NOISE IN RESISTANCES 165 



not produce n electrons, but may produce 0, 1, 2* • • etc. electrons, must 

 be the difference between (22) and (24), or 



if = leloBin^ - n'). (25) 



The quantity in parentheses is the mean square deviation in w.* 



If the input current has any noise components i\ , then the total noise 

 output component will be 



i' = fi-{-n¥„. (26) 



By applying (26) successively to stage after stage the noise output of a 

 multistage electron multiplier can be evaluated (if one knows (w^ — n^))J 

 Wonder is sometimes expressed that current can be noisier than shot 

 noise, in which the time of electron arrival is purely random. Obviously, 

 we can have more than shot noise only if there is something non-random 

 about the time of electron arrival, and the argument above discloses just 

 what this is; it is the arrival of electrons in bunches.** We can easily see 

 how erratic even large currents would be if electrons were bound together 

 in groups having a total group charge of a coulomb, all the electrons in a 

 group arriving simultaneously. Reverting to our shot noise formulas, 

 we may illustrate this by assuming a perfect multiplier with a shot noise 

 input, in which each input electron produces exactly N output electrons. 

 Arguing from the shot noise equation (17) and replacing e by Ne we should 

 expect an output noise current 



i"2 = 2{Ne)hB (27) 



where /i is the output current; we get exactly the same result by assuming 

 the input noise current squared amplified by N^ 



i^ = {2eIoB)m 



= 2{Ne) {NIo)B (28) 



= 2iNe) hB . 



*The mean square deviation is the sum with respect to n of the square of the devia- 

 tion from the mean value of «, n. 



S (« - nypn = S n^pn - 2n S «/»„ + ^2 2 />„. 



The summation in the first term is"«'7that in thT second term is n and that in the third 

 term is unity. Hence 



S (« - n)*pn = (n2 - n^). 



** Anything, (such as transit time difference for electrons within a bunch) which tends 

 to break up the bunches will reduce the noise— and the signal as well. Such noise reduc- 

 tion involves a return to a more nearly random flow. 



