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The super resolution of gamma-ray spectrum
NUCLEAR
INSTRUMENTS
A N D M E T H O D S 30 (I964) 2 2 4 - - 2 2 8 ;
~
NORTH-HOLLAND
PUBLISHING
CO.
THE S U P E R R E S O L U T I O N OF G A M M A - R A Y S P E C T R U M T. I N O U Y E
Central Research Laboratory, Tokyo Shibaura Electric Co., Ltd. Kawasaki, Japan Received 16 March 1964
The new method to analyze the gamma-ray spectrum was developed using the Fourier transforms. According to this method, the same result was obtained as the resolving power was effectively increased. This method was examined by numerical
simulation a n d the increase of resolution was proved. It was successfully applied to the determination of the energy difference between two g a m m a - r a y s from Zr 95 which h a d been impossible because of small energy difference.
1. Introduction
output. As input, we take here the spectrum of gammarays and as output the pulse height distribution. For the unit of variables of both functions the energy E is chosen. The input and output are related to each other by the linear relation. That is; if the input of q~t and ~o2 corresponds to the output of ~b1 and ~b2, respectively, then the input of a~pl +b~p 2 gives the output of a~bl + bOz. We assume also the invariable relation of the output functions form at a small shift of energy and that is the case when the energy shift is smaller than several times of the resolving power. In this case the response g(E), corresponding to the input o f f ( E ) , can be expressed by using h(E) which is the output corresponding to the input 6(E) as follows :
The gamma-ray spectrometry has many merits; the structure of the gamma-ray spectrum is simple and, accordingly, the measurement of that is very easily carried out by using a multichannel pulse height analyser. The preparation of samples to be analyzed is likewise easy. It is on account of these advantages that the gamma-ray spectrometry is widely used for the purpose of simply identifying the radionuclides in the research field of nuclear physics and radiochemistry. The faults of this technique are, however, that the resolving power of the measuring system is no more than about 7% so long as scintillation method is used for the detection of gamma-rays and, accordingly, it is difficult to distinguish two or more mixed radionuclides from which gamma-rays are emitted with a smaller energy difference than the resolving power of the measuring system, except if other techniques, for example, the analysis of decay curve or the chemical separation are employed. Particularly, when the half-lives of two or more radionuclides are of nearly the same value and the chemical separation of them is difficult, then, until now it was impossible to identify individually these radionuclides. This paper deals with the application of the Fourier transformation method to analyze the mixed gammaray spectra. By means of this method, the resolving power of the generally used gamma-ray spectrometer was effectively increased about ten times. This method was examined by the numerical simulation in which the output functions were constructed artificially and treated in the same way just as we treat the radiation spectrum. Then it was applied to the determination of the energy difference between two gamma-rays from Z r 95 which had been impossible because of small energy difference.
2. Theory We consider the whole system as a "black box" and take into account only the relation between input and
g(E) =
f(E')h(E - E')dE'
(1)
--o0
where 6(E) is the Dirac's delta function. This relation is alternatively expressed by using the Fourier transforms of these functions. That is: (2)
=
where F(co) =
exp(-
ie)E)f(E)dE, etc.
-ao
In the gamma-ray spectrometry, we have the information about above described g(E) and wish to know the function f(E) based on the knowledge about g(E). From the relation (2), we can know F(co) and by the inverse Fourier transforms we can have the inputf(E). The h(E) is the output function corresponding to the input of the delta function and it can be obtained by the observation of the monochromatic gamma-rays. This is a good function to express the characteristics of the system because the input function 6(E) equally includes all the components of frequency. In other words, H(~o) gives frequency characteristics of the system. On the contrary, the half width of the peak, which is usually adopted to express the resolving power, 224
THE SUPER
RESOLUTION
OF G A M M A - R A Y
is a numerical value and not sufficient to describe the characteristics of the system1). When two peaks come nearer to each other, then the shape of resultant output function has a smaller depth between them and when the energy difference between the peaks becomes smaller than the half width, the mixed spectra have no more two peaks but only one. This example may be
225
SPECTRUM
half width of the peak. In fact, if h(E) is gaussian, then the value H(2n/w) is about 3 ~ of H(0), where w is the half width. Therefore, the system contains much information to resolve the mixed spectra and by means of above described method, we can resolve the mixed gamma-ray spectra, even if the energy difference is smaller than the usually defined resolving power. ,xf = @
~E gf=
=
!
