COLOURS IN METAL GLASSES AND IN METALLIC FILMS. 403 



* 



We shall first confine attention to very thin films, defining very thin films to be 

 such that nd/\' may be treated as small, d being the thickness of the film, and X' the 

 wave-length of light in the film. 



It has already been noticed that equation (9), p. 393, does not hold for very thin films. 

 That equation is obtained by observing that the average action of its neighbours on 

 a particle is that due to a medium which is perfectly uniformly polarised in the 

 neighbourhood of the particle, and whose external boundary is that of the actual 

 medium, and whose internal boundary is a sphere of radius r () , equal to the smallest 

 distance between the centres of two particles. POISSON has shown that the effect 

 of such a uniformly polarised medium is equivalent to that of a surface distribution 

 over its internal and external boundaries. 



The medium actually present here can only be treated as uniformly polarised 

 throughout the region inside a sphere whose radius, ?,, is small compared with the 

 wave-length of light in the medium. When the outer boundary of the medium is 

 in all directions many wave-lengths distant from the particle under consideration, 

 the effect of the periodically varying polarisation outside r = r t can be allowed for 

 by neglecting the Poisson distribution on the outer boundary of the medium. 

 Consequently, in this case, the effect on any particle of the remaining particles 

 is that due to a Poisson distribution over the sphere r = ?, which leads to 

 equation (9). 



When the external boundary of the medium is, in any direction, at a very small 

 distance from the average particle, we are not justified in neglecting the Poisson 

 distribution over that boundary. In the case of a thin film of the medium in the 

 plane of xy it is, however, clear that when the electric force is parallel to that plane, 

 there is no Poisson distribution over the surfaces of the film. Consequently the film 

 has (complex) dielectric constants in the direction of the axes of x and ?/, which are 

 the same as for the medium in bulk. Omitting the accent in equation (10'), this 

 constant is given by 



The dielectric constant e', parallel to Oz, may be different from e ; if so, the film 

 behaves optically like a uniaxial crystal whose three (complex) dielectric constants 

 are e, e, e', the optic axis being normal to the film. 



8. Putting v = 1 in equations (12) and (13), we have 



where 



. =- ^-^2 *-s- O 3 ')- 



3 F 2 



