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A method for the separation of overlapping decaying peaks
NUCLEAR
INSTRUMENTS
A METHOD
AND
FOR
87(1970)305-306;
METHODS
THE
SEPARATION
0 NORTH-HOLLAND
OF OVERLAPPING
PUBLISHING
DECAYING
Co.
PEAKS
G. ADAM and J. KATRIEL Nuclear
Research
Centre, Negev, P.O.B.
9001, Beer-Sheva,
Israel
Received 2 July 1970 A computational
method for the separation of overlapping decaying gamma rays peaks is proposed. The method, advantage of the decay, overcomes some of the drawbacks of the standard peeling method.
In applications of nuclear gamma ray spectrometry one often encounters the problem of determining the activity of a gamma transition whose spectral peak is overlapped by a second peak due to another transition. The standard approach to this problem is the method of peeling’) which has some well known shortcomings. In the case of decaying sources we propose a different approach, which takes explicit advantage of the decay. In this method some of the drawbacks of the peeling method are avoided: one does not need to assume anything about the specific shape of each peak, no reference sources are needed, and the result is unambiguous. To outline the method let us consider the case of two partially or completely overlapping peaks, to be denoted A and B, having decay constants ;1, and i, respectively. We further assume that 1, #R,. The experimental part of the proposed method consists of performance of two consecutive pulse height analyses of the given sample, the time interval between these two measurements being t. It is assumed that t is chosen in such a way that it is of the order of magnitude of the half-lives involved, and sufficiently long compared to the measurement time. The contribution to the ith channel in the first spectrum can be written in the form Ci = C*,i + CB,i + Bi, (1) where C,,i and C,,i are the contributions of peaks A and B respectively and Bi is the corresponding background contribution. In the second spectrum the contributions to channel i will be Cl = C,,iexp(-R,t)+C,,ieXp(-&t)+Bi.
(2)
Subtracting eq. (2) from eq. (1) multiplied by exp ( -&z,t), and dividing the result by (exp( - A*z,t)exp( -&t)} we get {CieXp(--At)-Ci}/{exp(--At)-exp(-&t)} C,,i + {Biexp( -AAt)-Bl}/(exp(
The rhs consists background
B superimposed
on a
By = {Bi exp( - n,t) - Bi}/{exp( - A,t) - exp( - &t)}. B,? being a linear
combination of Bi and B; which are monotonous, approximately linear decreasing functions of i, is also necessarily of the same nature and therefore behaves as an effective background. The net peak B is then obtained employing the usual methods of separating a single peak from its background. A completely analogous treatment can be performed for peak A. The problem of two overlapping peaks is thus reduced to that of two separate peaks, each on an effective background. Obviously of the physical background existing for the overlapping peaks is negligible, then the effective background will also be of minor magnitude. It is obvious that By can obtain negative values, resulting in some esthetic discomfort. This can be eliminated by adding an arbitrary big enough constant to both sides of eq. (3), thus shifting the zero. Generalization of this method to the case of three or more peaks is straightforward. Let us assume that there are N peaks with decay constants 1,,;1,, . . . I,. The initial contribution (at t = 0) of peak k to channel i will be denoted by A,,i. We perform N pulse height analyses of the sample at times tl,t,,... tN and obtain the following set of equations for each channel i, Ci,j =
5
A,,iexp(-&tj)+Bi,j;
j = 1, ... N,
(4)
k=l
where Ci,j and Bi j are the total number of counts and background contribution respectively in channel i at thejth measurement. Solving this set of linear equations for Ak,i by Kramer’s method, one gets A,, i = [det(T,k)-
det(T,k)]/det(T),
(5)
where
= -AAt)-exp(
of the peak
which takes
-&t)}. T=
(3) The lhs of eq. (3) is easily computed from the measured spectra. One computes it for all the relevant channels. 305
exp(-l,t,) exp(-A,t,) . . . . . . . . . . . . . . exp(-l,t,)
exp(-&t,) exp(-&t,) . . . . . . . . . . . . . . exp(-I,t,)
. . . exp(-I$,) . . . exp(-&tz) . . . . . . . . . . . . . . . . . . . . . exp(-I&,) r
30(5
G. ADAM
A N D J. K A T R I E L
and To* and T k are the matrices obtained by interchanging the k th column of T with the columns (i~', ~) iN
and
(B~'~ t: \Bi, n /
respectively. Finally, let us write eq. (5) in the form d e t ( T k ) / d e t ( T ) = Ak, i + det(Tk)/det(T).
(6)
This is the generalization of the result, eq. (3), obtained previously. Again, the lhs can be directly calculated from the known data. The rhs is composed of a term
which is a linear combination of the physical backgrounds involved in the set of measurements, and therefore, as before, behaves as an effective background, and a term which is due to the k th component. Performing this analysis for all the relevant channels and for any desired k, one can construct the corresponding separated peak, superimposed on an effective background. Referenee
1) R. Van Lieshout, A. H. Wapstra, R. A. Ricci and R. K. Girgis, in Alpha, beta and gamma-ray spectroscopy (ed. K. Siegbahn; North-Holland Publ. Co., Amsterdam, 1966) p. 524.