12
14
j, ~k j ~1I
r
/
,\ \
\
// /
\\ \
r.
Fig. 1. S u p e r p o s i t i o n of two g a u s s i a n distributions. H a l f width: 11.8 channels.
seen in fig. 1. There the two gaussian distributions with the half width of 11.8 channels are superposed with different kind of intervals between them. We observe that the interval becomes smaller than the half width, then the resultant function has only one peak. From this point of view, the definition of the resolving power by the half width seems to be appropriate. But it cannot be said that the function of the mixed spectra contains no information to resolve these spectra. Concerning this problem, the following explanation would be illustrative. Suppose that the input function is sinusoidal taking the value between 0 and 2, as 1 + cos coE, then the following relation is obtained : (gmax -- gmin)/(g..... + gm~,) ---- [ /4(CO) I
(3)
where gm~x and gmln are the maximum and minimum value of g, respectively. Therefore, at the point where H(co) becomes zero, the ouput function cannot be resolved. But actually the function H(co) has a finite value beyond the frequency which corresponds to the
In most cases, the incident gamma-rays are the superposition of two monochromatic gamma-rays. That gives
f ( E ) = Aff~(E - E,) + A23(E - E2).
(4)
Then
G(co)/H(co) = F(co) = f exp ( - icoE)f(E) dE = A l e x p ( - icoE,) + A z e x p ( - i r n E z ) .
(5)
The absolute value of this function has minima at c o = ( 2 n - l ) T r / ] E , - E z[ and maxima at co = 2mz/f E 1 - E z [, where n is a positive integer. From the observation of these maximum and minimum values, we can decide the energy difference. From a simple algebra, we know that the maximum value is A, + A z and the minimum value is [A, - A z [ . Therefore the inverse Fourier transforms are not necessary to identify the nuclides. These results are easily extended to the case of mixed spectra of more than two radionuclides.
226
T. INOUYE
3. The numerical simulation using Gaussian distribution
W e m a d e the n u m e r i c a l s i m u l a t i o n to confirm the reliability o f this m e t h o d . There the f o r m o f p h o t o p e a k c o r r e s p o n d i n g to the i n c i d e n t m o n o c h r o m a t i c g a m m a - r a y was considered as a G a u s s i a n d i s t r i b u t i o n :
h(E) = ( l / x / 2 n a ) e x p ( - E2/2a2).
(6)
,o
Then the o u t p u t f u n c t i o n c o r r e s p o n d i n g to the two mixed g a m m a - r a y s with energy E 1 a n d E2, respectively, was expressed as
g(E) = (l/x/8rca) [ex p { - (E - E1)2/2a 2 } + + exp ( - ( E - Ez)Z/2a z }].
(7)
TABLE 1 Numerically simulated h(E). Ch.
Count/ch.
Ch.
1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2 4 11 26 58 122 246 476 886 1583 2716 4478 7094 10798 15790 22184
_
Count/ch.
Cb_
DO.
DO.
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
no.
29945 38837 48394 57938 66644 73654 78208 79788 78208 73654 66644 57938 48394 38837 29945 22184
o.5 Count/ch.
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
15790 10798 7094 4478 2716 1583 886 476 246 122 58 26 11 4 2 0
0
0,1
o.2
03
o4
o 5
,~6
0,7
o.8
o,~
~.o
11 O)
Fig. 3. Absolute value of F(e)). The energy E was expressed using the unit o f the channels in the pulse height analyser. W e p u t a = 5 getting thus the h a l f w i d t h o f 11.8 channels. The function h(E) was calculated for every integral value o f E as shown in table 1. In a similar way, we got such series o f s a m p l e d values o f g(E) for the energy differences o f 14, 12 ( a b o u t h a l f width), 10, 8, 6, 4, 2 a n d 1 a n d some o f t h e m are shown in fig. 1. Then f r o m these s a m p l e d values, we calculated n u m e r i c a l l y the F o u r i e r transforms. The calculation was carried o u t b y electronic c o m p u t e r I B M 7090 a c c o r d i n g to the modified F i l o n ' s formula2). The a b s o l u t e value o f thus o b t a i n e d H(~o) is shown in fig. 2. The I F(~o) ['s which were obtained b y the f o r m u l a I F(~o)[ = [ G(o~)/H(o~)[ are shown in fig. 3. T h e y are expected to have the form
ii
I
¢o /
to J
P
½l J e x p ( - i~E) { 3 ( E - E,) + 6 ( E - EE)} dE I =
c.t
= ½{2+2coso(E
1 - - E 2 ) } ¢.
