same values of kl, k2, k3 and different values of m'. Now consider a light- wave with electric vector (E., 0, 0), the direction of propagation being parallel to the z axis. We shall show that it is possible to obtain an approximate solution of Maxwell's equations for which Ex is a function of z only. The matrix element

(111213; m" I Ex (z) a/ax I klk2k3; m') (13)

is zero unless 11 = kl, 12= k2, and the corresponding expression

ik FX ax (14)

is periodic in the variables x, y with period equal to the lattice constant. We can therefore replace it by its average value over the crystal

____2 ek Li -+ dx dy =- (Ga2 J aQY dx dy, (15)

the surface integral vanishing on account of the periodicity of the integrand. If we write 13 k3 + ?1, the integrand of (15) is of the form

e27riAz/Ga V (xyz)




where v (xyz) is periodic in the lattice constant. Now, as we shall see, the important values of X are small compared with G, so that the exponential factor has a period large compared with the lattice constant and of the same order as the wave-length. Therefore we can replace v (xyz) by its average value, and leave the exponential as it is. We also neglect the variation of v (xyz) with A, and obtain finally for (14) the averaged value

e2 )riZ/Ga f1 dx dy dz, (Ga)3 aJJ

where 13= k3, the integration being taken over the whole crystal. In the same way we reduce the matrix element (13) to the approximate expression

e-2' Z EZ (z) dzdx dy dz,

where again 14 = kl, '= k2, 13 k3. The summation with respect to 1 in (10) is therefore now replaced by a summation with respect to ?. and a summation over the bands. If we neglect the dependence of vtk on X, the part containing X is proportional to

I E, e 27ri,z/C-a e-27rixz/Ga E. (z), dz-E.X (z), Ga A

and we obtain our final expression for the current

e2 aE? z h v1 2

rmG3L - (k) V22rd2MV2 Ij ?2 _ V2 ax h n,27 EC (z N (k) E(sin 2 (vlk + v)t sin 2ic(vZ - -v) t

2CM v k I Vl _V IVIk - 'V

x L|k& @'Td j 2 (16)

The charge density is zero since div J = 0. We have yet to verify that the important values of X are small. The

electric vector will depend on z through terms of the form e2,,ipzIc,

where q is a complex quantity whose modulus is of the order unity. The important values of ?/G are therefore of the order a v/c, which is about 10-4, while the values of k for the electrons are of the same order as G; so that we were justified in neglecting the variation of vl with A.

If we use the expression (16) for the current, we may, if we please, define a dielectric constant ? and a conductivity a by

JV O (C - 1) aEx + UExr

VOL. CLI.- A. U



282 A. H. Wilson

The Maxwell equations then have the usual solution

Ex Fe-2F e 27,Kz/c COS 2rcv (t - nz/c),

- Hl, =F' e2XKz/c cos 27v {(t-nz/c) , where

n2- K2 nKV - (a

and F'- F (n2K+i2)', tanyKl/n.

We shall use the quantities c and a to compare the theoretical and the experimental results.

The expressions (16) and (10) are not, of course, identical, (16) being the form of (10) when averaged over a small region, so that the approximations made are the same as those of the classical theory. Kronig and Groenewold (loc. cit.) have tried to solve the exact equations for a crystal whose thickness is small compared with the wave-length, in which case

Ex is constant in space to the first order and does not vary exponentially. They find deviations periodic in the lattice constant, which have a small effect. To calculate such deviations for a finite crystal would be very difficult, and they can play no important part when the wave-length is large compared with the lattice constant.

