Consider an Ising model in the presence of an external

Chapter 8, Problem 22P

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QUESTION:

Problem 22P

Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of −µBB if it points up and +µBB if it points down (where fm is the dipole’s magnetic moment). Analyze this system using the mean field approximation to find the analogue of equation. Study the solutions of the equation graphically, and discuss the magnetization of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions.

Equation:

Questions & Answers

QUESTION:

Problem 22P

Consider an Ising model in the presence of an external magnetic field B, which gives each dipole an additional energy of −µBB if it points up and +µBB if it points down (where fm is the dipole’s magnetic moment). Analyze this system using the mean field approximation to find the analogue of equation. Study the solutions of the equation graphically, and discuss the magnetization of this system as a function of both the external field strength and the temperature. Sketch the region in the T-B plane for which the equation has three solutions.

Equation:

ANSWER:

Step 1 of 3

The interaction energy of a single magnetic moment in uniform magnetic field.

                                                               

Here is the magnetic moment and is the magnetic field.

Every magnet has two orientation o=in the y=uniform magnetic field.

                                                               

Here, minus signs indicate that dipoles are aligned parallel and plus signs indicate that dipoles are aligned anti parallel.

The interaction energy between dipole and its neighbor is given as,

                                               

Here, is the average alignment of the neighbor.

The expression for partition function is given as,

                                                           

Further it can be written as,

                                                   

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