586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



and in terms of these one obtains the relations first derived by Polder: 



(4) 



Ho o-^l -\- a^) — I -Jr Ijaa Sgn p 



K _ — p 



Ho (t\1 +«") — !+ 2ja<T Sgn p' 

 where the function 



sgn p = + 1 p > 



= -1 p < 



a is the ratio of the natural precession frequency — | 7 1 i?^o to the 



Zir 



signal frequency, p is the ratio of a frequency ^^ | 7 1 Mo/ no , associated 



ZTT 



with the saturation magnetization Mo , to the signal frequency. Note 

 that a and p always have similar signs : if Ho is reversed, so is the satura- 

 tion magnetization. Equations (4) are true only for a fully saturated 

 sample. Therefore they hold good only for values of a greater than the 

 very small value corresponding to the amount of Ho reciuirecl to saturate 

 the sample. In practice that value of Ho is generally so small that this 

 restriction is trivial. In the text a number of formulae will appear 

 which apply "near a = 0". These are to be understood as applying near 

 the very small value of <x that corresponds to saturation. 



Equation (4) has an interesting implication with regard to the loss 

 parameter a. If a were zero, we would have 



= 1 - —^ — - and 



juo I — <x^ 



K p 



(5) 



Mo 



1 ■> 



1 — 0- 



and these equations describe the loss-free case. If in equations (5), a is 

 replaced by (a + ja sgn p), the resulting expressions check (4) to order a. 

 For small a, it follows that any propagation problem need be considered 

 for the loss-free case (5) only.* The first order change due to loss in any 

 formula so obtained can be deduced by differentiation of the formula 



* A form of the damping term in Equation (2), no less justified experimental!}' 



than the one used above, is — , : I M X —^ ). When this expression is used the 



11/1 V- dt ) 



permeabilities are exactly functions of the variable, <r -|- ja. sgn p. 