DISCUSSION OF THE EQUATIONS

3-The expression for the dielectric constant

I1- e2 N (k) h I ___lk _k d_ v 2

J wmG 3a3 k IV2 2r2rniV21 v2 -V2 ax17

can be written in the alternative form

mG3a3 k N(k) I 22m E 1 I 2dr)1 2 h2m 1 2

X v2) ''1* 1d- j (18) It is shown in the Appendix that

h 1 ~ ~kd '2 m ma2G2 a2W Ml 2-c2

, hd akI (19) 1 2 h2M I V iLk akd I = h2G ak 2m'S (9

where Win is the energy expressed as a function of the quantum numbers m', kL, k2, k3 used in ? 2.3, and m' distinguishes the band to which the state k belongs. The term in v2 in the dielectric constant is therefore zero only if EN (k) a2W/9k 2 vanishes. Replacing the summation by an




integration, we see that this happens if, and only if, aW/ ak1 vanishes at the limits of integration, that is if the electrons exactly fill an energy band, the solid then being an insulator.t For a metal on the other hand, this term is the most important one, and can be evaluated provided we know the energy levels. If the electrons move in a region of constant potential, as supposed in Sommerfeld's theory of a metal, we have

W 2 22 (k12 + k22 + k32) (20)

and noe 1 - ? 2 (21A)

where no = N/(G3a3) is the number of electrons per unit volume. This formula (21) is exact since the transition probabilities are zero for an electron under no forces. In an actual metal the energy will not be given by (20), and, if we neglect the second set of terms in (18), we obtain

? 1 n- (21B)

where m* is the effective mass of an electron. The difference between m* and m is a measure of the binding of the electrons in the metal, and in general m is less than m*. In the visible and ultra-violet most metals show little true absorption. The conductivity a ( nKV) iS small, and the metal is either totally reflecting, or else nearly transparent. It therefore seems at first sight as if m* could be determined from the frequency at which a metal becomes transparent, which is the frequency at which e changes sign. This is not quite correct since the neglected terms may give an important contribution to e. They do in fact give the ordinary static dielectric constant of an insulator, and there is no reason for supposing they are of a different order of magnitude in a metal, so that we have to replace the 1 in (21A) and (21B) by a number zo. There is an extra complication introduced by the fact that zo is not independent of frequency, and it is indeed obvious from an inspection of the equations that for high frequencies e must be given by (21A). However, zo is certainly greater than 1 provided v is less than any of the resonance frequencies v,. The increase in the effective mass of the electron and the increase in zo both have the same effect, that the wave-length at which a metal becomes transparent should be longer than that calculated from

t For the explicit expressions for the energy levels see, for example, Sommerfeld and Bethe, " Handbuch der Physik," 2nd ed., vol. 24/2, p. 398.

u 2



284 A. H. Wilson

(21A). This is actually found to be true.t As we shall see later, it is probable that the critical frequency for the alkalis is greater than the first resonance frequency, so that it is not easy to estimate the value of zo to be used, and all that we can hope for,from the experimental results is the order of magnitude of m*.

THE DISPERSION FORMULA FOR A METAL

4.1-To proceed further it is necessary to know the integrals

Mott and Zener have suggested tentatively that, near a resonance frequency v., e will behave like log 1 - v/v, 1. This cannot be so since we have shown that there are no infinities in the equations. To find the actual form of e we must discuss particular models.

Let the potential energy of an electron be expanded in the triple Fourier series

V (x y Z) = E Vay e2Tri(aX+ Y+yZ)/a (22) a,gy

Then we can obtain a first approximation to the wave functions by con- sidering the departures from uniformity in the potential to be small. The wave function (not normalized) are then

F1 ma2 Va.P e27ri(aX+9+yz)/a T 1k2101 h 2 fi 22 + p2 + y2 + 2 (oxkl + 3Pk2 + yk3)/Gf

x e 7ri(k1 +k2y+h3z)I (23)

With the wave function expressed in this form it is convenient to allow kl, k2, k3 to range over all values and not to be restricted to lie between ? 2G. The calculations with these wave functions have been carried out by Sergeiev and Tchernikovsky, so we shall quote their results. The integral

vanishes unless

11 kL + s,G, 12 k2 + s2G, 13 k3 + s3G,

where s1, S2, s3 are integers. Its value is then

47-rima s:_1 V, ,. h2 sI2 + S22 + s32 + 2 (s_tk4 + s2k2 + s3k3)/G

t Mott and Zener, ' Proc. Camb. Phil. Soc.,' vol. 30, p. 249 (1934).




