A system consisting of three spins in a line, each having s=(1)/(2), is coupled by nearest neigh

step 1 So if you consider the high temperature, KC, KC has to be less than or equal to E.

A system consisting of three spins in a line, each having...

A system consisting of three spins in a line, each having $s=\frac{1}{2}$, is coupled by nearest neighbor interactions (see Fig. 2.29). Fig. 2.29. Each spin has a magnetic moment pointing in the same direction as the spin, $\boldsymbol{\mu}=2 \mu \mathrm{s}$. The system is placed in an external magnetic field $H$ in the $z$ direction and is in thermal equilibrium at temperature $T$. The Hamiltonian for the system is approximated by an Ising model, where the true spin-spin interaction is replaced by a term of the form $J S_z(i) S_z(i+1)$ : $$ H=J S_z(1) S_z(2)+J S_z(2) S_z(3)-2 \mu H\left[S_z(1)+S_z(2)+S_z(3)\right], $$ where $J$ and $\mu$ are positive constants. (a) List each of the possible microscopic states of the system and its energy. Sketch the energy level diagram as a function of $H$. Indicate any degeneracies. (b) For each of the following conditions, write down the limiting values of the internal energy $U(T, H)$, the entropy $S(T, H)$, and the magnetization $M(T, H)$. 1) $T=0$ and $H=0$, 2) $T=0$ and $0<H \ll J / \mu$, 3) $T=0$ and $J / \mu \ll H$, 4) $J \ll k T$ and $H=0$. (c) On the basis of simple physical considerations, without doing any calculations, sketch the specific heat at constant field, $C_H(T, H)$ when $H=$ 0 . What is the primary temperature dependence at very high and very low $T$ ? (d) Find a closed form expression for the partition function $Q(T, H)$. (e) Find the magnetization $M(T, H)$. Find an approximate expression for $M(T, H)$ which is valid when $k T \gg \mu H$ or $J$.

A system consisting of three spins in a line, each having $s=\frac{1}{2}$, is coupled by nearest neighbor interactions (see Fig. 2.29).
Fig. 2.29.

Each spin has a magnetic moment pointing in the same direction as the spin, $\boldsymbol{\mu}=2 \mu \mathrm{s}$. The system is placed in an external magnetic field $H$ in the $z$ direction and is in thermal equilibrium at temperature $T$. The Hamiltonian for the system is approximated by an Ising model, where the true spin-spin interaction is replaced by a term of the form $J S_z(i) S_z(i+1)$ :
$$
H=J S_z(1) S_z(2)+J S_z(2) S_z(3)-2 \mu H\left[S_z(1)+S_z(2)+S_z(3)\right],
$$
where $J$ and $\mu$ are positive constants.
(a) List each of the possible microscopic states of the system and its energy. Sketch the energy level diagram as a function of $H$. Indicate any degeneracies.
(b) For each of the following conditions, write down the limiting values of the internal energy $U(T, H)$, the entropy $S(T, H)$, and the magnetization $M(T, H)$.
1) $T=0$ and $H=0$,
2) $T=0$ and $0<H \ll J / \mu$,
3) $T=0$ and $J / \mu \ll H$,
4) $J \ll k T$ and $H=0$.
(c) On the basis of simple physical considerations, without doing any calculations, sketch the specific heat at constant field, $C_H(T, H)$ when $H=$ 0 . What is the primary temperature dependence at very high and very low $T$ ?
(d) Find a closed form expression for the partition function $Q(T, H)$.
(e) Find the magnetization $M(T, H)$. Find an approximate expression for $M(T, H)$ which is valid when $k T \gg \mu H$ or $J$.

Key Concepts

Specific Heat

Specific heat at constant field measures how the internal energy of a system changes with temperature. It reveals crucial information about energy fluctuations and phase transitions by indicating how different energy levels become populated as temperature varies, offering insights into both low-temperature and high-temperature regimes.

Partition Function

The partition function is a central quantity in statistical mechanics that sums over the exponential factors of the energy eigenvalues weighted by temperature. It encodes the full thermodynamic information of the system and is used to derive other important quantities like internal energy, entropy, and magnetization.

Degeneracy

Degeneracy is the concept where different microscopic configurations share the same energy level. Recognizing degeneracies is important because they directly affect the entropy and the statistical weight of each energy level, influencing the overall thermodynamic properties.

Microscopic States and Energy Levels

Microscopic states refer to the distinct configurations that each spin can adopt, which collectively determine the overall energy of the system. Analyzing energy levels and their arrangement, including degeneracies, helps in understanding the system's behavior under varying conditions such as different temperatures and fields.

Thermodynamic Limits

Thermodynamic limits involve analyzing the behavior of physical quantities such as internal energy, entropy, and magnetization under extreme conditions like low and high temperatures or weak and strong magnetic fields. These limits provide insights into phase transitions, ground state properties, and thermal excitation effects.

Magnetic Field Effects

The presence of an external magnetic field affects the energy levels of magnetic spins by aligning or opposing the magnetic moments. This coupling leads to shifts in energy and influences thermodynamic properties like magnetization, making it essential to include in modeling magnetic systems.

Spin 1/2 Systems

Spin 1/2 systems are quantum systems where the intrinsic angular momentum of a particle can take one of two values, typically represented as 'up' or 'down.' These systems form the simplest nontrivial case in quantum magnetism and are fundamental in building more complex spin chain models.

Ising Model

The Ising model is a mathematical model used in statistical mechanics to describe interacting spins on a lattice. It simplifies true spin–spin interactions by considering only nearest-neighbor interactions and discrete spin states, often used to study phase transitions and critical phenomena in magnetic systems.

Hamiltonian in Magnetic Systems

The Hamiltonian in magnetic systems encapsulates both the spin–spin interactions and the interaction of spins with an external magnetic field. It provides the energy landscape of the system, and its formulation is crucial for determining microscopic states, energy eigenvalues, and subsequent thermodynamic behavior.

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