INSTRUMENTS
A METHOD
AND
FOR
87(1970)305-306;
METHODS
THE
SEPARATION
0 NORTH-HOLLAND
OF OVERLAPPING
PUBLISHING
DECAYING
Co.
PEAKS
G. ADAM and J. KATRIEL Nuclear
Research
Centre, Negev, P.O.B.
9001, Beer-Sheva,
Israel
Received 2 July 1970 A computational
method for the separation of overlapping decaying gamma rays peaks is proposed. The method, advantage of the decay, overcomes some of the drawbacks of the standard peeling method.
In applications of nuclear gamma ray spectrometry one often encounters the problem of determining the activity of a gamma transition whose spectral peak is overlapped by a second peak due to another transition. The standard approach to this problem is the method of peeling’) which has some well known shortcomings. In the case of decaying sources we propose a different approach, which takes explicit advantage of the decay. In this method some of the drawbacks of the peeling method are avoided: one does not need to assume anything about the specific shape of each peak, no reference sources are needed, and the result is unambiguous. To outline the method let us consider the case of two partially or completely overlapping peaks, to be denoted A and B, having decay constants ;1, and i, respectively. We further assume that 1, #R,. The experimental part of the proposed method consists of performance of two consecutive pulse height analyses of the given sample, the time interval between these two measurements being t. It is assumed that t is chosen in such a way that it is of the order of magnitude of the half-lives involved, and sufficiently long compared to the measurement time. The contribution to the ith channel in the first spectrum can be written in the form Ci = C*,i + CB,i + Bi, (1) where C,,i and C,,i are the contributions of peaks A and B respectively and Bi is the corresponding background contribution. In the second spectrum the contributions to channel i will be Cl = C,,iexp(-R,t)+C,,ieXp(-&t)+Bi.
(2)
Subtracting eq. (2) from eq. (1) multiplied by exp ( -&z,t), and dividing the result by (exp( - A*z,t)exp( -&t)} we get {CieXp(--At)-Ci}/{exp(--At)-exp(-&t)} C,,i + {Biexp( -AAt)-Bl}/(exp(
The rhs consists background
B superimposed
on a
By = {Bi exp( - n,t) - Bi}/{exp( - A,t) - exp( - &t)}. B,? being a linear
combination of Bi and B; which are monotonous, approximately linear decreasing functions of i, is also necessarily of the same nature and therefore behaves as an effective background. The net peak B is then obtained employing the usual methods of separating a single peak from its background. A completely analogous treatment can be performed for peak A. The problem of two overlapping peaks is thus reduced to that of two separate peaks, each on an effective background. Obviously of the physical background existing for the overlapping peaks is negligible, then the effective background will also be of minor magnitude. It is obvious that By can obtain negative values, resulting in some esthetic discomfort. This can be eliminated by adding an arbitrary big enough constant to both sides of eq. (3), thus shifting the zero. Generalization of this method to the case of three or more peaks is straightforward. Let us assume that there are N peaks with decay constants 1,,;1,, . . . I,. The initial contribution (at t = 0) of peak k to channel i will be denoted by A,,i. We perform N pulse height analyses of the sample at times tl,t,,... tN and obtain the following set of equations for each channel i, Ci,j =
5
A,,iexp(-&tj)+Bi,j;
j = 1, ... N,
(4)
k=l
where Ci,j and Bi j are the total number of counts and background contribution respectively in channel i at thejth measurement. Solving this set of linear equations for Ak,i by Kramer’s method, one gets A,, i = [det(T,k)-
det(T,k)]/det(T),
(5)
where
= -AAt)-exp(
of the peak
which takes
-&t)}. T=
(3) The lhs of eq. (3) is easily computed from the measured spectra. One computes it for all the relevant channels. 305
exp(-l,t,) exp(-A,t,) . . . . . . . . . . . . . . exp(-l,t,)
exp(-&t,) exp(-&t,) . . . . . . . . . . . . . . exp(-I,t,)
. . . exp(-I$,) . . . exp(-&tz) . . . . . . . . . . . . . . . . . . . . . exp(-I&,) r
30(5
G. ADAM
A N D J. K A T R I E L
and To* and T k are the matrices obtained by interchanging the k th column of T with the columns (i~', ~) iN
and
(B~'~ t: \Bi, n /
respectively. Finally, let us write eq. (5) in the form d e t ( T k ) / d e t ( T ) = Ak, i + det(Tk)/det(T).
(6)
This is the generalization of the result, eq. (3), obtained previously. Again, the lhs can be directly calculated from the known data. The rhs is composed of a term
which is a linear combination of the physical backgrounds involved in the set of measurements, and therefore, as before, behaves as an effective background, and a term which is due to the k th component. Performing this analysis for all the relevant channels and for any desired k, one can construct the corresponding separated peak, superimposed on an effective background. Referenee
1) R. Van Lieshout, A. H. Wapstra, R. A. Ricci and R. K. Girgis, in Alpha, beta and gamma-ray spectroscopy (ed. K. Siegbahn; North-Holland Publ. Co., Amsterdam, 1966) p. 524.