In [1]:
%pylab inline
#import matplotlib.pyplot as plt

#matplotlib.rcParams['text.usetex'] = True

from scipy.integrate import * # az integráló rutinok betöltése

import warnings
warnings.filterwarnings('ignore')

Populating the interactive namespace from numpy and matplotlib

The mean square displacement in a three-dimensional crystal at finite temperature using the Debye model:

\begin{eqnarray} \langle\mathbf{u}^2(\mathbf{R})\rangle &=& \frac{9 \hbar^2}{4 k_{\mathrm{B}}\Theta_\mathrm{D}M}\, {\left(\frac{T}{\Theta_\mathrm{D}}\right)}^2\, \int_0^{\Theta_\mathrm{D}/T}\, x \left[\frac{1}{e^x-1}+\frac{1}{2}\right]\, dx \nonumber \\ &=& \alpha^2 a^2 \frac{\Theta_\mathrm{D}}{T_m}\, {\left(\frac{T}{\Theta_\mathrm{D}}\right)}^2\,\int_0^{\Theta_\mathrm{D}/T}\, x \left[\frac{1}{e^x-1}+\frac{1}{2}\right]\, dx, \nonumber \end{eqnarray}

where $a$ is the lattice constant, $T_m$ is the melting temperature and $\alpha \approx 1/4$ according to Lindemann criterion for melting.

For high temperature ($T \gg \Theta_\mathrm{D}$):

$\langle\mathbf{u}^2(\mathbf{R})\rangle \approx \frac{9 \hbar^2}{k_{\mathrm{B}}\Theta_\mathrm{D}M}\, \frac{T}{\Theta_{\mathrm{D}}}\, \left[1+\frac{1}{36} {\left(\frac{\Theta_{\mathrm{D}}}{T}\right)}^2 + \dots\right].$

For low temperature ($T \ll \Theta_\mathrm{D}$):

$\langle\mathbf{u}^2(\mathbf{R})\rangle \approx \frac{9 \hbar^2}{4 k_{\mathrm{B}}\Theta_\mathrm{D}M}\, \left[1+\frac{2\pi^2}{3} {\left(\frac{T}{\Theta_{\mathrm{D}}}\right)}^2 + \dots\right].$

See Eq. (12.3.9) in Jenő Sólyom: Fundamentals of the Physics of Solids, Volume 1: Structure and Dynamics (A modern szilárdtestfizika alpajai I. A szilárd testek szerkezete és dinamikája) see here on page 412.

In [2]:
# Az abra kimentesehez az alabbiakat a plt.show()  ele kell tenni!!! 

#savefig('fig_rainbow_p2_1ray.pdf');  # Abra kimentese
#savefig('fig_rainbow_p2_1ray.eps');  # Abra kimentese

# Abra es fontmeretek
xfig_meret= 12   #    12 nagy abrahoz
yfig_meret= 8   #   12 nagy abrahoz
xyticks_meret= 15  #  20 nagy abrahoz
xylabel_meret= 20  #  30 nagy abrahoz
legend_meret= 20   #  30 nagy abrahoz
In [3]:
def u2(t):
    res = t*(1/(exp(t)-1)+1/2)
    return res
In [23]:
Np = 100

Tmax=2.7
pici = 0.001

T = linspace(pici,Tmax,Np)
Tl = linspace(pici,Tmax/5,Np)
Tg = linspace(0.4,Tmax)
u2Tg = Tg*(1+1/Tg/Tg/36)
u2Tl = 1/4*(1+2*pi**2/3*Tl*Tl)

u2int=[]
for tt in T:
    u2int.append(tt*tt*quad(u2,pici,1/tt)[0])

#label1=r"$\langle\mathbf{u}^2(\mathbf{R})\rangle \approx \frac{T}{\Theta_{\mathrm{D}}} \
#\left[1+\frac{1}{36} {\left(\frac{\Theta_{\mathrm{D}}}{T}\right)}^2 + \dots\right], T \gg \Theta_\mathrm{D}$"

#label2=r'$\langle\mathbf{u}^2(\mathbf{R})\rangle \approx \
#                          \frac{1}{4}\left[1+\frac{2\pi^2}{3} {\left(\frac{T}{\Theta_{\mathrm{D}}}\right)}^2 + \
#                          \dots\right], T \ll \Theta_\mathrm{D}$'

label1=r'$\langle\mathbf{u}^2(\mathbf{R})\rangle$'+ '  for  ' + r'$T \gg \Theta_\mathrm{D}$'
label2=r'$\langle\mathbf{u}^2(\mathbf{R})\rangle$'+ '  for  ' + r'$T \ll \Theta_\mathrm{D}$'


figsize(xfig_meret,yfig_meret)
plot (T, u2int,color='r')
plot (Tg,u2Tg, label=label1)
plot (Tl,u2Tl, color='g', label=label2)

xlabel(r'$T/\Theta_{\mathrm{D}}$',fontsize=xylabel_meret)
ylabel(r'$\langle\mathbf{u}^2(\mathbf{R})\rangle$',fontsize=xylabel_meret)

legend(loc='upper left',fontsize=legend_meret)

cim= 'The mean square displacement in units of  ' + r'$\frac{9 \hbar^2}{k_{\mathrm{B}}\Theta_\mathrm{D}M}$' +"\n"
title(cim,fontsize=20)


xlim(0,Tmax)

grid();

#savefig('Fig_Debye_fajho.pdf');  # Abra kimentese

r$\langle\mathbf{u}^2(\mathbf{R})\rangle \approx \frac{T}{\Theta_{\mathrm{D}}} \ \left[1+\frac{1}{36} {\left(\frac{\Theta_{\mathrm{D}}}{T}\right)}^2 + \dots\right], T \gg \Theta_\mathrm{D}$

In [ ]: