Solution to Set 3: Difference between revisions
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<math> H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i </math> | <math> H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i </math> | ||
The answer becomes <math> H\approx \frac{1}{2}\lambda \mu_0 \textbf{M}^{2} - \sum_{i}\lambda \mu_0 \textbf{\mu_i} </math> | The answer becomes | ||
<math> H \approx \frac{1}{2} \lambda \mu_0 \textbf{M}^{2} - \sum_{i}\lambda \mu_0 \mu_i \textbf{M} </math> | |||
where | |||
* <math>\lambda = \frac{2 \sum_{j \neq i} \jmath_{i j}} {N \mu_0 g^2 \mu_b^2}</math> | |||
* <math>\mu_i = -g \mu_b \textbf{S}_i</math> | |||
==Part 2== | ==Part 2== | ||
Revision as of 20:34, 3 February 2009
This assignment corresponds to Section 8.3 and 8.4 in the Solid State Physics book. It deals with the topics of Magnetic Order.
Given Informatin
- Classical Ising antiferromagnet on a ”bipartite” lattice given by Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i } ,
- Bipartite lattice: a lattice with 2 sublattices, so that each spin on sublattice A interacts only with some spin on the other sublattice B. In an antiferromagnetic state, each sublattice assumes a uniform magnetization.
- Magnetization for Lattice A: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_A = < S^{(A)} > }
- Magnetization for Lattice B: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_B = < S^{(B)} > }
- Average magnetization: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = \frac{1}{2} (m_A + m_B ),}
- "Staggered” magnetization: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} = \frac{1}{2} (m_A - m_B ),} (Note: It's the difference between the two sublattices)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 1\;} for perfect ferromagnetic order
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger} = 1} for perfect antiferromagnetic order
Part 1
(1) Use Weiss mean-field decoupling to replace one of the spins in the Hamiltonian by its thermal average. The Weiss field experienced by a given spin is then proportional to the sublattice magnetization on the other sublattice. Write down self-consistent equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_A\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_B\;} , and express them through the order parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger}} .
pg 222 of Solid State Physics
The systematic way of replacing the spin operator in the Heisenberg Hamiltonian by their average values in order to obtain the Weiss molecular field approximation is to insert Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i = < S > + (S_i - < S >) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_j = < S > + (S_j - < S >) } into Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{J}{2} \sum_{<ij>} S_i S_j -− h\sum_i S_i }
The answer becomes
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H \approx \frac{1}{2} \lambda \mu_0 \textbf{M}^{2} - \sum_{i}\lambda \mu_0 \mu_i \textbf{M} }
where
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \frac{2 \sum_{j \neq i} \jmath_{i j}} {N \mu_0 g^2 \mu_b^2}}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i = -g \mu_b \textbf{S}_i}
Part 2
(2) Assume that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = 0\;} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 0\;} , and solve the mean-field equations by expanding in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{\dagger}} . Determine the Neel (ordering) temperature, and calculate the order-parameter exponent.
Part 3
(3) Now consider a small external field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h > 0\;} , so that both order parameters can assume a nonzero value (Note: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\;} will be small). By keeping only the leading terms in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\;} , calculate the uniform spin susceptibility Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi = \partial m/ \partial h} , as a function of temperature. Plot Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi\;} as a function of temperature, and show that it has a cusp around Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_N\;} .
Part 4
(4) Imagine adding a ”staggered” external field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^{\dagger}} , which would be positive on sublattice A, but would be negative on sublattice B. Concentrate on the system with no uniform field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (h = 0)\;} , and determine the behavior of the staggered susceptibility Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^{\dagger}= \partial m^{\dagger} / \partial h^{\dagger} } . Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^{\dagger}} blows up at the Neel temperature.