{"cells": [{"cell_type": "markdown", "metadata": {"tags": ["module-htg"]}, "source": ["# Sm-Nd Decay\n", "[High-Temperature Geochemistry](module-htg) \n", "```{index} Sm-Nd decay\n", "```"]}, {"cell_type": "code", "execution_count": 3, "metadata": {"tags": ["hide-input"]}, "outputs": [], "source": ["# import relevant modules\n", "\n", "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import pandas as pd\n", "from IPython.display import display\n", "from math import log10, floor"]}, {"cell_type": "code", "execution_count": 4, "metadata": {"tags": ["hide-input"]}, "outputs": [], "source": ["# create our own functions\n", "\n", "# function to round a value to a certain number of significant figures\n", "def round_to_n_sf(value, no_of_significant_figures):\n", " value_rounded = round(value, no_of_significant_figures-1-int(floor(log10(abs(value)))))\n", " if value_rounded == int(value_rounded): \n", " value_rounded = int(value_rounded)\n", " return value_rounded\n", " \n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["## Sm-Nd Decay System\n", "\n", "$${^{147}Sm \\longrightarrow {^{143}Nd} + \\alpha} \\qquad t_{\\frac{1}{2}} = 106\\,Gyr$$\n", "\n", "Both $Sm$ (samarium) and $Nd$ (neodymium) are Rare Earth Elements ($REEs$). In nature, both elements generally occur in dispersed form, with typical concentrations in mantle and crustal rocks of less than $\\sim20\\,ppm$. \n", "\n", "Most $REEs$, including $Sm$ and $Nd$, occur as trivalent ($3+$) ions with ionic radii that decrease systematically with increasing atomic number, so $Sm$ has a smaller ionic radius than $Nd$.\n", "\n", "Both $Nd$ and $Sm$ are moderately incompatible elements, but $Nd$ is slightly more incompatible than $Sm$ during mantle melting because it has a slightly larger ionic radius. \n", "\n", "The $REEs$ are generally considered to be relatively resistant toward mobilization by fluids \u2013 they are fluid-immobile elements. \n", "\n", "## Dating of Terrestrial Rocks\n", "\n", "The continental crust in general and siliceous rocks in particular have low and relatively uniform $Sm/Nd$ ratios (= parent/daughter ratio), so the $Sm$-$Nd$ system is not particularly suitable for dating such rocks. On the other hand, mafic and ultramafic rocks have variable and high $Sm/Nd$ ratios, so they are good candidates for $Sm$-$Nd$ dating. \n", "\n", "Due to the fluid immobility of the $REEs$, $Sm$-$Nd$ ages and initial $Nd$ isotope compositions are not very sensitive to weathering and metamorphism. \n", "\n", "Notice how the $Sm$-$Nd$ system is opposite to the $Rb$-$Sr$ decay system in many ways! \n"]}, {"cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": ["# Sm-Nd decay equation - very similar to the Rb-Sr one\n", "# each return depends on what we want to find from the equation\n", "def Sm_Nd_decay_equation(Nd143_Nd144_ratio, initial_Nd143_Nd144_ratio, Sm147_Nd144_ratio, t):\n", " decay_const_Sm = 6.54 * 10**-12 # yr^-1 # decay constant of Sm-147\n", " if Nd143_Nd144_ratio == '?':\n", " return initial_Nd143_Nd144_ratio + Sm147_Nd144_ratio*(np.exp(decay_const_Sm*t)-1)\n", " elif initial_Nd143_Nd144_ratio == '?':\n", " return Nd143_Nd144_ratio - Sm147_Nd144_ratio*(np.exp(decay_const_Sm*t)-1)\n", " elif Sm147_Nd144_ratio == '?':\n", " return (Nd143_Nd144_ratio - initial_Nd143_Nd144_ratio)/(np.exp(decay_const_Sm*t)-1)\n", "\n"]}, {"cell_type": "markdown", "metadata": {}, "source": ["## Problem Set 5\n", "\n", "### Question 1\n", "\n", "A pigeonite basalt ($12039$, $19$) from the Moon yielded the following results:"]}, {"cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [{"data": {"text/html": ["\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Samples $${^{147}Sm}/{^{144}Nd}$$ $${^{143}Nd}/{^{144}Nd}$$
Whole rock0.2090.513142
Plagioclase0.17270.512365
Pyroxene0.24340.513861
