The purpose of this appendix is to determine the effect of the scale-of-two on the variance of the count-rate indication. The method used consists of determining the distributed portion of the power spectral density of this indication and comparing it with the power spectral density of the indication that would be obtained without a scale-of-two. Since the variance is the integral of the power spectral density, a comparison of the variances can be obtained in this manner.

In this derivation it is assumed that the average counting rate is constant. The power. spectral density of a signal, S(t), is given by

. G(o;) = f ф(5) cos w 6d6 , (A-l)

ir J о

where ф(6) is the correlation function of S(t) and is defined as

ф(6) = ave [S (t) S(t+ б)] . (A-2)

Now the signal, S(t), can be considered as consisting of two parts: a d-c component generated by those diode pumps that are saturated, and a second part composed of a linear superposition of pulses from the unsaturated diode pumps. We will neglect the first part since it only contributes an impulse function at f = o, representing a d-c component of S(t).

The second part can be written as

co

S2(t)

where the kth pulse starts at tk and where v(t) describes a pulse that starts at t = o. This equation can be written in a form suitable for statistical analysis in the following manner. Divide the time axis into small equal intervals of length At; then the onset of a pulse during the nth interval will produce a signal at time t of v (t — n At), and the total signal due to all of the

A-l

previous pulses will be. T

S2^ = 2v^’nAt^’ (A-4)

n

where the summation includes only those intervals during which a pulse starts. If we define a random variable 77 , which equals one if a pulse starts during the 11th time«interval and equals zero if no pulse starts during the nin time interval, then the total signal can be written as

where the summation now includes all the time intervals that precede t.

Likewise,

. (t + 6 )/At.

S2(t + 6) = ^ ^ 77m v (t + б — m At) , ,r (A-6)

m = -00 ‘ • ‘

and hence

t/At	(t + 6)/At
S2(t)S2(t+6) = yt	^ ^ t)n7]m v(t — nAt) v(t + 6 — miit) .	(A-7)
n = -00	m = —00 ■

For a stationary random process (constant average counting rate in this case), the time average of a function is equal to the ensemble average, so we can write.

Ф(6) = .<S2(t). S2(t + 6)> , • (A-8)

where the brackets <> indicate expected value at time t, or ensemble average. Hence,

t/At (t + 6)/At IK

d{6) = y<

. n = -00 m = — .rl

since the average of a sum of random variables is equal to the sum of the averages.

A-2

It is convenient at this point to make a change of variables.

L-*i I — — a a = o b = o IA t

oO 00

We now must evaluate < n/ f V / t я >> anc* we use the fact that the

( —-a] ( —+ — — b] ‘

At J V At At j

expected value of a random variable is the sum of the products of each value the variable can as­sume and the probability of the variable assuming that value. Since 77 or any product of 77’s can assume only the values one or zero, and the zeros will not contribute to the expected value, we need consider only the value unity and the probability of assuming this value.

The first case to consider is — — a = — + — — b; i. e., both time intervals are one

and the same. A pulse will start in the ( — — a V*1 interval if a neutron is detected during this

. — v* /

■ interval and if the scale-of-two output is in its less positive state during the preceding interval. Hence, .

At At

for a = b——— . The first factor on the right-hand side of Equation (A-12) is the probability

At

that the scale-of-two output is in its less positive state, and the second factor is the probability of detecting a neutron during the/ — — aV*1 interval; к is the counting efficiency, Ф is the flux,

W )

and hence кФ. is the counting rate at the input of the scale-of-two.

GE АР — 4900 Г

If — — a precedes — + . —— b (і. e., if a > b — б / At), then it is necessary that a

• At. . 1’"’ At At ‘ ‘ ■■ ‘ ■

pulse start in the first of these two intervals, an odd number of neutron detections occur between them, and a neutron detection occur in the second of these two intervals. Hence,

for a>b — — . The first factor on the right-hand side of Equation (A-13) is the probability

. At ‘ ■ • .

l*~i) ‘(І. Ц At / At At /

that a pulse starts in the first of these two intervals, the second factor is the probability of an odd number of neutron detections occurring between them, and the third factor is the probability of detecting a neutron in the second of these two intervals. This can be written as

Likewise, if — + — — b precedes — — a (i. e., if a < b — б /At), then At At At.

