Diffusion plane source

The mathematical translation of the plane-source problem is as follows. Initially, there is a finite amount of mass M but very high concentration at a = 0, i.e., the density or concentration at a = 0 is defined to be infinite (which is unrealistic but merely an abstraction for the case in which initially the mass is concentrated in a very small region around a = 0). The initial condition is not consistent with that required for Boltzmann transformation. Hence, other methods must be used to solve the case of plane-source diffusion. Because this is the classical random walk problem, the solution can be found by statistical treatment as the following Gaussian distribution ... 

If the plane source is on the surface of a semi-infinite medium, the problem is said to be a thin-film problem. The diffusion distance stays the same, but the same mass is distributed in half of the volume. Hence, the concentration must be twice that of Equation 3-45a ... 

AS. 1.1 Plane source for one-dimensional infinite medium (-oo[Pg.570]

Figure 9.11 shows a typical diffusion penetration curve for tracer self-diffusion into a dislocated single crystal from an instantaneous plane source at the surface [17]. In the region near the surface, diffusion through the crystal directly from the surface source is dominant. However, at depths beyond the range at... 

Beeause the eoneentration of metal ions at the interface is far in excess of that in the bulk of the solution, diffusion into the solution begins. Since the source of the diffusing ions is an ion layer parallel to the plane electrode, it is known as a plane source and since the diffusing ions are produced in an instantaneous pulse, a fuller deseription of the souree is eontained in the term instantaneous plane source. 

As the ions from the instantaneous plane source diffuse into the solution, their concentration at various distances will change with time. The problem is to calculate the distance and time variation of this concentration. 

This is the solution to the instantaneous-plane-source problem. When n/n,otai is plotted against X for various times, one obtains curves (Fig. 4.31) that show how the ions injected into the x = 0 plane at f = 0 (e.g., ions produced at the electrode in an impulse of metal dissolution) are distributed in space at various times. At any particular time t, a semi-bell-shaped distribution curve is obtained that shows that the ions are mainly clustered near the x = 0 plane, but there is a spread. With increasing time, the spread of ions increases. This is the result of diffusion, and after an infinitely long time, there are equal numbers of ions at any distance. 

Fig. 4.34. When diffusion occurs from an instantaneous plane source (set up, e.g., by a pulse of electrode dissolution), then 68% of the ions produced in the pulse lie between x = 0 and x = x g, after the time t.

Fig. 4.35. If it were possible for diffusion to occur in the +xand -xdirections from an instantaneous plane source at x= 0, then one-third of the diffusing species would lie between x = 0 and x= +x g and a similar number would lie between x=Oandx=-xj g.

When the current, or flux, is a single impulse, an instantaneous-plane source for diffusion is set up and the concentration variation is given by... 

Ions are pumped into a system electrochemically. At r = 0, a short burst of dissolution of an electrode is caused, giving rise to totai ions, which then begin to diffuse away from the source. Seek in the text the appropriate equation by which one may know the number of ions at a distance x and time t. This is a plane-source problem. Thus, Cu ions could be dissolved from a Cu plate filling the end of a tube of solution. The question is how many ions would have diffused... 

TPS (Transient Plane Source) technique has been shown to be effective method to measure the thermal conductivity, diffusivity of rare earth oxide powder such as gadolinium oxide, samarium oxide, and yttrium oxide. The details of the measutrement are described in Ref 66. The experimental results of effective thermal conductivity as well as thermal diffusivity of the above described three rare earth oxides are tabulated in Table 5-7. 

The time-dependent temperature distribution in a transient experiment is governed by Eq. 4, and usually the related parameter, thermal diffusivity. is obtained. However, under certain circumstanees the solution to the heat equation contains the thermal conductivity as well as the thermal diffusivity, and by choosing a suitable method the diffusivity can be eliminated from the answer. The more important methods are the line and plane source heater methods and arc described below. These arc not Standard methods, but they can be used where speed is more imp .>rtant than absolute accuracy, to give a conductivity value more quickly than the Standard methods. They can also be used to compare a range of materials. 