To the first order the energy levels are those of a free electron, so

h vlk =2ma2 {s12 +A s2 + S32 + 2 (slkl + s2k2 + s3k3)/G}. (25)

We now change to variables i, ^, C defined by (R, , C) = 2-(kl, k2, k3)/G, and replace the sums by integrals.t We also assume that the temperature is not too high, so that the electron gas is completely degenerate, and all the electrons occupy the IN lowest energy levels, which are within a sphere of a certain radius po in the , e, 4 space. We then have N (k) = 2 inside this sphere, and N (k) = 0 outside it. When the integrations have been carried out we find the two alternative expressions for ? derived from (17) and (18) respectively,

C=1 4mae E 2 2 I2 log VS(2) + Si2yVs im2a73 h44 s1~s 123 S S

~~-(v8(') - ~ Vs (2) - V v +2 -i(vS(l)-V) (9S(2) vlog "S(J) V-(v.?) +v) (vS(2) +V) 10g 'fs+ $

or (26) noe1 +4mae2 s

IV2 h hv Rsls S3s

V 2) (V (2) _ (1)) (VS(2) + VS1M)) 2

V 2loV x Y28123 V2 (og 2v8 VS V v82 /+ S 1

2)lgV SASS3 v 2 g (1) Iv (2) 4 s s 14 (1)

-i (-(1)- v) (vSt2)- v) log Vs( ) v -i (vs8l + v) (v8t2>+ v) log 'Vs+ 3 2

where S2 s2 + S 2 + S32

and hs hs

= 2rhma2 (rs - Po)' (2) =2r sa2 (S + Po)' (28)

The latter form for C is preferable theoretically in that it groups together those terms which remain finite as v tends to zero.

The conductivity is also easily evaluated. Since v is always positive, we have

Ae2h EN tp - d 2sin 2i (v,- v) t 8vT3m2vG3a3 *k x Vlk - v

t The integers k are the most convenient quantities for describing the states. For calculations with particular models, on the other hand, we require continuous variables, so that we use the variables ~, - , 4 in the limiting case of very large G.



286 A. H. Wilson

which is different from zero only if v lies in one of the ranges (vS'), vs(2)).

It is easily shown by using polar co-ordinates in the i, , 4 space that a = rmae2 s12 V2 (<S2) -3)

(- )) v(1) I<v vs2). (29)

The curves for s and a are very similar to the anomalous dispersion curves for gases, except that the " anomalies " are spread over the range from v (') to v (2) which is a few volts. This large spread results in the absorption band being very unsymmetrical, there being a steep rise on the low-frequency side and a long tail on the high-frequency side.

4.21-In comparing theory with experiment it is simplest to consider a, since the frequencies v.('.) and v<(2) can be immediately deduced. Unfortunately v,(2) lies too far in the ultra-violet to be observed, and the experimental results only give v,(') and the frequency for which a is a maximum. Fujioka came to the conclusion that the theoretical and experimental value for 1s(l) differ enormously-he found for silver VS1)- 0 077 x 1015 theoretically, the experimental value being 0 9 X 1015 -and it has been suggested in many places that this shows that the wave functions are highly inaccurate. The discrepancy is not, however, anything like so great as supposed by Fujioka. His calculations were for a simple cubic lattice, whereas silver has a face-centred cubic lattice. Now if

V Va,6) e27ri(_X+ht+-yZ)/a

cq3y

is the potential of a simple cubic lattice, the potential of the corresponding face-centred cubic lattice formed by four inter-penetrating simple cubic lattices is

fi V e2 C(-X+P?+yz)/a -ri { 1 + ( -1- + + eXi(Y+>)}

X ve2 C27ri(-.x+PV+z)/a (30)

Therefore V1oo and Vllo are both zero, V,,, being the first non-zero Fourier coefficient, so that the first energy zone in E, , 4 space is bounded by the planes