(8)
,o s
ic •
05
/O
Fig. 2. Absolute value of simulated H(o~),
/ 5
Except the case o f A E = E 2 - E1 = 1, they coincide with this function. F o r the d e t e r m i n a t i o n o f the energies, these functions should have the m i n i m u m at co = n/AE and this is observed if AE > 3 is satisfied. F o r the cases o f A E = 2 a n d 1, by the e x t r a p o l a t i o n , we can k n o w the p o i n t where F(~o) becomes zero. In
THE SUPER RESOLUTION OF GAMMA-RAY SPECTRUM table 2, the estimated values by the formula A E = n/~o o are c o m p a r e d with the originally given values, where ~oo is the first zero point o f F(¢o). They show a remarkable agreement. All these ] F(to)['s differ from the analytically calculated value when the energy frequency exceeds from about 1.0. This critical value is estimated by the condition of H(og)/H(O) ~ 10 -5 and this is the result that the initial sampled values have the error of 10 -5 for the integrated value. F r o m these results, we can decide the energy difference even when the energy difference is less than 1/10 of the usually defined resolving power. TABL~ 2 Comparison between given A E and estimated AE. Originally ] given A E I
F r o m these results it may be stated that the resolving power is effectively increased by this method.
,o /<
(2.)
e,~
Estimated ~o
AE
0.225 0.263 0.314 0.393 0.524 0.786 1.057" 1.684" 4.745*
14.00 12.00 10.00 8.00 6.00 4.00 3.00 2.00 1.00
227
¢
13.96 11.94 10.01 7.99 6.00 4.00 2.97 1.87 0.66
0,I
0,2
(P3
04
0,5
06
c, 7
~o
0,7
/.0
cO
Fig. 4. Absolute value of F(~o). 4. The application to the determination of the energy difference between two gamma-rays from Zr s5
* Estimated by extrapolation. N o w we show that the abundance ratio can be also accurately obtained by this method. As described in the preceding part, we got the sampled value o f g(E) --- A t exp { -- (E -- E1)2/50 ) +
+ A2 exp { - (E - E2)2/50 }
(9)
for every integral value of E. The combinations of A 1, A2 and A E are shown in table 3. The functions I r(co) [ = I A1 exp ( - i c o E 0 + A2 exp ( - i o g E z ) I were obtained in the same way and shown in fig. 4. The estimated values of A 1, A2 and A E by this method are shown in table 3. They also show a good agreement with the originally given values.
The two gamma-rays with energy of 757 keV and 724 keV, respectively, cannot be observed separately because o f their small energy difference (about the half o f the usually defined resolving power), so the separation o f this spectrum has been impossible by usual method. We applied our method to the determination o f the energy difference between these two gamma-rays. The decay scheme o f Zr 95 s) is shown in fig. 7. By means of the anion exchange c h r o m a t o g r a p h i c method 4) of the chloro-fluoro-complex, Zr 95 was exXIO 3
I0 q.
@0 I
I
~7 $
O O
4
T~ABLE 3
Comparison between given and estimated values. Case
]
Originally given
,
Estimated
2
i
(1) (2) (3)
d
] r
6.00 6.00 4.00
1
1: 2 1: 5 3: 10
i I
6.00 6.00 4.08
1 : 2.00 1: 5.00 1 : 3.36
\ *%.
o 200
250
Fig. 5. The function g(E) of Zr95.
c~an.e.tO
228
T. I N O U Y E
tracted from the Z r 9 5 - N b 9s mixed solution which had been distributed by the Oak Ridge National Laboratory. The observed gamma-ray spectrum of Zr 95 is x tt~ ~ 10
a
<
also carried out by the electronic computer IBM 7090 with high accuracy. Fig. 7 shows the function I F(o))I obtained by the same procedure explained in the preceding section. This function has the minimum at o) = 0.397 which gives the energy difference of 7.92 channels = 27.3 keV. This value coincides with that reffered to in the literature but a little smaller. But the decision by this method would be more accurate. 5. Discussions
6
0
.."
