A new, "quantum mechanical" algorithm is proposed for automatic photopeak
location in gamma-ray spectra from semiconductor and scintillator detectors.Comment: 9 pages, LaTex, without figures. Sample program and figures (by FAX)
  can be sent on reques

Abbondanno

Baba

Balian

Beschastnov

Blok

Bukin

Campbell

Dupont

Feller

Giftcioglu

Helmer

Hillman

Jorch

Koshelo

Mariscotti

Mcnelles

Misra

Morales

Phillips

Robertson

Romanovskii

Roscoe

Routti

Sekine

Slaviĉ

Takeda

Z.K. Silagadze

Zimmermann

English

arXiv

A new, "quantum mechanical" algorithm is proposed for automatic photopeak location in gamma-ray spectra from semiconductor and scintillator detectors

Silagadze, Z K

CERN Document Server

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A new algorithm for automatic photopeak searches
Z.K. Silagadze
Budker Institute of Nuclear Physics
630 090, Novosibirsk, Russia
Abstract
A new, "quantum mechanical" algorithm is proposed for automatic photopeak lo-
cation in gamma-ray spectra from semiconductor and scintillator detectors.
1 Introduction
A great variety of programs can be found in the literature for automatic
gamma-ray spectrum analysis with computers [1]. An important ingredient
of every such program is some peak nding procedure. Usually the following
two characteristics of a local maximum of continuous function are used in
peak searching algorithms:
1) Near the maximum the curve is convex and so its second derivative becomes
negative, having a minimum value around the maximum of the peak.
2) While passing a peak the rst derivative changes sign.
The use of the second derivative for automatic photopeak location was
proposed by Mariscotti [2]. The real spectra, however, can have signicant
uctuations because of the discrete and statistical nature of the data. So a
discrete analog of the second derivative (the second dierence) dd
i
= N
i+1
 
2N
i
+ N
i 1
should be used, N
i
being the pulse heights of the spectrum in
channel i.
To reduce further the inuence of the statistical uctuations, this second
dierence can be averaged and after n iterations replaced by the generalized
second dierence
dd
i
(n;m) =
i+m
X
j
n
=i m
j
n
+m
X
j
n 1
=j
n
 m
  
j
2
+m
X
j
1
=j
2
 m
=
X
C
j
N
j
Its standard statistical deviation is sd
i
(n;m) =
q
P
C
2
j
N
j
and therefore not
1
only dd
i
(n;m) < 0 condition is demanded for the peak area location but also
a big enough value of




dd
i
(n;m)
sd
i
(n;m)




.
In [3] some modication of this method was used. Instead of smoothed
second dierence one can consider the sum dd
i
(k) =
P
i+k
j=i k
C
j
N
j
where the
C
j
coecients are chosen in such a way that dd
i
(k) < 0 near the peak. For
example in [3] they take
C
j
=
j
2
  p
2
p
4
exp
0
@
 
j
2
2p
2
1
A
=
d
2
dx
2
exp
0
@
 
x
2
2p
2
1
A






x=j
;
p being some parameter to be optimized. Note that near the peak
X
C
j
N
j

Z
1
 1
d
2
dx
2
2
4
exp
0
@
 
x
2
2p
2
1
A
3
5
N(x) dx =
Z
1
 1
exp
0
@
 
x
2
2p
2
1
A
d
2
N(x)
dx
2
< 0 :
In [4] the smoothed rst derivative was used for photopeak searches. The
peak is identied at the point where the rst derivative changes sign from
positive to negative.
One can smooth not the rst or second derivatives but the original data
and establish the sign of the rst derivative by simply comparing consecutive
channels. To increase statistical reliability, one demands not only N
i
> N
i 1
,
but also N
i+1
> N
i
[5]. As is claimed in [6], for low statistics the best result
is given by the following smoothing

N
i
=
1
9
(N
i+2
+ 2N
i 1
+ 3N
i
+ 2N
i+1
+N
i+2
) :
One more method for photopeak position nding, although somewhat
complex, is tting the whole spectrum or its part by appropriate analytic
function. A lot of various functions were suggested for this purpose, some of
them modeling even Compton continuum [7]. If resolution is very good, as it
is usually for semiconductor detectors, so that one has almost ideal gaussian
peaks and the background in the peak area is nearly constant, it is possible to
t successive small sections of the data by Gaussian function and identify the
peak position where the amplitude of the tted Gaussian becomes signicant
[8].
In this note a new and simple enough algorithm is proposed for photopeak
searches, which is completely dierent from the above described methods.
2
2 The idea
The proposed method is based on a very simple but nice idea, which can
be explained as follows. Suppose we place a small ball on the edge of the
irregular potential wall:
Classical ball will stop in front of the rst obstacle. But if it is a quantum one
and can penetrate through narrow barriers it still goes down to the potential
wall bottom and oscillates there.
For automatic photopeak nding this idea can be realized, for example, in
such a way. Let us take a point somewhere on the right slope of the peak and
let this point can jump left or right by one channel, or remain on its position,
so that the probability to jump left is proportional to exp
"
N
i 1
 N
i
p
N
i 1
+N
i
#
and to
jump right exp
"
N
i+1
 N
i
p
N
i+1
+N
i
#
. This point will climb up quickly enough to the
photopeak and will oscillate on the top. If now we follow up some amount
of jumps and calculate the mean position, this just gives us the photopeak
location.
To smear the "quantum ball" even more, one can take for the probability
to jump from channel i to channel i-1 :
P
i;i 1

m
X
k=1
exp
2
4
A
k
N
i k
 N
i
p
N
i k
+N
i
3
5
;
where A
k
are some numerical constants. Changing them and the number m,
one can govern the penetrating ability of the ball.
3 Realization via discrete Markov chain
In fact the above described set of channels and transition probabilities can
be treated as a nite Markov chain [9, 10].
3
For simplicity let us assume that from any state (channel) in this Markov
chain only the closest left and right neighbour states can be reached with
nonzero probabilities in one step, and that the probability for any state to
remain unchanged is zero. Thus the transition probability matrix for this
chain looks like
P =
0
B
B
B
B
B
B
B
B
B
@
0 1 0 0 0   
P
21
0 P
23
0 0   
0 P
32
0 P
34
0   
       
