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Chapter 2: Problem 12

A lattice contains \(N\) normal lattice sites and \(N\) interstitial lattice sites. The lattice sites are all distinguishable. \(N\) identical atoms sit on the lattice, \(M\) on the interstitial sites, and \(N-M\) on the normal sites ( \(N \gg M>1\) ). If an atom occupies a normal site, it has energy \(E=0\). If an atom occupies an interstitial site, it has energy \(E=\varepsilon\). Compute the internal energy and heat capacity as a function of temperature for this lattice.

Short Answer

Expert verified
Internal energy: \( U = M \varepsilon \frac{e^{-\varepsilon/(k_BT)}}{1+e^{-\varepsilon/(k_BT)}} \). Heat capacity: \( C = M \frac{\varepsilon^2}{k_BT^2} \frac{e^{-\varepsilon/(k_BT)}}{(1+e^{-\varepsilon/(k_BT)})^2} \).

Step by step solution

01

Understand the System

We have a lattice with two types of sites: normal sites and interstitial sites. There are N normal lattice sites and N interstitial lattice sites, both of which are distinguishable. We have N identical atoms where M atoms occupy the interstitial sites and N-M atoms occupy the normal sites.
02

Identify Energies

Atoms on normal sites have energy E=0, and atoms on interstitial sites have energy E=\(\varepsilon\).
03

Calculate Total Internal Energy

The total internal energy U is the sum of the energies of all atoms. As atoms on normal sites contribute zero energy and M atoms on interstitial sites contribute an energy of \(\varepsilon\): \[ U = M \varepsilon \]
04

Evaluate Partition Function

To compute the heat capacity, the partition function Z is needed. For each interstitial site, the canonical partition function considering both available states (energy 0 and \(\varepsilon\)) is: \[ Z = 1 + e^{-\varepsilon/(k_BT)} \]
05

Compute Average Energy per Atom

The average energy per atom on interstitial sites considering the probability distribution is: \[\langle E \rangle = \varepsilon \frac{e^{-\varepsilon/(k_BT)}}{Z} = \varepsilon \frac{e^{-\varepsilon/(k_BT)}}{1+e^{-\varepsilon/(k_BT)}} \] For M atoms, the total energy is then: \[ U = M \varepsilon \frac{e^{-\varepsilon/(k_BT)}}{1+e^{-\varepsilon/(k_BT)}} \]
06

Calculate Heat Capacity

Heat capacity can be computed by differentiating the internal energy with respect to the temperature T: \[ C = \frac{\partial U}{\partial T} \] Given the internal energy expression: \[ C = M \varepsilon \frac{\partial}{\partial T} \left( \frac{e^{-\varepsilon/(k_BT)}}{1+e^{-\varepsilon/(k_BT)}} \right) \] Applying the chain rule and simplifying: \[ C = M \varepsilon \left( -\frac{\varepsilon}{k_BT^2} \right) \frac{1}{(1+e^{-\varepsilon/(k_BT)})^2} \] This simplifies to: \[ C = M \frac{\varepsilon^2}{k_BT^2} \frac{e^{-\varepsilon/(k_BT)}}{(1+e^{-\varepsilon/(k_BT)})^2} \]

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Key Concepts

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

Lattice Sites
In this problem, we deal with a lattice that has two types of sites: normal and interstitial. Both types are distinguishable and each has exactly N sites. Think of these sites as 'places' where atoms can sit. Normal sites are just that—normal, with zero energy. Interstitial sites are special spots that give the atom an energy of \(\truevarepsilon\). Atoms can move between these sites, and our aim is to understand the energy properties of this system.
Partition Function
The partition function is a key concept in statistical mechanics. It's a sum of all possible states of a system, weighted by their energy and the temperature of the system. Here, the partition function Z for atoms choosing between a normal site (energy 0) and an interstitial site (energy \(\truevarepsilon\)) is: \[ Z = 1 + e^{-\truevarepsilon/(k_B T)} \] This equation means we are considering the relative probability of an atom being in either of the two possible energy states based on temperature T.
Heat Capacity Calculation
Heat capacity measures how much energy the lattice can store up as its temperature changes. We begin by finding the internal energy U. For M atoms on interstitial sites, the internal energy U is: \[ U = M \varepsilon \frac{e^{-\varepsilon/(k_B T)}}{1+ e^{-\varepsilon/(k_B T)}} \] To get the heat capacity C, we differentiate U with respect to T: \[ C = \frac{\truepartial U}{\truepartial T} \] Leading to: \[ C = M \varepsilon \left( -\frac{\varepsilon}{k_B T^2}\right) \frac{1}{(1+e^{-\truevarepsilon/(k_B T)})^2}\]Simplifying, we obtain: \[ C = M \frac{\varepsilon^2}{k_B T^2} \frac{e^{-\truevarepsilon/(k_B T)}}{(1+ e^{-\varepsilon/(k_B T)})^2 \]
Internal Energy
Internal energy (U) is the total energy of the atoms in the lattice. At normal sites, atoms contribute zero energy. At interstitial sites, atoms contribute an energy \(\truevarepsilon\). Total internal energy for all M atoms on interstitial sites is given by: \[ U = M \varepsilon \frac{e^{-\varepsilon/(k_B T)}}{1+ e^{-\varepsilon/(k_B T)}} \] This formula combines the energy contributions from all states an atom can be in, weighted by their respective probabilities.

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Most popular questions from this chapter

Consider a lattice with \(N\) spin- 1 atoms with magnetic moment \(\mu\). Each atom can be in one of three spin states, \(S_{z}=-1,0,+1\). Let \(n_{-1}, n_{0}\), and \(n_{1}\) denote the respective number of atoms in each of those spin states. Find the total entropy and the configuration which maximizes the total entropy. What is the maximum entropy? (Assume that no magnetic field is present, so all atoms have the same energy. Also assume that atoms on different lattice sites cannot be exchanged, so they are distinguishable.)

In how many ways can five red balls, four blue balls, and four white balls be placed in a row so that the balls at the ends of the row are the same color?

Find the number of permutations of the letters in the word, MONOTONOUS. In how many ways are four O's together? In how many ways are (only) 3 O's together?

A system consists of \(N\) noninteracting, distinguishable two-level atoms. Each atom can exist in one of two energy states, \(E_{0}=0\) or \(E_{1}=\varepsilon\). The number of atoms in energy level, \(E_{0}\), is \(n_{0}\) and the number of atoms in energy level, \(E_{1}\), is \(n_{1}\). The internal energy of this system is \(U=n_{0} E_{0}+n_{1} E_{1}\). (a) Compute the multiplicity of microscopic states. (b) Compute the entropy of this system as a function of internal energy. (c) Compute the temperature of this system. Under what conditions can it be negative? (d) Compute the heat capacity for a fixed number of atoms, \(N\).

A deck of cards contains 52 cards, divided equally among four suits, Spades (S), Clubs (C), Diamonds (D), and Hearts (H). Each suit has 13 cards which are designated: \((2,3,4,5,6,7,8,9\), \(10, \mathrm{~J}, \mathrm{Q}, \mathrm{K}, \mathrm{A}\) ). Assume that the deck is always well shuffled so it is equally likely to receive any card in the deck, when a card is dealt. (a) If a dealt hand consists of five cards, how many different hands can one be dealt (assume the cards in the hand can be received in any order)? (b) If the game is poker, what is the probability of being dealt a Royal Flush (10, J, Q, K, and A all of one suit)? (c) If one is dealt a hand with seven cards, and the first four cards are spades, what is the probability of receiving, at least one additional spade?
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