Ph 801 — Exercise 7
1. The average energy required for the production of a:
a.  in a plastic scintillator is 100 eV;
b. an electron-ion pair in air is 30 eV;
c. an electron-hole pair in Si is 3.65 eV;
d. a quasi-particle (break up of a Cooper pair in a superconductor)
is 1 meV.
What is the relative energy resolution in these counters for a stopping 10
KeV particle assuming Poisson statistics (and neglecting the Fano effect*).
(*) The fluctuations of secondary particle production (like electron-ion pairs)
are smaller than might be expected according to Poisson distribution. This is
a consequence of the fact that the total energy loss is constrained by the
fixed energy of the incident particle. This leads to a standard error  =
sqrt(FN), when N = total number of produced secondaries and the Fano
factor is smaller than 1.
2. Calculate the mean and the standard deviation of a uniform
distribution f(x) = 1/(B-A) for A ≤ x ≤B (the case in which A = 0 and
B = 1 is called uniform distribution).
Suggested readings:
http://pdg.lbl.gov/2005/reviews/probrpp.pdf
http://pdg.lbl.gov/2005/reviews/statrpp.pdf
Suggested solution:
1. The energy resolution is determined by the fluctuations of the number
of produced particles. If W is the energy required for the production
of a pair, the energy resolution is:
E N
N
1



E
N
N
N
N  E /W 
E
1
W


E
E
N
a) 10% b) 5.5% c) 1.9% d) 3.2e-4

2. Let x be the possible outcome of an observation that can take any value in
a continuous range, the probability that the measurement’s outcome lies
between x and x+dx is f(x,), where f is the probability density function
(p.d.f.) which may depend on one or more parameters . The p.d.f. is always
normalized to unit area.
The expectation value of any function u(x) is
E[u(x)] 



u(x) f (x)dx
The nth moment of a random variable is

 n  E[x n ] 

x
n
f (x)dx

The mean is the first order moment  = 1 and the variance is:
 2  V[x]   2   2


The mean of f(x) = 1/(B-A) for A ≤ x ≤B is


 xf (x)dx  

B
A
x
B2  A2 A  B
dx 

B A
2(B  A)
2
and the standard deviation is

 2   2  2 



 x 2 f (x)dx   xf (x)dx 


2

B

A
B x

x2
dx  
dx 
B A
A B  A 
2
B  A  4(B  A )(B  A)  3(B 2  A 2 ) 2 (B  A)[4B 3  4 A 3  3(B  A) 2 (B  A)]
B A
 


 
3(B  A) 2(B  A) 
12(B  A) 2
12(B  A) 2
3
3
2
2
2
3
3
(B  A)[4B 3  4 A 3  (B  A) 2 (B  A)] 4B 3  4 A 3  3(B 2  A 2  2AB)(B  A) 4B 3  4 A 3  3(B 2  A 2  2AB)(B  A)



12(B  A) 2
12(B  A)
12(B  A)
B 3  A 3  3B 2 A  3A 2 B  6AB 2  6A 2 B B 3  A 3  3AB 2  3A 2 B  6AB 2  6A 2 B B 3  A 3  3AB 2  3A 2 B (B  A) 2



12(B  A)
12(B  A)
12(B  A)
12