Gustafsson, S.E. (1991) Transient plane source techniques for thermal conductivity and thermal diffusivity measurements of solid materials. Rev. Sci. Instrum., 62, 797-804. 

Various other methods have been described for the determination of thermal conductivity. Capillarity has been used to measure the thermal conductivity of LDPE, HOPE and PP at various temperatures and pressures [30]. A transient plane source technique has been applied in a study of the dependence of the effective thermal conductivity and thermal diffusivity of polymer composites [31]. 

The self-diffusion coefficient of calcium in single crystals of calcium oxide was measured at 1465 to 1760C using vapor-deposited thin films of radioactive Ca O and boundary conditions for diffusion from a plane source into a semi-infinite medium. The temperature dependence of the diffusivity was expressed as ... 

In order to demonstrate the application of this general method to neutron diffusion in infinite media, the plane-source and line-source distribution problems (Secs. 5.2a and 5.2c) will be treated using (5.97). 

If the adjoining regions are both diffusion materials, an estimate can be obtained for the albedo through the use of the partial-current relations. Consider, as an introductory example, the case of a plane source of strength qa neutrons per unit area per unit time placed on the face of a semi-infinite slab. If the origin of the coordinate system is at the source plane, the general solution of the diffusion eciuation has the form of Eq. (5.78), namely,... 

The cross sections in Eq. (P6.1) refer to the diffusion medium, and the source function S(x) may be defined so as to include all the remaining neutron sources and sinks in the medium. The function (x]x ) is the plane-source kernel and gives the flux at x due to a unit source at x The final result for 0ui(x) is to include only the nuclear constants of the system, So, and t. 

The solution of the heat conduction equation given in sect. 10.3.II of this reference for an instantaneous plane source of strength Q parallel to the plane z - 0 can be used to solve the particular diffusion problem. The reader can verify it by substituting solution (3.2.15) into equation (3.2.12) and seeing that it is satisfied. 

This experiment is a measurement of thermal-neutron diffusion length in H2O. This measurement utilizes reactor leakage neutrons as a plane source, indium foils as thermal-neutron detectors, and cadmium absorption coupled with the indium resonance as an epithermal neutron detector. The thermal-neutron diffusion length is evaluated from the parameters of a cadmium-covered indium distribution. 

B. Infinite Plane Source in Infinite Medium Assuming that the source is infinite in the x and y planes, then at any given value of z from the source the flux will be independent of x and y, so that will be d / dz, and the diffusion equation can be written... 

D. Effect of Finite Dimensions If it were possible for us to duplicate experimentally an infinite plane source in an infinite medium, we could determine the thermal-neutron diffusion length from a set of measurements of the spatial variation of the neutron flux. From Eq. (9), we see that... 

If the neutrons from an infinite plane source diffuse into a slab of infinite extent but of finite thickness, by the same arguments used in the infinite dimension case, we find the general solution given by Eq. (3) to be applicable. To evaluate the arbitrary constants A and C, it is necessary to introduce an additional boundary condition that the flux shall vanish at the extrapolated boundary. If a is the finite dimension of the slab including this extrapolation distance, then... 

In the x-ray portion of the spectmm, scientific CCDs have been utilized as imaging spectrometers for astronomical mapping of the sun (45), galactic diffuse x-ray background (46), and other x-ray sources. Additionally, scientific CCDs designed for x-ray detection are also used in the fields of x-ray diffraction, materials analysis, medicine, and dentistry. CCD focal planes designed for infrared photon detection have also been demonstrated in InSb (47) and HgCdTe (48) but are not available commercially. 

Experimental measurements of DH in a-Si H using SIMS were first performed by Carlson and Magee (1978). A sample is grown that contains a thin layer in which a small amount (=1-3 at. %) of the bonded hydrogen is replaced with deuterium. When annealed at elevated temperatures, the deuterium diffuses into the top and bottom layers and the deuterium profile is measured using SIMS. The diffusion coefficient is obtained by subtracting the control profile from the annealed profile and fitting the concentration values to the expression, valid for diffusion from a semiinfinite source into a semi-infinite half-plane (Crank, 1956),... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

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