•
l/
"?' • •
200
Fig. 6. The function
h(E)
250 cho~necO obtained from Ce 139rn.
shown in fig. 5. As the function h(E), we used the gamma-rays from Ce139m(E7 = 740 keV) which has the mono-energetic gamma-ray with the energy near those of Zr 95 and the observed output function h(E)is shown in fig. 6. The calculation of the Fourier transforms was I.O
5
t2~
The sampling theorem says that: if the f u n c t i o n f ( x ) contains no higher frequency components than n/w t h e n f ( x ) ' s at every sampled value o f x with the interval of W decide the function exactly. In the case above described, the higher frequency components than co= ~/channel are usually less than 10 -5, so these sampled values at every integral value of E, which are just observed by the pulse height analyser, determines the originally given spectrum almost exactly. Therefore, nonintegral value of the channel number has meaning. The limit of this method for the determination of energy difference is decided by the condition how much frequency components can be neglected. This condition depends on the distribution of H(m) and the statistical fluctuation of the counts per channel. The accuracy of the numerical computation of the Fourier transforms was examined in many cases. In fig. 2 the Fourier transform of gaussian distribution shows a good agreement with the analytically obtained value which is parabolic form in semi-log table when ~o < 1.0 is satisfied. The Filon's formula was compared with other formulas such as Simpson's formulaS). As a result, it was proved that the Filon's formula is most exact and accurate enough for this method. Here we have limited the application of this method to the gamma-ray spectrometry but it can be applied to any other radiation spectrometries. The author would like to thank Dr. H. K a m o g a w a and Dr. J. Terada for their helpful discussions. He also wish to acknowledge the collaboration of Dr. I. Fujii and Mr. H. Muto in the experiment and Mr. !. Watanabe in the numerical calculation. References
0/
o,Y
03
0.4
o5
oJ
Fig. 7. Absolute value of F(oJ) obtained from Zr 95 and the level scheme of Zr95.
1) T. Inouye and T. Yoshioka, Japan Conf. Radioisotopes, 5 No. 3 (1963) 133; Nucl. Sci. Abstr., 17:30103 (1963). 2) E. A. Flinn Jour. ACM, 7 (1960) 181. 3) p. W. King, Rev. Mod. Phys., 26 (1954) 327. 4) j. L. Hague et. al., Jour. Research N. B. S., 62 (1959) 11. 5) 1. Watanabe, Jo-ho Shori (in Japanese) (to be published).
INSTRUMENTS
A N D M E T H O D S 30 (I964) 2 2 4 - - 2 2 8 ;
~
NORTH-HOLLAND
PUBLISHING
CO.
THE S U P E R R E S O L U T I O N OF G A M M A - R A Y S P E C T R U M T. I N O U Y E
Central Research Laboratory, Tokyo Shibaura Electric Co., Ltd. Kawasaki, Japan Received 16 March 1964
The new method to analyze the gamma-ray spectrum was developed using the Fourier transforms. According to this method, the same result was obtained as the resolving power was effectively increased. This method was examined by numerical
simulation a n d the increase of resolution was proved. It was successfully applied to the determination of the energy difference between two g a m m a - r a y s from Zr 95 which h a d been impossible because of small energy difference.
1. Introduction
output. As input, we take here the spectrum of gammarays and as output the pulse height distribution. For the unit of variables of both functions the energy E is chosen. The input and output are related to each other by the linear relation. That is; if the input of q~t and ~o2 corresponds to the output of ~b1 and ~b2, respectively, then the input of a~pl +b~p 2 gives the output of a~bl + bOz. We assume also the invariable relation of the output functions form at a small shift of energy and that is the case when the energy shift is smaller than several times of the resolving power. In this case the response g(E), corresponding to the input o f f ( E ) , can be expressed by using h(E) which is the output corresponding to the input 6(E) as follows :
The gamma-ray spectrometry has many merits; the structure of the gamma-ray spectrum is simple and, accordingly, the measurement of that is very easily carried out by using a multichannel pulse height analyser. The preparation of samples to be analyzed is likewise easy. It is on account of these advantages that the gamma-ray spectrometry is widely used for the purpose of simply identifying the radionuclides in the research field of nuclear physics and radiochemistry. The faults of this technique are, however, that the resolving power of the measuring system is no more than about 7% so long as scintillation method is used for the detection of gamma-rays and, accordingly, it is difficult to distinguish two or more mixed radionuclides from which gamma-rays are emitted with a smaller energy difference than the resolving power of the measuring system, except if other techniques, for example, the analysis of decay curve or the chemical separation are employed. Particularly, when the half-lives of two or more radionuclides are of nearly the same value and the chemical separation of them is difficult, then, until now it was impossible to identify individually these radionuclides. This paper deals with the application of the Fourier transformation method to analyze the mixed gammaray spectra. By means of this method, the resolving power of the generally used gamma-ray spectrometer was effectively increased about ten times. This method was examined by the numerical simulation in which the output functions were constructed artificially and treated in the same way just as we treat the radiation spectrum. Then it was applied to the determination of the energy difference between two gamma-rays from Z r 95 which had been impossible because of small energy difference.
