mirror of
https://github.com/CoolProp/CoolProp.git
synced 2026-01-15 00:48:18 -05:00
293 lines
23 KiB
Plaintext
293 lines
23 KiB
Plaintext
{
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"metadata": {
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"name": ""
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},
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"nbformat": 3,
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"nbformat_minor": 0,
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"worksheets": [
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{
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"cells": [
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{
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"cell_type": "heading",
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"level": 1,
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"metadata": {},
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"source": [
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"Derivation and Conversion of Planck-Einstein Terms"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Start with a generalized Helmholtz energy from \n",
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"$$\\alpha^0 = a_k\\log\\left[c_k + d_k\\exp\\left(\\frac{b_k\\tau}{T_c}\\right)\\right]$$\n",
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"\n",
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"The ideal-gas specific heat can be obtained from \n",
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"$$\\frac{c_p^0}{R} = -\\tau^2\\alpha^0_{\\tau\\tau}$$\n",
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"\n",
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"After some math shown below (and some manual grouping):\n",
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"$$\\frac{c_p^0}{R} = - \\frac{a_{k} (b_{k}\\tau/T_c)^{2} c_k d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}}{ \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)^2} $$\n",
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"\n",
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"Two important identities:\n",
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"$$ \\left(\\frac{1}{\\cosh x}\\right) = \\frac{2e^{-x}}{1+e^{-2x}}$$\n",
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"$$ \\left(\\frac{1}{\\sinh x}\\right) = \\frac{2e^{-x}}{1-e^{-2x}}$$\n",
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"which yield\n",
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"$$ \\left(\\frac{x}{\\cosh x}\\right)^2 = \\frac{4x^2e^{-2x}}{(1+e^{-2x})^2} = \\frac{(-2x)^2e^{-2x}}{(1+e^{-2x})^2}$$\n",
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"$$ \\left(\\frac{x}{\\sinh x}\\right)^2 = \\frac{4x^2e^{-2x}}{(1-e^{-2x})^2} = \\frac{(-2x)^2e^{-2x}}{(1-e^{-2x})^2}$$"
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]
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"from __future__ import division\n",
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"from sympy import *\n",
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"init_session(quiet=True)\n",
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"init_printing()"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"IPython console for SymPy 0.7.5 (Python 2.7.2-32-bit) (ground types: python)\n"
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]
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}
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],
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"prompt_number": 4
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},
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{
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"cell_type": "code",
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"collapsed": false,
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"input": [
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"a_k,b_k,c_k,d_k,tau,T_c = symbols('a_k,b_k,c_k,d_k,tau,T_c',real=True)\n",
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"alpha0 = a_k*log(c_k + d_k*exp(b_k*tau/T_c))\n",
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"print 'No derivatives'\n",
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"display(alpha0)\n",
