By Rolf E. Hummel
This conscientiously revised 3rd variation at the electric, optical, magnetic, and thermal houses of fabrics stresses thoughts instead of mathematical formalism. Many examples from engineering perform supply an figuring out of universal units and methods.
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Example text
The product V0 b is called the potential barrier strength. 62) with b yields yb = ~ J(V b)b. 63) 0 Since V0 b has to remain finite (see above) and b ~ 0 it follows that yb becomes very small. For a small yb we obtain (see tables of the hyperbolic functions) ยท cosh(yb) ~ 1 and sinh(yb) ~ yb. 47), and Fig. 61) reads as follows: m2 V0 b sin aa +cos aa =cos ka. 66) sin aa P - - +cos aa =cos ka. 45). We notice that the boundary conditions lead to an equation with trigonometric functions. Therefore, only certain values of a are possible.
28 I. 10. One-dimensional periodic potential distribution for a crystal (muffin tin potential). 45) need to be solved simultaneously, a task which can be achieved only with considerable mathematical effort. 48) (Bloch function), where u(x) is a periodic function which possesses the periodicity of the lattice in the x-direction. 2), but changes periodically with increasing x (modulated amplitude). Of course, u(x) is different for various directions in the crystal lattice. 67). 48) twice with respect to x provides d2t/J (d2u du .
7. Electron wave meeting a potential barrier. ) is 'I' Since the probability 1/11/1* can never be larger than one (certainty), 1/1 11 --+ oo is no solution. To avoid this, C has to go to zero: c--+ 0. 39) which reveals that the 1/1-function decreases in region II exponentially, as shown in Fig. 7. , for a large potential barrier, V0 . If the potential barrier is only moderately high and relatively narrow, the electron wave may continue on the opposite side of the barrier. This penetration of a potential barrier by an electron wave is called "tunneling" and has important applications in solid state physics (tunnel diode, field ion microscope).