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Chapter 5: Problem 9

Heat capacity of a solid Debye's theory of solids gives the heat capacity of a solid at temperature \(T\) to be $$ C_{V}=9 V \rho k_{B}\left(\frac{T}{\theta_{D}}\right)^{3} \int_{0}^{\otimes_{D} / T} \frac{x^{4} \mathrm{e}^{x}}{\left(e^{x}-1\right)^{2}} \mathrm{~d} x $$ where \(V\) is the volume of the solid, \(\rho\) is the number density of atoms, \(k_{B}\) is Boltzmann's constant, and \(\theta_{D}\) is the so-called Debye temperature, a property of solids that depends on their density and speed of sound. a) Write a Python function \(\mathrm{cv}(\mathrm{T})\) that calculates \(C_{V}\) for a given value of the temperature, for a sample consisting of 1000 cubic centimeters of solid aluminum, which has a number density of \(\rho=6.022 \times 10^{28} \mathrm{~m}^{-3}\) and a Debye temperature of \(\theta_{D}=428 \mathrm{~K}\). Use Gaussian quadrature to evaluate the integral, with \(N=50\) sample points. b) Use your function to make a graph of the heat capacity as a function of temperature from \(T=5 \mathrm{~K}\) to \(T=500 \mathrm{~K}\).

Short Answer

Expert verified
Define constants and an integrand function. Write a function to compute heat capacity. Plot heat capacity vs temperature using the defined function.

Step by step solution

01

Understand the Given Formula

The heat capacity of a solid is given by the formula: \[C_{V}=9 V \rho k_{B}\left(\frac{T}{\theta_{D}}\right)^{3} \int_{0}^{\theta_{D} / T} \frac{x^{4} \,\mathrm{e}^{x}}{\left(\mathrm{e}^{x}-1\right)^{2}} \, \mathrm{d} x \]where \(V\) is the volume of the solid, \(\rho\) is the number density of atoms, \(k_{B}\) is Boltzmann's constant, and \(\theta_{D}\) is the Debye temperature.
02

Set Up Known Constants

For a sample of 1000 cubic centimeters of solid aluminum, the constants are:- Volume, \(V = 1000 \times 10^{-6} = 0.001\) cubic meters- Number density, \(\rho = 6.022 \times 10^{28} \, \mathrm{m}^{-3}\)- Debye temperature, \(\theta_{D} = 428 \, \mathrm{K}\)- Boltzmann's constant, \(k_{B} = 1.38064852 \times 10^{-23} \, \mathrm{J/K}\)
03

Import Required Python Libraries

Use NumPy for numerical operations and SciPy for Gaussian quadrature integration:```pythonimport numpy as npimport scipy.integrate as integrateimport matplotlib.pyplot as plt```
04

Define the Integrand Function

The integrand of the given formula is:\(f(x) = \frac{x^{4}\mathrm{e}^{x}}{(\mathrm{e}^{x} - 1)^{2}}\)This can be defined in Python as:```pythondef integrand(x): return (x**4 * np.exp(x)) / ((np.exp(x) - 1)**2)```
05

Define the Heat Capacity Function

Write the function \( cv(T) \) to compute heat capacity:```pythondef cv(T): V = 0.001 # volume in cubic meters rho = 6.022e28 # number density in m^-3 k_B = 1.38064852e-23 # Boltzmann's constant in J/K theta_D = 428 # Debye temperature in K integrand_upper_limit = theta_D / T integral, _ = integrate.fixed_quad(integrand, 0, integrand_upper_limit, n=50) return 9 * V * rho * k_B * (T / theta_D)**3 * integral```
06

Generate Temperature Range

Create an array of temperature values from 5K to 500K:```pythontemperatures = np.linspace(5, 500, 100)```
07

Calculate Heat Capacities

Apply the \( cv(T) \) function to the temperature range to calculate heat capacities:```pythonheat_capacities = [cv(T) for T in temperatures]```
08

