Behavior of MAG Standard Problem No . 2 in theSmall

1 Behavior of MAG Standard Problem No. 2 in the Small Parti le Limit M. J. Donahue, D. G. Porter, R. D. M Mi hael National Institute of Standards and Te hnology, Gaithersburg, MD 20899 J. Ei ke Institute for Magneti s Resear h, George Washington University, Washington, DC 20052 Abstra t| For a uniformly magnetized re tangular partile with dimensions in the ratio 5 : 1 : 0.1, the oer ive and swit hing elds in the (1; 1; 1) dire tion are determined to be H =Ms = 0:057069478 and Hs=Ms = 0:057142805. Previous miromagneti omputations of oer ive and swit hing elds that did not approa h these values for small parti les are analyzed. It is shown that the disagreement was primarily due to a disparity in the method of al ulating demagnetization energy. Corre ted simulations are shown to agree with analyti ally determined values. I. Introdu tion When solutions to the rst MAG standard problem failed to show good agreement, [1℄ a simpler standard problem was designed to examine the details of how di erent numeri al te hniques yield di erent solutions. The se ond MAG standard problem onsiders a re tangular parti le with dimensions L : d : t in the ratio 5 : 1 : 0.1. Only ex hange and magnetostati energy terms are onsidered. The oer ive eld along the (1; 1; 1) dire tion is to be al ulated as a fun tion of the ratio of parti le size to ex hange length lex = (2A= 0M2 s )1=2. Here A is the ex hange sti ness oeÆ ient in J/m and Ms is the saturation magnetization in A/m. Published solutions [2℄, [3℄, [4℄ show mu h better agreement than the results from the rst problem. It was expe ted that for a small enough parti le size, exhange energy would dominate, and the oer ive eld predi ted by all al ulations would onverge to the oer ive eld of a uniformly magnetized parti le. As seen in Fig. 1 however, signi ant di eren es were observed for small simulated parti les. In this paper we provide analyti values of the oer ive and swit hing elds in the small parti le limit. Our previous al ulations [3℄ (labeled \OOMMF 1.0" in Fig. 1) are examined in detail to determine and orre t the sour es of error when simulating small parti les. New solutions are omputed by a orre ted solver. [5℄ II. Small Parti le Theory In this se tion we analyze the equations of our mi romagneti model in the small parti le limit. The intent is to examine whether the numeri al methods used in our miromagneti simulations behave properly in this limit. The intent is not to predi t the physi al behavior of small magneti parti les. Many important in uen es on the physi al 0.045 0.050 0.055 0.060 0.065