PHZ3400-11 Problem Set 3: Difference between revisions
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''' Problem | ''' Problem 3 - Hysteresis ''' | ||
Consider the same equation os state for <math>T = 0.5 Jz < T_c</math>. Solve numerically the equation os state starting at large field (say <math>h = 3</math>, and plot the solution as a function of <math>h</math>. You should find that the magnetization, which is close to <math>m=1</math> for large field, decreases as the field is rediced, but remains finite at <math>h=0+</math>. Continue plotting it by further reducing the field, until you reach the (negative) [http://en.wikipedia.org/wiki/Hysteresis coercive field] <math>h_{-}</math>, where the positive solution can no longer be found. Repeat the calculation starting at very large negative fields and finding the negative magnetization solution, which can be found all the way to the (positive) coercive field <math>h_{+}</math>. In the field range between the coercive fields, there are two solutions and we find the [http://en.wikipedia.org/wiki/Hysteresis hysteresis loop], which is typically seen at any first-order transition. | Consider the same equation os state for <math>T = 0.5 Jz < T_c</math>. Solve numerically the equation os state starting at large field (say <math>h = 3</math>, and plot the solution as a function of <math>h</math>. You should find that the magnetization, which is close to <math>m=1</math> for large field, decreases as the field is rediced, but remains finite at <math>h=0+</math>. Continue plotting it by further reducing the field, until you reach the (negative) [http://en.wikipedia.org/wiki/Hysteresis coercive field] <math>h_{-}</math>, where the positive solution can no longer be found. Repeat the calculation starting at very large negative fields and finding the negative magnetization solution, which can be found all the way to the (positive) coercive field <math>h_{+}</math>. In the field range between the coercive fields, there are two solutions and we find the [http://en.wikipedia.org/wiki/Hysteresis hysteresis loop], which is typically seen at any first-order transition. | ||
Revision as of 16:48, 13 February 2011
Problem 1 - Critical Behavior of a Ferromagnet
a) Show that the mean-field equation of state for a ferromagnetic Ising model is given by (use units 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 k_B = 1} ; 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 \beta = 1/T} ):
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 = \tanh \left( \beta Jz m + \beta h \right)} .
b) Consider 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 solve this equation numerically (using Mathematica, MAPLE, or FORTRAN), and 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 m(T)} . At 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 < T_c = Jz} , you will find three solutions. Pick the one with positive magnetization and plot it as a function of temperature. Show the printout of your computer plot as a part of the homework.
c) By expanding the RHS in powers of 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} , to cubic order, solve analytically for the magnetization close to 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_c} . Plot the expression you obtain on the same graph as the full numerical solution (dashed lines). The two curves should agree only close to 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_c}
d) Now choose a small but finite external magnetic 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} . Solve the above equation numerically, again picking the positive solution. You should find that transition is "rounded" and the magnetization remains finite at any temperature.
e) Now chose 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 = T_c = Jz} , and expand the above equation to linear order 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 to cubic order 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} . Solving 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} as a function of 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} , show that at the critical point
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 \propto h^{1/\delta}} , where the critical exponent 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 \delta = 3} .
Problem 2 Gap to low temperature excitations
At 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=0} , our equation of state in zero external field gives fully saturated 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=1} (consider the positive solution). At very small but finite temperature, there is a very small temperature correction of the form
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 \delta m(T) = 1- m(T) \propto \exp \{-E_g /T \} } . Determine the energy of the gap 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 E_g } to low-temperature excitations, in terms of the interaction 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 J} , and the coordination number 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 z} . How does it compare to the energy to perform a single spin-flip from the ground state (all spins aligned)?
Problem 3 - Hysteresis
Consider the same equation os state 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 T = 0.5 Jz < T_c} . Solve numerically the equation os state starting at large field (say 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 = 3} , and plot the solution as a function of 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} . You should find that the magnetization, which is close to 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 large field, decreases as the field is rediced, but remains finite at 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+} . Continue plotting it by further reducing the field, until you reach the (negative) coercive 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_{-}} , where the positive solution can no longer be found. Repeat the calculation starting at very large negative fields and finding the negative magnetization solution, which can be found all the way to the (positive) coercive 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_{+}} . In the field range between the coercive fields, there are two solutions and we find the hysteresis loop, which is typically seen at any first-order transition.