37rt El 0 C ?, (31)

and by certain other planes at greater distances from the origin. The first energy zone for a simple cubic lattice is -7W < i, -, 4 < 7. Now there are four electrons per unit cell in a face-centred lattice, and so po (127c2)1/3, giving for silver (a 4 08 x 10-8 cm)

Vs 27h 2 {3 - 31/2 (12n2)1I3}

0 65 x 1015,



T1he Optical Properties of Solids 287

which is about two-thirds of the experimental value. We have so far only taken the first approximation to the energy, and if we included the second order terms the value of v (1) would be increased, and there is no doubt that there is very little discrepancy between theory and experiment. The value of v.2) has to be obtained from the frequency v0 at which a is a maximum. The relation between v (2) and v0 is

v (2) - 2vov,(1) -~V2 (32) s 3 (1)-2(32)

It is therefore impossible to determine vSj2) unless both vsl) and v0 are known with great accuracy, which is not so. Measurements further' into the ultra-violet would not help to determine Vs2), since the thresholds for other absorption bands occur at much smaller frequencies than V,s2 . For the same reasons it is not possible to find the exact magnitude of V,,1, but it must be about 1 volt in order to give the correct maximum value for a, and this is quite a reasonable figure. I

4.22-The alkalis form body-centred lattices, so that for them the unit cell contains two atoms and the first non-zero Fourier coefficient is Vllo. The first energy zone is bounded by twelve planes such as

27 i i i B 0. (33)

We then have po -(6-r2)1I3, giving the following values for v,('): lithium 7.4 x 1014, sodium 4 8 x 1014, potassium 3 3 x 1014. These figures may be incorrect by a factor of 2, but even so internal photoelectric absorption ought to take place before the metals become transparent by e becoming positive; but measurements to test this are lacking. It is perhaps worth remarking that we should expect the calculated and experimental values of vs(l) to agree better for silver than for the alkalis. The reason for this is that our assumption that the energy levels are those of a free electron is most likely to' be true for energy levels far away from the boundaries of the energy zones. From (31) and (33) we see that the ratio of the maximum linear' extension of occupied levels in i, , 4 space to the width of the first energy zone is 0 98 for a simple cubic lattice, 0 88 for a body-centred lattice, and 0 81 for a face-centred lattice. We should therefore expect the theory to be quite incorrect for a simple cubic lattice, and to be best for a face-centred lattice.

THE DISPERSION FORMULA FOR AN INSULATOR

5-It is possible to discuss insulators either from the point of view of single-electron wave functions or from the point of view adopted in the



288 A. H. Wilson

theory of ferromagnetism. We shall adopt the first method here, since we are mainly interested in the photoelectric conductivity. When the electrons in a solid are tightly bound, as in an insulator, the wave functions are of the following form.t For wave functions derived from an atomic s state

S (4 - C) E e'a01?02+93)/a qg1g2, (34)

where Xg,gg2.3 is a spherically symmetrical function centred round the lattice point (gl, g2, g3). The energy is

W, WE - C) - W2p (cos + cos + cos (35)

oc and P1 being positive constants. The wave functions derived from an atomic p state have a threefold degeneracy. They are given by

i (0 -i4 e- z e1+2+Cg3)a Xglg2g3 (36) 91f/293

where XZ,g2g3 is of the form xfj(r) or yf (r) or zf (r). If we consider only the first form, then

Wv(R ) W2- 2?+ 2Y2 cos E-22 (cos? + cos), (37)

where oc2, P2, Y2 are positive constants, P2 being less than 31 but of the same order. All the states (37) have greater energy than any of the states (35). The matrix element

is zero unless Z is of the form xf(r) and unless i - - = It is then equal to

E ZXg1g2g93a 19293 dT 1.Y29f3 aX

on making the approximations used in deriving the wave functions, and this we put equal to C, a constant independent of i, , C. We also have

hvtk W2 - (X2 - W1 + cc1 + 2 (Y2 + PI) COS

+ 2 (p,- 2) (COS + COS ) or

vl vo + X cos + F (cos + cos), (38)

t Bloch, 'Z. Physik,' vol. 52, p. 555 (1929); A. H. Wilson, 'Proc. Roy. Soc.,' A vol. 133, p. 458 (1931).




where ?< and ,u are positive constants, X being greater than ,u, and (v - - 2 p) being positive.

