单层石墨莫比乌斯纳米带的形成判据和能量变化

莫比乌斯带是一种无向的几何面,从带面上任何一点出发都可到达带面上任何一点.可以通过对一个二维矩形带的扭转对接得到莫比乌斯带.单层石墨纳米材料Graphene的发现为研究莫比乌斯带提供了一个真实的物理模型.然而与纯粹几何面不同,在Graphene莫比乌斯带结构中碳原子分布是离散的,由于平面Graphene的平衡键长约为1.42,在形成莫比乌斯带后键长将发生一定变化,这就使得并非对于任意数目的碳原子Graphene平面都会形成相应的莫比乌斯带结构.这里有两个基本问题需要回答：（ⅰ）平面Graphene形成莫比乌斯带需要满足什么条件.（ⅱ）当平面Graphene形成莫比乌斯带之后,由于键长键角的变化引起的能量变化是怎样的.本文第一次系统地研究了这一体系.根据Lindemann判据,我们发现当平面Graphene的横纵单元数之比小于7时,不存在满足我们假设的莫比乌斯带,当Graphene的横纵单元数之比大于50时,存在满足我们假设的莫比乌斯带.当Graphene的横纵单元数之比处于7～50之间时,我们给出了判别莫比乌斯带存在性的计算方法.应用遗传算法我们发现,随着原子数的增加,由平面Graphene变化到莫比乌斯带所产生的弹性形变能是呈指数递减的.

The formation criterion and energy change of graphene Mobius nano strip

Mobius strip is a well-known geometric surface which has been attracting a lot of attention in mathematics, physics and chemistry. The discovery of graphene sheet provides us a real physical model to study Mobius strip. However, different from the mathematical strip, the distribution of carbon atoms is discrete with the bond length of 1.42 A. So not every graphene sheet with an arbitrary size can form a Mobius strip. This fact arises the following two basic questions： （i） what is the size requirement for the graphene sheet to form a Mobis strip while keeping the bonding of carbon atoms？ （ii） What is the elastic energy change for the geometric transformation？ Based on Lindemann criterion and Genetic algorithm, we carry out a systematic study of graphene Mobius nano strip. We find that when the ratio of the transverse element number and longitude element number in the Graphene sheet is less than 7, we can＇t get a Mobius strip, while we can when the ratio is more than 50. When the ratio is between 7 and 50, we show a numerical method for the answer. Furthermore, we find that the elastic energy caused by the transformation decreases with the increase of the ratio of the transverse element number and longitude element number in the graphene sheet.