{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
Radioactive Decay Chains and the Bateman Equation
\n", "\n", "
Part II Particle & Nuclear Physics
\n", "\n", "Many radioactive decay chains exist in nature and relative abundances of substances within these chains can be used to the date the object. E.g. Uranium-238 decay chain:\n", "\n", "\n", "\n", "For these chains, the ith isotope obeys the following decay rate equation:\n", "\n", "
$\\frac{d N_i}{dt} = -\\lambda_i N_i + \\lambda_{i-1}N_{i-1}$
\n", "\n", "The process above in the chain produces it at a rate $\\lambda_{i-1}N_{i-1}$ as it decays at a rate $\\lambda_i N_i$. The general solution to this set of equations is the Bateman equation." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#NAME: Bateman Equation\n", "#DESCRIPTION: Radioactive decay chains and the Bateman equation.\n", "\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "\n", "%matplotlib notebook\n", "\n", "from IPython.display import display\n", "import ipywidgets as widgets\n", "from ipywidgets import interact, FloatSlider, IntSlider, fixed" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Decay_cycle: This function takes in the number of time steps (N_t), the total number of nuclei (N), the number of links in decay chain (N_d), an array containing the probability that each daughter nucleii will decay (P) and the initial data array (data).\n", "\n", "It then cycles through the time, nuclei and decay links comparing a array of random numbers between 0 and 1 (rand) to the probability of decay of a given nucleus. If the random number is less than the decay probability then the decay occurs and the nucleus moves along one step in the decay chain." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def decay_cycle(N_t,N,N_d,P,data):\n", "\tfor i in range(1,N_t+1):\n", "\t\trand = np.random.rand(N)\n", "\t\tfor j in range(N):\n", "\t\t\tfor k in range(N_d):\n", "\t\t\t\tif rand[j] < P[k]:\n", "\t\t\t\t\tif data[j,i] == k: \n", "\t\t\t\t\t\tdata[j,i:] = data[j,i:] + 1.0\n", "\t\t\t\t\t\tbreak\n", "\treturn data" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Counter: This function counts the number of each type of nucleus of the total N nuclei at a given time step. E.g. if there are 3 type 0 nuclei at t = 0 and 2 type 0 and 1 type 1 at t = 1 then counter will return [3,0,...] and [2,1,...] respectively." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def counter(N_t,N_d,data):\n", "\t# Counted nucleii\n", "\tcount = np.zeros((N_d,N_t+1))\n", "\tfor i in range(N_d):\n", "\t\tcount[i,:] = np.sum(data.astype('int64') == i,axis = 0)\n", "\treturn count" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Creates and initialises parameters for a chain containing four decays of N nuclei" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [], "source": [ "global N,N_t,P\n", "\n", "# Number of nucleii\n", "N = 10000\n", "\n", "# Number of daughters in chain\n", "N_d = 4\n", "\n", "# Probability of each decay\n", "P = np.zeros((N_d))\n", "P[0] = 0.25\n", "P[1] = 0.025\n", "P[2] = 0.075\n", "P[3] = 0.001\n", "\n", "# Number of time steps\n", "N_t = 25" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This section contains plotting and widget elements for controlling the input parameters." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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