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BELL SYSTEM TECHNICAL JOURNAL 



in Fig. 26& and with Pauli's principle. The arrangement of lowest 

 energy is shown in Fig. 26c; electrons have shifted from the band of 

 plus spin to states of lower total energy in the band of minus spin until 

 the two bands are filled to the same energy level, indicated by the 

 solid horizontal line. As the figure shows, the number of electrons 

 shifted will be the number lying in the energy range bE = tx^H}^ 



(a) 



-SPIN + SPIN 



(b) 



-SPIN + SPIN 



(c) 



SPIN + SPIN 



— iN(E) ^N{E)-* 



NUMBER OF QUANTUM STATES AND ELECTRONS PER UNIT ENERGY 



Fig. 26 — The paramagnetism of free electrons. 



(a) Distribution of electrons in energy. 



(b) Displacement of levels by a magnetic field. 



(c) Distribution of electrons in energy in a magnetic field. 



The number of states, bN, lying in this energy range in the band of 

 plus spin, which contains of course half the states in the band, is ac- 

 cording to equation (3) 



m = }4N{Ei)8E = y2NiEi)npH. 

 The magnetic moment of these states is 



8M+ = - fi^m = - y2N{Ei)ix^''H. 



(13) 



(14) 



The minus sign occurs because the angular momentum and the mag- 

 netic moment of an electron are in opposite directions; the states of 

 plus spin have minus moments in Fig. 26. 



The electrons that occupied these states before the field was applied 

 now occupy states with minus spin and produce a magnetic moment of 



bM- = }4N{E,)tx^m. 



(15) 



Hence the minus band gains a plus moment and the plus band loses a 



" We have here assumed that the fractional change in N{E) in the interval n^H 

 is negligible; this assumption is reasonable. For a field of 10,000 gauss, fi^H is only 

 5.77 X 10~^ ev while £i — £o is of the order of several ev. 