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"print 'First partial w.r.t. tau'\n",
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"display(diff(alpha0, tau, 1).simplify())\n",
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"print 'Second partial w.r.t. tau'\n",
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"display(diff(alpha0, tau, 2).simplify())\n",
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"print latex(diff(alpha0, tau, 2).simplify()*(-tau**2))\n",
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"print 'Third partial w.r.t. tau'\n",
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"display(diff(alpha0, tau, 3).simplify())"
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],
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"language": "python",
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"metadata": {},
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"outputs": [
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"No derivatives\n"
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]
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},
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{
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"latex": [
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"$$a_{k} \\log{\\left (c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}} \\right )}$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAoBAMAAAChsh0EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNK64R/yf8j+av+zHorG6yAAAAABJRU5ErkJggg==\n",
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"text": [
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" \u239b b_k\u22c5\u03c4\u239e\n",
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" \u239c \u2500\u2500\u2500\u2500\u2500\u239f\n",
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" \u239c T_c \u239f\n",
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"a_k\u22c5log\u239dc_k + d_k\u22c5\u212f \u23a0"
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]
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},
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"First partial w.r.t. tau\n"
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]
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},
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{
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"latex": [
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"$$\\frac{a_{k} b_{k} d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c} \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)}$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAABMBAMAAACIdNgoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nm+3fsAlYAAAAAElFTkSuQmCC\n",
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"text": [
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" b_k\u22c5\u03c4 \n",
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" \u2500\u2500\u2500\u2500\u2500 \n",
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" T_c \n",
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" a_k\u22c5b_k\u22c5d_k\u22c5\u212f \n",
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"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
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" \u239b b_k\u22c5\u03c4\u239e\n",
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" \u239c \u2500\u2500\u2500\u2500\u2500\u239f\n",
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" \u239c T_c \u239f\n",
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"T_c\u22c5\u239dc_k + d_k\u22c5\u212f \u23a0"
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]
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},
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"Second partial w.r.t. tau\n"
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]
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},
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{
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"latex": [
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"$$\\frac{a_{k} b_{k}^{2} c_{k} d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c}^{2} \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)^{2}}$$"
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],
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"metadata": {},
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"output_type": "display_data",
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"png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAABQBAMAAAB/pCPzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNFxuru+P1H8Bv/Dea92wCuIAAAAASUVORK5CYII=\n",
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"text": [
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" b_k\u22c5\u03c4 \n",
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" \u2500\u2500\u2500\u2500\u2500 \n",
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" 2 T_c \n",
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"a_k\u22c5b_k \u22c5c_k\u22c5d_k\u22c5\u212f \n",
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"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
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" 2\n",
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" \u239b b_k\u22c5\u03c4\u239e \n",
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" \u239c \u2500\u2500\u2500\u2500\u2500\u239f \n",
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" 2 \u239c T_c \u239f \n",
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"T_c \u22c5\u239dc_k + d_k\u22c5\u212f \u23a0 "
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]
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},
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{
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"output_type": "stream",
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"stream": "stdout",
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"text": [
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"- \\frac{a_{k} b_{k}^{2} c_{k} d_{k} \\tau^{2} e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c}^{2} \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)^{2}}\n",
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"Third partial w.r.t. tau\n"
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]
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},
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{
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"latex": [
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"$$\\frac{a_{k} b_{k}^{3} c_{k} d_{k} \\left(c_{k} - d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right) e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c}^{3} \\left(c_{k}^{3} + 3 c_{k}^{2} d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}} + 3 c_{k} d_{k}^{2} e^{\\frac{2 b_{k}}{T_{c}} \\tau} + d_{k}^{3} e^{\\frac{3 b_{k}}{T_{c}} \\tau}\\right)}$$"
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],
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"metadata": {},
|
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"output_type": "display_data",
|
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"png": "iVBORw0KGgoAAAANSUhEUgAAAZYAAABUBAMAAABTgQyuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
|
|
"text": [
|
|
" \u239b b_k\u22c5\u03c4\u239e b_k\u22c5\u03c4 \n",
|
|
" \u239c \u2500\u2500\u2500\u2500\u2500\u239f \u2500\u2500\u2500\u2500\u2500 \n",
|
|
" 3 \u239c T_c \u239f T_c \n",
|
|
" a_k\u22c5b_k \u22c5c_k\u22c5d_k\u22c5\u239dc_k - d_k\u22c5\u212f \u23a0\u22c5\u212f \n",
|
|
"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
|
|
" \u239b b_k\u22c5\u03c4 2\u22c5b_k\u22c5\u03c4 3\u22c5b_k\u22c5\u03c4\u239e\n",
|
|
" \u239c \u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n",
|
|
" 3 \u239c 3 2 T_c 2 T_c 3 T_c \u239f\n",
|
|
"T_c \u22c5\u239dc_k + 3\u22c5c_k \u22c5d_k\u22c5\u212f + 3\u22c5c_k\u22c5d_k \u22c5\u212f + d_k \u22c5\u212f \u23a0"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 16
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Aly-Lee starts as\n",
|
|
"$$\\frac{c_p^0}{R_u} = A + B\\left(\\frac{C/T}{\\sinh(C/T)}\\right)^2 + D\\left(\\frac{E/T}{\\cosh(E/T)}\\right)^2$$\n",
|
|
"and generalized Plank-Einstein term is given by\n",
|
|
"$$\\frac{c_p^0}{R_u} = - \\frac{a_{k} b_{k}^{2} c_{k} d_{k} \\tau^{2} e^{\\frac{b_{k} \\tau}{T_{c}}}}{T_{c}^{2} \\left(c_{k} + d_{k} e^{\\frac{b_{k} \\tau}{T_{c}}}\\right)^{2}}$$\n",
|
|
"\n",
|
|
"Constant is separated out, and handled separately. sinh part can be expanded as\n",
|
|
