12 BELL SYSTEM TECHNICAL JOURNAL 



The component of magnetization parallel to H is 



/ = /gcos (00 - 0), 



where Is is the saturation magnetization. By starting with half of the 

 line of magnets pointing in a direction opposite to that of the other 

 half, the initial magnetization is zero and an unmagnetized or demag- 

 netized material is simulated. Thus a magnetization curve and a 

 hysteresis loop of this assemblage are obtained by plotting H against /. 

 Such a plot is shown in Fig. 7(b), with the scale of H determined by 

 the magnitudes of fXA and r. The curves are obviously similar to those 

 for real materials. 



Limitations of Ewing's Theory 



So far, this calculation is equivalent to what Ewing did over four 

 decades ago. But now we know the crystal structure of iron and in 

 particular the distances between the atoms. We also know the mag- 

 netic moment of each iron atom and know, therefore, the value of 

 IJ-AJf^ which determines the scale of H. Using the appropriate values 

 fiA = 2.0 X 10"^" erg/gauss and r = 2.5 X 10~^ cm, the coercive force 

 He for Ijr = 0.1 is found to be 4600 oersteds. This is affected some- 

 what by the ratio Ijr, but in any case He is found to be of this order of 

 magnitude unless Ijr is very close to unity. This magnitude of He is 

 greater by a factor of 10^ than the lowest value obtained experimen- 

 tally, 0.01. Similarly the initial permeability, /xo, according to the 

 model is about unity while observed values for iron range from 250 to 

 20,000. Adjustment of Ijr to higher values decreases /xo. 



This calculation of the magnetization curve and hysteresis loop are 

 based on a very much idealized model, and it is difficult to estimate 

 the error to which it may lead. One factor that has been completely 

 neglected is the fluctuation in energy. A much better approximation 

 would be to calculate the magnetic potential energy of a group of 

 magnets arranged in space in the same way that the iron (or nickel) 

 atoms are arranged in a crystal. This has been done by Mahajani * 

 who showed that application of Eq. (1) with / = (but summed to 

 account for the effects of all magnets in the structure) leads to the 

 result that the magnetic potential of the space array is independent 

 of d, in other words one orientation of the dipoles is as stable as any 

 other and the magnetization curve would go to saturation in infinitesi- 

 mal fields no matter in what direction H might be applied. If I is 

 finite, the stable positions of the magnets are parallel to the body- 

 diagonals of the cube which is the unit of the crystal structure, and 