2. Theory We consider the whole system as a "black box" and take into account only the relation between input and
g(E) =
f(E')h(E - E')dE'
(1)
--o0
where 6(E) is the Dirac's delta function. This relation is alternatively expressed by using the Fourier transforms of these functions. That is: (2)
=
where F(co) =
exp(-
ie)E)f(E)dE, etc.
-ao
In the gamma-ray spectrometry, we have the information about above described g(E) and wish to know the function f(E) based on the knowledge about g(E). From the relation (2), we can know F(co) and by the inverse Fourier transforms we can have the inputf(E). The h(E) is the output function corresponding to the input of the delta function and it can be obtained by the observation of the monochromatic gamma-rays. This is a good function to express the characteristics of the system because the input function 6(E) equally includes all the components of frequency. In other words, H(~o) gives frequency characteristics of the system. On the contrary, the half width of the peak, which is usually adopted to express the resolving power, 224
THE SUPER
RESOLUTION
OF G A M M A - R A Y
is a numerical value and not sufficient to describe the characteristics of the system1). When two peaks come nearer to each other, then the shape of resultant output function has a smaller depth between them and when the energy difference between the peaks becomes smaller than the half width, the mixed spectra have no more two peaks but only one. This example may be
225
SPECTRUM
half width of the peak. In fact, if h(E) is gaussian, then the value H(2n/w) is about 3 ~ of H(0), where w is the half width. Therefore, the system contains much information to resolve the mixed spectra and by means of above described method, we can resolve the mixed gamma-ray spectra, even if the energy difference is smaller than the usually defined resolving power. ,xf = @
~E gf=
=
!
12
14
j, ~k j ~1I
r
/
,\ \
\
// /
\\ \
r.
Fig. 1. S u p e r p o s i t i o n of two g a u s s i a n distributions. H a l f width: 11.8 channels.
seen in fig. 1. There the two gaussian distributions with the half width of 11.8 channels are superposed with different kind of intervals between them. We observe that the interval becomes smaller than the half width, then the resultant function has only one peak. From this point of view, the definition of the resolving power by the half width seems to be appropriate. But it cannot be said that the function of the mixed spectra contains no information to resolve these spectra. Concerning this problem, the following explanation would be illustrative. Suppose that the input function is sinusoidal taking the value between 0 and 2, as 1 + cos coE, then the following relation is obtained : (gmax -- gmin)/(g..... + gm~,) ---- [ /4(CO) I
(3)
where gm~x and gmln are the maximum and minimum value of g, respectively. Therefore, at the point where H(co) becomes zero, the ouput function cannot be resolved. But actually the function H(co) has a finite value beyond the frequency which corresponds to the
In most cases, the incident gamma-rays are the superposition of two monochromatic gamma-rays. That gives
f ( E ) = Aff~(E - E,) + A23(E - E2).
(4)
Then
G(co)/H(co) = F(co) = f exp ( - icoE)f(E) dE = A l e x p ( - icoE,) + A z e x p ( - i r n E z ) .
(5)
The absolute value of this function has minima at c o = ( 2 n - l ) T r / ] E , - E z[ and maxima at co = 2mz/f E 1 - E z [, where n is a positive integer. From the observation of these maximum and minimum values, we can decide the energy difference. From a simple algebra, we know that the maximum value is A, + A z and the minimum value is [A, - A z [ . Therefore the inverse Fourier transforms are not necessary to identify the nuclides. These results are easily extended to the case of mixed spectra of more than two radionuclides.
226
T. INOUYE
3. The numerical simulation using Gaussian distribution
W e m a d e the n u m e r i c a l s i m u l a t i o n to confirm the reliability o f this m e t h o d . There the f o r m o f p h o t o p e a k c o r r e s p o n d i n g to the i n c i d e n t m o n o c h r o m a t i c g a m m a - r a y was considered as a G a u s s i a n d i s t r i b u t i o n :
h(E) = ( l / x / 2 n a ) e x p ( - E2/2a2).