Plot the Heat Capacity vs Temperature Graph

Plot the graph using Matplotlib:```pythonplt.plot(temperatures, heat_capacities)plt.xlabel('Temperature (K)')plt.ylabel('Heat Capacity (J/K)')plt.title('Heat Capacity vs Temperature for Solid Aluminum')plt.show()```

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Debye Theory
Debye theory is a fundamental concept in solid-state physics. It explains the heat capacity of solids based on the vibrational modes of atoms within the solid. Instead of treating atoms individually, Debye theory considers the collective motion of atoms as vibrational waves or phonons. This theory is crucial for understanding how specific heat changes with temperature.
At low temperatures, the heat capacity of a solid is proportional to \(T^3\), while at higher temperatures, it approaches a constant value. This temperature dependence matches experimental observations and is successfully explained by the Debye model.
Important parameters in Debye theory include the volume of the solid (V), the number density of atoms (\(\rho\)), Boltzmann's constant (\(k_B\)), and the Debye temperature (\(\theta_D\)). The Debye temperature is particularly significant, as it characterizes the highest vibrational mode within the solid.
Gaussian Quadrature
Gaussian quadrature is a numerical method used to approximate definite integrals, especially when they cannot be solved analytically. This method is highly efficient because it provides accurate results using fewer sample points compared to other numerical integration techniques.
In the context of the given problem, Gaussian quadrature is utilized to evaluate the integral part of the heat capacity formula. By choosing an appropriate number of sample points (N=50), we can achieve precise results for the integral:
  • \(f(x) = \frac{x^{4}\text{e}^{x}}{(\text{e}^{x} - 1)^{2}}\)
  • Integration limits are from 0 to \(\theta_D/T\)
Using the SciPy library in Python, we can easily implement Gaussian quadrature to solve the integral efficiently.
Python Programming
Python is a versatile and powerful programming language widely used for scientific computing. In this exercise, we harness Python's capabilities to calculate the heat capacity of a solid. Here's a step-by-step breakdown:
First, we import necessary libraries:
import numpy as np
import scipy.integrate as integrate
import matplotlib.pyplot as plt

Next, we define the integrand function based on the given formula:
def integrand(x):
   return (x**4 * np.exp(x)) / ((np.exp(x) - 1)**2)
We then write the main function to compute heat capacity:
def cv(T):
   V = 0.001  # volume in cubic meters
   rho = 6.022e28  # number density in m^-3
   k_B = 1.38064852e-23  # Boltzmann's constant in J/K
   theta_D = 428  # Debye temperature in K
   integrand_upper_limit = theta_D / T
   integral, _ = integrate.fixed_quad(integrand, 0, integrand_upper_limit, n=50)
   return 9 * V * rho * k_B * (T / theta_D)**3 * integral
Numerical Integration
Numerical integration is a key concept when analytical solutions are difficult or impossible to obtain. It includes various techniques to approximate the value of integrals using computational methods.
In this exercise, we specifically use Gaussian quadrature within the Python programming language to perform numerical integration. This provides an efficient and accurate way to calculate the integral within the heat capacity formula:
integral, _ = integrate.fixed_quad(integrand, 0, integrand_upper_limit, n=50)

Gaussian quadrature involves weighting function values at specific points (nodes) within the integration domain. The result is a weighted sum that approximates the integral's value. By selecting N=50 points, we ensure the balance between accuracy and computational efficiency.
Thermodynamic Properties
Thermodynamic properties are crucial for understanding and predicting the behavior of materials under various conditions. The heat capacity of a solid, as highlighted in this exercise, is one such property.
  • It measures the amount of heat required to raise the temperature of a given quantity of the material by one degree.
  • Also, It plays a vital role in applications involving thermal management and material science.
Through the lens of the Debye theory, we gain insight into how the vibrational modes of atoms within a solid contribute to its specific heat.
By programming a function in Python to calculate and graph the heat capacity of aluminum, we not only apply theoretical concepts but also understand practical computational tools in thermodynamics.

