- 作者: 黃振麟; 王嘉申
- 作者服務機構: 國立臺灣大學物理學系
- 中文摘要: 在固態物理的理論中,我們時常面對下面形式的積分: I=∫F(k) d k式中k為第一布里淵區(The first Brillouin zone)內的向量,積分範圍遍及該區的全域內。在電子系問題中,例如要求出電子狀態密度、磁化率、電介質應度函數等時,被積分函數F(k)為帶能量E和波函數的函數;在聲子系問題中,例如要決定頻率分佈、比熱、熱膨脹係數時,F(k)則為晶格振動頻率ν的函數。F(k)的決定通常須經由一複雜的判別行列方程式的解。 在本文中,我們以實例證示,筆者前此所導出的固體頻率分佈計算怯(修改的Houston法)能適用於計算這種積分I,並且其結果令人滿意。我們再由這個方法的思想演進,推出適合於電子計算機的兩種計算法,一為根據數字積分公式的方式。另一為Monte Ca-rlo計算方式。前者若利用性能較佳的積分公式,例如Gauss公式之類,則我們可導出類似於Singhal展開式的展開法,以少數的展開項可獲得極好的結果。後者能使我們的計算手續成為頭尾一貫,一切祇聽由電子計算機的操動即可。 我們再就Monte Carlo計算中的困難,即決定被積分函數F(k)動時無法避免的繁雜手續,提出一個利用Lagrange內插公式的解決辦法。這個辦法和Gilat-Raubenheimer的線型內插法,Mueller等人的二次型內插決以及Cooke-Wood.的結台內插法形成一個明顯的對照,而能節省大批的計算時問。
- 英文摘要: In solid state physics we are frequently faced with the following type ofintegrals: I=∫F(K)dk,where k is a vector in the first Brillouin zone, and the integration is takenover the entire region of the zone. In problems of electronic systems such as todetermine electronic densities of state, magnetic susceptibilities, and dielectric-responsefunctions, the intergrands F (K) are functions of band energies E and of wavefunctions; in those of phonon systems, e.g. to find frequency distribrutions, specificheats, coefficients of thermal expansion, they are functions of frequencies ν oflattice vibrations. A typical method of calculating F(k) would involve solutionsof secular determinantal equations. In this paper we illustrate with an actual example that a method of calculatingfrequency distributions of solids proposed early by us is also applicable to evaluatethe integral I and to give a very satisfactory result. Then extending the idea ofthis method, we present here two kinds of method which are most convenient forprogramming of electronic computers. One is based on formulas of numericalquadrature, and the other is on a scheme of Monte Carlo calculation. In theformer method, if the formula with high degree of accuracy, e.g. the Gauss Schemeis used, then it leads to an expansion formula which resembles that of Singhal andyields an accurate result with few terms. In the latter one, it gives us a unifiedprocedure which involves only an operation of electronic computers. Further, a method to overcome a difficulty inevitable in the Monte Carlocalculation i.e. a cumbersome procedure for calculating the integrand F(k) isproposed. The method originates from the interpolation formula of Lagrange, andis comparable with a linear interpolation scheme of Gilat and Raubenheimer, aquadratic interpolation scheme of Mueller et. al., and with a combined linear andquadratic interpolation method of Cooke and Wood. It may save for us a lot ofcomputation time.
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