"], "text/plain": [""]}, "metadata": {}, "output_type": "display_data"}], "source": ["# create a dataframe to show the obtained results\n", "samples = [\"Whole rock\", \"Plagioclase\", \"Pyroxene\"]\n", "Sm147_Nd144_ratio = [0.2090, 0.1727, 0.2434]\n", "Nd143_Nd144_ratio = [0.513142, 0.512365, 0.513861]\n", "\n", "dict1 = {'Samples' : samples,\n", " '$${^{147}Sm}/{^{144}Nd}$$' : Sm147_Nd144_ratio,\n", " '$${^{143}Nd}/{^{144}Nd}$$' : Nd143_Nd144_ratio}\n", "df1 = pd.DataFrame(dict1)\n", "df1.loc[:, '$${^{147}Sm}/{^{144}Nd}$$'] = df1['$${^{147}Sm}/{^{144}Nd}$$'].map('{:g}'.format)\n", "df1.loc[:, '$${^{143}Nd}/{^{144}Nd}$$'] = df1['$${^{143}Nd}/{^{144}Nd}$$'].map('{:g}'.format)\n", "display(df1.style.hide_index())"]}, {"cell_type": "markdown", "metadata": {}, "source": ["a) Plot the data in an isochron diagram. Scale the y-axis from about ${^{143}Nd}/{^{144}Nd}$ = $0.5080$ to $0.5150$ and the x-axis from ${^{147}Sm}/{^{144}Nd}$ = $0$ to about $0.3$.\n", "\n", "b) Determine the age and initial ${^{143}Nd}/{^{144}Nd}$ ratio of this rock from the slope and y-intercept of the isochron.\n", "\n", "c) Calculate the $Nd$ isotope ratio of CHUR for the age of rock.\n", "\n", "d) Express the initial $Nd$ isotope composition of the rock as an $\\epsilon_{Nd}$ value relative to $CHUR$. What does the initial $\\epsilon_{Nd}$ value tell you, if you assume that the Moon as a whole has chondritic $Sm/Nd$ and $Nd$ isotope ratios?"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Solution:\n", "\n", "a) See below"]}, {"cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [{"data": {"text/plain": [""]}, "execution_count": 8, "metadata": {}, "output_type": "execute_result"}, {"data": {"image/png": 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k+ppyc125Lb9w4UIaNGhA6dKlKV68OGeffTbDhw8Hgs/BkCFD8hSviOTMHV57DdLSgqmHP/kEnn4aSpaMd2SRU5+Bvbj11mCEqGhKTYVQ7W2uLVmyhKFDh/Laa69x9dVX8/7771O3bl3efvttpk6dSokSJbjxxhsZMmQI1157LQD9+/fn0EMPZfPmzZxyyik0b94cgMWLFzNgwIC/Vf336dOHMWPGMGHCBMqWLcv9998PwBNPPME333zD3GxuSGZmJsuWLaNy5coAVK9ene+//54dO3aQkpLCHXfcwbPP7jk69JlnnrnHLIu7PPPMM5x//vkA/PTTT1Ss+NdglRUqVOCrr776W5lI9lu+fDlz5syhfv362d7b7GQt89BDD/HAAw+wevVq5syZw6hRo3I4QvYivS4zo3HjxpgZnTp1omPHjvssf9lll9GjRw/Wrl3L/vvvz+jRo3c3L5x88snMnDkzT/GKyL798Qd07Ajvvgvnnw9vvglHHpljsUJHyUCCqFKlCqmpqQDUrVuX5cuX88cffzBr1ixOOeUUIGi7L1++/O4yvXv33v3X4cqVK1myZAlHHHEElSpVokGDBlGJ69dff91jJsJixYpRvXp1FixYwJIlSzjmmGN2/7W9y+TJk3M8rmczgZZl8yxOTvtt2LCB5s2b8/zzz3PQQQfleN69lTnrrLNwd5577jkmTpxISkrKHmXq16/P1q1b2bBhA7/99tvuf6snn3ySCy64INfXNXXqVI466ihWr15No0aNOOmkk3bHkF35qlWrcvfdd9OoUSMOPPBAatWqRfHQkGYpKSmULFmSP//8kzJlykR0D0QkZ9OnB/0BMjLgiSeC6YeLJWh9u5KBvcjrX/AFZb/99tv9PiUlhc2bN+PutG3blscff/xv+0+cOJHPP/+c6dOnU7p0aRo2bMiWLVsA9jqlcV7sv//+u4+7S4MGDZg6dSqvvPIKY8aM+VuZSGoGKlSosEcTRkZGBkcdddTfyuxrv23bttG8eXNat27NFRHO/rG3MvPnz+fnn3+mbNmy2X6h7vrrfuLEiQwcOJCBAwdme/xIr2vXuvLly3P55ZczY8YMzjrrrH2W79ChAx06dADg3nvvpUKFCrv327p1K6VKlYroHojIvu3cCU8+GcwpULEiTJ4MUfr7Km4SNIcRgPPOO4/33nuP1atXA/Dbb7+xYsUKANatW8chhxxC6dKlWbRoEV9++WWez1OmTJlsv7wBDjnkEHbs2LFHQtCgQQPuu+8+Lr/8co4++ui/lZk8eTJz587922tXIgBwyimnsGTJEn744QcyMzMZNmwYl1xyyd+Otbf93J0OHTpQtWpVbr/99oiuc29lfv75Z1q3bs3IkSM54IAD+PTTTyM6XnYiua6NGzfuvt8bN27ks88+2/3Uw77K7/oc/Pjjj3zwwQe0bBkM2Ll27VrKlStHiRIl8hy3iAR+/hkaN4Z774XmzYOnBRI9EQAlAwmtWrVqPPLIIzRu3JiaNWvSqFEjfv75ZwCaNGnC9u3bqVmzJvfff3++mgUOO+wwTj/9dE4++eRsOxA2btyYKVOm7F4+6aST2G+//bj77rvzfM7ixYvz0ksvccEFF1C1alWuvvpqqlevvnv7RRddxKpVq/a639SpUxk0aBDjx48nNTWV1NRURo/+axTsli1bcuqpp7J48WIqVKhAv3799lrmiiuu4Nlnn6Vq1arcf//99OzZs0Cua9c1/fLLL5xxxhnUqlWLevXq0bRpU5o0aZJj+ebNm1OtWjUuvvhiXn75ZQ455BAAJkyYwEUXXZTnmEUk8MknUKsWTJsGr78Ow4ZBWCtpQrPs2iCTQVpamqenp++xbuHChVStWjVOESWuOXPm8NxzzzFo0CAAunbtyimnnELbtm3jHJkAXHHFFTz++OOceOKJ2W7X515k3zIz4Z574LnnoEaNIAmoVi3eUeWemc1y97TstqlmQPKtdu3anHPOOXz//fecdNJJbN68WYlAIZGZmclll12210RARPZt6VI47bQgEbjxRvjqq8RMBHKiDoQSFe3btwdg0aJFcY5EwpUsWXL3o6YikjuDB0OXLlCiBHzwAVx+ebwjKjiqGRAREQmzYUMwsVCbNsH4MHPnFu1EAJQMiIiI7DZnDtStC4MGBVMPT5gAxxwT76gKnpIBERFJeu7wwgvBY4IbN8L48fDQQ1A8SRrTk+QyI+fu2Y4IJ1IUJevTRCLh1qyBdu3g44/h4ouhf38oWzbeUcWWagbClCpVirVr1+oXpCQFd2ft2rUamVCS2oQJwdgBY8dC794wcmTyJQKgmoE9VKhQgYyMDNasWRPvUERiolSpUnsMWyySLLZvD5oBHn0U/vlPGD066CyYrJQMhClRogRVqlSJdxgiIlKAVqyA1q1h6tSgeaB3bwjN2J60lAyIiEjS+OAD6NABduyAt96C0BQeSU99BkREpMjbvDkYQKh5czj++OARQiUCf1EyICIiRdqCBVCvHvTpA926Bc0Dxx0X76gKFzUTiIhIkeQOr70Gt94KZcrAmDFwwQXxjqpwUs2AiIgUOX/8Af/6F3TqBGecAfPmKRHYFyUDIiJSpEyfHjwmOHw4PPlkUCNwxBHxjqpwUzIgIiJFwo4d8NhjcOaZYAZTpsBdd0ExfdPlSH0GREQk4a1aFcwyOH580Dzw6qvwj3/EO6rEoWRAREQS2ujR0LZtMMHQ669D+/ZBzYBETpUnIiKSkLZuhdtvh6ZN4aijYNasYEAhJQK5p5oBERFJOEuWQIsWMHs23HQTPPMMaM6tvFMyICIiCWXw4GA0wRIlgicGLrss3hElPjUTiIhIQvjzz6BvQJs2ULt2MHaAEoHoUDIgIiKF3uzZULduUCvw4IPBUwMVK8Y7qqJDyYCIiBRa7vD889CgAWzaFCQBPXtCcTVyR5Vup4iIFEpr1sB11wWPDl5yCfTvD4cdFu+oiibVDIiISKEzYQLUqgWffw4vvggjRigRKEgxTwbMrImZLTazpWbWPZvtDc1snZnNDb0eCNvW38xWm9k3ezn2nWbmZla2IK9BREQKxvbtcN99cN55cNBB8NVX0LWrxg4oaDFtJjCzFOBloBGQAcw0s1Hu/m2WXSe7e7NsDjEQeAl4M5tjVwwd98eoBi0iIjGxYgW0agXTpgWjCPbuDQccEO+okkOsawbqAUvdfZm7ZwLDgEsjLezuk4Df9rK5F3AX4PmOUkREYur994OZBufPh7fegn79lAjEUqyTgaOBlWHLGaF1WZ1qZvPM7BMzq57TQc3sEuAnd5+Xw34dzSzdzNLXrFmTq8BFRCT6Nm+Gzp3hyivhhBNgzhxo2TLeUSWfWCcD2bX6ZP1LfjZQyd1rAS8CI/Z5QLPSQA/ggX3tB+Dufd09zd3TypUrF1nEIiJSIBYsgFNOCWYY7NYtmHL4uOPiHVVyinUykAGEDxNRAVgVvoO7r3f3DaH3o4ESOXQIPA6oAswzs+WhY842syOiGbiIiESHe5AApKUFjw+OGQNPPQUlS8Y7suQV63EGZgInmFkV4CegBdAqfIfQl/gv7u5mVo8gYVm7twO6+3ygfFj55UCau/8a/fBFRCQ/fv8dbrgh6CPQqBG8+SYcoT/d4i6mNQPuvh3oCnwKLATecfcFZtbZzDqHdrsS+MbM5gG9gRbu7gBmNhSYDpxoZhlm1iGW8YuISC4NGQKVK0OxYkw74gpST9jIyJHw5JNBjYASgcLBQt+zSSctLc3T09PjHYaISNE1ZAh07MiOTVt4gu48yEMcYysZ+uBi6j/YJN7RJR0zm+Xuadlt0wiEIiJSMHr0YNWmf9CIsdzHo1zFu8zxVOoP6JxzWYkpzU0gIiIF4uMVJ3MdA9hEafrRnnYMCB4p+/HPeIcmWahmQEREomrrVrj9dmjGRxzFKtJJo/2uRADgmGPiGZ5kQ8mAiIhEzZIlcNpp0KsXdG20mK/2P4eqLPprh9Kl4dFH4xegZEvJgIiIRMWgQVCnDixfHswy+OJnJ1LqtRehUqVgpqFKlaBvX2jdOt6hShbqMyAiIvny559w001BMnDWWTB4MFTcNbxc69b68k8AqhkQEZE8mzUrqA0YMgR69oTx48MSAUkYSgZERCTX3IN+AaeeGkw2NGECPPggpKTEOzLJCzUTiIhIrqxZA9ddB6NHw6WXBtMNH3ZYvKOS/FDNgIiIRGz8eKhVC8aNg5deguHDlQgUBUoGREQkR9u2QY8ecP75cNBB8NVXQadBy25iekk4aiYQEZF9Wr4cWrWC6dOhfXvo3RsOOCDeUUk0KRkQEZG9eu89uP562LkThg6FFi3iHZEUBDUTiIjI32zaBJ06wVVXwYknwty5SgSKMiUDIiKyh2++gXr1gsEC77oLpkyBY4+Nd1RSkNRMICIiQDB2wKuvwm23BZ0EP/0UGjeOd1QSC6oZEBERfv89aBLo0iUYUvjrr5UIJBMlAyIiSW7qVEhNhZEj4amn4JNP4PDD4x2VxJKSARGRJLVjRzCb8NlnQ/HiQVLQrRsU0zdD0lGfARGRJLRqFVxzTTCnQMuW0KdP0E9AkpOSARGRJPPxx8HcAps2Qf/+wXuNJJjcVBkkIpIktm4NnhRo1gyOPjqYfrhdOyUCopoBEZGk8N13waBBc+bAzTcHHQVLlYp3VFJYKBkQESni3nwTbrwR9tsveGLgkkviHZEUNmomEBEpov78E9q0gbZtoW5dmDdPiYBkT8mAiEgRNGsW1KkDb70FDz0E48dDhQrxjkoKKyUDIiJFyM6d8NxzcOqpsGULTJwIDzwAKSnxjkwKM/UZEBEpIlavDh4T/OQTuOwy6NcPDj003lFJIlDNgIhIETBuHNSqFTQHvPwyfPCBEgGJnJIBEZEEtm0b3HsvNGoEhxwCM2YETw5o7ADJDTUTiIgkqOXLg6GEv/wSOnSAF16AAw6Id1SSiFQzICKSCIYMgcqVg1mEKlfm3Vsmk5oK334Lw4bB668rEZC8U82AiEhhN2QIdOwImzaxif25bcU99H3xTOof9ytDx5alSpV4ByiJTjUDIiKFXY8esGkT31CdU5hJXzpxN08weVsDJQISFaoZEBEp5HzFj7xKJ26jF/9gHZ/RiEZ8DivVS1CiQ8mAiEgh9vvvcH3pj/lg04VcwBjeoC2HszrYeMwx8Q1Oigw1E4iIFFJTp0JqKozaegFPl7iX0Vz0VyJQujQ8+mhc45OiI+bJgJk1MbPFZrbUzLpns72hma0zs7mh1wNh2/qb2Woz+yZLmf+Y2deh/T8zs6NicS0iIgVhxw545BE4+2woXhymTS/GnQOqU6zSMcEAApUqQd++0Lp1vEOVIsLcPXYnM0sBvgMaARnATKClu38btk9D4E53b5ZN+bOADcCb7n5y2PqD3H196P0tQDV377yvWNLS0jw9PT3f1yQiEk0//QTXXBPMKdCqFfz3v3DQQfGOSooCM5vl7mnZbYt1zUA9YKm7L3P3TGAYcGmkhd19EvBbNuvXhy0eAMQuwxERiZKPPgqGFJ4xAwYMgMGDlQhIbMQ6GTgaWBm2nBFal9WpZjbPzD4xs+qRHNjMHjWzlUBr4IGc9hcRKSy2boVbb4WLL4aKFWH27GDCIQ0pLLES62Qgu4921r/iZwOV3L0W8CIwIpIDu3sPd68IDAG6Zntys45mlm5m6WvWrIk8ahGRAvLdd8F0wy+8ALfcAtOnw4knxjsqSTaxTgYygIphyxWAVeE7uPt6d98Qej8aKGFmZXNxjreA5tltcPe+7p7m7mnlypXLXeQiIlH25ptQpw78+COMGhUkBKVKxTsqSUaxTgZmAieYWRUzKwm0AEaF72BmR5gFlWNmVi8U49p9HdTMTghbvARYFNWoRUSi6M8/oU0baNsW0tJg3rygiUAkXmI66JC7bzezrsCnQArQ390XmFnn0PY+wJVAFzPbDmwGWnjokQczGwo0BMqaWQbwoLv3A54wsxOBncAKYJ9PEoiIxEt6OrRoAT/8AA8/HEw/nJIS76gk2cX00cLCRI8Wikgs7dwJzz8P3bvDEUfAW2/BGWfEOypJJvt6tFDDEYuIFLDVq4OnAz75BC6/PJhu+NBD4x2VyF80HLGISAEaNy4YO2D8eHjlFXj/fSUCUvgoGRARKQDbtgX9ARo1gkMOCQYS6tJFYwdI4aRmAhGRKFu+HFq2hC+/hBtugF694IAD4h2VyN4pGRARiaJ33w0SAHd4+224+up4RySSMzUTiIhEwaZN0LFj8OV/0kkwd64SAUkcSgZERPJp/nw45ZTgKYHu3WHyZKhSJd5RiUROzQQiInnkDn36wO23w8EHw2efwfnnxzsqkdxTzYCISB789htceSXceCM0bBgMKaxEQBKVkgERkVyaMgVSU+HDD+GZZ+Djj6F8+XhHJZJ3SgZERCK0Ywc88gicfTaULAnTpsEdd0Ax/SaVBKc+AyIiEfjpJ7jmGpg4EVq1gv/+Fw46KN5RiUSHkgERkRx89FEwt8CWLTBwIFx7rUYSlKJFlVsiInuxdSvceitcfDFUrAizZkHbtkoEpOhRMiAiko3vvoNTT4UXXoB//zsYWvjEE+MdlUjBUDOBiEgYd3jzTbjpJihVKnhioFmzeEclUrBUMyAiEvLnn9CmTdA/IC0tGDtAiYAkAyUDIiJAejrUrg1Dh8LDD8O4cXD00fGOSiQ2lAyISFLbuROefRZOOw0yM+GLL+D++yElJd6RicRORMmAmV1oZkcVdDAiIrG0ejU0bQp33hk0B8ydC2ecEe+oRGIv0g6EzYH/mNnhwCJgLjAv9HOhu+8okOhERArI558H/QN+/x1eeQU6d9Yjg5K8IqoZcPfr3T0NeBb4DvgBOAeYAawouPBERKJr2za45x5o3BgOOQRmzoQuXZQISHLL7aOF7dy91q4FM3sF6BbdkEREomzIEOjRg+UrjJYl3+fLzDrccAM8/zyULh3v4ETiL7cdCNebWd1dC+4+C/hndEMSEYmiIUOgY0feXXEKqcxhYeaxvF2yDX3PHqJEQCQktzUD7YHBZvYtMAuoAWyLelQiIlGy6Z7/cOumXrxGRxownaG0pHLmCugxGVq3jnd4IoVCrmoG3H0JcBowGjgcWAhcVABxiYjk2/z5cMrK93md67mHx5jEWVTe1c3pxx/jG5xIIRJRzYCZHePuPwKEnhx4N/QSESl03KFPH7j9djg4pRyf7WjM+Yzbc6djjolPcCKFUKTNBGPMrDxBTcB84OvQz/nuvr6gghMRya3ffoPrr4fhw+HCC2Fgs0mU7zYdNoXtVLo0PPpo3GIUKWwiSgbcvZqZlQSqE/QTqAlcBtQ0s63uXqXgQhQRicyUKdCqFfzvf/DMM3DbbVCs2JXwj63Qo0fQNHDMMUEioP4CIrtF3IHQ3TOBOWa2FNgMlAVOIKghEBGJmx074LHHoGdPqFIFpk0LJhrarXVrffmL7EOkfQZOBJoCzYBywFhgCNAxlCSIiMTFTz/BNdfAxInB9/0rr8BBB8U7KpHEEmnNwEJgDvAEMMrdtxZcSCIikfnwQ2jXDrZsgYED4dprNZKgSF5E+mhhF2A60BVYaWYLzewdM7vfzC4rsOhERLKxdSv8+99wySVBF4DZs6FtWyUCInkVaQfCV8OXzawCQSfCGgSTGI2IemQiItlYvBhatAhmGPz3v+HJJ2G//eIdlUhiy+0IhAC4ewaQQTD4kIhIgXOHN96Arl2hVKmgiaBZs3hHJVI05JgMmNmuEQYNuB54zd2VBIhIzKxfH8ws+NZb0LAhDB4MRx8d76hEio5I+gw8TDC+QFmgdOiniEhMzJwJderAsGHwn//A558rERCJtkiSgbOAA4EtwAJ3f7NgQxIRgZ07g4GDTjsNtm2DSZPgvvsgJSXekYkUPTkmA+6+yd0fBH4lGGwoX8ysiZktNrOlZtY9m+0NzWydmc0NvR4I29bfzFab2TdZyjxtZovM7GszG25mB+c3ThGJn9WroWlT6NYteGJg7lw4/fR4RyVSdOWYDJjZRaF+A6WAamF9CHLNzFKAl4ELgWpASzOrls2uk909NfR6OGz9QKBJNvuPBU5295rAd8A9eY1RROLr88+hVq1gEKH//hfeew8OOSTeUYkUbbHuM1APWOruy0IjFw4DLo20sLtPAn7LZv1n7r49tPglUCEfMYpIHGzbBvfcA40bw6GHBn0FOnfW2AEisRDrPgNHAyvDljNC67I61czmmdknZlY9l+doD3yS1wBFJPZ++AHOPBOeeAJuuCFIBE4+Od5RiSSP3PYZ2JTT/jnILsf3LMuzgUruXgt4kVwMaGRmPYDtBPMmZLe9o5mlm1n6mjVrIj2siBSgt9+G1FRYtAjeeQdefTWYYVhEYifS4Yhx93Hufp+Z3Z2P82UAFcOWKwCrspxnvbtvCL0fDZQwsxybJsysLcFESq3dPWuCsevYfd09zd3TypUrl9drEJEo2LQpqAVo0QKqVw86CV51VbyjEklOkQw69E74IpAKPJnH880ETjCzKsBPQAugVZbzHQH84u5uZvUIEpa1OcTYBLgbONvd81t7ISIF7OuvgyRg0aKgn8BDD0GJEvGOSiR5RTIc8Xp3v37Xgpn9N68nc/ftZtYV+BRIAfq7+wIz6xza3ge4EuhiZtsJHmVssesvfTMbCjQEyppZBvCgu/cDXgL2A8Za0NvoS3fvnNc4RaRguAdPCNx+e/CEwNixcN558Y5KRGwvNep/7WBWxd1/CFs+1N3/1qM/0aSlpXl6enq8wxBJGr/9BtdfD8OHw4UXBlMOly8f76hEkoeZzXL3tOy2RdKB8AczO9VCf3IXhURARGJr8uSgk+BHH8GzzwY/lQiIFB6RdiBsC8wys2Fmdl2oXV9EZJ927ICHHw4mF9pvP5g2LWgiKBZx12URiYWIpjDe1f5uZicRjB440Mz+AUwAxgBT3X1HgUUpIgknIwOuuQa++CL4+corUKZMvKMSkexElJ+b2YVmdpS7L3L3Xu7eBDgXmAJcBXxVkEGKSGIZNSoYUjg9Hd54AwYNUiIgUphFVDMANAf+Y2aHA4uAucC80M9bVSsgIgBbtsBdd8GLL0Lt2sG0w//8Z7yjEpGcRFQz4O7Xh3ogPkswEdAPwDnADGBFwYUnIoli8WJo0CBIBG69FaZPVyIgkigirRnYpV1omGAAzOwVoFt0QxKRROIePCbYtWswjPCHH0KzZvGOSkRyI7d9etebWd1dC+4+C1DuL5Kk1q+H1q2hfXuoXx/mzVMiIJKIclsz0B4YbGbfArOAGsC2qEclIoXezJnBkMIrVsAjj0D37pCSEu+oRCQvclUz4O5LgNOA0cDhwELgogKIS0QKqZ074emn4bTTYPv24NHBHj2UCIgksohqBszsGHf/ESD05MC7oZeIJJFffoG2beHTT6F5c3jttWCOARFJbJE2E4wxs/IENQHzga9DP+e7+/qCCk5ECo+xY6FNG1i3Dvr0gY4dIRikXEQSXaQjEFYzs5JAdYJ+AjWBy4CaZrbV3asUXIgiEk/btsF998FTT0G1avD553DyyfGOSkSiKeIOhO6eCcwxs6UEUwuXBU4gqCEQkSJo2TJo2RJmzAhqAnr1Ch4fFJGiJdI+AycCTYFmQDlgLDAE6BhKEkSkiBk2DDp1CpoC3nkHrroq3hGJSEGJtGZgITAHeAIY5e5bCy4kEYmnjRvh3/+Gfv3g1FPhrbegcuV4RyUiBSnSZKALQV+BrsDLZraWUAdCgk6EIwomPBGJpa+/hn/9Kxha+N57oWdPKFEi3lGJSEGLtAPhq+HLZlaBoBNhDYJJjEZEPTIRiRn3YIrhO+6AQw8Nnhw477x4RyUisZJjMmBmnYG6wDigNfCxu/cBMggGHxKRBPbbb8FwwiNHwkUXBfMMlCsX76hEJJYiqRk4F/gXMNndzzCzPgUck4jEyOTJ0KpVMJjQc88FfQWK5XbGEhFJeJH8t1/r7g48GVpW50GRBLdjBzz0EDRsCKVKBdMN33abEgGRZBVJzcALAO7+YWj5g4ILR0QKWkZGMNPgpEnBiIIvvwxlysQ7KhGJpxyTAXdflGX5i4ILR0QK0qhR0K4dbN0Kb7wB114b74hEpDDIsVLQzC4KvZqa2XAz0yyFIglmyxa4+Wa49FKoVAlmz1YiICJ/iaSF8GGCOQnKAqVDP0UkQSxaBA0awEsvwa23Bv0D/vnPeEclIoVJJMnAWcCBwBZggbu/WbAhiUg0uMOAAVC3Lvz0E3z0UTC3wH77xTsyESlsckwG3H2Tuz8I/ApsKviQRCS/1q0LOgm2bw/168O8edC0abyjEpHCKuIHidx9nLvfZ2Z3F2RAIpI/M2ZA7drB5EKPPBKMJnjUUfGOSkQKs0hGIHwnfBFI5a8xB0SkkNi5E555Bnr0CL78J02C006Ld1QikggiGWdgvbtfv2vBzP5bgPGISB788kvwdMBnn0Hz5vDaa3DIIfGOSkQSRSTNBI9mWe5REIGISN589hnUqhXUBPTpA+++q0RARHInkg6EP2RZ/q3gwhGRSGVmwt13wwUXwGGHwcyZ0KkTmMU7MhFJNBFNYSwihcuyZdCyZdBZsFOnYJKh0qXjHZWIJColAyIJZtiwv2oA3n0Xrrwy3hGJSKLTHGUiCWLjRujQIagROPnkYOwAJQIiEg1KBkQSwLx5kJYWjCjYowd88UUwx4CISDQoGRApxNyDOQXq1w9GFfz882AgoeJq4BORKNKvFJFCau3aoFlg5Ei46CIYOBDKlYt3VCJSFMW8ZsDMmpjZYjNbambds9ne0MzWmdnc0OuBsG39zWy1mX2TpcxVZrbAzHaaWVosrkOkIE2aBKmpMHp08KTARx8pERCRghPTZMDMUoCXgQuBakBLM6uWza6T3T019Ho4bP1AoEk2+38DXAFMinLIIjG1Ywc89BCccw6UKhVMN3zbbRo7QEQKVqybCeoBS919GYCZDQMuBb6NpLC7TzKzytmsXxg6XvQiFYmxlSvhmmuCWoE2beDll6FMmXhHJSLJINbNBEcDK8OWM0LrsjrVzOaZ2SdmVj1aJzezjmaWbmbpa9asidZhRfJt5MigWWDWLHjzzeClREBEYiXWyUB2f7p7luXZQCV3rwW8CIyI1sndva+7p7l7Wjk1wEohsGUL3HwzXHYZVK4Mc+YEtQIiIrEU62QgA6gYtlwBWBW+g7uvd/cNofejgRJmVjZ2IYrExqJF0KBB8OjgbbfBtGlwwgnxjkpEklGsk4GZwAlmVsXMSgItgFHhO5jZERZq/DezeqEY18Y4TpEC4w79+0PduvDTT/Dxx8ETA/vtF+/IRCRZxTQZcPftQFfgU2Ah8I67LzCzzmbWObTblcA3ZjYP6A20cHcHMLOhwHTgRDPLMLMOofWXm1kGcCrwsZl9GsvrEonUunXQqlUwfkD9+sHIghddFO+oRCTZWeh7NumkpaV5enp6vMOQJDJjBrRoAT/+CA8/HEw/nJIS76hEJFmY2Sx3z3YsHg1HLFLAdu6Ep56C008PxhGYNAnuvVeJgIgUHhqOWKQA/fILXHstfPYZNG8Or70GhxwS76hERPakmgGRAvLZZ1CzZlAT0KcPvPuuEgERKZyUDIhEWWYm3HUXXHBBMJ/AzJnQqZOGFBaRwkvNBCJRtGwZtGwZdBbs1Cl4ZLB06XhHJSKyb0oGRKJk6NAgAUhJCZoErrwy3hGJiERGzQQi+bRxYzBuQKtWUKMGzJ2rREBEEouSAZF8mDs3GElwwADo0QO++AIqVYp3VCIiuaNkQCQP3IM5BerXh/Xr4fPP4ZFHoLga3kQkAelXl0gurV0L7dvDqFHBUMIDBwZPDYiIJCrVDIjkwhdfQK1a8Mkn0KsXfPSREgERSXxKBkQisH079OwJ554bPCr45Zdw660aO0BEigY1E4jkYOVKaN0aJk8OhhZ+6SUoUybeUYmIRI+SAZF9GDkS2rWDbdtg0CC45pp4RyQiEn1qJhDJxpYt0LUrXHYZVKkCs2crERCRokvJgEgWCxcGjwy+/DLcfjtMmwYnnBDvqERECo6aCURC3KF/f7jllqCT4McfB48OiogUdaoZEAHWrQsmGLr+emjQAObNUyIgIslDyYAkva++gtq14b334LHH4LPP4Kij4h2ViEjsKBmQpLVzJzz5JJxxRvB+0iS4555g1kERkWSiPgOSlP73v2DMgLFjgxkGX3sNDj443lGJiMSHagYk6Xz6aTCk8OTJ8Oqr8M47SgREJLkpGZCkkZkJd90FTZpA+fKQng4dO2pIYRERNRNIUvj+++BpgZkzoXNneO452H//eEclIlI4KBmQIm/oUOjUKegY+N570Lx5vCMSESlc1EwgRdbGjdC+PbRqBTVrwty5SgRERLKjZECKpLlzoW5dGDgQ7rsPJk6ESpXiHJSISCGlZECKFHd48cVgboE//4Rx4+A//4HiahATEdkr/YqUImPt2qBZYNQoaNYMBgyAsmXjHZWISOGnmgEpEr74Ihg7YMwYeP75ICFQIiAiEhklA5LQtm+HBx+Ec88NZhqcPh3+/W+NHSAikhtqJpCEtXIltG4djCTYti289BIceGC8oxIRSTxKBiQhjRgR9A/Ytg0GDw6SAhERyRs1E0hC2bIFbroJLr8cjj0W5sxRIiAikl9KBiRhLFwYPDL4yitwxx0wbRocf3y8oxIRSXxqJpBCzx369YNbbgn6BIweDRdeGO+oRESKDtUMSKG2bl0wwdANN8Bpp8G8eUoERESiLebJgJk1MbPFZrbUzLpns72hma0zs7mh1wNh2/qb2Woz+yZLmUPNbKyZLQn9PCQW1yIF68svITU1mFzo8cfhs8/gyCPjHZWISNET02TAzFKAl4ELgWpASzOrls2uk909NfR6OGz9QKBJNvt3B8a5+wnAuNCyJKidO+HJJ+HMM4PlKVOge3copnosEZECEetfr/WApe6+zN0zgWHApZEWdvdJwG/ZbLoUeCP0/g3gsnzGKXHyv//BBRcEX/6XXx48LdCgQbyjEhEp2mKdDBwNrAxbzgity+pUM5tnZp+YWfUIjnu4u/8MEPpZPv+hSqyNGRNMNTx1KvTtC2+/DQcfHO+oRESKvlgnA9kNEutZlmcDldy9FvAiMCJqJzfraGbpZpa+Zs2aaB1W8ikzE7p1CzoGHn44pKcHHQY1pLCISGzEOhnIACqGLVcAVoXv4O7r3X1D6P1ooISZ5TTlzC9mdiRA6Ofq7HZy977unubuaeXKlcvrNUgULV0Kp58OzzwDXbrAjBlQLbteJCIiUmBinQzMBE4wsypmVhJoAYwK38HMjjAL/iY0s3qhGNfmcNxRQNvQ+7bAyKhGLQXirbegTp0gIXj//WAwof33j3dUIiLJJ6bJgLtvB7oCnwILgXfcfYGZdTazzqHdrgS+MbN5QG+ghbs7gJkNBaYDJ5pZhpl1CJV5AmhkZkuARqFlKaQ2bIB27YJhhGvWDMYOuOKKeEclIpK8LPQ9m3TS0tI8PT093mEknblz4V//giVL4L774IEHoLjGwRQRKXBmNsvd07Lbpie3JSbcoXfvYG6BDRtg/Hh4+GElAiIihYF+FUuB+/XXYLrhDz+EZs1gwAAom1OXUBERiRnVDEiBmjgRatWCTz+FF16AUaOUCIiIFDZKBqRAbN8e9Ac499xgpsEvvwxmHdTYASIihY+aCSTqfvwxeFJgyhS47jp48cUgIRARkcJJyYBE1fDh0KFDUDMweHCQFIiISOGmZgKJis2b4cYbg/ECjjsumGBIiYCISGJQMiD59u23wSOD//0v3HFHMNHQccfFOyoREYmUmgkkz9zh9dfh3/8O+gSMHh1MNiQiIolFNQOSJ3/8EYwk2LFjMNHQvHlKBEREEpWSAcm16dOhdu2gs+ATTwRjCBx5ZLyjEhGRvFIyIBHbuRMefxzOPDNYnjwZ7r4biulTJCKS0NRnQCLy88/Qpg2MGwdXXw2vvgoHHxzvqEREJBqUDEiOPvkE2rYNJhh67bVgHAGNJCgiUnSoglf2KjMzeFTwoovgiCMgPR2uv16JgIhIUaOaAcnW0qXQogXMmhUMJvTMM7D//vGOSkRECoKSAfmbwYOhSxcoUQI++AAuvzzeEYmISEFSM4HstmFDMLFQmzaQmgpz5yoREBFJBkoGBAjmEqhbFwYNCqYenjABjjkm3lGJiEgsKBlIcu7wwgvQoAFs3Ajjx8NDD0FxNSCJiCQN/cpPYmvWQLt28PHHcPHF0L8/lC0b76hERCTWVDOQpCZODPoFjB0LvXvDyJFKBEREkpWSgSSzfTvcfz+ce24w0+BXX8HNN2vsABGRZKZmgiSyYgW0bg1TpwZPDbz4YpAQiIhIclMykCQ++CAYRnjHDhgyBFq1indEIiJSWKiZoIjbvDkYQKh5czj++OARQiUCIiISTslAEbZgAdSrB336wJ13Bs0Dxx0X76hERKSwUTNBEeQezC54661Qpkww62CTJvGOSkRECivVDBQxf/wB//oXdOoEp58O8+YpERARkX1TMlCETJ8ejB0wfDg88QR8+mkw9bCIiMi+KBkoAnbsgMcegzPPDMYLmDIF7r4biulfV0REIqA+Awlu1Sq49loYNy5oHnj1VfjHP+IdlYiIJBIlAwls9Gho2zaYYOj116F9e40kKCIiuaeK5ASUmQl33AFNm8JRR8GsWcGAQkoEREQkL1QzkGCWLIGWLYME4Kab4JlnoFSpeEclIiKJTMlAAhk8OBhNsESJ4ImByy6Ld0QiIlIUqJkgAfz5Z9A3oE0bqF07GDtAiYCIiESLkoFCbvZsqFs3qBV48EEYPx4qVox3VCIiUpTEPBkwsyZmttjMlppZ92y2NzSzdWY2N/R6IKeyZlbLzKab2Xwz+9DMDorV9RQUd3j+eWjQADZtCpKAnj2huBp2REQkymKaDJhZCvAycCFQDWhpZtWy2XWyu6eGXg9HUPZ1oLu71wCGA90K+FIK1Jo1cPHFcNttwVDC8+bB2WfHOyoRESmqYl0zUA9Y6u7L3D0TGAZcGoWyJwKTQu/HAs2jGHNMTZgAtWrB2LHQuzeMHAmHHRbvqEREpCiLdTJwNLAybDkjtC6rU81snpl9YmbVIyj7DXBJ6P1VQMK1qm/fDvfdB+edBwcdBF99BTffrLEDRESk4MU6Gcjuq82zLM8GKrl7LeBFYEQEZdsDN5nZLKAMkJntyc06mlm6maWvWbMmt7EXmBUrgmaARx+Fdu2CMQRSU+MdlYiIJItYJwMZ7PlXewVgVfgO7r7e3TeE3o8GSphZ2X2VdfdF7t7Y3esCQ4Hvszu5u/d19zR3TytXrly0rilf3n8/+OKfPx/eegv69YMDDoh3VCIikkxinQzMBE4wsypmVhJoAYwK38HMjjALKsfNrF4oxrX7Kmtm5UM/iwH3AX1idD15tnkzdO4MV14JJ5wAc+YEIwuKiIjEWkwfVHP37WbWFfgUSAH6u/sCM+sc2t4HuBLoYmbbgc1AC3d3INuyoUO3NLObQu8/AAbE7qpyb8GCYIbBBQugWzd45BEoWTLeUYmISLKy4Hs2+aSlpXl6enpMz+kOffvCrbcGnQTffBMuuCCmIYiISJIys1nunpbdNo1AGCO//w5XXx00DZx5ZjB2gBIBEREpDJQMxMC0aUEnwREj4MknYcwYOOKIeEclIiISUDJQgHbsgMceg7POgpQUmDIF7roLiumui4hIIaKR7gvIqlXBLIPjx0OLFtCnD/zjH/GOSkRE5O+UDBSAjz+G664LJhjq1y8YSEgjCYqISGGlCuso2roVbr8dmjWDo46C9HRo316JgIiIFG6qGYiSJUuC5oDZs6FrV3j6aShVKt5RiYiI5EzJQBQMGgQ33hgMHDRiBFwa6TyMIiIihYCSgXxyh9GjoXZtGDIEKibcfIkiIpLslAzkkxm8/jrstx8U190UEZEEpK+vKNAsgyIiksj0NIGIiEiSUzIgIiKS5JQMiIiIJDklAyIiIklOyYCIiEiSUzIgIiKS5JQMiIiIJDklAyIiIklOyYCIiEiSUzIgIiKS5JQMiIiIJDklAyIiIknO3D3eMcSFma0BVkTxkGWBX6N4vESn+/EX3Ys96X7sSffjL7oXe4r2/ajk7uWy25C0yUC0mVm6u6fFO47CQvfjL7oXe9L92JPux190L/YUy/uhZgIREZEkp2RAREQkySkZiJ6+8Q6gkNH9+IvuxZ50P/ak+/EX3Ys9xex+qM+AiIhIklPNgIiISJJTMpADM2tiZovNbKmZdc9mu5lZ79D2r82sTqRlE1E+78dyM5tvZnPNLD22kReMCO7HSWY23cy2mtmduSmbaPJ5L5Lxs9E69H/kazObZma1Ii2biPJ5P4rU5yOCe3Fp6D7MNbN0Mzsj0rJ55u567eUFpADfA8cCJYF5QLUs+1wEfAIY0AD4KtKyifbKz/0IbVsOlI33dcT4fpQHTgEeBe7MTdlEeuXnXiTxZ+M04JDQ+wv1uyP7+1HUPh8R3osD+asZvyawqKA/G6oZ2Ld6wFJ3X+bumcAw4NIs+1wKvOmBL4GDzezICMsmmvzcj6Iox/vh7qvdfSawLbdlE0x+7kVRFMn9mObuv4cWvwQqRFo2AeXnfhQ1kdyLDR769gcOADzSsnmlZGDfjgZWhi1nhNZFsk8kZRNNfu4HBB/oz8xslpl1LLAoYyc//8ZF7fOR3+tJ9s9GB4IatbyUTQT5uR9QtD4fEd0LM7vczBYBHwPtc1M2L4pH4yBFmGWzLuvjF3vbJ5KyiSY/9wPgdHdfZWblgbFmtsjdJ0U1wtjKz79xUft85Pd6kvazYWbnEHz57WoXLmqfDcjf/YCi9fmI6F64+3BguJmdBfwHOD/SsnmhmoF9ywAqhi1XAFZFuE8kZRNNfu4H7r7r52pgOEGVVyLLz79xUft85Ot6kvWzYWY1gdeBS919bW7KJpj83I+i9vnI1b9vKOk5zszK5rZsrsS7M0VhfhHUnCwDqvBXZ43qWfZpyp4d5mZEWjbRXvm8HwcAZcLeTwOaxPuaCvp+hO3bkz07EBapz0c+70VSfjaAY4ClwGl5vZeJ8srn/ShSn48I78Xx/NWBsA7wU+h3aoF9NtRMsA/uvt3MugKfEvTi7O/uC8ysc2h7H2A0QQ/6pcAmoN2+ysbhMqImP/cDOJygyguCD/Rb7j4mxpcQVZHcDzM7AkgHDgJ2mtmtBL1/1xelz0d+7gXBzGxJ99kAHgAOA14JXft2d09L4t8d2d4PitjvjgjvRXPgWjPbBmwG/uVBZlBgnw2NQCgiIpLk1GdAREQkySkZEBERSXJKBkRERJKckgEREZEkp2RAREQkySkZEBERSXJKBkRERJKckgGRJGVmx5pZPzN7L2zdAaHJYJqFls80sz5m9rqZTYtHDHtbJyLRo2RAJEl5MA1qhyyr7wbeCdtnsrt3Bj4C3gAwsyPNbJiZpZvZd2Y2IbfnNrNXzez0SGLIbp2ZdTIzN7OqYesWmlnlvZyvn5k1zW2cIslCyYCIAGBm5wPfAr9ks7kVMDT0fhAwPDR07j+BW/JwuvoEc9bnGMNe4qoJzCWYCwMz249g2NoVezlfKsE47iKSDc1NICK7nEMwEUw1YLOZjXb3nWZ2DLAuNJ9CCtAQuGZXIXefv+u9mb1L8KWdSjC7WmugI8GkVZPdvUPor/nv3H1HJDHsZV0N4AmgM/AMUB1YGBq/HTP7J9Af+Efo5xHunpH/WyRSNCkZEElSZnYY8ChQ28zucfceofXXAb+6+87Qrh2AAQDuvsPMPgfmmdkI4E13nxp22BrAdHfvamYPA/0Ikoe1wC9mdiNwITAmFzH8bZ2ZVQNGAQ+Y2T9C550f2q84MBjo6u4zzOwVYFH07pxI0aOJikQkVyyYPu504BKgC9DG3UeYWSlgOXBU6Av7HmCHuz8VKvcTwfzrY4B2HpqjPg/nrwh87u4nmtnzBFPa1ge+d/dXzOxqoKm7tw3tfxNwrLvfkferFina1GdARHLFA1Pc/S5gGEH7PQRV9bPDahRqAV8BmFkFYBWwP3BwXhOBkJqEagEIpsxuSljNQGj7rLD966L+AiL7pGRARCJmZheYWcnQ+/LAGcDY0OYa7PmlWxP4OvS+Vuj9OUCunz7IIvyL/wvgTPZMENYCJ4dirAu0JOhsKCJ7oWRARHLjSmChmc0jeNzwfnefHtpWg9CXbqjJYH93/z20bVdisLu/QD7sTgbcfWvofaa7/xHaPghINbO5wF3AH8DCfJ5TpEhTnwERiRkzmw3Ud/dt8Y5FRP6iZEBERCTJqZlAREQkySkZEBERSXJKBkRERJKckgEREZEkp2RAREQkySkZEBERSXJKBkRERJKckgEREZEkp2RAREQkyf0fVosk63pIkj8AAAAASUVORK5CYII=\n", "text/plain": ["