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Most popular questions from this chapter

Electric tield of a charge distribution: Suppose we have a distribution of charges and we want to calculate the resulting electric field. One way to do this is to first calculate the electric potential \(\phi\) and then take its gradient. For a point charge \(q\) at the origin, the electric potential at a distance \(r\) from the origin is \(\phi=q / 4 \pi \epsilon_{0} r\) and the electric field is \(\mathbf{E}=-\nabla \phi\). a) You have two charges, of \(\pm 1 \mathrm{C}, 10 \mathrm{~cm}\) apart. Calculate the resulting electric potential on a \(1 \mathrm{~m} \times 1 \mathrm{~m}\) square plane surrounding the charges and passing through them. Calculate the potential at \(1 \mathrm{~cm}\) spaced points in a grid and make a visualization on the screen of the potential using a density plot. b) Now calculate the partial derivatives of the potential with respect to \(x\) and \(y\) and hence find the electric field in the \(x y\) plane. Make a visualization of the field also. This is a little trickier than visualizing the potential, because the electric field has both magnitude and direction. One way to do it might be to make two density plots, one for the magnitude, and one for the direction, the latter using the "hsv"

5.17 The gamma function: A commonly occurring function in physics calculations is the gamma function \(\Gamma(a)\), which is defined by the integral $$ \Gamma(a)=\int_{0}^{\infty} x^{n-1} \mathrm{e}^{-x} \mathrm{~d} x . $$ There is no closed-form expression for the gamma function, but one can calculate its value for given \(a\) by performing the integral above numerically. You have to be careful how you do it, however, if you wish to get an accurate answer. a) Write a program to make a graph of the value of the integrand \(x^{n-1} \mathrm{e}^{-x}\) as a function of \(x\) from \(x=0\) to \(x=5\), with three separate curves for \(a=2,3\), and 4 , all on the same axes. You should find that the integrand starts at zero, rises to a maximum, and then decays again for each curve. b) Show analytically that the maximum falls at \(x=a-1\). c) Most of the area under the integrand falls near the maximum, so to get an accurate value of the gamma function we need to do a good job of this part of the integral. We can change the integral from 0 to \(\infty\) to one over a finite range from 0 to 1 using the change of variables in Eq. (5.67), but this tends to squash the peak towards the edge of the \([0,1]\) range and does a poor job of evaluating the integral accurately. We can do a better job by making a different change of variables that puts the peak in the middle of the integration range, around \(\frac{1}{2}\). We will use the change of variables given in Eq. (5.69), which we repeat here for convenience: $$ z=\frac{x}{c+x} . $$ For what value of \(x\) does this change of variables give \(z=\frac{1}{2}\) ? Hence what is the appropriate choice of the parameter \(c\) that puts the peak of the integrand for the gamma function at \(z=\frac{1}{2}\) ? d) Before we can calculate the gamma function, there is another detail we need to attend to. The integrand \(x^{n-1} \mathrm{e}^{-x}\) can be difficult to evaluate because the factor \(x^{2-1}\) can become very large and the factor \(\mathrm{e}^{-x}\) very small, causing numerical overflow or underflow, or both, for some values of \(x\). Write \(x^{n-1}=\mathrm{e}^{(a-1) \ln x}\) to derive an alternative expression for the integrand that does not suffer from these problems (or at least not so much). Explain why your new expression is better than the old one. e) Now, using the change of variables above and the value of \(c\) you have chosen, write a user-defined function gamma (a) to calculate the gamma function for arbitrary argument \(a\). Use whatever integration method you feel is appropriate. Test your function by using it to calculate and print the value of \(\Gamma\left(\frac{3}{2}\right)\), which is known to be equal to \(\frac{1}{2} \sqrt{\pi} \simeq 0.886\). f) For integer values of \(a\) it can be shown that \(\Gamma(a)\) is equal to the factorial of \(a-\) 1. Use your Python function to calculate \(\Gamma(3), \Gamma(6)\), and \(\Gamma(10)\). You should get answers closely equal to \(2 !=2,5 !=120\), and \(9 !=362880\).