0     0 1 0
1
C
C
C
C
C
C
C
C
C
A
: (1)
Where we shall take
P
i;i1
= A
i
m
X
k=1
exp
2
4
N
ik
 N
i
p
N
ik
+N
i
3
5
; (2)
A
i
normalization constant is dened from the P
i;i 1
+ P
i;i+1
= 1 condition.
This Markov chain has a very simple invariant distribution [10]:
u
2
=
P
12
P
21
u
1
; u
3
=
P
12
P
23
P
32
P
21
u
1
;    ; u
n
=
P
12
P
23
  P
n 1;n
P
n;n 1
P
n 1;n 2
  P
21
u
1
; (3)
u
1
being dened from the normalization condition
n
X
i=1
u
i
= 1 : (4)
Now this invariant distribution has very sharp peaks which correspond
to a local maximums in the original spectrum. This is illustrated by Fig.1.
It is much simpler to identify peaks by some computer program in the u
n
distribution than in the original data. Fig.2 gives a clear example of this
(note the logarithmic scale on the vertical axis).
In fact this method of photopeak nding can work even for very low statis-
tics, as Fig.3 shows.
4 Photopeak tting
After the photopeak is located by the above described algorithm, one can t
it by gaussian function and extract its other characteristics. Fig.2 and Fig.3
show that our "quantum"algorithm not only gives the peak position, but can
also provide useful limits in which it is worthwhile to t it by Gaussian.
4
For good enough gaussian peaks one can use a non-iterative method for
the fast tting [11]. Suppose we want to t a Gaussian to the spectrum
region from channelm
1
to channelm
2
. Let us form an array S(i) = ln
N
i
N
i+m
 
ln
N
i 1
N
i+m 1
, where m is some integer number. If we have a pure Gaussian
N
i
= A exp
2
4
 
(i  p)
2
2
2
3
5
;
then S(i) =
m

2
doesn't depend on i. In fact this will not be the case because
of statistical uctuations and we should nd a constant C which minimizes
the sum
m
2
 m
X
i=m
1
+1
[S(i)   C]
2
D
2
(i)
; D
2
(i) =
1
N
i
+
1
N
i+m
+
1
N
i 1
+
1
N
i+m 1
;
where D(i) is a standard deviation for the casual quantity S(i). The solution
of this linear minimization problem is well known
C =
2
4
m
2
 m
X
i=m
1
+1
S(i)
D
2
(i)
3
5

2
4
m
2
 m
X
i=m
1
+1
1
D
2
(i)
3
5
: (5)
If we disregard statistical uctuations of D(i), the statistical error in deter-
mination of C is
C =
2
4
m
2
 m
X
i=m
1
+1
1
D
2
(i)
3
5
 1=2
:
So we get
 =
s
m
C
;  =

3
2m
C (6)
The parameter m can be determined from the condition  to be minimal.
Let us take as an estimation D(i) =
2
p
N
i
. Then  
1
m
p
m
2
 m
1
 m
, which is
minimal for
m =
2
3
(m
2
 m
1
) : (7)
Analogously one can nd the position and amplitude of the tted peak by
considering arrays P (i) = ln
N
i
N
i+m
=
m(2i 2p+m)
2
2
and Q(i) = lnN
i
= lnA  
(i p)
2
2
2
. The results are
p =
2
6
4
m
2
 m
X
i=m
1
+1
i  

2
m
P (i) +
m
2
DP (i)
2
3
7
5

2
4
m
2
 m
X
i=m
1
+1
1
DP (i)
2
3
5
DP (i)
2
=

2
m
2
"
4()
2
P
2
(i) + 
2
 
1
N
i
+
1
N
i+m
!#
; (8)
5
and
lnA =
2
6
4
m
2
X
i=m
1
lnN
i
+
(i p)
2
2
2
DN(i)
2
3
7
5

2
4
m
2
X
i=m
1
1
DN(i)
2
3
5
DN(i)
2
=
1
N
i
+
 


!
2
 
i  p

!
4
: (9)
where m is dened by eq.7 and
p
p



relation was assumed while deriving
the last equation.
The degree of t between the calculated and observed spectra can be
further analyzed by means of some quantitative criteria, suggested in the
literature [12].
5 Conclusions
It seems to us that the proposed algorithm is simple and eective enough to
be recommended for practical applications.
An earlier version of it was used with excellent results for uniformity stud-
ies of about 1650 NaJ crystals for SND detector [13], and also for vacuum
phototriodes testing [14].
Acknowledgements
The author is grateful to V.B. Golubev, S.I. Serednyakov and A.N. Peryshkin
for useful discussions.
6
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7
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8
Figure captures
Fig.1 | Invariant probability distribution u(i) for single gaussian peak, m=3.
Fig.2 | Invariant probability distribution for peak + background, m=3.
Fig.3 | The same as on Fig.2 for low statistics.
9


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