By Arto Salomaa
Salomaa A. Formal languages (AP, [1987)(ISBN 0126157502)
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Extra resources for Formal languages
The oscillations visible, due to the trigonometric functions in Eq. (98), are known as Friedel oscillations and are a consequence of the wave nature of the electron. The value of xi is found by demanding global charge neutrality, or ð1 ½re ðxÞ À ri ðxÞdx ¼ 0 ð99Þ À1 A bit of manipulation shows that Eq. 554 Ion −4 −2 0 2kF(x–xo)/p FIGURE 9. Electron density compared to the bulk value and the nature of Friedel Oscillations at the surface. 35749 A. Evaluating the dipole resulting from the charge distribution in Figure 9 requires a bit more finesse.
Minimization of the energy with respect to the constraint Ð that rðrÞdr ¼ N ¼ constant, analogous to the procedures leading to Eq. (12), serves to relate ro to the effective one‐body potential under which an electron in the material is considered to move (Jones and March, 1985). The variation of Vxc from deep in the bulk of the material to the vacuum outside the surface allows for the determination of the largest component of the work function. , U0 þ DU without the kinetic energy term), or Vxc ðrÞ ¼ dExc ½rðrÞ : drðrÞ ð83Þ Technically, Eq.
The free electron result, as expected. In the opposite limit, when ko ) 1, solutions exist only when tan ðbkÞ %À2k=ko ! 0, or k % jp for integer j, which is the square well limit. For intermediate values of ko, the discrete energy levels of the square well merge into the continuum levels of the free electron, as shown in Figures 4 and 5. 18 KEVIN L. JENSEN The consequences of the previous treatment indicate that the wave function of electrons above the potential barriers more or less mimics free electron wave functions and that the extent of the band gap is dependent on the magnitude of the potential barrier.
Formal languages by Arto Salomaa