The Stefan-Boltzmann constant The Planck theory of thermal radiation tells us that in the (angular) frequency interval \(\omega\) to \(\omega+d \omega\), a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to \(I(\omega)\) d \(\omega\), where $$ I(\omega)=\frac{\hbar}{4 \pi^{2} c^{2}} \frac{\omega^{3}}{\left(\mathrm{e}^{\hbar \omega / k_{a} T}-1\right)} $$ Here \(h\) is Planck's constant over \(2 \pi, c\) is the speed of \(h\) ght, and \(k_{B}\) is Boltzmann's constant. a) Show that the total rate at which energy is radiated by a black body per unit area, over all frequencies, is $$ W=\frac{k_{B}^{4} T^{4}}{4 \pi^{2} c^{2} h^{3}} \int_{0}^{\infty} \frac{x^{3}}{\mathrm{e}^{x}-1} \mathrm{~d} x . $$ b) Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is. c) Even before Planck gave his theory of thermal radiation around the turn of the 20 th century, it was known that the total energy \(W\) given off by a black body per unit area per second followed Stefan's law: \(W=\sigma T^{4}\), where \(\sigma\) is the StefanBoltzmann constant. Use your value for the integral above to compute a value for the Stefan-Boltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.

Diffraction gratings: Light with wavelength \(\lambda\) is incident on a diffraction grating of total width \(w\), gets diffracted, is focused with a lens of focal length \(f\), and falls on a screen: Theory tells us that the intensity of the diffraction pattern on the screen, a distance \(x\) from the central axis of the system, is given by $$ I(x)=\left|\int_{-w / 2}^{w / 2} \sqrt{q(u)} \mathrm{e}^{i 2 \pi x u / \lambda f} \mathrm{~d} u\right|^{2} $$ where \(q(u)\) is the intensity transmission function of the diffraction grating at a distance \(u\) from the central axis, i.e., the fraction of the incident light that the grating lets through. a) Consider a grating with transmission function \(q(u)=\sin ^{2} \alpha u\). What is the separation of the "slits" in this grating, expressed in terms of \(\alpha\) ? b) Write a Python function \(q(u)\) that returns the transmission function \(q(u)=\sin ^{2} \alpha u\) as above at position \(u\) for a grating whose slits have separation \(20 \mu \mathrm{m}\). c) Use your function in a program to calculate and graph the intensity of the diffraction pattern produced by such a grating having ten slits in total, if the incident light has wavelength \(\lambda=500 \mathrm{~nm}\). Assume the lens has a focal length of 1 meter and the screen is \(10 \mathrm{~cm}\) wide. You can use whatever method you think appropriate for doing the integral. Once you've made your choice you'll also need to decide the number of sample points you'll use. What criteria play into this decision? Notice that the integrand in the equation for \(I(x)\) is complex, so you will have to use complex variables in your program. As mentioned in Section 2.2.5, there is a version of the math package for use with complex variables called cmath. In particular you may find the exp function from cmath useful because it can calculate the exponentials of complex arguments. d) Create a visualization of how the diffraction pattern would look on the screen using a density plot (see Section 3.3). Your plot should look something like this: e) Modify your program further to make pictures of the diffraction patterns produced by gratings with the following profiles: i) A transmission profile that obeys \(q(u)=\sin ^{2} \alpha d \sin ^{2} \beta u\), with \(\alpha\) as before and the same total grating width \(w\), and \(\beta=\frac{1}{2} \alpha\). ii) Two "square" slits, meaning slits with \(100 \%\) transmission through the slit and \(0 \%\) transmission everywhere else. Calculate the diffraction pattern for non-identical slits, one \(10 \mu \mathrm{m}\) wide and the other \(20 \mu \mathrm{m}\) wide, with a \(60 \mu \mathrm{m}\) gap between the two.

Create a user-defined function \(f(x)\) that returns the value \(1+\frac{1}{2} \tanh 2 x\), then use a central difference to calculate the derivative of the function in the range \(-2 \leq x \leq 2\). Calculate an analytic formula for the derivative and make a graph with your numerical result and the analytic answer on the same plot. It may help to plot the exact answer as lines and the numerical one as dots. (Hint: In Python the tanh function is found in the math package, and it's called simply tanh.)
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