"$$B\\left(\\frac{C/T}{\\sinh(C/T)}\\right)^2 = \\frac{B(-2C/T)^2\\exp(-2C/T)}{(1-\\exp(-2C/T))^2}$$\n",
|
|
"where\n",
|
|
"$$a_k = B$$\n",
|
|
"$$b_k = -2C$$\n",
|
|
"$$c_k = 1$$\n",
|
|
"$$d_k = -1$$\n",
|
|
"\n",
|
|
"cosh part can be expanded as\n",
|
|
"$$D\\left(\\frac{E/T}{\\cosh(E/T)}\\right)^2 = \\frac{D(-2E/T)^2\\exp(-2E/T)}{(1+\\exp(-2E/T))^2}$$\n",
|
|
"where\n",
|
|
"$$a_k = -D$$\n",
|
|
"$$b_k = -2E$$\n",
|
|
"$$c_k = 1$$\n",
|
|
"$$d_k = 1$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"collapsed": false,
|
|
"input": [
|
|
"B, D, T = symbols('B, D, T', real = True, positive = True)\n",
|
|
"# From Aly-Lee:\n",
|
|
"term1 = ((D/T)/sinh(D/T))**2\n",
|
|
"# Planck Einstein\n",
|
|
"term2 = (B/T)**2*exp(B/T)/(exp(B/T)-1)**2\n",
|
|
"diff1 = term2 - term1\n",
|
|
"display(diff1.rewrite(exp).subs(B, 2*D).simplify())\n",
|
|
"term2 = (2*D/T)**2*exp(2*D/T)/(exp(2*D/T)-1)**2\n",
|
|
"display(term2)\n",
|
|
"print 'which is the same as the Aly-Lee sinh term with a_k = C, b_k = 2*D, c_k = -1, d_k = 1'\n",
|
|
"\n",
|
|
"B, F, T = symbols('B, F, T', real = True, positive = True)\n",
|
|
"# From Aly-Lee:\n",
|
|
"term1 = ((F/T)/cosh(F/T))**2\n",
|
|
"# Planck Einstein\n",
|
|
"term2 = (B/T)**2*exp(-B/T)/(exp(-B/T)+1)**2\n",
|
|
"diff2 = term2 - term1\n",
|
|
"display(diff2.rewrite(exp).subs(B, 2*F).simplify())\n",
|
|
"# So Aly-Lee cosh term can be directly converted to Plank-Einstein2 term with b_k = 2*F, and c=1"
|
|
],
|
|
"language": "python",
|
|
"metadata": {},
|
|
"outputs": [
|
|
{
|
|
"latex": [
|
|
"$$0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
|
|
"text": [
|
|
"0"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$\\frac{4 D^{2} e^{\\frac{2 D}{T}}}{T^{2} \\left(e^{\\frac{2 D}{T}} - 1\\right)^{2}}$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAABLBAMAAACsFlnLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAEZ0lEQVRYCe1YXYgbVRT+spufSSZ/KIj6kog/BUWMqFREMfRFii4NC4X2oeyKiqJi\nB5Qi+pBBUBQRIwjionQoVTEihoUudKFuwCIIlV1KK4rGjeybL13buurWOp77N5PJ3EkG42Mv5J5z\nvu+7Z+7czL05EyBmu/6ohant9s8x5VpZvVhDooWspWXjgS3zElJV5Jx48gHVkV0dFeVrKHSRq6o4\nrs1ZhZbSHreQtpDsqjiunbYSf0it+SOwBFwdd6inS/cz/8hg8WUHJ5F5yONGOlcRe8CdOTPDvoL8\nJal90K3l32+3veUZmQJ3M/ovwPib7HRjtFjPph4nPLFN3S30OSJFJVc1CYw0L75LdIotwzkbRnWk\nNop0WI7kJnVNB7/iF09nLHxx64IXjXIMi+Uo1ak7t5b44NTDwPLeHWzEFK7BMeaMbcfAclTYUs45\nadfdRHEnXmLD8ngLXzJnbHN4jrkyCdf7XH2oa94ghp0VZlyfL/Mce5huS4i/e3VfR3h3CjOuXwbP\ncZF0xp9CfFmNMdlCx2jf9npbZ5H4naRLdaHfVsOKLeWNtfSMpX4j1W6pfJpi7n5639ixSkD3kaVZ\nb1QlsB8m/24VH8Medb8vHL7cOyO/C5rEza/HGHZFcmUFolZAHZP/3cbciVET+F/w3E23TZznE/ww\ncY6dqFiTJnkOzf6kOYDVzuQ57tGmMGwtHARPyDCiLtofVOsjdT6/oqWLz2jhYfAAB4pOsTzMULyk\nwESbip472p/3NSI6tm0GX3fqpG5NT6shSWyQCHhPAQGb4QXSQdcNoCJIqOIJWax0MAu0NSqCenqY\nodmqz30F/OSXQj7OvUr4JtTuaVqe1qS7eBJ4xAMCTrYRCFmgds+NPnOiA3MeiZqPDHrG/GDEfbV7\nnvAYc8cy8nW84E/Mo5jjVa7kbs2sX+wdduTuybBfa9Gm3Qv4ZqH9mYp9m+0wn37LZaMXgXQNU7bc\nPcamIqJtZg/P4Z8cBar/qjD6AN89UzVvrHfCDqU1d63zHHOe8lrg7S5SHflWMV33mGhnjudYtZSi\nT7VbB0VA7J5SVRFkoypSkaPSHZA+xny5e0oNFpgfzbKLRFWkIseKzaSiZc4zK3ePIDasjENYVEUq\ncojLsbE0Y/Z+ALl7mjb5mWexWGYg9BWpzOFwCe8K3i6jkN9k4am9bwrer0iP38XaOxyWOapCw3p6\nPPxWsckvrUkgoiIN56DHw2/8XkqOBNSJ5/PcC98LPR5+W2mQn3YAZhFRkcocNpOIJp5cGfAppFpY\ntBStsSJHs6uo3G73Q35NAaTXmJ3dZ4tQ299+8AFGV/pall6gWhFEGF7thDGO5OYjiDD8cRgSiPc+\nGyXw8RnfDXqmfGkJotooWhmZfTiP90/AMAG8Fob0SC569Ve6+iEhNG2HIAXQ8xWvfT1C9ugIbpC6\ndzAY8t8YiiNCY9R8k42IUUH4kBWMA5HpnzsBPBiY9wfjoej5UVdQ2qStPK3NlLVwEBz88+BfPbFP\nimrffr4AAAAASUVORK5CYII=\n",
|
|
"text": [
|
|
" 2\u22c5D \n",
|
|
" \u2500\u2500\u2500 \n",
|
|
" 2 T \n",
|
|
" 4\u22c5D \u22c5\u212f \n",
|
|
"\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
|
|
" 2\n",
|
|
" \u239b 2\u22c5D \u239e \n",
|
|
" \u239c \u2500\u2500\u2500 \u239f \n",
|
|
" 2 \u239c T \u239f \n",
|
|
"T \u22c5\u239d\u212f - 1\u23a0 "
|
|
]
|
|
},
|
|
{
|
|
"output_type": "stream",
|
|
"stream": "stdout",
|
|
"text": [
|
|
"which is the same as the Aly-Lee sinh term with b_k = 2*D\n"
|
|
]
|
|
},
|
|
{
|
|
"latex": [
|
|
"$$0$$"
|
|
],
|
|
"metadata": {},
|
|
"output_type": "display_data",
|
|
"png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB5wID+0cG/gsMfN8Z5BUY+L4wzDdg\nYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxylQEA9FISlFfRJtkAAAAASUVORK5C\nYII=\n",
|
|
"text": [
|
|
"0"
|
|
]
|
|
}
|
|
],
|
|
"prompt_number": 10
|
|
}
|
|
],
|
|
"metadata": {}
|
|
}
|
|
]
|
|
} |