Riseborough Peter S
Temple University, Philadelphia, Pennsylvania 19122, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jan;85(1 Pt 1):011129. doi: 10.1103/PhysRevE.85.011129. Epub 2012 Jan 18.
We have calculated the lowest energy quantized breather excitations of both the β and the α Fermi-Pasta-Ulam monoatomic lattices and the diatomic β lattice within the ladder approximation. While the classical breather excitations form continua, the quantized breather excitations form a discrete hierarchy labeled by a quantum number n. Although the number of phonons is not conserved, the breather excitations correspond to multiple bound states of phonons. The n=2 breather spectra are composed of resonances in the two-phonon continuum and of discrete branches of infinitely long-lived excitations. The nonlinear attributes of these excitations become more pronounced at elevated temperatures. The calculated n=2 breather and the resonance of the monoatomic β lattice hybridize and exchange identity at the zone boundary and are in reasonable agreement with the results of previous calculations using the number-conserving approximation. However, by contrast, the breather spectrum of the α monoatomic lattice couples resonantly with the single-phonon spectrum and cannot be calculated within a number-conserving approximation. Furthermore, we show that for sufficiently strong nonlinearity, the α lattice breathers can be observed directly through the single-phonon inelastic neutron-scattering spectrum. As the temperature is increased, the single-phonon dispersion relation for the α lattice becomes progressively softer as the lattice instability is approached. For the diatomic β lattice, it is found that there are three distinct branches of n=2 breather dispersion relations, which are associated with three distinct two-phonon continua. The two-phonon excitations form three distinct continua: One continuum corresponds to the motion of two independent acoustic phonons, another to the motion of two independent optic phonons, and the last continuum is formed by propagation of two phonons that are one of each character. Each breather dispersion relation is split off the top from of its associated continuum and remains within the forbidden gaps between the continua. The energy splittings from the top of the continua rapidly increase, and the dispersions rapidly decrease with the decreasing energy widths of the associated continua. This finding is in agreement with recent observations of sharp branches of nonlinear vibrational modes in NaI through inelastic neutron-scattering measurements. Furthermore, since the band widths of the various continua successively narrow as the magnitude of their characteristic excitation energies increase, the finding is also in agreement the theoretical prediction that breather excitations in discrete lattices should be localized in the classical limit.
我们已经在阶梯近似下计算了β和α费米-帕斯塔-乌拉姆单原子晶格以及双原子β晶格的最低能量量子化呼吸子激发。虽然经典呼吸子激发形成连续谱，但量子化呼吸子激发形成由量子数n标记的离散层级。尽管声子数不守恒，但呼吸子激发对应于声子的多个束缚态。n = 2的呼吸子谱由双声子连续谱中的共振以及无限长寿命激发的离散分支组成。这些激发的非线性属性在高温下变得更加明显。计算得到的n = 2呼吸子与单原子β晶格的共振在区域边界处发生杂化并交换特性，并且与先前使用数守恒近似的计算结果合理吻合。然而，相比之下，α单原子晶格的呼吸子谱与单声子谱发生共振耦合，无法在数守恒近似下进行计算。此外，我们表明，对于足够强的非线性，α晶格呼吸子可以通过单声子非弹性中子散射谱直接观测到。随着温度升高，当接近晶格不稳定性时，α晶格的单声子色散关系逐渐变软。对于双原子β晶格，发现存在三个不同的n = 2呼吸子色散关系分支，它们与三个不同的双声子连续谱相关。双声子激发形成三个不同的连续谱：一个连续谱对应于两个独立声学声子的运动，另一个对应于两个独立光学声子的运动，最后一个连续谱由一个每种类型的两个声子的传播形成。每个呼吸子色散关系从其相关连续谱的顶部分裂出来，并保留在连续谱之间的禁带内。从连续谱顶部的能量分裂迅速增加，并且色散随着相关连续谱的能量宽度减小而迅速减小。这一发现与最近通过非弹性中子散射测量在NaI中观察到的非线性振动模式的尖锐分支一致。此外，由于各种连续谱的带宽随着其特征激发能量大小的增加而相继变窄，这一发现也与离散晶格中呼吸子激发在经典极限下应局域化的理论预测一致。
