當single domain 鐵磁粒子之內能為一角度之函數(外加磁場與磁軸，磁矩與磁軸) ，而當此一函數恰為雙位階型式。則兩位階間傳輸機率可用Brownian motion 中的Fokker-Planck equation 或 Krmers' approximation method 來解，最後再利用master equation 合併雙向傳輸機率。
本文之理論模擬均以時變磁場條件下之變化為主。首先討論時變磁場下粒子磁矩對應外加磁場的關係。再進一步討論在固定磁場下，以類似於時變磁場中的統計法來探討粒子在磁場中轉動的特性。
此外，值得注意的是在本研究之中，master equation的數值計算方式，有別於一般的Euler method。而是採用back-Euler method來避免數值計算的發散情形。

A ferromagnetic particle with single domain, at some
kinds of applied field (at some angle or strangth), the
particle's free energy would be two state model. The
rate of barrier crossing could be solve by Fokker-Planck
equation .And use master equation to find out the Total
rate between two potential well.
In this thysis, we use the upper method to simulate
particle's magnetic moment under time varying magnetic
field at fixed angle or fixed magnetic applied rotate
the particle.
In numerical method, we use the back Euler method
to prevent the divergence of the calculation.

CONTENTS OF FIGURES …………………………………………………...…...2
ABSTRACT……………………………………………………………………………... 4
Chapter I. Introduction ……………………………………………………………….5
Chapter II. Theory and Methods…………………………………………………..6
Chapter III. Analysis and calculation of Hysteresis Loop with delta fixed…………………………………………………………………………………...17
Chapter IV. Analysis and calculation of Hysteresis loop with delta not fixed……………………………………………………………………………………...30
Chapter V. Conclusion of Hysteresis……………………………………………53
Reference ………………………………………………………………………………57

