In the mean field approximation to the Ising lattice, the order parameter \((s\rangle\), satisfies the equation \((s)=\tanh \left(\langle s) T_{\mathrm{c}} / T\right)\), where \(T_{\mathrm{c}}=v \varepsilon / 2 k_{\mathrm{B}}\) with \(\varepsilon\) the strength of the coupling between lattice sites and \(v\) the number of nearest neighbors. (a) Show that s has the following temperature dependence: (i) \(\langle s\rangle \approx 1-2 e^{-2 T_{c} / T}\) if \(T \sim 0 \mathrm{~K}\), and (ii) \(\langle s\rangle \approx \sqrt{3\left(1-T / T_{\mathrm{c}}\right)}\) if \(T \sim T_{c^{\prime}}\) (b) Compute the jump in the heat capacity at \(T=T_{c^{-}}\)(c) Compute the magnetic susceptibility, \(X_{T, N}(B=0)\), in the neighborhood of \(T=T_{c}\) for both \(T>T_{\mathrm{c}}\) and \(T

Step by step solution

01

Understand the given order parameter equation

Start with the provided equation for the order parameter: \[ \langle s \rangle = \tanh \left( \langle s \rangle \frac{T_c}{T} \right) \]where \(T_c\) is given as \( T_{\text{c}} = \frac{v \varepsilon}{2 k_{\text{B}}} \), and \( \varepsilon \) is the coupling strength between lattice sites, \( v \) is the number of nearest neighbors.

02

Expand \( \tanh \) for small \( T \) (Part a(i))

Consider the limit \( T \to 0 \). For small arguments, the \( \tanh x \approx x \), assuming \( \langle s \rangle \approx 1 \). Substitute this into the order parameter equation to get: \[ \langle s \rangle = \tanh \left( \frac{ T_c}{T} \right) \approx 1 - 2 e^{-2 T_c / T} \]Thus, proving the expression.

03

Approximate for temperatures near \(T_c\) (Part a(ii))

Now, consider the limit \( T \approx T_c \). Near \( T_c \), we can use the approximation \( \tanh x \approx x - \frac{x^3}{3} \). Substitute into the order parameter equation: \[ \langle s \rangle = \frac{T_c}{T} \langle s \rangle - \frac{1}{3} \left( \frac{T_c}{T} \langle s \rangle \right)^3 \] Simplify to find: \[ \langle s \rangle^2 \approx 3 \left( 1 - \frac{T}{T_c} \right) \] Hence, \[ \langle s \rangle \approx \sqrt{3 \left( 1 - \frac{T}{T_c} \right)} \]

04

Calculate the jump in heat capacity (Part b)

The heat capacity jump at the critical temperature \( T_c \) can be found using the change in energy. At \( T_c \), the jump in specific heat \( C \) is proportional to the derivative of the internal energy with respect to \( T \). The internal energy has a form near \( T_c \):\[ E \propto \left( 1 - \frac{T}{T_c} \right)^2 \] The discontinuity in heat capacity is: \[ \Delta C = - T_c \frac{d}{dT} E \approx 2 T_c \ln(1 - \frac{T}{T_c}) \] Compute this limit.

05

Derive magnetic susceptibility (Part c)

Magnetic susceptibility can be derived from the free energy near \( T_c \). For \( T > T_c \) and \( T < T_c \):\[ \chi_T = - \frac{\partial^2 F}{\partial B^2} \large|_{B=0} \] For temperatures just above \( T_c \),\[ \chi_T \propto \left( \frac{1}{T - T_c} \right) \] and just below \( T_c \), \[\chi_T \propto - \left( \sqrt{3 \left( 1 - \frac{T}{T_c} \right)} \right)^2 \] Identify and calculate the exponents.

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Key Concepts

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

Ising Model

The Ising Model is a mathematical model used in statistical mechanics to understand phase transitions, such as the transition from a magnetized to a non-magnetized state in ferromagnetic materials. It considers a lattice of spins that can be either up (+1) or down (-1). Nearest neighbors interact with each other with a coupling strength, \(\varepsilon\), which tries to align them. The model is crucial for explaining critical phenomena, where small changes in temperature induce significant changes in the system's properties.

Order Parameter

The order parameter quantifies the degree of order across a phase transition. In the Ising Model, the order parameter, \(\langle s \rangle \), represents the average spin value. It varies from high values (near +1 or -1) at low temperatures, indicating ordered phases, to zero at high temperatures in disordered phases. Represented mathematically, the order parameter helps in understanding how a system transitions between different phases and provides insights about the symmetry properties of these phases.

Critical Temperature

The critical temperature, \(\ T_c \ \), is the temperature at which a phase transition occurs. For the Ising Model, it is the temperature where the lattice transforms from the magnetized (ordered) phase to the disordered phase. It is given by \(\ T_{\text{c}} = \frac{\text{v} \varepsilon }{2 k_{\text{B}}} \), where \(\ k_{\text{B}} \ \) is Boltzmann constant, \(\ v \ \) is number of nearest neighbors, and \(\ \varepsilon \ \) is the coupling strength. Identifying \(\ T_c \ \) is essential for understanding at what temperature significant structural changes occur in the material.

Heat Capacity

Heat capacity indicates how much heat energy is required to change a system's temperature. Near the critical temperature, the heat capacity shows a 'jump' due to the latent heat associated with the phase transition. For the Ising Model, this jump can be calculated by analyzing the change in internal energy. It states that the specific heat capacity, \(\ C \ \), is proportional to the derivative of the internal energy with respect to temperature, reflecting sudden energy changes at the critical point.

Magnetic Susceptibility

Magnetic susceptibility, \(\ \chi \ \), measures how much a material will become magnetized in an applied magnetic field. In the vicinity of the critical temperature \(\ \ T_c \ \), the susceptibility diverges, meaning the system becomes infinitely sensitive to the magnetic field. For temperatures above \(\ \ T_c \ \), \(\ \chi \ \) typically grows as \(\ \ \chi \propto 1/(T - T_c) \), while below \(\ \ T_c \ \), \(\ \ \chi \propto \left( 1 - \frac{T}{ T_c} \right)^{- \gamma} \ \). This growth near \(\ \ T_c \ \) is a crucial characteristic of phase transitions.

Critical Exponent

Critical exponents describe the behavior of physical quantities near the critical point. They are a key aspect of critical phenomena theory. In our context, critical exponents define how the order parameter, heat capacity, and susceptibility diverge near \(\ T_c \ \). For example, susceptibility \(\ \ \chi \ \) typically has a critical exponent \(\ \gamma \ \ \) such that \(\ \ \chi \propto (T - T_c)^{-\frac {1}{2}} \ \). These exponents are universal, meaning they depend only on the system's dimensionality and symmetry rather than its microscopic details.