In an insulator all the energy levels of the s band are occupied so that N (k) = 2 for all values of i, ', 4 lying between -7r, t. It is therefore obvious from (19) and (35) that the term in e involving v-2 is zero, since

.W/aR vanishes at the limits of integration. We therefore have

e2hC2 d[ 4 di d , 87C6m2a3 JjJ vk (vtk2 - v2)

if we assume that there is only one p band which gives a large contribution to e. The integrand may be written

1T1~~+ 1 + 140) V2 { Vk 2 (V1k + V) 2 (V1k

- V)(

'The first two terms cannot produce anomalies in e and give a negative scontribution which varies monotonically with v. The third term is the interesting one, and we shall concentrate our attention on it.

We must be somewhat careful in calculating the integrals, since the principal value of

l2r dO Jo 1 + e cosO

is zero when lel > 1. We must therefore satisfy ourselves that an integral is not identically zero before we start approximating.

The first integration is easily carried out, and

d~~ d-~~ d~cos i d~d (41) ir 7r

_i Ik - V IJu o+ CS + pco CO 4- )2 - }S(1

provided I vo + t cos B + t cos -vI > X, and is zero otherwise. The next integration involves elliptic functions, so instead of finding the exact value we approximate by expanding the cosines in the neighbourhood of -I, and use plane polar co-ordinates r, . We then have for (41)

?27C r dr X (42) 2JV{(vo 2 - v + 1tr2)2 -

In order to obtain the correct energy limits for the band, the limits of integration are 0, 2 /2, but since we must have

lvo + t cos n + t cosC- vI >X



290 A. H. Wilson

in addition, the lower limit for I r2 is the greater of 0 and v- (v0- X-2 ), and the upper limit for - r2 is the less of 4p and vo - (v + X-2p). The integral (42) has therefore different expressions for different energy ranges, and the expressions depend on the relative magnitudes of X and t. We have no very certain knowledge of X and t, but we should expect X to be greater than 2 p, and t may be fairly small, since it is the difference of two quantities of the same order. If then we put 24f(v) for (42), we have to consider five different ranges for v, and we find the following expressions -forf (v).

V < Vo - 2-

f(v) klog vo + 2t- IV + {vo + 2~t- V)2 -

X2} f (v) = log V 2 - { -F )2 -x2} ' (43A)

vo+ X - 2 t < v < vo + X + 2 t

f(v) - log! [vo + 2p- - v + -{(vo + - 2 - )]. (43B)

Ivo- X + 2,u < v < vo + X - 2p

f (v) = 0. (43c)

Ivo + X -2pt< v < vo + X + 2pt [L~ ?

f (v) = -log [-,/f(vo - 2 p- V)2 _ 2} - (vo - 2p- v)]. (43D)

vo + X + 2t ,< V

f(v) = - llog vo 2 - v+ {(vo 2- V)2- X2} (43E) f () p Iv0- 2~t- v +V{(,vo - 2v- V)2 - X2}

We therefore have + e2hC2 8-e,;m a 3,V f (v) g (v)

where g (v) is derived from the first two terms of (40) and is given by

g (v) =2log vo + 2V + \{(v0-+ 2V)2- X2}

log vo + 2p + v + V'{(vO + 2p + v)2 - 2

p vo - 2p + v + \{(vo- 2 + v)2- X2}

The graph off(v) is given in fig. 1; f (v) is continuous, but its differential coefficient is discontinuous. The greatest and least values off (v) depend




very much on t, being + (8/X7t)4 for small values of t. On the other hand, g (v) is a positive, increasing, bounded function of v which does not depend much upon ,t. The dielectric constant e is therefore positive for small values of v and greater than unity. For larger values of v anomalies occur and s may even become negative, provided ,u is small enough or C is large enough. Assuming reasonable values for these constants, e will actually become negative provided the anomalous ranges do not lie too far in the ultra-violet. For an insulator the maximum numerical values of e occur at the edges of the absorption band, whereas for a metal they are situated in the interior of the band.

f (V)

V0+X-2.L V0+X+26

V0-X-4#L V0-X+-2L VU\

FIG. 1

To evaluate a we change from i to a new variable x defined by

x = V0 + X cos X + t (cos n + cos C)-,

and we obtain the integral

In sin 2xt dxd dU -7r d dE JJ?sini x J /?_(V0 + ,UCOS _T+ ,UCOSC -,V)2}'

-7r

ranges for which - X-o v + t (cos n + cos C) > 0

or v0 + X v + t (cos n + cos <) ?0

being excluded, since the integration over x gives a non-zero result only



292 A. H. Wilson

if the upper and lower limits of x are positive and negative respectively. The further integration proceeds as before, and we find

e2hC2 ( (44) 167i 4M2a3IV ()

where a1 (v) is given by the following expressions:

'-? -2 2 < v < v0 -X + 2,

a1 (v) = k cos-vo2, v (45A)

vo- X + 2p < v < vo + X -2,

51(v)=1 s vo+2)-v_1 sin'0-2sl-n (45B) 11 ~ XX

vo + X - 2p < v < vo + X + 2,

a1 (v) - cos-1 v - - (45c)

For other ranges a, (v) is zero. The curve a, (v) is symmetrically placed with respect to the curve f(v), but a is not, its maximum being displaced to smaller frequencies.

The dispersion curve of an insulator is so peculiar that it would be most interesting to see if it is verified by experiment, but unfortunately the requisite data are lacking. Most of the experiments have been concerned with the internal photoelectric effect, and not with n and K themselves. The relation between the number of photoelectrons produced and the optical constants is discussed in ? 6.

THE ABSORPTION OF LIGHT

6-The number of electrons excited per second to the state I is

X N (k) [CAkl12,

where CkI is given by (7B), SO that the total number of electrons excited is

M 2 N(k) a k *2i?kd: 2sin 27r (v - v) t m 2

N (k cAx 7X_ dv(6

This gives the number of electrons produced in the whole solid, and it is not strictly possible to speak of the number of electrons produced in a




small unit volume, or of the absorption at a point, but we may define the former by

em2c2V IaI22 Nk(k) +* kd 2 sin 27(vlk- )tV) t

where V is the volume of the solid. As we shall see, this gives the correct value for the total number of electrons produced, and it would be the form we should obtain if we considered an isolated small portion of the solid. To show that (46) and (47) give the same result we have to transform (46) by the method adopted in ? 2. The quantity ax* depends only on z, and by following the procedure of ? 2, and neglecting the momentum of the light-wave, we easily obtain

." (4) 2 dk Ns(k) 2a (vji - v) t M = -m2 2V

la dr E N (k) ,1

9 a

(46B)

so that our definition of the number of electrons produced in a small region is a consistent definition. The relation between ,u and a is

82 v a 12aa. (48)

Now from the Maxwell theory we have a nKv, and the mean energy flux is 27nv2 I ax l2/c, while the energy absorbed is ,uhv, so that the energy absorbed per, unit volume per unit energy incident on the volume is

4JCK V

C

showing that K is the absorption coefficient. The energy absorbed per unit volume, on the other hand, is proportional to nK, while the total energy absorbed in a portion of the solid of unit area and of thickness d is

cn F2 (1 -47rKd

where F is the electric vector immediately inside the surface on which the light is incident.