(.L-» ”( .L* A At / At At /

— 2кФ

Substitution of Equations (A-12). (A-14), and (A-15) into (A-11) yields

0° ‘ • .

Ф(6) = 0, 5 к Ф At v (a At) v (a At t 6) 4

a = о.

-2кф/ь-а — — — lAt ^ ^ 0. 5 (кФ At)^ * ~ e———— ——— v (a At) v (b At) .

$(*)

which becomes, as At — o,

This last Equation (A-20) can be rearranged to

ф(6) = 0. 5кФ ] v(x) v(x + 6) dx

J Л

0. 5кФ f	oo. v(x) dx Э	2
	• oo		"-X+ 6 —
(0.5кФ)2′	J е-2кФ(х + б)	J е2кфуу(у) dy 0

dx

(A-21)

The second term on the right-hand side of Equation (A-21) contributes to the power spectral density an impulse function at f = o. (It represents the d-c component of the pulses that make up S(t)). We will neglect this term and write the following as the correlation function of the a-c component of the signal:

(A-22)

The first term on the right-hand side of Equation (A-22) isv ‘ the cor relation, function of the a-c component of a signal composed of a linear superposition of pulses of the form v(t), arriving, randomly at an average rate of 0. 5кФ ; the second term, therefore, is the contribution to the spectrum resulting from the regularizing action of the scale-of-two.

To illustrate the magnitude of this effect, let

. v(t) = vQe’a;ot. (A-23)

and calculate: (a) the power spectral density of the signal obtained with the scale-of-two present, and (b) the power spectral density of the signal obtained without the scale-of-two present and with half the average counting rate. The first calculation is performed by combining Equations (A-l), (A-23), and (A-22); the second by combining Equations (A-l), (A-23), and the first term of (A-22). The results are: ‘ .

with the scale-of-two, present,

without the scale-of-two.

A comparison of these two equations shows: (a) Ga(u-‘) is one-half G^^’) between the frequencies zero and /2 к Ф, and (b) Ga(w) equals G^(<^’) between the frequencies 2кФ and infinity. Hence, the integral of G^u,’) is less than twice the integral of G (<*:)• and the variance obtained by neglecting the scale-of-two is too large by less than a factor of 2.

BY AN AMOUNT (1/P)E

Rate of Exceeding (1/P)T+ (1/P)E

0.050 nepers sec 0.060 nepers sec 0.100 nepers sec 0.125 nepers sec 0.150 nepers sec 0.175 nepers sec 0. 200 nepers sec

[2]

01) (a’L+ °’) (a’ H+ “l)

But the first term on the right, above, is the value of the signal without reactor noise, so the second term is the contribution made by the reactor noise. Hence, the reactor-noise contribution expressed as a fraction of the "correct” signal is the ratio of the second to the first term: i. e.,

[3] 40 feet, of 50-ohm cable, 125 feet of 75-ohm (RG-6) cable terminated in 5000-ohms,

• electronic system. frequency breakpoint at 5 and 9 kc, 100 volts applied.

[4] 40 feet of 75-ohm cable, 85 feet of 185-ohm. (RG-114) cable terminated in 5000 ohms, electronic — system. frequency breakpoints at 8 and 18 kc, 100 volts applied.-

[5] 85 feet ofr 185-ohm cable (RG-114) cable terminated in 5000 ohms, electronic system ■frequency breakpoints at 8 and 60 kc, 400 volts applied. ■

[6]The rise time of the measuring device was 11. 7 nsec.

[7]The sample chamber is chamber No. 17, which was later made a part of detector assembly No. 2.

[8]Also called a multi-range mean square voltage monitor (MMSVM).