(6)
,o
Then the o u t p u t f u n c t i o n c o r r e s p o n d i n g to the two mixed g a m m a - r a y s with energy E 1 a n d E2, respectively, was expressed as
g(E) = (l/x/8rca) [ex p { - (E - E1)2/2a 2 } + + exp ( - ( E - Ez)Z/2a z }].
(7)
TABLE 1 Numerically simulated h(E). Ch.
Count/ch.
Ch.
1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
2 4 11 26 58 122 246 476 886 1583 2716 4478 7094 10798 15790 22184
_
Count/ch.
Cb_
DO.
DO.
18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
no.
29945 38837 48394 57938 66644 73654 78208 79788 78208 73654 66644 57938 48394 38837 29945 22184
o.5 Count/ch.
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
15790 10798 7094 4478 2716 1583 886 476 246 122 58 26 11 4 2 0
0
0,1
o.2
03
o4
o 5
,~6
0,7
o.8
o,~
~.o
11 O)
Fig. 3. Absolute value of F(e)). The energy E was expressed using the unit o f the channels in the pulse height analyser. W e p u t a = 5 getting thus the h a l f w i d t h o f 11.8 channels. The function h(E) was calculated for every integral value o f E as shown in table 1. In a similar way, we got such series o f s a m p l e d values o f g(E) for the energy differences o f 14, 12 ( a b o u t h a l f width), 10, 8, 6, 4, 2 a n d 1 a n d some o f t h e m are shown in fig. 1. Then f r o m these s a m p l e d values, we calculated n u m e r i c a l l y the F o u r i e r transforms. The calculation was carried o u t b y electronic c o m p u t e r I B M 7090 a c c o r d i n g to the modified F i l o n ' s formula2). The a b s o l u t e value o f thus o b t a i n e d H(~o) is shown in fig. 2. The I F(~o) ['s which were obtained b y the f o r m u l a I F(~o)[ = [ G(o~)/H(o~)[ are shown in fig. 3. T h e y are expected to have the form
ii
I
¢o /
to J
P
½l J e x p ( - i~E) { 3 ( E - E,) + 6 ( E - EE)} dE I =
c.t
= ½{2+2coso(E
1 - - E 2 ) } ¢.
(8)
,o s
ic •
05
/O
Fig. 2. Absolute value of simulated H(o~),
/ 5
Except the case o f A E = E 2 - E1 = 1, they coincide with this function. F o r the d e t e r m i n a t i o n o f the energies, these functions should have the m i n i m u m at co = n/AE and this is observed if AE > 3 is satisfied. F o r the cases o f A E = 2 a n d 1, by the e x t r a p o l a t i o n , we can k n o w the p o i n t where F(~o) becomes zero. In
THE SUPER RESOLUTION OF GAMMA-RAY SPECTRUM table 2, the estimated values by the formula A E = n/~o o are c o m p a r e d with the originally given values, where ~oo is the first zero point o f F(¢o). They show a remarkable agreement. All these ] F(to)['s differ from the analytically calculated value when the energy frequency exceeds from about 1.0. This critical value is estimated by the condition of H(og)/H(O) ~ 10 -5 and this is the result that the initial sampled values have the error of 10 -5 for the integrated value. F r o m these results, we can decide the energy difference even when the energy difference is less than 1/10 of the usually defined resolving power. TABL~ 2 Comparison between given A E and estimated AE. Originally ] given A E I
F r o m these results it may be stated that the resolving power is effectively increased by this method.
,o /<
(2.)
e,~
Estimated ~o
AE
0.225 0.263 0.314 0.393 0.524 0.786 1.057" 1.684" 4.745*
14.00 12.00 10.00 8.00 6.00 4.00 3.00 2.00 1.00
227
¢
13.96 11.94 10.01 7.99 6.00 4.00 2.97 1.87 0.66
0,I
0,2
(P3
04
0,5
06
c, 7
~o
0,7
/.0
cO
Fig. 4. Absolute value of F(~o). 4. The application to the determination of the energy difference between two gamma-rays from Zr s5
* Estimated by extrapolation. N o w we show that the abundance ratio can be also accurately obtained by this method. As described in the preceding part, we got the sampled value o f g(E) --- A t exp { -- (E -- E1)2/50 ) +
+ A2 exp { - (E - E2)2/50 }
(9)
for every integral value of E. The combinations of A 1, A2 and A E are shown in table 3. The functions I r(co) [ = I A1 exp ( - i c o E 0 + A2 exp ( - i o g E z ) I were obtained in the same way and shown in fig. 4. The estimated values of A 1, A2 and A E by this method are shown in table 3. They also show a good agreement with the originally given values.
The two gamma-rays with energy of 757 keV and 724 keV, respectively, cannot be observed separately because o f their small energy difference (about the half o f the usually defined resolving power), so the separation o f this spectrum has been impossible by usual method. We applied our method to the determination o f the energy difference between these two gamma-rays. The decay scheme o f Zr 95 s) is shown in fig. 7. By means of the anion exchange c h r o m a t o g r a p h i c method 4) of the chloro-fluoro-complex, Zr 95 was exXIO 3
I0 q.
@0 I
I
~7 $
O O
4
T~ABLE 3
Comparison between given and estimated values. Case
]
Originally given
,
Estimated
2
i
(1) (2) (3)
d
] r
6.00 6.00 4.00
1
1: 2 1: 5 3: 10
i I
6.00 6.00 4.08
1 : 2.00 1: 5.00 1 : 3.36
\ *%.
o 200
250
Fig. 5. The function g(E) of Zr95.
c~an.e.tO
228
T. I N O U Y E
tracted from the Z r 9 5 - N b 9s mixed solution which had been distributed by the Oak Ridge National Laboratory. The observed gamma-ray spectrum of Zr 95 is x tt~ ~ 10
a
<
also carried out by the electronic computer IBM 7090 with high accuracy. Fig. 7 shows the function I F(o))I obtained by the same procedure explained in the preceding section. This function has the minimum at o) = 0.397 which gives the energy difference of 7.92 channels = 27.3 keV. This value coincides with that reffered to in the literature but a little smaller. But the decision by this method would be more accurate. 5. Discussions
6
0
.."
•
l/
"?' • •
200
Fig. 6. The function
h(E)
250 cho~necO obtained from Ce 139rn.
shown in fig. 5. As the function h(E), we used the gamma-rays from Ce139m(E7 = 740 keV) which has the mono-energetic gamma-ray with the energy near those of Zr 95 and the observed output function h(E)is shown in fig. 6. The calculation of the Fourier transforms was I.O
5
t2~
The sampling theorem says that: if the f u n c t i o n f ( x ) contains no higher frequency components than n/w t h e n f ( x ) ' s at every sampled value o f x with the interval of W decide the function exactly. In the case above described, the higher frequency components than co= ~/channel are usually less than 10 -5, so these sampled values at every integral value of E, which are just observed by the pulse height analyser, determines the originally given spectrum almost exactly. Therefore, nonintegral value of the channel number has meaning. The limit of this method for the determination of energy difference is decided by the condition how much frequency components can be neglected. This condition depends on the distribution of H(m) and the statistical fluctuation of the counts per channel. The accuracy of the numerical computation of the Fourier transforms was examined in many cases. In fig. 2 the Fourier transform of gaussian distribution shows a good agreement with the analytically obtained value which is parabolic form in semi-log table when ~o < 1.0 is satisfied. The Filon's formula was compared with other formulas such as Simpson's formulaS). As a result, it was proved that the Filon's formula is most exact and accurate enough for this method. Here we have limited the application of this method to the gamma-ray spectrometry but it can be applied to any other radiation spectrometries. The author would like to thank Dr. H. K a m o g a w a and Dr. J. Terada for their helpful discussions. He also wish to acknowledge the collaboration of Dr. I. Fujii and Mr. H. Muto in the experiment and Mr. !. Watanabe in the numerical calculation. References
0/
o,Y
03
0.4
o5
oJ
Fig. 7. Absolute value of F(oJ) obtained from Zr 95 and the level scheme of Zr95.
1) T. Inouye and T. Yoshioka, Japan Conf. Radioisotopes, 5 No. 3 (1963) 133; Nucl. Sci. Abstr., 17:30103 (1963). 2) E. A. Flinn Jour. ACM, 7 (1960) 181. 3) p. W. King, Rev. Mod. Phys., 26 (1954) 327. 4) j. L. Hague et. al., Jour. Research N. B. S., 62 (1959) 11. 5) 1. Watanabe, Jo-ho Shori (in Japanese) (to be published).