It is obvious from the results of the last paragraph that the photoelectric response in an insulator is determined much more by the refractive index than by the absorption coefficient. Now, as we have seen, the absorption coefficient is increasing rapidly as we pass through an absorption band frGm the low-frequency side, while the refractive index is large near the threshold and diminishes rapidly in the interior of the band. It therefore seems possible that this is the explanation of the falling off of the photo-



294 A. H. Wilson

electric response in the interior of the absorption band. A fact which supports this hypothesis is that the phenomenon is only observed in idiochromatic crystals such as diamond, which have an abnormally high refractive index in the visible. For these bodies the quantity C must be large and the first resonance frequency must be small, so that the conditions which tend to produce high values of K and low values of n in the first absorption band are satisfied in these substances.

APPENDIX

A proof of the "f-sum rule " for a metal has been given by Sommerfeld and Bethe.t Their proof is based upon a definition of the oscillator strength for zero frequency, a concept which has much to be said against it. The following is a more rigorous proof.

Let (L + W) [4] -8h2 2 +(W-V) -O (49)

be the wave equation for an electron moving in a crystal field with cubic symmetry, so that V is periodic with period a. The wave functions are those given in ? 2.3, or, changing the notation slightly,

t]q;m (x y z) - e;V+V+?z)l/ u~q,m' (x y z), (50)

where um' has period a, and - < i, n . The functions u,?m' and are orthogonal, and the energy Wnqm' is a continuous function of

X, -, 4 for each fixed value of m'. We shall usually omit i, , 4 explicitly and write 4m' and urn.

To obtain a " sum rule" it is necessary to transform the infinite sum

IJ4S,* 3f'dT IJ * amdT| jv ___v______ ax___ _ _ ax

rn"#m' TmWm" - m I "/m' Wm/n - Wm(5

into an expression involving quantities connected with the state m' alone. The integration in (51) is taken over a unit cell, and the wave functions are normalized in the cell. The usual methods of proof do not apply here, since they depend on Q and its derivatives vanishing on the boundary of the region of integration. We must therefore proceed differently. The first step is to get rid of the energy explicitly in (51). Write (t . r) for (ix + -jy + z) and d.ifferentiate (49) with respect to i, obtaining

FL r) [=au a (L ? Wm +M))L~Mei(t .r)/a in' ? (L ? Wi) [Mt']~a 0. (52)

t ' Handbuch der Physik,' 2nd ed., vol. 24/2, p. 378.




If we multiply this by tPm * the resulting expression is periodic in a, so that by integrating over a unit cell and using Green's theorem we find

4h2 Jm * -au) (+W L(r)Ia aUm,j 4n" 7:mt| 4'a- dr Ml=-X * (L + W, . ) [e i( $' ) dc

- _J ei(f(r)Ia ' m (L + W,.) [ ,,*I dt

=(Wm" t W.) U. u^*am d,(53)

the surface integrals vanishing on account of the periodicity of the integrand. The expression (51) now becomes

R_____ Fm uml *d aum dma au na' d IUrn" T'' (d Um" R -I ih2 ml'm, J ax J ai ih.J 2 x aZ

(54) where R denotes the real part, since we may add to the left-hand side the term arising from im" = m', which is zero because

Jum' a rn d- + * aU., dT = J14)m 2dr =

This last expression can be transformed still further. If we differentiate (52) with respect to i we find

F. a2Um,2ix aUn,' x2 (L + W)

i i(, r)/a a + 2ix ei(J r)Ia a

i(C.r)/a aWm' aUm' 2X aWm' a2W + ~ @ me + 2i a a M + ai2 im'-0.

Subtract 2ix/a times (52), multiply by 4m'* and integrate over the unit cell. We then obtain

,a2WM I h2 h" aWm a um d iai2 47flma2 2+ -ma2 a/ 3' Um !dt

ih2 juU* ua d J m* (L + W) el\t r;Ia a'm] d 2it~2ma axO h dI + R 2ia aum* a dm), (55) 47r%t? ax 7

so that

___ dax 47-2 ma2 6W,,W, 2, 2m Wm"_Wm, h2 a '2 (56)

which is the i'f-sum rule."



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The Optical Properties of Solids