NATIONAL RESEARCH NUCLEAR UNIVERSITY MEPhI
(Moscow Engineering Physics Institute)
M.M. Maslov, K.P. Katin, L.A. Openov
INTRODUCTION TO PHYSICS OF SECOND-ORDER
MAGNETIC PHASE TRANSITIONS
This textbook is recommended by UMO “Nuclear Physics and Technologies” as a textbook for students of higher educational institutions of Russia
Moscow 2015
UDC 537.9
LBC 22.317
M31
Maslov M.M., Katin K.P., Openov L.A. Introduction to Physics of Second-Order Magnetic Phase Transitions: Textbook. M.: NRNU MEPhI, 2015. – 88 p.
The present textbook provides a brief introduction to basic physics of second-order phase transitions by the specific examples of paramagnetic-ferromag- netic/antiferromagnetic transitions in the solid state system of local magnetic moments. Topics covered include the classification of materials by their magnetic properties; mean-field approach to calculation of various magnetic characteristics; summary of the phenomenological Landau model. This textbook is intended to students and researchers interested in theoretical aspects of magnetic phase transitions.
Prepared in the Programs framework of the creation and development of NRNU MEPhI.
Reviewer: Dr. Mikhail A. Remnev, Senior Researcher at All-Russia Research Institute of Automatics (VNIIA)
ISBN 978-5-7262-2195-3 |
© National Research Nuclear University MEPhI |
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(Moscow Engineering Physics Institute), 2015 |
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CONTENT |
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Preface...................................................................................................... |
5 |
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1. |
Atomic magnetic moment.................................................................... |
6 |
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2. |
Physical quantities characterizing the magnetic |
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properties of matter .................................................................................. |
8 |
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3. |
Classification of materials for their magnetic properties ..................... |
9 |
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4. |
Isolated local magnetic moment in an external magnetic field.......... |
10 |
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5. |
Interplay of quantum mechanics and statistics................................... |
11 |
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6. |
A system of noninteracting local magnetic moments in an external |
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magnetic field. Curie law ....................................................................... |
13 |
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7. |
Effective Weiss field.......................................................................... |
15 |
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8. |
Exchange interaction.......................................................................... |
18 |
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9. |
Interaction of two local magnetic moments ....................................... |
21 |
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10. |
Heisenberg model and Ising model.................................................. |
22 |
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11. |
Mean-field approximation in the Ising model.................................. |
24 |
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12. |
Curie-Weiss equation and Curie-Weiss law..................................... |
27 |
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13. |
Ferromagnetic transition in the Ising model. Curie temperature. |
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Order parameter ..................................................................................... |
29 |
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14. |
Temperature dependence of the ferromagnetic |
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order parameter in the Ising model ........................................................ |
31 |
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15. |
Ground and excited states of a ferromagnet in the Ising model....... |
34 |
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16. |
Free energy of a ferromagnet in the Ising model ............................. |
36 |
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17. |
Free energy of a ferromagnet near the critical temperature ............. |
42 |
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18. |
Spontaneous symmetry breaking |
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at the paramagnetic-ferromagnetic transition......................................... |
45 |
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19. |
Phenomenological Landau theory of second-order |
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phase transitions..................................................................................... |
49 |
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20. |
Heat capacity of the Ising ferromagnet |
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in the mean-field approximation ............................................................ |
51 |
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21. |
Magnetic susceptibility of the Ising ferromagnet |
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in the mean-field approximation ............................................................ |
54 |
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22. |
Critical exponents ............................................................................ |
56 |
23. |
Exact solution of the Ising model in one dimension ........................ |
57 |
24. |
Short-range and long-range orders. Correlations. Fluctuations ....... |
63 |
25. |
Correlation function of a ferromagnet in the Ising model................ |
65 |
26. |
Heat capacity of a ferromagnet |
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in the Ising model with account for fluctuations.................................... |
72 |
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27. |
Magnetic susceptibility of a ferromagnet |
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in the Ising model with account for fluctuations.................................... |
73 |
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28. |
Ising model for antiferromagnets. Mean-field approximation. |
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Neél temperature.................................................................................... |
75 |
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29. |
Magnetic susceptibility of the Ising antiferromagnet |
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in the mean-field approximation ............................................................ |
80 |
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30. |
Spin waves in the Heisenberg model ............................................... |
83 |
Literature................................................................................................ |
87 |
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A phase transition is the transformation of a thermodynamic system from one phase (or state) of matter to another one. During a phase transition certain properties change as a result of the change of some external condition, such as temperature, pressure, or others. Examples of phase transitions include: the transitions between the solid, liquid, and gaseous phases of a single component; the transition between the ferromagnetic and paramagnetic phases of magnetic materials at the Curie point; changes in the crystallographic structure such as between ferrite and austenite of iron; order-disorder transitions such as in alpha- titanium aluminides; the emergence of superconductivity in certain metals and ceramics when cooled below a critical temperature; quantum condensation of bosonic fluids (Bose–Einstein condensation); the breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled, etc.
Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables. This condition generally stems from the interactions of a large number of particles in a system, and does not appear in systems that are too small. At the phase transition point (for instance, boiling point) the two phases of a substance, liquid and vapor, have identical free energies and therefore are equally likely to exist. Below the boiling point, the liquid is the more stable state of the two, whereas above the gaseous form is preferred.
In the modern classification scheme, phase transitions are divided into two broad categories. First-order phase transitions are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy per volume. During this process, the temperature of the system will stay constant as heat is added: the system is in a “mixed-phase regime” in which some parts of the system have completed the transition and others have not. Familiar examples are the melting of ice or the boiling of water. Second-order phase transitions are also called continuous phase transitions. They are characterized by a divergent susceptibility, an infinite correlation length, and a power-law decay of correlations near criticality. Examples of sec-
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