1. Peter Hanggi，Peter Talkner， "Reaction-rate theory: fifty years after Kramers"，Rev. Mod. Phys.Vol.62，No.2，April 1990，pp.251-341
2. H. Risken，"The Fokker-Planck Equation Methods of Solution and Applications"，Springer-Verlag，New York，1984
3. C. W. Gardiner，"Handbook of Stochastic Methods-for physics，chemistry and the natural sciences"，2nd edition，Springer-Verlag，New York，1985
4. W. F. Brown，"Therma; Fluctions of a Single-Domain Particle"，Phys. Rev. 130：1677 ，1963
5. T. L. Gilbert，"Magnetism；Magnetic Resonance"， Phys. Rev. 100，1243，D6，1955(the magnetic moment equation)
6. A. M. Jayannavar，" Brownian-Motion of Spins - Generalized Spin Langevin Equation"，Z. Phys. B----Condensed Matter，82，1991，pp153-156
7. I. Klik ，L.Gunther，"Rotation of Magnetization in Unison and Langevin-Equations for a Large Spin"，J. Stat. Phys，157，Vol.66NOS1/2，1992，pp635-645
8. I. Klik，L.Gunther，"1st-Passage-Time Approach to Overbarrier Relaxation of Magnetization"，J. Stat. Phys. Vol. 60NOS3/4，1990，pp473-484
9. Richard E. Wilde，Surjit Singh，"Statistical Mechanics (Fundamental and ModernApplications)"，1997，pp.256-pp.267
10. S. Chandrasekhar，"The Escape of Particle over potential Barriers"，Physica 7，284，1940
11. I. N. Levine，"Physical Chemistry"，Allyn and Bacon，Boston，1983，3rded. Sec. 17.7
12. I. Klik and Y. D. Yao，X. Yan，C. R. Chang，"Magnetic viscosity effevt in ac susceptibility measurements"，Phy. Rev. B，Vol.57，Num.1， 1998， pp92-95
13. I. Klik，C. R. Chang，H. L. Huang，" Dynamics of Micromagnetic Measurements"，Phy. Rev. B-condensed matter，1993，Vol.47，Iss 14, pp 8605-8612
14. W. T. Coffey，D. S. F. Crothers，J. L. Dormann，Yu. P. Kamykov，E. C. Kennedy，and W. Wernsdorfer，"Thermally Activated Relaxation Time of a Single Domain Ferromagnetic Particle Subjected to a Uniform Field at an Oblique Angle to the Easy Axis：Comparison with Experimental Observations"，Phy. Rev. Letters，Vol.80，No.25，22，June，1998
15. N. G. Van Kampen，"Stochastic process in Physics and Chemistry"，North -Holland，1981
16. I. Klik，L. Gunther，" Thermal Relaxation over a Barrier in Single Domain Ferromagnetic Particles"，Vol.67，Iss 9，J. App. Phy.，1990，pp 4505-4507
17. S. H. Charap，" Magnetic Viscosity in Recording Media"，Vol. 63, Iss 6，pp 2054-2057，J. App. Phy. 1988
18. J. F. Smyth，S. Schultz，D. R. Fredkin，D. P. Kern，S. A. Rishton，H. Schmid， M. Cali and T. R. Koehler，" Hysteresis in Lithographic Arrays of Permalloy Particles - Experiment and Theory"，J. App. Phy.，1991，Vol.69，Iss 8，pp 5262-5266
19. Geoffrey Bate，" Particulate Recording Materials"， PROCEEDINGS OF THE IEEE 1986，Vol.74，Iss 11，pp 1513-1525
20. S. Nieber，H. Kronmuller，" Micromagnetics of Critical Fields and Elementary Catastrophe-Theory"，Phy. Status Sol. B-Basic research，1991，Vol.165，Iss 2， pp 503-515
21. Ching-Ray Chang，" Micromagnetic Studies of Coherent Rotation with Quartic Crystalline Anisotropy"，J. App. Phy. 1991，Vol.69，Iss 4，pp 2431-2439
22. Amikam Aharoni，"The Agitation of Single Domain Particles"，Phy. Rev.，20. July. 1964，Vol.135，Num.2A，ppA447
23. Amikam Aharoni，"Effect of a Magnetic Field on the Superparamagnetic Relaxation Time"， Phy. Rev.，10. Jan. 1969，Vol.177，Num. 2，pp 793
24. W. T. Coffey，D. S. F. Crothers，J. L. Dormann，L. J. Geoghegan，Yu. P. Kamykov，A. W. Wickstead，"Effect of an Oblique magnetic field on the superparamagnetic relaxation time"，Phy. Rev. Letters，Vol.52，No.22，1，Dec.，1995
25. M. Lederman，D. R. Fredkin，R. O'Barr，and S. Schultz，M. Ozaki，"Measurement of thermal switching of the magnetization of single domain particles"，J. App. Phys. ，Vol.75，No.10，15. May 1994，pp6217
26. M. Lederman，and S. Schultz，M. Ozaki，"Measurement of the Dynamics of the Magnetization Reversal individual Single-Domain Ferromagnetic Particles"，Phy. Rev. Letters，Vol.73，No.14，3. Oct. 1994，pp1986
27. O. Kubo，E. Ogawa，"Barium ferrite particles for high density magnetic recording"，J. M. M. M.，Vol.134，1994，p376-381
28. T. L. Templeton and A. S. Arrott，"Rotational hysteresis in metal particle recording media：Remanent moment vectors"，J. App. Phys.，Vol.81，No.8，15. April 1997 ，pp3797
29. Herbert B. Callen and Theodore A. Welton，"Irreversibility and Generalized Noise"，Phy. Rev.，Vol.83，No.1，1. July 1951，pp34
30. J. C. Slonczewski，"Conductance and exchange coupling of two ferromagnets separated by tunneling barrier"，Phy. Rev. B，Vol.39，No.10，1. April 1989，pp6995
31. M. P. Sharrock，St. Paul，"Time-Dependent Magnetic Phenomena and Particle-size Effects in Recording Media"，IEEE Trans. Magn.，Vol.26，No.1，Jan. 190，pp193
32. I. Klik，Y. D. Yao，"Initial susceptibility of noninteracting particulate media"，J. M. M. M.，Vol.186，1998，pp233-238
33. W. H. Press，S. A. Teukolsky，W. T. Vetterling，B. P. Flannery，"Numerical Recipes in Fortran-The Art of Scientific Computing" 2nd edition，cambridge，1986
34. I. Klik，Y. D. Yao，"Eigen spectrum of Brown's Fokker-Planck equation for corherent rotation of magnetization"，J. M. M. M. ，Vol.182，1998，pp335-340
35. I. Klik，Y. D. Yao，C. R. Chang，"Chain Formation，Thermal Relaxation aand Noise In Magnetic Recording"，IEEE Trans. Magn.，Vol.34，No.2，March 1998，pp358-362
36. V. L. Sobolev，I. Klik，C. R. Chang，H. L. Huang，J. Appl. Phys.，Vol.75，No.10，15 May 1994，pp5794-5796
37. I. Klik，C. R. Chang，"Thermal Relaxation in arrays of coupled ferromagnetic particles"，Phy. Re