"]}, "metadata": {"needs_background": "light"}, "output_type": "display_data"}], "source": ["# Question 1a\n", "\n", "# set figure size\n", "plt.figure(figsize=(8,6))\n", "# Plot data points\n", "plt.plot(Sm147_Nd144_ratio, Nd143_Nd144_ratio, 'ro', label=\"Data points\")\n", "# plot isochron by fitting a polynomial degree 1 - ie a straight line.\n", "poly_coeffs=np.polyfit(Sm147_Nd144_ratio, Nd143_Nd144_ratio, 1)\n", "p1 = np.poly1d(poly_coeffs)\n", "slope = poly_coeffs[0] # e^(\\lambda t) - 1\n", "y_intercept = poly_coeffs[1] # initial Nd143/Nd144 ratio\n", "x = np.linspace(0, 0.3, 10) # Sm147/Nd144 ratio\n", "plt.plot(x, p1(x), 'b', label=\"linear fit ($y = %gx + %g$)\" % (round_to_n_sf(slope, 3), round_to_n_sf(y_intercept, 3)))\n", "# label and title the plot\n", "plt.xlabel('${^{147}Sm}/{^{144}Nd}$')\n", "plt.ylabel('${^{143}Nd}/{^{144}Nd}$')\n", "plt.title('Isochron plot', fontsize=14)\n", "plt.legend(loc='best', fontsize=10)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["b) Now we have already got the slope ($m$) and y-intercept ($b$) of the isochron, so we can estimate the age ($t$) and initial $Nd$-$143/Nd$-$144$ ratio ($({^{143}Nd}/{^{144}Nd})_0$) since\n", "\n", "$$m = e^{\\lambda t} - 1 \\quad \\longrightarrow \\quad t = \\frac{1}{\\lambda}\\ln(m+1)$$\n", "$$b = \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_0$$"]}, {"cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["The age implied by the isochron is 3.20e+09 yr.\n", "The initial Nd-143/Nd-144 ratio is 0.508713.\n"]}], "source": ["# Question 1b\n", "\n", "decay_const_Sm = 6.54 * 10**-12 # yr^-1 # decay constant of Sm-147\n", "t = (1/decay_const_Sm) * np.log(slope + 1) # age\n", "b = y_intercept # initial Nd143/Nd144 ratio\n", "# print answers\n", "print(\"The age implied by the isochron is %.2e yr.\" % t)\n", "print(\"The initial Nd-143/Nd-144 ratio is %.6f.\" % b)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["As the distribution and evolution of $Sm$ and $Nd$ in the Earth are not explained in this page where we focus on quantitative parts, we will recall the concepts of CHUR and $\\epsilon_{Nd}$ from the lecture slide before attempting questions c and d.\n", "\n", "CHUR (Chondritic Uniform Reservoir) is defined by the average present-day $Sm/Nd$ ratio and $Nd$ isotope composition of chondritic meteorites. It is representative for the $Nd$ isotope composition and evolution of the bulk Earth and bulk silicate Earth. The present-day $Nd$ isotope composition of chondrites is $({^{143}Nd}/{^{144}Nd})_{CHUR} = 0.512638$. The average $({^{147}Sm}/{^{144}Nd})_{CHUR} = 0.1967$. \n", "\n", "It is convenient to consider past and present variations in $Nd$ isotope compositions relative to the isotopic evolution of CHUR. This is done using the $\\epsilon$ notation: \n", "\n", "$$\\epsilon_{Nd} = \\frac{({^{143}Nd}/{^{144}Nd}) - ({^{143}Nd}/{^{144}Nd})_{CHUR}}{({^{143}Nd}/{^{144}Nd})_{CHUR}} \\times 10^4$$\n", "\n", "$\\epsilon_{Nd}$ values denote relative differences in $Nd$ isotope compositions (relative to CHUR) in parts per $10,000$."]}, {"cell_type": "markdown", "metadata": {}, "source": ["c) From b), the age of the basalt is $3.20\\,Gyr$. So, in this question, we are going to calculate $({^{143}Nd}/{^{144}Nd})_{CHUR}$ at $3.20\\,Gya$ using the $Sm$-$Nd$ decay equation (very similar to $Rb$-$Sr$ decay equation):\n", "\n", "$$\\frac{^{143}Nd}{^{144}Nd} = \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_0 + \\frac{^{147}Sm}{^{144}Nd}(e^{\\lambda t} - 1)$$\n", "\n", "In this case:\n", "\n", "$\\quad {^{143}Nd}/{^{144}Nd} = ({^{143}Nd}/{^{144}Nd})_{CHUR,\\,present} = 0.512638$\n", "\n", "$\\quad ({^{143}Nd}/{^{144}Nd})_0 = ({^{143}Nd}/{^{144}Nd})_{CHUR,\\,3.20\\,Gyr} =$ haven't know yet\n", "\n", "$\\quad {^{147}Sm}/{^{144}Nd} = ({^{147}Sm}/{^{144}Nd})_{CHUR,\\,present} = 0.1967$\n", "\n", "$\\quad \\lambda = 6.54 \\times 10^{-12}\\,yr^{-1}$\n", "\n", "$\\quad t = 3.20 \\times 10^9\\,yr$"]}, {"cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["The Nd isotope ratio of CHUR at 3.20 Gya is 0.508478.\n"]}], "source": ["# Question 1c\n", "\n", "# Given values\n", "Nd143_Nd144_ratio = 0.512638\n", "Sm147_Nd144_ratio = 0.1967\n", "t = 3.2 * 10**9 # yr\n", "initial_Nd_ratio_CHUR = Sm_Nd_decay_equation(Nd143_Nd144_ratio, '?', Sm147_Nd144_ratio, t) # Nd isotope ratio of CHUR at 3.20 Gya\n", "# print answer\n", "print(\"The Nd isotope ratio of CHUR at 3.20 Gya is %g.\" % round_to_n_sf(initial_Nd_ratio_CHUR, 6))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["d) \n", "\n", "From b), the initial $Nd$ isotope ratio of the rock is $0.508713$.