Instantaneous Localized Sources in Infinite Media

Equation 4.40 gives the solution for one-dimensional diffusion from a point source on an infinite line, an infinite thin line source on an infinite plane, and a thin planar source in an infinite three-dimensional body (summarized in Table 5.1). Corresponding solutions for two- and three-dimensional diffusion can easily be obtained by using products of the one-dimensional solution. For example, a solution for three-dimensional diffusion from a point source is obtained in the form 

Solution Type Symmetric Part of V2 Fundamental Solution 

Point source in ID Line source in 2D Plane source in 3D 

The form of the solution for one-dimensional diffusion is illustrated in Fig. 5.3. The solution c(x,t) is symmetric about x = 0 (i.e., c(x,t) = c(—x,t)). Because the flux at this location always vanishes, no material passes from one side of the plane to the other and therefore the two sides of the solution are independent. Thus the general form of the solution for the infinite domain is also valid for the semi-infinite domain (0 x oo) with an initial thin source of diffusant at x = 0. However, in the semi-infinite case, the initial thin source diffuses into one side rather than two and the concentration is therefore larger by a factor of two, so that 

Equation 5.18 offers a convenient technique for measuring self-diffusion coefficients. A thin layer of radioactive isotope deposited on the surface of a flat specimen serves as an instantaneous planar source. After the specimen is diffusion annealed, the isotope concentration profile is determined. With these data, Eq. 5.18 can be written 

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