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UBC Theses and Dissertations

Sur la theorie de l'aimantation spontanee d'une substance ferromagnetique aux basses temperatures Banville, Marcel


In 1930, Bloch derived a formula for the temperature-dependence of the spontaneous magnetisation of a ferromagnetic substance (in the sense of the well-known Heisenberg model) valid asymptotically as the temperature T tends to zero. In 1936, Kramers rederived Bloch's formula using an entirely new approximate method. In 1937, Opechowski applied Kramers’ method to obtain, in addition to Bloch's T³′²-term in the expression for the magnetisation, two additional terms, in T² and T⁵′². In 1956, Dyson found a rigourous method for dealing with this problem. His result shows that there is no T²-term and the T⁵′²-term has a coefficient different from that found by Opechowski. In this thesis, some possibilities are investigated of modifying Kramers' method. In particular, the question is considered, which assumptions in Kramers' method are responsible for the above mentioned discrepancies. In Kramers' method, the partition function of the Heisenberg model is identified with the largest term in its power series expansion. The calculation of the largest term is in turn reduced to a certain random walk problem. This reduction of the problem to a random walk problem involves certain assumptions which we have not tried to modify in this thesis. What is new is a careful discussion of, and improvement on the solution of the random walk problem. The improved method of solving this problem leads to a cubic equation in P¹′², where P is a certain parameter with no single physical meaning. In chapters 6 and 7, a first approximation is obtained by omitting the term in P³′². The resulting quadratic equation in P¹′² leads to an expression for the spontaneous magnetisation containing no term in T² as in Dyson's formula. The solution of the complete cubic equation unfortunately leads to an expression for the spontaneous magnetisation, in which the term in T² reappears again. One obtains again Opechowski's result, except for a small modification of the coefficient of the T⁵′²-term; this is due to a better approximation for the factorials occuring in the calculations. This fact shows that Kramers’ random walk problem constitutes too crude an approximation of the actual problem. After the writing of this thesis was completed, Professor Opechowski found a way of modifying Kramers' method. The calculation of the partition function in the modified method is reduced to a slightly different random walk problem. The expression for the spontaneous magnetisation becomes then identical with Dyson's up to the T⁷′²-term inclusive.

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