一维单原子链的热容量推导

The Deduction of Heat Capacity in One-dimension Single Atom Chain

在固体的热容量理论中,德拜模型是一种很好的近似,但对于固体中的晶体而言,其热容量的推导可采用更为精确的方法,而不用引入更多的假设,即只考虑原子间的简谐作用,引入声子的概念,再利用色散关系和玻色-爱因斯坦统计,便可得出热容量.一维单原子链是最简单的情形,利用该方法得到的结论为:在温度T→∞时,其热容量CV=R;在T→0时,CV∝T与以前理论中严格成立的部分相一致.

Among the theories about the heat capacity of solids,the Debye Model is an excellent approximation.But for crystal,the deduction of its heat capacity can adopt a more precise method without introducing more assumptions.That is,just considering simple harmonic interactions between the nearest atoms,introducing the ＂phonon＂ concept and taking advantage of dispersion relation and Bose-Einstein statistics to obtain heat capacity.Using that method in one-dimension single atom chain comes to conclusions that when T→∞,CV=R and whenT→0,CV∝T,which corresponds to the strictly correct portions of previous theories and demonstrates the validity of such method.