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On counting precision and the S2B criterion
International JournalofAppliedRadiationandIsotopes,1969, Vol. 20, p. 205. Peqwnon Press. Printed in Northern Ireland
On Counting the
Precision
and
t,=Tz/s+b/(z/s+b+ -= Td/s + b(2/s
S2/BCriterion
A RECENT letter by PORGE@ sounds a timely warning against indiscriminate use of the S2/B criterion for the choice or adjustment of counting equipment, particularly when S is not < B. A reference suggests that a previous publication(2) by the present correspondent advocates the wide use of this criterion. In fact the paper cited is not directly concerned with it, but relates to the signal counting rate necessary for measurement with a given precision in the presence of background, when a limited time T is available. It is used in determining the total amount of radioactivity needed in a practical tracer experiment once the detector sensitivity and background are known. The two communications are thus complementary and if an apparent error in Porges’ derivation is corrected, they can also be shown to be compatible. Substitution for C, and C, in equation (2) of the letter(l) leads to a dimensionless expression for the relative variance: v2 = 1 + c&/s
(1)
62 = vyst, = s-y 1 + d/S)
(2)
or alternatively
where 6 is the relative standard deviation in s. For highest precision it is this last parameter which has to be minimised and not the relative variance, so PORGES’ subsequent argument is valid if v2 is replaced by cY2. Now, when a limited time T is available and this is divided to obtain the minimum relative standard deviation 6, the corrected expression (1) for v2 can be shown to reduce to PUTMAN’sexpressiont2) z/rb
With the optimum allocation of counting times to minimise 6:
=
z/b + 1/(62/T)
CL= 1 +
6) + b -
l/s)/s
t,/t, = (d/s + b + %6)/z/’
= s/&(z/s
+ b -
d/s)
Also v2 = S2st, and
B/S
= b/s
Hence, substituting in (1) : STl/s
-
+ b(l/s
-
+ 6 -
4:)
=l+v%/(&+b---z/b) --
= 2/s + b/(ds
+ b -
d/6)
Whence a2T(l/s
+ b -
46,”
= 1
From which expression (3) follows. Different assumptions are made about the relative counting times for determining the signal and background rates, however. In practice the time available for background determination is seldom unlimited, but for repetitive measurements the same background figure can be used more than once: this would lead to some modification of PUTMAN’s arguments(2). On the other hand the assumption that in scintillation counters a single high-precision background determination can be used indefinitely, even in the conditions described by Porges, is misleading, since it overlooks long-term variations in circuit parameters and the emissivity of PM photocathodes. Wantage Research Laboratory Wantage, Berks.
J. L.
PWMAN
References (3)
where b is the background count rate and (s + b) the signal plus background rate.
205
1. PORGES K. G. Int. J. a@l. Radiat. Isotopes 19, 711
2.
(1968).
PUTMAN J.
L., ibid. 13, 99 and 222 (1962).
On Counting the
Precision
and
t,=Tz/s+b/(z/s+b+ -= Td/s + b(2/s
S2/BCriterion
A RECENT letter by PORGE@ sounds a timely warning against indiscriminate use of the S2/B criterion for the choice or adjustment of counting equipment, particularly when S is not < B. A reference suggests that a previous publication(2) by the present correspondent advocates the wide use of this criterion. In fact the paper cited is not directly concerned with it, but relates to the signal counting rate necessary for measurement with a given precision in the presence of background, when a limited time T is available. It is used in determining the total amount of radioactivity needed in a practical tracer experiment once the detector sensitivity and background are known. The two communications are thus complementary and if an apparent error in Porges’ derivation is corrected, they can also be shown to be compatible. Substitution for C, and C, in equation (2) of the letter(l) leads to a dimensionless expression for the relative variance: v2 = 1 + c&/s
(1)
62 = vyst, = s-y 1 + d/S)
(2)
or alternatively
where 6 is the relative standard deviation in s. For highest precision it is this last parameter which has to be minimised and not the relative variance, so PORGES’ subsequent argument is valid if v2 is replaced by cY2. Now, when a limited time T is available and this is divided to obtain the minimum relative standard deviation 6, the corrected expression (1) for v2 can be shown to reduce to PUTMAN’sexpressiont2) z/rb
With the optimum allocation of counting times to minimise 6:
=
z/b + 1/(62/T)
CL= 1 +
6) + b -
l/s)/s
t,/t, = (d/s + b + %6)/z/’
= s/&(z/s
+ b -
d/s)
Also v2 = S2st, and
B/S
= b/s
Hence, substituting in (1) : STl/s
-
+ b(l/s
-
+ 6 -
4:)
=l+v%/(&+b---z/b) --
= 2/s + b/(ds
+ b -
d/6)
Whence a2T(l/s
+ b -
46,”
= 1
From which expression (3) follows. Different assumptions are made about the relative counting times for determining the signal and background rates, however. In practice the time available for background determination is seldom unlimited, but for repetitive measurements the same background figure can be used more than once: this would lead to some modification of PUTMAN’s arguments(2). On the other hand the assumption that in scintillation counters a single high-precision background determination can be used indefinitely, even in the conditions described by Porges, is misleading, since it overlooks long-term variations in circuit parameters and the emissivity of PM photocathodes. Wantage Research Laboratory Wantage, Berks.
J. L.
PWMAN
References (3)
where b is the background count rate and (s + b) the signal plus background rate.
205
1. PORGES K. G. Int. J. a@l. Radiat. Isotopes 19, 711
2.
(1968).
PUTMAN J.
L., ibid. 13, 99 and 222 (1962).