末端间矢量

${\vec {R}}$${\vec {R}}$是聚合物的末端间矢量

末端间矢量是从聚合物的一个末端指向另一个末端的矢量，定义为

${\vec {R}}={\vec {R_{N}}}-{\vec {R_{O}}}=\sum _{i=1}^{N}{\vec {r_{i}}}$${\vec {R}}={\vec {R_{N}}}-{\vec {R_{O}}}=\sum _{i=1}^{N}{\vec {r_{i}}}$

均方末端间矢量定义为

${\begin{aligned}\langle R^{2}\rangle &=\langle \sum _{i}{\vec {r_{i}}}\cdot \sum _{j}{\vec {r_{j}}}\rangle \\&=\sum _{i}\sum _{j}\langle {\vec {r_{i}}}\cdot {\vec {r_{j}}}\rangle \\&=\sum _{i}\langle {\vec {r_{i}}}\cdot {\vec {r_{i}}}\rangle +2\sum \sum _{i<j}\langle {\vec {r_{i}}}\cdot {\vec {r_{j}}}\rangle \\\end{aligned}}$${\begin{aligned}\langle R^{2}\rangle &=\langle \sum _{i}{\vec {r_{i}}}\cdot \sum _{j}{\vec {r_{j}}}\rangle \\&=\sum _{i}\sum _{j}\langle {\vec {r_{i}}}\cdot {\vec {r_{j}}}\rangle \\&=\sum _{i}\langle {\vec {r_{i}}}\cdot {\vec {r_{i}}}\rangle +2\sum \sum _{i<j}\langle {\vec {r_{i}}}\cdot {\vec {r_{j}}}\rangle \\\end{aligned}}$

参考文献

• The mean square end-to-end vector - W.J. Briels

参见

• 聚合物
• 自由连接链
• 蠕虫状链