Substrate heating by transmitted radiation

Another consideration in the production of thin films by photochemical processes is that the fraction of the beam which is not used in photodecomposition will heat any substrate on which it is desired to form the film. The power of the light source which can be used for photodecomposition in the gaseous phase only is therefore limited by the transmission of energy. Clearly this transmitted beam represents a constant source of energy which 

Here Q is the amount of heat contained per unit volume in the substrate, x is the distance down into the substrate, and t is the time of irradiation. The solutions of this equation depend on the physical conditions of the irradiation. Thus if the surface is subject to a constant energy supply, Qo, the solution is 

erfcx is the error function complement of x and ierfc is its inverse. The physical properties are represented by a, the thermal diffusivity, which is equal to K/pCp, where k is the thermal conductivity, p is the density and Cp, the specific heat capacity at constant pressure. The surface temperature during this irradiation, Ts, at x = 0, is therefore 

The analytical methods for solving the Fourier equation, in which Q and T are functions of the spatial co-ordinate and time, include a change of variables by combination, and in the more general case the use of Laplace transforms. 

To find the temperature distribution, the first procedure requires a change of variables to the variable, z, defined by the equation 

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