The molar free energy of a spin system can be written ϕ(T, H)=ϕ0(T) -(1)/(2) J m^2 +(1)

The molar free energy of a spin system can be written ϕ(T,...

The molar free energy of a spin system can be written $$ \begin{aligned} \phi(T, H)=\phi_{0}(T) &-\frac{1}{2} J m^{2} \\ &+\frac{1}{2} k_{\mathrm{B}} T[(1+m) \ln (1+m)+(1-m) \ln (1-m)]-m H \end{aligned} $$ where $J$ is the interaction strength, $m$ is the net magnetization per mole, $\phi_{0}(T)$ is the molar free energy in the absence of a net magnetization, $H$ is an applied magnetic field, $k_{\mathrm{B}}$ is Boltzmann's constant, and $T$ is the temperature. (a) Compute the critical temperature (called the Curie temperature). (b) Compute the linear magnetic susceptibility of this system. (Hint: Only consider temperatures in the neighborhood of the critical point where $m$ is small.)

The molar free energy of a spin system can be written
$$
\begin{aligned}
\phi(T, H)=\phi_{0}(T) &-\frac{1}{2} J m^{2} \\
&+\frac{1}{2} k_{\mathrm{B}} T[(1+m) \ln (1+m)+(1-m) \ln (1-m)]-m H
\end{aligned}
$$
where $J$ is the interaction strength, $m$ is the net magnetization per mole, $\phi_{0}(T)$ is the molar free energy in the absence of a net magnetization, $H$ is an applied magnetic field, $k_{\mathrm{B}}$ is Boltzmann's constant, and $T$ is the temperature. (a) Compute the critical temperature (called the Curie temperature). (b) Compute the linear magnetic susceptibility of this system. (Hint: Only consider temperatures in the neighborhood of the critical point where $m$ is small.)

Key Concepts

Mean Field Approximation

The mean field approximation simplifies many-body problems by replacing interactions between particles with an average or 'mean' effect. In the context of magnetic systems, it allows one to compute thermodynamic properties such as the free energy and critical temperature by averaging over the interactions between spins. This approximation is instrumental in deriving analytical results and understanding the basic physics of phase transitions in complex systems.

Free Energy Expansion and Landau Theory

Expanding the free energy in powers of a small order parameter, as done in Landau theory, is a powerful method to analyze phase transitions. This approach allows one to identify the coefficients of the expansion that change sign at the critical point, thereby determining the onset of spontaneous order. It is widely used to study second-order (continuous) phase transitions and to derive expressions for quantities such as the critical temperature and the linear response coefficients like magnetic susceptibility.

Curie Temperature

The Curie temperature is the critical temperature at which a ferromagnetic material undergoes a phase transition to a paramagnetic state. At temperatures below the Curie temperature, the system exhibits spontaneous magnetization, while above this temperature the thermal agitation overcomes the magnetic ordering. This concept is fundamental in understanding how magnetic ordering in materials changes as a function of temperature and plays a crucial role in the study of phase transitions.

Magnetic Susceptibility

Magnetic susceptibility is a measure of how much a material becomes magnetized in response to an applied magnetic field. Near the critical point or phase transition, the susceptibility often shows divergent or enhanced behavior due to the increased fluctuations of the magnetic order parameter. It provides key insights into the stability of magnetic phases and the nature of the phase transition.

Transcript

00:01 Consider a simple thermodynamic system in which particle can occupy only two states which are lower and whose energy is defined as zero and the upper state with the energy, eu.

00:28 First, carry out the sum with only two states.

00:34 Integration is certainly not valid giving the particle sum which is giving the particles average energy or we can say average particle energy e bar and plot the result as a function of temperature plot as a function of temperature for b explain qualitatively why it should behave as a does.

01:14 And for c part, this system can be used as a model of paramagnetism where individual atoms, magnetic moments can either be aligned or unaligned with an external magnetic field, giving a low or high energy respectively.

01:30 Then describe how average alignment or anti -alignment, alignment depends on temperature.

01:47 Depends on temperature.

01:50 Does it make sense in order to find the average particle energy e bar for two particles with the lower energy and upper energy the equation is given by summation of e n exponential raise to the power e n by k b t from n and submission of e raised to the power minus e n by kbt.

02:25 Here kb is the boozman constant t is the temperature, en is the energy of aneth state or nth level.

02:34 Now for average energy of the two particles the sum is used using 0 for lower n and u for the upper n.

02:44 So the energy will be given by e not exponential raised to the power e0 by kb t plus e u e raised to the power e u by k b t divided by e raise to the power minus e not by k b into t plus e raise to the power minus e u by k b t then use that the value of e not is zero so we have the equation 0 into e -raise to the power minus e0 by kb t this term will become 1 plus eu e -raise to the power minus e u by kb t divided by e -raise to the power minus 0 by kb t plus e raise to the power minus e u by kb t this term will be simplified as eu, e raised to the power minus eu by kvt divided by 1 plus e raise to the power minus eu by kvt.

04:07 This is the average energy of two particles...

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