Exercise_4:Decay Simulation

python physics

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摘要

• 本次作业是根据两个衰变方程，设定不同的初始条件，如$N_{A}=100$，$N_{B}=0$，τ=1.利用Python编程计算出两种核子数随时间变化的关系. 证明随时间的推移NA和NB会趋向于常数，且它们的导数趋向于0.

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背景介绍

• 学会如何运用计算机解决物理问题.

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正文

题目：

有两类核子A和B，其衰变的过程可以相互影响，相互转化，因而达到一种平衡的状态。给出了核子对于时间的微分方程：

$$\frac{dN_{A}}{dt}=\frac{dN_{B}}{τ}-\frac{dN_{A}}{τ}$$

$$\frac{dN_{B}}{dt}=\frac{dN_{A}}{τ}-\frac{dN_{B}}{τ}$$

其时间常数相同，τ取为1s.通过解这个方程组可以得到,$N_{A}$和$N_{B}$为常数且$N_{A}$=$N_{B}$. 现通过程序来模拟计算这个结果.

解答

代码

import pylab as pl
class uranium_decay:
    def __init__(self, number_of_nuclei_A = 100,\
    number_of_nuclei_B = 0,time_constant_1 = 1,\
    time_constant_2 = 1, time_of_duration = 8,\
    time_step = 0.05):
        # unit of time is second
        self.n1_uranium = [number_of_nuclei_A ]
        self.n2_uranium = [number_of_nuclei_B ]
        self.t = [0]
        self.tau1 = time_constant_1
        self.tau2 = time_constant_2
        self.dt = time_step
        self.time = time_of_duration
        self.nsteps = int(time_of_duration // time_step + 1)
        print("Initial number of nuclei A ->", number_of_nuclei_A)
        print("Initial number of nuclei B ->", number_of_nuclei_B)
        print("Time constant A ->", time_constant_1)
        print("Time constant B ->", time_constant_2)
        print("time step -> ", time_step)
        print("total time -> ", time_of_duration)
    def calculate(self):
        for i in range(self.nsteps):
            tmp1 = self.n1_uranium[i] - self.n1_uranium[i] /self.\
            tau1 * self.dt+self.n2_uranium[i] / self.tau2 * self.dt
            tmp2 = self.n2_uranium[i] - self.n2_uranium[i] /self.\
            tau2 *self.dt+self.n1_uranium[i] / self.tau1 * self.dt
            self.n1_uranium.append(tmp1)
            self.n2_uranium.append(tmp2)
            self.t.append(self.t[i] + self.dt)
    def show_results(self):
        pl.plot(self.t, self.n1_uranium)
        pl.plot(self.t, self.n2_uranium)
        pl.xlabel('time ($s$)')
        pl.ylabel('Number of Nuclei ')
        pl.legend(['$N_A$','$N_B$'])
        pl.show()
a = uranium_decay()
a.calculate()
a.show_results()

运行结果

• 运行
  

• 模拟
  

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结论

• 当初始$N_{A}=100$,$N_{B}=0$，τ=1时，τ足够大，使得$N_{A}=N_{B}=50$,且$\frac{dN_{A}}{dt}=0$，$\frac{dN_{B}}{dt}=0$，符合运行结果.

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致谢

Lesson plan of Cai Hao:Chapter 1