On the Vibration of Disordered Linear Lattice. III
On the Vibration of Disordered Linear Lattice. III
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关于无序线性晶格的振动。
DOI：
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发表时间：
1957
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影响因子：
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通讯作者：
J. Hori
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作者：
J. Hori
用一种只需要较少量数值工作的方法近似计算了同位素双组分无序晶格的特征频谱。一个基于微扰理论的论证形式化地表明，完全随机晶格的谱与由平均质量原子组成的虚规则晶格的谱是相同的，除了在带的边缘和带外。我们首先研究了这种说法在多大程度上是有效的，并得到了结果，即轻原子的浓度越小，其频谱可以被认为与虚规则晶格的谱近似相同的频域越大。接下来，我们通过仅对该区域应用矩迹方法，计算了上述陈述不成立的频带边缘附近的频谱。结果表明，当轻原子的浓度与重原子的浓度相当或大于重原子的浓度时，虚规则晶格的能带边缘位置只有一个可能的圆形最大值，而当轻原子的数量越少时，出现杂质带，随着轻原子的浓度越小，其与主能带的分离越明显。两个结果都是自然的，只要谱接近泊松晶格的谱，因为较轻的原子变少了。在以前的论文中，我们用传递矩阵的方法处理了含同位素杂质的线性无序晶格的本征频率分布问题。特征频率由方程Trace H=2获得，其中H是传递矩阵的乘积。当杂质是随机分布时，必须研究痕量II的统计量。在杂质密度无穷小的极限情况下，即“泊松晶格”，我们可以计算出它的分布函数，由此我们可以得出结论，在只含一个杂质原子的晶格中，杂质频带以杂质频率为中心出现了一个极窄的杂质带，而带内频率的分布与规则晶格相同。施密德)用另一种但类似的方法处理了这个问题，并得出了同样的结论。他还得到了杂质带密度分布函数的近似表达式。然而，在杂质密度有限的情况下，即对于“非泊松”晶格，不可能获得这样的透视结果。我们能得到的最多是平均特征频率方程(Trace H) =2。由此推导出的特征频谱与的特征频谱一致
Eigenfrequency spectrum of isotopic two-component disordered lattice has been calculated approximately by a method which requires only a comparatively small amount of numerical work. An argument based on perturbation theory shows formally that the spectrum of completely random lattice is the same as that of virtual regular lattice composed of atoms with average mass, except at the edge and outside of the band. We have first investigated how far this statement is valid and obtained the result that the smaller the concentration of lighter atoms, the larger the frequency domain in which the spectrum can be regarded as approximately the same as that of virtual regular lattice. Next, we have calculated the spectrum in the neighborhood of the edge of the band where the above statement does not hold, by applying the moment-trace method only to that region. The result is that when the concentration of lighter atoms is comparable with or larger than that of heavier atoms, there is only one presumably rounded maximum at the position of the band-edge of virtual regular lattice, whereas when the number of lighter atom becomes smaller, there appears an impurity band, its separation from the main band coming out the more distinct, as the con centration of lighter atoms gets smaller. Both results are natural provided the spectrum is to approach that of Poisson lattice as the lighter atoms become few. In previous papersl) we treated the problem of eigenfrequency distribution of linear disordered lattices containing isotopic impurities by the method of transfer matrix. Eigenfrequencies were obtained from the equation Trace H=2, where H is a product of transfer matrices. When impurities are randomly distributed, we must investigate the statistics of the quantity Trace II. In the limiting case of infinitely small density of the impurities, i.e., in the case of "Poisson lattice ", it was possible to calculate its distribution function, from which we could conclude that there appears an extremely narrow impurity band centering about the impurity f.requency of the lattice containing only one impurity atom, while the distribution of in-band frequencies remains the same as that of regular lattice. Schmide) treated the problem by another but similar method and reached the same conclusion. He moreover obtained an approximate expression for the density distribution function of the impurity band. In the case of finite density of impurities, i.e., for "non-Poisson" lattice , however, it is impossible to obtain such a perspective result. What we could obtain was at most the average eigenfrequency equation (Trace H) =2. The eigenfrequency spectrum derived from this proved to be identical with that of