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Method of evaluating delayed coincidence experiments
NUCLEAR
INSTRUMENTS
AND
METHODS
9 (1960)
149-153; N O R T H - I : O L L A N D
PUBLISHING
CO.
METHOD OF EVALUATING DELAYED COINCIDENCE EXPERIMENTS R. S. W E A V E R a nd R. E. B E L L
Radiation Laboratory, McGill University, Montreal R e c e i v e d 25 J u l y 1960
F r e q u e n t l y in d e l a y e d c o i n c i d e n c e w o r k a r e s o l u t i o n c u r v e is m e a s u r e d w i t h good a c c u r a c y , b u t no e x a c t l y c o m p a r a b l e p r o m p t r e s o l u t i o n c u r v e is a v a i l a b l e . T h e c e n t r o i d - s h i f t m e t h o d of e v a l u a t i n g t h e l i f e t i m e T c a n n o t be a p p l i e d bec a u s e t h e zero of t i m e (centroid of t h e p r o m p t curve) is n o t k n o w n . If z is close to t h e i n s t r u m e n t a l l i m i t , i t c a n n o t be m e a s u r e d fro m t h e l o g a r i t h m i c slope of t h e m e a s u r e d r e s o l u t i o n curv e. \Ve t h e n r e q u i r e an o b j e c t i v e m e a s u r e of t h e a s y m m e t r y of t h e m e a s u r e d curve, c a l c u l a b l e w i t h o u t r e f e r e n c e to a n y o t h e r curve. Such a q u a n t i t y is t h e t h i r d
m o m e n t of t h e m e a s u r e d c u r v e a b o u t i t s own c e n t r o i d . \Ve s h o w t h a t t h e n o r m a l i z e d t h i r d m o m e n t is e q u a l to (2z a + e), w h e r e ~ is t h e n o r m a l i z e d t h i r d m o m e n t of tile p r o m p t c urve , u s u a l l y s m a l l a nd f r e q u e n t l y n e g l i g i b l e . Thus T c a n be e v a l u a t e d w i t h a t m o s t o n l y r o u g h k n o w l e d g e of tile p r o m p t curve. T h e McGill IB M 650 c o m p u t e r has be e n p r o g r a m m e d to p e r f o r m t h e a r i t h m e t i c a l work, w hi c h is o t h e r w i s e laborious. An a p p l i c a t i o n to a n a c t u a l e x a m p l e is shown. E s t i m a t e s are g i v e n of t h e a c c u r a c y to be e x p e c t e d from the method.
1. Introduction
delayed resolution curves. Their method thus depends on accurate knowledge of the prompt curve, and in their experiments they adopt a fast cycling procedure to minimize drifts.
In measuring short lifetimes by the delayed coincidence method, the usual procedure is to measure the time resolution curve for the radiation in question (the "delayed" curve), and a curve of the time resolution of the apparatus for a source of radiation of very short lifetime (the " p r o m p t " curve). Ideally the prompt curve should be measured with a source identical in all respects to the delayed source, except for the lifetime. Such a source is almost never available. The position of the centroid of the prompt curve defines the zero of time for the experiment. Unfortunately this centroid position is sensitive to the detailed properties of the prompt source, and is also subject to instrumental drifts. These effects m a y cause large errors in evaluating the lifetime by the centroid-shift method of Bay 1) or the area analysis method of NewtonZ), if the lifetime being measured is close to the instrumental limit. Birk, Goldring and Wolfsona) have recently introduced a method of analysis making detailed use of the moments of the prompt and
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I I t I
I I TIME
1) Z. Bay, P h y s . R e v . 77 (1950) 419. 2) T. D. N e w t o n , P h y s . R e v . 78 (1950) 490. 3) M. Birk, G. G o l d r i n g a n d Y. Wolfson, P h y s . R e v . 116 (1959) 730.
T
Fig. l. N o r m a l i z e d p r o m p t a n d d e l a y e d c oi nc i de nc e curves. The p r o m p t c u r v e is d r a w n w i t h a b r o k e n l i ne b e c a u s e w e a s s u m e t h a t i t h a s n o t be e n o b s e r v e d a c c u r a t e l y . 149
150
R.S.
WEAVER
AND
2. Derivation of Moment Relations We now show t h a t it is possible to e v a l u a t e the required lifetime from the shape of the delayed c u r v e with o n l y rough knowledge of the shape of the p r o m p t curve. The relative positions of the centroids of the two curves do not e n t e r the analysis. In fig. 1, P(x) is the u n o b s e r v e d true p r o m p t curve and F(x) is the m e a s u r e d delayed curve, b o t h normalized to unit area a n d p l o t t e d on a logarithmic scale as a function of x, the d e l a y time. The true zero of t i m e coincides with the centroid of the u n o b s e r v e d p r o m p t curve P(x), a n d on this scale the centroid of F(x) lies at + ~, where z is the m e a n life to be e v a l u a t e d . B a y 1) has shown t h a t the following relation holds a m o n g the m o m e n t s of the two curves:
Mr(F) = ~
r!
lc:o k!(r--k)!
Mr-k(P) M~- (w)
(1)
where Mr(F) is the rth m o m e n t of F(x),
Mr(F)
=
['°° xr F(x) dx d-- oo
N o w let us t a k e m o m e n t s a b o u t the (calculable) centroid of the delayed curve. W e denote such m o m e n t s b y the letter N r a t h e r t h a n the previous M. Since the eentroid of F(x) lies at + ~, we now h a v e Nr(V)
= f2
( x - - T) r F ( x ) d x .
(8)
E x p a n d i n g (8) for the cases r = I, 2, 3, and s u b s t i t u t i n g k n o w n values, we get NI(F) = 0
(definition of centroid of F)
(9)
N2(F) = a 2 + ,~
(10)
N3(F) = 2z:~ + ~.
(11)
T h u s the procedure is to calculate the centroid of the observed delayed curve F(x), a n d a d o p t it as the origin of x, ensuring t h a t (9) is obeyed. T h e n upon c o m p u t i n g N3(F) f r o m the e x p e r i m e n t a l data, we h a v e = { N 3 ( F ) - - , }at " (12) 2
(2) Here e, the third m o m e n t of the p r o m p t curve
and ~o(x) = [z -1 e x p ( - - x / z ) ] is the r a d i o a c t i v e d e c a y law for the r a d i a t i o n of m e a n life z. W e h a v e at once b y i n t e g r a t i o n Mn(oJ) -- u!vn; in particular Mt
R. E. BELL
P(x) a b o u t its own centroid, is usually small a n d f r e q u e n t l y negligible, so t h a t for m a n y cases it is sufficient to write
N3(F)
(a~) = r
M ~ (~o) = Ma(~o)
232
(3)
-- 638 ,
and b y definition of the zero of time, Mt (P) =: O.
(4)
W e also define M2(P) a 2 (this does not i m p l y t h a t P ( x ) i s a Gaussian), and M s ( P ) s. N o w writing out B a y ' s relation (1) for r = 1,2 and 3, we h a v e M1 (f) = T (5) M2 (F) -- a2 + 2r2
(6)
M3 (F) -- 3a2T + 6~3 + e .
(7)
These are the first three m o m e n t s of the delayed curve a b o u t the zero of time, i.e. a b o u t the (unknown) centroid of the p r o m p t curve.
In a n y case, rough knowledge of P(x) will give a value of e ( ~ M3(P)) sufficiently accurate to m a k e a correction. E v e n if e is as large as ½ of Na(F), the error in z caused b y ignoring it alt o g e t h e r is only a b o u t I0 %.
3. Sample Computations T h e e x p e r i m e n t a l d a t a usually consist of t o t a l counts in a n u m b e r of t i m e channels, recorded in the m u l t i c h a n n e l a n a l y z e r t h a t f o r m s the o u t p u t of a t i m e - t o - a m p l i t u d e converter. T h e a r i t h m e t i cal operations are simple, b u t laborious if performed b y hand. The McGill I B M 650 c o m p u t e r has been p r o g r a m m e d to do the calculations. T h e o u t p u t d a t a consists of the t o t a l a r e a of the curve, and the normalized values of the first three m o m e n t s .
DELAYED
COINCIDENCE
As an e x a m p l e , we use the d a t a f r o m m e a s u r e m e n t of the half life of the 1507 excited s t a t e of Gd 156, published b y Bell J o r g e n s e n 4) (fig. 2). In this case the half-life .
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I
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the keV and was
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GdlS6:DECAYOF 4+ STATE AT 1507keY
10 3
-10
r,/2:(1.88*-O.1O),dO s O
t~ o
PROMPT (-3.3)
-7 O t.)
lo
0
10
20
DELAY FROMCENTRE OF PROMPT CURVE ClO-~°s UNITS) Fig. 2. A d e l a y e d r e s o l u t i o n c u r v e m e a s u r e d by Bell and Jorgensen4), w i t h an a p p r o x i I n a t e p r o m p t c u r v e for comparison, used in s a m p l e c a l c u l a t i o n s by the t h i r d m o m e n t method.
r e a d f r o m the slope of the exponential d e c a y of the d e l a y e d curve, a n d was given as (1.88 _k 0.10) x 10 10 s. T h e assigned error of a b o u t 5 % was m a i n l y due to causes o t h e r t h a n statistical. The p r o m p t c u r v e in the d i a g r a m is an e x a m p l e of the case discus.~;ed above, representing the app r o x i m a t e shape, but not the exact position, of the true p r o m p t curve. The half-life of this state is short enough to p e r m i t a useful application of the m o m e n t procedure, and long enough to provide i n d e p e n d e n t m e a n s of d e t e r m i n a t i o n of the half life. A c o m p u t e r calculation of the third m o m e n t of the delayed curve yielded a result e q u i v a l e n t to a half-life of 1.88 x 10 10s, the exact a g r e e m e n t with the published value being fortuitous. The calculated third m o m e n t of the a) R. E. Bell and ?,I. H. J o r g e n s e n , Nucl. Phys. 12 (1959) 413
EXPERIMENTS
151
p r o m p t curve was 0 . 1 2 % of t h a t of the delayed curve, so t h a t the error caused b y neglecting it in calculating T was only 0.04%. This implies t h a t the rough practical limit m e n t i o n e d a b o v e (e ½N3(F)) would h a v e been reached if the c o m p u t e d third m o m e n t of the d e l a y e d curve h a d been a b o u t 300 times smaller, i.e. if the half life h a d been a b o u t 3 x 10 -11 s. As the c o m p u t e r lends itself to fast calculations of such quantities as m o m e n t s , various r e a r r a n g e m e n t s of the actual e x p e r i m e n t a l d a t a were tried. F o r example, the leading edge of the delayed coincidence curve has a slight tail at low values of time, dne to i n s t r u m e n t a l resolution difficulties; at low values of counts on both leading and trailing edges the c u r v e has poor statistics. T h e c u r v e can be s m o o t h e d out, and new values of counts in a channel read from this s m o o t h e d curve. Using these "smoothed"values, a value of T was o b t a i n e d differing b y less t h a n 0.1% f r o m the value o b t a i n e d with the original figures. In a n o t h e r test, successively m o r e and more values at the e x t r e m e s of the c u r v e were eliminated, and the value of T o b t a i n e d was found to be quite insensitive. F o r example, the curve for Gd is6 (fig. 2) has values e x t e n d i n g o v e r more t h a n 4 decades. If all values m o r e than 3 decades below the p e a k are neglected, the value of T o b t a i n e d is only 4 % lower t h a n the " c o r r e c t " value. T h u s even a rough e x t r a p o l a t i o n of the 3decade curve to correct for this neglect will give an error m u c h less t h a n 4 % . The second m o m e n t of the p r o m p t curve, as c o m p u t e d from the delayed curve, was 5.76 units; the second m o m e n t , as found directly from the a p p r o x i m a t e p r o m p t curve itself, was 5.43 units.
4. Errors The .~tatistical errors t h a t one should assign to these values of a and r could h a v e been comt>uted b y h a v i n g the machine c a r r y the s t a n d a r d deviations of the individual points through tile calculation. However, in order to give the error in closed form, we now assume t h a t the p r o m p t curve is a Gaussian of s t a n d a r d deviation a. (The result is not sensitive to this assumption.) In t h a t case it is easily shown (see Appendix) t h a t
159
R. S. WEAVER AND R. E. BELL
the fractional s t a n d a r d deviation in the comp u t e d value of the lifetime is ZlV ( ~ 2 ) ~ [~/A(1~ ) 6 r
+ 3(-~)4 -[- 9(-~)2 @ 53]½3j
where A is the t o t a l of all the counts in the delayed curve. Applied to our present e x a m p l e , this e s t i m a t e gives a s t a n d a r d deviation in v of 1.3%. In an e x p e r i m e n t like this, a s y s t e m a t i c error of a b o u t 3 % is assigned to c o v e r i n s t r u m e n t calibration and linearity; the statistical error in this case is therefore negligible. In our e x a m p l e , if the half life b e c a m e smaller, the fractional s t a n d a r d deviation would rise, reaching 20 % for a h a l f l i f e o f ~ 3 × 10 - u s e c .
5. Concluding Remarks A n o t h e r use of this t h i r d - m o m e n t technique would be in the "self-comparison" m e t h o d of Bell, G r a h a m and Petch5). The two d e l a y e d a) R. E. Bell, R. L. Graham, and H. F. Petch, Can. J. Phys. 30 (1952) 35.
curves o b t a i n e d would h a v e a difference between their third m o m e n t s of 4~a, while the third m o m e n t of the p r o m p t curve would c o n t r i b u t e nothing, to first order; hence the lower limit of m e a s u r e m e n t of the half life could be s o m e w h a t decreased. T h r o u g h o u t this p a p e r we h a v e a s s u m e d t h a t the delayed c u r v e corresponds to a single exponential decay, in p a r t i c u l a r t h a t chance coincidences h a v e been s u b t r a c t e d and t h a t no p r o m p t events are occurring in the delayed curve. T h e neglected presence of either kind of e v e n t will cause errors, as it does in the centroid-shift case. A m o m e n t analysis can be m a d e to t a k e account of either t y p e of event, as shown b y Birk el al. 3) for the centroid-shift case, but the work is laborious and d e p e n d e n t on the exact circumstances of the m e a s u r e m e n t . One of us (R.S.W.) expresses his t h a n k s to the N a t i o n a l Research Council of C a n a d a for a S t u d e n t s h i p held during the course of this work.
Appendix CALCULATION OF FRACTIONAL STANDARD DEVIATION
To calculate the s t a n d a r d deviation in our c o m p u t e d r, we assume t h a t the n u m b e r of counts, F(x) dx, in a t i m e channel of w i d t h dx has a s t a n d a r d deviation IF(x) dx I ~. This error c o n t r i b u t e s an error in the t h i r d m o m e n t of a m o u n t (x - - r ) 3 IF(x) dx] i. Combining the various channel contributions q u a d r a t i c a l l y and replacing the s u m o v e r channels b y an integral, we get
[f:o
= If2
dC
,
the second factor, for normalization, being just A, the total n u m b e r of c o u n t s in the curve. E x p a n d i n g this expression a n d m a k i n g use of eq. (1) a n d eq. (3), we get
if
A(Na) = ~ - -
]
M6(P) + 15z ~ M4(P) + 40raM3(P) + 135. 4 M2(P) + 2 6 4 r 5 M l ( P ) + 265~ 6 ½.
If P has been observed, the m o m e n t s given here m a y be c o m p u t e d . I n order to give a definite f o r m for the error, however, we now assume t h a t P is a Gaussian with s t a n d a r d deviation a; i.e. M2(P) = a 2, M4(P) = 3o 4, M6 ( P ) = 15a~, a n d all odd m o m e n t s are zero. W e t h e n get
A(Na) = ~
if
]
15a~ +45~2o 4 + 135~4a2 + 265~6 ½-
This is the t o t a l error in 2v3: hence the fractional statistical error can be expressed as
aj_
i
(5)~
53]~
DELAYED
COINCIDENCE
EXPERIMENTS
153
In fig. 3 is plotted a curve of (Av/z)~ vs the ratio of a/v, and also the ratio of W~/T½where W½ is the full width at half m a x i m u m of the prompt curve and TI is the half life being measured. If A = 10 000 counts, then the fractional error is read directly as percent.
2
4
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I0
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20
IOO
I0
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0
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2
[%]
3
4
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6
Fig. 3, Fractional s t a n d a r d d e v i a t i o n in T p l o t t e d as a function of a/~: (lower abscissa) or A is t h e total n u m b e r of c o u n t s in t h e o b s e r v e d resolution curve.
W}/T}
(upper abscissa).
INSTRUMENTS
AND
METHODS
9 (1960)
149-153; N O R T H - I : O L L A N D
PUBLISHING
CO.
METHOD OF EVALUATING DELAYED COINCIDENCE EXPERIMENTS R. S. W E A V E R a nd R. E. B E L L
Radiation Laboratory, McGill University, Montreal R e c e i v e d 25 J u l y 1960
F r e q u e n t l y in d e l a y e d c o i n c i d e n c e w o r k a r e s o l u t i o n c u r v e is m e a s u r e d w i t h good a c c u r a c y , b u t no e x a c t l y c o m p a r a b l e p r o m p t r e s o l u t i o n c u r v e is a v a i l a b l e . T h e c e n t r o i d - s h i f t m e t h o d of e v a l u a t i n g t h e l i f e t i m e T c a n n o t be a p p l i e d bec a u s e t h e zero of t i m e (centroid of t h e p r o m p t curve) is n o t k n o w n . If z is close to t h e i n s t r u m e n t a l l i m i t , i t c a n n o t be m e a s u r e d fro m t h e l o g a r i t h m i c slope of t h e m e a s u r e d r e s o l u t i o n curv e. \Ve t h e n r e q u i r e an o b j e c t i v e m e a s u r e of t h e a s y m m e t r y of t h e m e a s u r e d curve, c a l c u l a b l e w i t h o u t r e f e r e n c e to a n y o t h e r curve. Such a q u a n t i t y is t h e t h i r d
m o m e n t of t h e m e a s u r e d c u r v e a b o u t i t s own c e n t r o i d . \Ve s h o w t h a t t h e n o r m a l i z e d t h i r d m o m e n t is e q u a l to (2z a + e), w h e r e ~ is t h e n o r m a l i z e d t h i r d m o m e n t of tile p r o m p t c urve , u s u a l l y s m a l l a nd f r e q u e n t l y n e g l i g i b l e . Thus T c a n be e v a l u a t e d w i t h a t m o s t o n l y r o u g h k n o w l e d g e of tile p r o m p t curve. T h e McGill IB M 650 c o m p u t e r has be e n p r o g r a m m e d to p e r f o r m t h e a r i t h m e t i c a l work, w hi c h is o t h e r w i s e laborious. An a p p l i c a t i o n to a n a c t u a l e x a m p l e is shown. E s t i m a t e s are g i v e n of t h e a c c u r a c y to be e x p e c t e d from the method.
1. Introduction
delayed resolution curves. Their method thus depends on accurate knowledge of the prompt curve, and in their experiments they adopt a fast cycling procedure to minimize drifts.
In measuring short lifetimes by the delayed coincidence method, the usual procedure is to measure the time resolution curve for the radiation in question (the "delayed" curve), and a curve of the time resolution of the apparatus for a source of radiation of very short lifetime (the " p r o m p t " curve). Ideally the prompt curve should be measured with a source identical in all respects to the delayed source, except for the lifetime. Such a source is almost never available. The position of the centroid of the prompt curve defines the zero of time for the experiment. Unfortunately this centroid position is sensitive to the detailed properties of the prompt source, and is also subject to instrumental drifts. These effects m a y cause large errors in evaluating the lifetime by the centroid-shift method of Bay 1) or the area analysis method of NewtonZ), if the lifetime being measured is close to the instrumental limit. Birk, Goldring and Wolfsona) have recently introduced a method of analysis making detailed use of the moments of the prompt and
/
'X
/
t
P(x I - ~ l I
] I
I I I I
I I I
t
!
I
I
I
I I
I
I I I |
I I t I
I I TIME
1) Z. Bay, P h y s . R e v . 77 (1950) 419. 2) T. D. N e w t o n , P h y s . R e v . 78 (1950) 490. 3) M. Birk, G. G o l d r i n g a n d Y. Wolfson, P h y s . R e v . 116 (1959) 730.
T
Fig. l. N o r m a l i z e d p r o m p t a n d d e l a y e d c oi nc i de nc e curves. The p r o m p t c u r v e is d r a w n w i t h a b r o k e n l i ne b e c a u s e w e a s s u m e t h a t i t h a s n o t be e n o b s e r v e d a c c u r a t e l y . 149
150
R.S.
WEAVER
AND
2. Derivation of Moment Relations We now show t h a t it is possible to e v a l u a t e the required lifetime from the shape of the delayed c u r v e with o n l y rough knowledge of the shape of the p r o m p t curve. The relative positions of the centroids of the two curves do not e n t e r the analysis. In fig. 1, P(x) is the u n o b s e r v e d true p r o m p t curve and F(x) is the m e a s u r e d delayed curve, b o t h normalized to unit area a n d p l o t t e d on a logarithmic scale as a function of x, the d e l a y time. The true zero of t i m e coincides with the centroid of the u n o b s e r v e d p r o m p t curve P(x), a n d on this scale the centroid of F(x) lies at + ~, where z is the m e a n life to be e v a l u a t e d . B a y 1) has shown t h a t the following relation holds a m o n g the m o m e n t s of the two curves:
Mr(F) = ~
r!
lc:o k!(r--k)!
Mr-k(P) M~- (w)
(1)
where Mr(F) is the rth m o m e n t of F(x),
Mr(F)
=
['°° xr F(x) dx d-- oo
N o w let us t a k e m o m e n t s a b o u t the (calculable) centroid of the delayed curve. W e denote such m o m e n t s b y the letter N r a t h e r t h a n the previous M. Since the eentroid of F(x) lies at + ~, we now h a v e Nr(V)
= f2
( x - - T) r F ( x ) d x .
(8)
E x p a n d i n g (8) for the cases r = I, 2, 3, and s u b s t i t u t i n g k n o w n values, we get NI(F) = 0
(definition of centroid of F)
(9)
N2(F) = a 2 + ,~
(10)
N3(F) = 2z:~ + ~.
(11)
T h u s the procedure is to calculate the centroid of the observed delayed curve F(x), a n d a d o p t it as the origin of x, ensuring t h a t (9) is obeyed. T h e n upon c o m p u t i n g N3(F) f r o m the e x p e r i m e n t a l data, we h a v e = { N 3 ( F ) - - , }at " (12) 2
(2) Here e, the third m o m e n t of the p r o m p t curve
and ~o(x) = [z -1 e x p ( - - x / z ) ] is the r a d i o a c t i v e d e c a y law for the r a d i a t i o n of m e a n life z. W e h a v e at once b y i n t e g r a t i o n Mn(oJ) -- u!vn; in particular Mt
R. E. BELL
P(x) a b o u t its own centroid, is usually small a n d f r e q u e n t l y negligible, so t h a t for m a n y cases it is sufficient to write
N3(F)
(a~) = r
M ~ (~o) = Ma(~o)
232
(3)
-- 638 ,
and b y definition of the zero of time, Mt (P) =: O.
(4)
W e also define M2(P) a 2 (this does not i m p l y t h a t P ( x ) i s a Gaussian), and M s ( P ) s. N o w writing out B a y ' s relation (1) for r = 1,2 and 3, we h a v e M1 (f) = T (5) M2 (F) -- a2 + 2r2
(6)
M3 (F) -- 3a2T + 6~3 + e .
(7)
These are the first three m o m e n t s of the delayed curve a b o u t the zero of time, i.e. a b o u t the (unknown) centroid of the p r o m p t curve.
In a n y case, rough knowledge of P(x) will give a value of e ( ~ M3(P)) sufficiently accurate to m a k e a correction. E v e n if e is as large as ½ of Na(F), the error in z caused b y ignoring it alt o g e t h e r is only a b o u t I0 %.
3. Sample Computations T h e e x p e r i m e n t a l d a t a usually consist of t o t a l counts in a n u m b e r of t i m e channels, recorded in the m u l t i c h a n n e l a n a l y z e r t h a t f o r m s the o u t p u t of a t i m e - t o - a m p l i t u d e converter. T h e a r i t h m e t i cal operations are simple, b u t laborious if performed b y hand. The McGill I B M 650 c o m p u t e r has been p r o g r a m m e d to do the calculations. T h e o u t p u t d a t a consists of the t o t a l a r e a of the curve, and the normalized values of the first three m o m e n t s .
DELAYED
COINCIDENCE
As an e x a m p l e , we use the d a t a f r o m m e a s u r e m e n t of the half life of the 1507 excited s t a t e of Gd 156, published b y Bell J o r g e n s e n 4) (fig. 2). In this case the half-life .
.
.
.
I
.
.
.
.
~ _
I
.
.
.
.
!
.
.
.
the keV and was
.
GdlS6:DECAYOF 4+ STATE AT 1507keY
10 3
-10
r,/2:(1.88*-O.1O),dO s O
t~ o
PROMPT (-3.3)
-7 O t.)
lo
0
10
20
DELAY FROMCENTRE OF PROMPT CURVE ClO-~°s UNITS) Fig. 2. A d e l a y e d r e s o l u t i o n c u r v e m e a s u r e d by Bell and Jorgensen4), w i t h an a p p r o x i I n a t e p r o m p t c u r v e for comparison, used in s a m p l e c a l c u l a t i o n s by the t h i r d m o m e n t method.
r e a d f r o m the slope of the exponential d e c a y of the d e l a y e d curve, a n d was given as (1.88 _k 0.10) x 10 10 s. T h e assigned error of a b o u t 5 % was m a i n l y due to causes o t h e r t h a n statistical. The p r o m p t c u r v e in the d i a g r a m is an e x a m p l e of the case discus.~;ed above, representing the app r o x i m a t e shape, but not the exact position, of the true p r o m p t curve. The half-life of this state is short enough to p e r m i t a useful application of the m o m e n t procedure, and long enough to provide i n d e p e n d e n t m e a n s of d e t e r m i n a t i o n of the half life. A c o m p u t e r calculation of the third m o m e n t of the delayed curve yielded a result e q u i v a l e n t to a half-life of 1.88 x 10 10s, the exact a g r e e m e n t with the published value being fortuitous. The calculated third m o m e n t of the a) R. E. Bell and ?,I. H. J o r g e n s e n , Nucl. Phys. 12 (1959) 413
EXPERIMENTS
151
p r o m p t curve was 0 . 1 2 % of t h a t of the delayed curve, so t h a t the error caused b y neglecting it in calculating T was only 0.04%. This implies t h a t the rough practical limit m e n t i o n e d a b o v e (e ½N3(F)) would h a v e been reached if the c o m p u t e d third m o m e n t of the d e l a y e d curve h a d been a b o u t 300 times smaller, i.e. if the half life h a d been a b o u t 3 x 10 -11 s. As the c o m p u t e r lends itself to fast calculations of such quantities as m o m e n t s , various r e a r r a n g e m e n t s of the actual e x p e r i m e n t a l d a t a were tried. F o r example, the leading edge of the delayed coincidence curve has a slight tail at low values of time, dne to i n s t r u m e n t a l resolution difficulties; at low values of counts on both leading and trailing edges the c u r v e has poor statistics. T h e c u r v e can be s m o o t h e d out, and new values of counts in a channel read from this s m o o t h e d curve. Using these "smoothed"values, a value of T was o b t a i n e d differing b y less t h a n 0.1% f r o m the value o b t a i n e d with the original figures. In a n o t h e r test, successively m o r e and more values at the e x t r e m e s of the c u r v e were eliminated, and the value of T o b t a i n e d was found to be quite insensitive. F o r example, the curve for Gd is6 (fig. 2) has values e x t e n d i n g o v e r more t h a n 4 decades. If all values m o r e than 3 decades below the p e a k are neglected, the value of T o b t a i n e d is only 4 % lower t h a n the " c o r r e c t " value. T h u s even a rough e x t r a p o l a t i o n of the 3decade curve to correct for this neglect will give an error m u c h less t h a n 4 % . The second m o m e n t of the p r o m p t curve, as c o m p u t e d from the delayed curve, was 5.76 units; the second m o m e n t , as found directly from the a p p r o x i m a t e p r o m p t curve itself, was 5.43 units.
4. Errors The .~tatistical errors t h a t one should assign to these values of a and r could h a v e been comt>uted b y h a v i n g the machine c a r r y the s t a n d a r d deviations of the individual points through tile calculation. However, in order to give the error in closed form, we now assume t h a t the p r o m p t curve is a Gaussian of s t a n d a r d deviation a. (The result is not sensitive to this assumption.) In t h a t case it is easily shown (see Appendix) t h a t
159
R. S. WEAVER AND R. E. BELL
the fractional s t a n d a r d deviation in the comp u t e d value of the lifetime is ZlV ( ~ 2 ) ~ [~/A(1~ ) 6 r
+ 3(-~)4 -[- 9(-~)2 @ 53]½3j
where A is the t o t a l of all the counts in the delayed curve. Applied to our present e x a m p l e , this e s t i m a t e gives a s t a n d a r d deviation in v of 1.3%. In an e x p e r i m e n t like this, a s y s t e m a t i c error of a b o u t 3 % is assigned to c o v e r i n s t r u m e n t calibration and linearity; the statistical error in this case is therefore negligible. In our e x a m p l e , if the half life b e c a m e smaller, the fractional s t a n d a r d deviation would rise, reaching 20 % for a h a l f l i f e o f ~ 3 × 10 - u s e c .
5. Concluding Remarks A n o t h e r use of this t h i r d - m o m e n t technique would be in the "self-comparison" m e t h o d of Bell, G r a h a m and Petch5). The two d e l a y e d a) R. E. Bell, R. L. Graham, and H. F. Petch, Can. J. Phys. 30 (1952) 35.
curves o b t a i n e d would h a v e a difference between their third m o m e n t s of 4~a, while the third m o m e n t of the p r o m p t curve would c o n t r i b u t e nothing, to first order; hence the lower limit of m e a s u r e m e n t of the half life could be s o m e w h a t decreased. T h r o u g h o u t this p a p e r we h a v e a s s u m e d t h a t the delayed c u r v e corresponds to a single exponential decay, in p a r t i c u l a r t h a t chance coincidences h a v e been s u b t r a c t e d and t h a t no p r o m p t events are occurring in the delayed curve. T h e neglected presence of either kind of e v e n t will cause errors, as it does in the centroid-shift case. A m o m e n t analysis can be m a d e to t a k e account of either t y p e of event, as shown b y Birk el al. 3) for the centroid-shift case, but the work is laborious and d e p e n d e n t on the exact circumstances of the m e a s u r e m e n t . One of us (R.S.W.) expresses his t h a n k s to the N a t i o n a l Research Council of C a n a d a for a S t u d e n t s h i p held during the course of this work.
Appendix CALCULATION OF FRACTIONAL STANDARD DEVIATION
To calculate the s t a n d a r d deviation in our c o m p u t e d r, we assume t h a t the n u m b e r of counts, F(x) dx, in a t i m e channel of w i d t h dx has a s t a n d a r d deviation IF(x) dx I ~. This error c o n t r i b u t e s an error in the t h i r d m o m e n t of a m o u n t (x - - r ) 3 IF(x) dx] i. Combining the various channel contributions q u a d r a t i c a l l y and replacing the s u m o v e r channels b y an integral, we get
[f:o
= If2
dC
,
the second factor, for normalization, being just A, the total n u m b e r of c o u n t s in the curve. E x p a n d i n g this expression a n d m a k i n g use of eq. (1) a n d eq. (3), we get
if
A(Na) = ~ - -
]
M6(P) + 15z ~ M4(P) + 40raM3(P) + 135. 4 M2(P) + 2 6 4 r 5 M l ( P ) + 265~ 6 ½.
If P has been observed, the m o m e n t s given here m a y be c o m p u t e d . I n order to give a definite f o r m for the error, however, we now assume t h a t P is a Gaussian with s t a n d a r d deviation a; i.e. M2(P) = a 2, M4(P) = 3o 4, M6 ( P ) = 15a~, a n d all odd m o m e n t s are zero. W e t h e n get
A(Na) = ~
if
]
15a~ +45~2o 4 + 135~4a2 + 265~6 ½-
This is the t o t a l error in 2v3: hence the fractional statistical error can be expressed as
aj_
i
(5)~
53]~
DELAYED
COINCIDENCE
EXPERIMENTS
153
In fig. 3 is plotted a curve of (Av/z)~ vs the ratio of a/v, and also the ratio of W~/T½where W½ is the full width at half m a x i m u m of the prompt curve and TI is the half life being measured. If A = 10 000 counts, then the fractional error is read directly as percent.
2
4
6
8
I0
12
14
16
IO
20
IOO
I0
•
0
I
I
L
I
I
I
I
2
[%]
3
4
5
I
6
Fig. 3, Fractional s t a n d a r d d e v i a t i o n in T p l o t t e d as a function of a/~: (lower abscissa) or A is t h e total n u m b e r of c o u n t s in t h e o b s e r v e d resolution curve.
W}/T}
(upper abscissa).