Calculated Characteristics of Three Types of Beta Gages 
Al Sz S. 1 
Type of Gage (amp) (ua/mg/cm?) (ua/wt. %) bz /x bc/c o./X o./c (sec) 
Thickness-by- 
transmission 1.72 X 10-9 3.3 _ 6.1 X 1078 _ 1.6 X 1073 _ 12.9 
Thickness-hv- 
reflection 1.85 X 107!° 5.8 — 6.9 X 1073 — 1.9 X 10-3 _— 12.7 
Concentration-by- 
reflection 4.8 x 107! _ alfa’ — 2.3 * 107? — 0.66 X 10-2 10.4 
associated number of ion pairs (electric 
charge) that appear in the ionization 
chamber. The larger of the two 
factors determines the gage’s over-all 
accuracy. 
The accuracy of the measurement of 
thickness or concentration is related to 
the uncertainty in the meter reading by 
5M se = OM 
Ss. or C= S. 
In the case of a ratemeter-type cir- 
cuit, the fractional standard deviation 
of the ionization current due to the 
random nature of radioactive decay 
is (6) 
O1 -_ 1 x 1 
IT VS2NRC VW 2IRC/qav 
Since there is a statistical fluctuation 
in the charge produced by an indi- 
vidual beta particle, the expected 
fractional standard deviation of I is 
somewhat greater than that given 
above, and is 
ore (Yin) 1 
I Gao J ~/2TRC/qavg 
The standard deviation of the meter 
reading due to statistical fluctuations is 
IRgm ( Vv Wun) 
V 2IRC/qavg Javg 
The fractional standard deviations 
of thickness and concentration are 
6x = 
ou =o;Rgn= 
oz _ Om = Jm (ae V (a2)aec 
z zS, zs 2C Qavg 
and 
core TRQove V (4?) avg 
Cc on 3c avg 
Over-all accuracy. In the deter- 
mination of thickness and concentra- 
tion with the three types of gages, the 
fractional uncertainties associated with 
inherent meter accuracy (6’s) and 
statistical fluctuations (0’s) of the 
radiation source are 
Thickness-by-transmission 
bz _ 5M _ ete26M 
xz Sz ~ pat] oRGm ohgm 
z sit" (Vie) 
TZ pat V21oRC/qavg \ avo 
Thickness-by-reflection 
6x _ ee? 5M 
z wert pRgn{(Za4/Ze)" — 1] 
ea alee 
Oz 
((Z4/Zp)" — lle? 
TZ pyx[(Za/Za)" — 1]-V21eRC/qarg 
(~ Dn) 
avg 
Concentration-by-re flection 
5c 5M 
¢  IohgmkK(Z — Z,)ick 
ao  WK(Z—ZJick& +1 
© RK(Z — Z,)ict \/2ToRC/ dave 
(~ Wier) 
qavg 
The gross accuracy of measurement 
is determined principally by the larger 
of the two uncertainties. More pre- 
cisely, the over-all fractional uncer- 
tainty is the root-mean-square value 
of the two fractional uncertainties. 
In general, the meter accuracy uncer- 
tainty is proportional to the source 
strength, the amplification, and the 
load resistance; but the source fluc- 
tuation uncertainty is-inversely propor- 
tional to the square-root of both source 
strength and time constant RC. In 
an optimum design these variables 
would be chosen so that the two uncer- 
tainties are about equal. 
Response Time 
In the design of automatic controls 
that rely on beta-gage measurements, 
the gage’s time of response is impor- 
tant. The equilibrium time for a rate 
meter to go from a rate of 0 to N 
particles per second is (6) 
Tt = RC[}4 In (2NRC) + 0.394] 
where 7 is the time required for the ex- 
ponentially decreasing difference be- 
tween the actual and true counting rate 
reading to be less than the probable 
error due to statistical fluctuations. 
It is, however, more rational to 
define equilibrium time in terms of the 
standard error rather than the prob- 
able error, and 
tT = 14 RC In (2NRC) 
For thickness gage applications it is 
interesting to estimate the equilibrium 
time in going from counting rate N, 
to N» or ionization current J, to Jo. 
The equilibrium time (defined in terms 
of the standard error) in this case is 
1 2(N2 — Ni)? 
or 
= ag ty [202 = RO 
oars RC |n i ze 
It may be noted that 7 is zero when 
= ty = WV 12/VS2RC/ Garg 
This occurs when the change in ioniza- 
tion current in going from J, to J; is 
equal to the standard deviation ex- 
pected for an ionization current of 
magnitude I>. 
In terms of AJ and J, where thickness 
or concentration changes from 0 to 
x or c, the equilibrium time is 
2(AI)? | 
I qavg 
One of the first steps in the design of 
a beta gage consists of choosing a time 
constant RC such that suitable equilib- 
rium times are obtained for the changes 
in I involved in making the measure- 
ment of thickness or concentration. 
= 1.15 RC loge | 
Example 
To illustrate the application of the 
quantitative relationships to the evalu- 
ation of an industrial beta gage, con- 
167 
