91Ó°ÊÓ

$$ \psi_{h}(t, m)=-h m+q(t)+r(t) m^{2}+s(t) m^{4}+u(t) m^{6} $$ with \(u(t)\) a fixed positive constant. Minimize \(\psi\) with respect to the variable \(m\) and examine the spontaneous magnetization \(m_{0}\) as a function of the parameters \(r\) and \(s\). In particular, show the following: \({ }^{21}\) (a) For \(r>0\) and \(s>-(3 u r)^{1 / 2}, m_{0}=0\) is the only real solution. (b) For \(r>0\) and \(-(4 u r)^{1 / 2}0\) and \(s=-(4 u r)^{1 / 2}, m_{0}=0\) or \(\pm(r / u)^{1 / 4}\). Now, the minimum of \(\psi\) at \(m_{0}=0\) is of the same height as the ones at \(m_{0}=\pm(r / u)^{1 / 4}\), so a nonzero spontaneous magnetization is as likely to occur as the zero one. (d) For \(r>0\) and \(s<-(4 u r)^{1 / 2}, m_{0}=\pm m_{1}\) - which implies a first-order phase transition (because the two possible states available here differ by a finite amount in \(m\) ). The line \(s=-(4 u r)^{1 / 2}\), with \(r\) positive, is generally referred to as a "line of first-order phase transitions." (e) For \(r=0\) and \(s<0, m_{0}=\pm(2|s| / 3 u)^{1 / 2}\). (f) For \(r<0, m_{0}=\pm m_{1}\) for all \(s\). As \(r \rightarrow 0, m_{1} \rightarrow 0\) if \(s\) is positive. (g) For \(r=0\) and \(s>0, m_{0}=0\) is only solution. Combining this result with (f), we conclude that the line \(r=0\), with \(s\) positive, is a "line of second- order phase transitions," for the two states available here differ by a vanishing amount in \(m\). The lines of first-order phase transitions and second-order phase transitions meet at the point \((r=0, s=0)\), which is commonly referred to as a tricritical point (Griffiths, 1970).

Step by step solution

01

Minimise \(\psi\) with respect to \(m\)

The function is minimum when its derivative is equal to zero. Hence, differentiate \(\psi_{h}(t, m)\) with respect to \(m\), equate it to zero and solve for \(m\) to find the solutions that minimize \(\psi_{h}(t, m)\). Note that \(m_{0}\) is the spontaneous magnetization i.e. the minimum of \(\psi\) with respect to \(m\).

02

Analyze for \(r>0\) and \(s>-(3 u r)^{1 / 2}\)

Substitute these conditions into the solutions obtained in Step 1 to obtain the specific solutions that satisfy these conditions and show that \(m_{0}=0\) is the only real solution.

03

Analyze for \(r>0\) and \(-(4 u r)^{1 / 2}

Substitute these conditions and show that \(m_{0}=0\) or \(\pm m_{1}\) are the solutions. Also, prove that the minimum of \(\psi\) at \(m_{0}=0\) is lower than the minima at \(m_{0}=\pm m_{1}\), so \(m_{0}\) should be 0.

04

Analyze for \(r>0\) and \(s=-(4 u r)^{1 / 2}\)

Repeat the analysis of Step 3 for these specific conditions to show that \(m_{0}=0\) or \(\pm(r / u)^{1 / 4}\). Also, prove that the minimum of \(\psi\) at \(m_{0}=0\) is as likely to occur as the non-zero ones.

05

Continue the analysis for remaining conditions

Continue substituting the conditions as given in the problem to solve for \(m_{0}\) for all the respective cases as in the earlier steps.

06

Understand the phase transition implications

Understand and explain the implications of these solutions, in terms of physical concepts such as first and second order phase transitions and the concept of a tricritical point.

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Key Concepts

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

Spontaneous Magnetization

Spontaneous magnetization is a concept in physics where a material, like a ferromagnet, becomes magnetized without an external magnetic field. This happens because of the alignment of individual magnetic moments in the material due to thermal agitation reducing at lower temperatures. In the context of the minimization function described, spontaneous magnetization is represented by the variable \( m_0 \), which is found at the minimum of the potential \( \psi_h(t, m) \).
If \( m_0 = 0 \), the system is in its unmagnetized state. If \( m_0 eq 0 \), spontaneous magnetization occurs, indicating that the system has adopted a stable alignment of magnetic moments even in the absence of an external field. This can be dependent on parameters such as \( r \) and \( s \) as shown in different regions of the phase diagram.
Here, we note the importance of these parameters, as they not only define whether spontaneous magnetization occurs, but also help to understand the behavior of the system near critical regions.

First-order Phase Transition

A first-order phase transition involves a sudden change in physical properties, such as density or magnetization, during the transition between phases. In this discussion, it happens when the parameter conditions like \( r > 0 \) and \( s < -(4ur)^{1/2} \) lead to solutions of \( m_0 = \pm m_1 \). This indicates a significant change in magnetization (\( m \)) without gradual transformation.
The system jumping between states indicates a discontinuity in the order parameter, in this case, the magnetization, which corresponds with the importance of \( m \). During this transition, properties like latent heat can be observed, as the material absorbs or releases energy to rearrange the magnetic moments.
Such transitions tend to have a clear boundary or line in the phase diagram, known as the line of first-order phase transitions. This behavior shows the sharp distinction between different states of the system under certain conditions.

Second-order Phase Transition

Unlike first-order transitions, second-order phase transitions occur more smoothly, without drastic changes like latent heat. These transitions are marked by a continuous change in the order parameter, magnetization here, identified by the conditions where \( r = 0 \) and \( s > 0 \). This means the system's properties vary softly across the transition.
In these conditions, the magnetic moments align gradually, causing the magnetization \( m_0 \) to change more continuously as opposed to a sudden flip. Hence, the solutions show \( m_0 = 0 \) is the sole possibility, indicating the system smoothly traverses from one phase to another as temperature changes.
Second-order transitions are critical for understanding phenomena like critical points, where critical fluctuations occur. Such transitions are characterized by parameters that shift the system's balance softly, often graphed without a visible phase boundary.

Tricritical Point

A tricritical point represents a unique location on a phase diagram where lines of first-order and second-order phase transitions converge. At the point where \( r = 0 \) and \( s = 0 \), both types of transitions might be observed, granting distinct critical behaviors.
The presence of a tricritical point indicates an overlap where both the discontinuous and continuous transitions can coexist, proving essential for theoretical physics in mapping complex phase diagrams. This point helps in defining regions of stability and instability among different phases.
Understanding the tricritical point is vital because it offers insights into phenomena where a slight modification in parameters can shift the nature of a transition. As such, it represents a crossover between different phases and their distinct types of transitions, simplifying the transition landscape for both theoreticians and experimental observers.