\n", "\n", "From c), the initial $Nd$ isotope ratio of CHUR is $0.508478$.\n", "\n", "Calculate the initial value of $\\epsilon_{Nd}$ (at $t = 3.20\\,Gyr$)"]}, {"cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["The initial differences in Nd isotope compositions (relative to CHUR) of the rock in parts per 10,000 is 4.62.\n"]}], "source": ["# function to calculate epsilon Nd\n", "def epsilon_Nd(Nd_ratio_rock, Nd_ratio_CHUR):\n", " return (Nd_ratio_rock - Nd_ratio_CHUR)/Nd_ratio_CHUR * 10000\n", "\n", "\n", "# Question 1d\n", "\n", "# Given values\n", "initial_Nd_ratio_rock = 0.508713\n", "initial_Nd_ratio_CHUR = 0.508478\n", "# print answer\n", "print(\"The initial differences in Nd isotope compositions (relative to CHUR) of the rock in parts per 10,000 is %g.\" \\\n", " % round_to_n_sf(epsilon_Nd(initial_Nd_ratio_rock, initial_Nd_ratio_CHUR), 3))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["The positive $\\epsilon_{Nd}$ initial value means that the rock comes from a depleted source. The depleted source is probably the lunar mantle, which was depleted by the formation of the lunar crust."]}, {"cell_type": "markdown", "metadata": {}, "source": ["### Question 2\n", "\n", "Neodymium model ages or crustal residence ages are obtained by calculating the intersection of the $Nd$ isotope evolution of a rock sample with the $Nd$ isotope evolution of a reservoir (RES) that is assumed to have a composition akin to CHUR (for $t_{CHUR}$) or a depleted mantle (DM) composition (for $t_{DM}$).\n", "\n", "The respective Nd isotope evolutions evolution curves are given by:\n", "\n", "Sample: \n", "\n", "$$\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Sam} = \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Sam,0} + \\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{Sam}(e^{\\lambda t} - 1)$$\n", "\n", "CHUR or DM reservoir: \n", "\n", "$$\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Res} = \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Res,0} + \\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{Res}(e^{\\lambda t} - 1)$$\n", "\n", "The intersections is obtained by equating $\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Sam,0} = \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Res,0}$\n", "\n", "After rearranging this yields for $t$:\n", "\n", "$$t = \\frac{1}{\\lambda} \\ln\\left[\\frac{\\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Sam} - \\left(\\frac{^{143}Nd}{^{144}Nd}\\right)_{Res}}{\\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{Sam} - \\left(\\frac{^{147}Sm}{^{144}Nd}\\right)_{Res}} + 1\\right]$$\n", "\n", "Calculate the $t_{DM}$ for a sample characterized by ${^{147}Sm}/{^{144}Nd} = 0.102$ and ${^{143}Nd}/{^{144}Nd} =\n", "0.511552$. The depleted mantle is assumed be characterized by ${^{147}Sm}/{^{144}Nd} = 0.222$ and ${^{143}Nd}/{^{144}Nd} = 0.513114$."]}, {"cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [{"name": "stdout", "output_type": "stream", "text": ["The Nd model age is 1.98e+09 yrs.\n"]}], "source": ["# Question 2 \n", "\n", "# decay constant of Sm-147\n", "decay_const = 6.54 * 10**-12 # yr^-1 \n", "# isotopic ratios of the sample\n", "Sm147_Nd144_ratio_sample = 0.102\n", "Nd143_Nd144_ratio_sample = 0.511552\n", "# isotopic ratios of the depleted mantle\n", "Sm147_Nd144_ratio_DM = 0.222\n", "Nd143_Nd144_ratio_DM = 0.513114\n", "# calculate t\n", "t = (1/decay_const) * np.log(((Nd143_Nd144_ratio_sample - Nd143_Nd144_ratio_DM)/(Sm147_Nd144_ratio_sample - Sm147_Nd144_ratio_DM)) + 1)\n", "# print answer\n", "print(\"The Nd model age is %.2e yrs.\" % t)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["## References\n", "\n", "- Lecture slide and Practical for Lecture 5 of the High-Temperature Geochemistry module"]}], "metadata": {"celltoolbar": "Tags", "kernelspec": {"display_name": "Python 3", "language": "python", "name": "python3"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 3}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.8"}}, "nbformat": 4, "nbformat_minor": 4}