## Derivation in Time Domain

Since the voltage at the output of the amplifier is given by

 V(t) t/At = 2 ^n v ft ■ 11 At) • ‘ (3-3) n = -°°

then

 v2(t) Jl t/At • ’ ■ У] . Vm Vn V(t — mAt) V (t — IlAt) . (3-81) m = — ^ n =-<*■• since the expectation of the sum of a number of random variables is equal to the sum of their expectations.   If the average fission rate in the reactor is F and the counting efficiency of the detector is e counts per fission, then the probability of detecting a neutron during the n^ time interval is F є At. and the probability of detecting no neutron during the n*^ time interval is 1 — Fe At. So the expected value of ^ nm^2 is

The expected value of v?>n has two terms: one due to accidental pairs, and one due to coupled

pairs. An accidental pair is caused by a count registering in the m*’*1 time interval due to a

neutron in a family from one primary source neutron, and a count registering in the n*1*1 time

interval due to a neutron in an entirely separate family from another independent primary neutron.

A coupled pair is caused by counts in the mth and ntfl time intervals due to neutrons that can be

(2)    traced back to some common ancestor; i. e.. one fission (refer to de Hoffman’ ‘ for a discussion of accidental and coupled pairs). So the expected value of rm r is

probability of registering a count in the time interval, conditional probability that if a count is registered in the iL

min time interval an accidental count will be registered in the n*’*1 time interval, and.

conditional probability that if a count is registered in the m*11 time interval a coupled count will be registered in the nth time interval. de Hoffmann^ that  P (count in m) (count in n) + = (F€ At)2 + Ffc2 vlv-1) c-a (tg — 4) (At)2

. 2o Tf2

where

v = number of neutrons emitted per fission,

a = Rossi alpha = v (1 — Kp)/~f •

tg-tj = time between mm and nl time intervals.

Kp = promp’t neutron multiplication factor, and Tf = mean life of neutron for fission;

and Orndoff^ gives.    P (count in m) ^Рд (count in n) + Pf, (count in n) J

where

t = time between m**1 and n^ time intervals.

: ‘ r0 = mean life of neutron including absorption and leakage, and

о = Rossi alpha = (1 — Kp)/~0   But since

these two expressions are equivalent within a factor of Kp (which is essentially unity in a critical reactor). We will use Equation (3-87) and write, by combining Equations (3-85) with (3-8.7),

<VmVt>= F £ Д t [ F £ At + Aee’a (mAt ‘ n At) At] ,

for m > n

= F £ At [ F£ At + A£e-° (nAt" mAt)At] for m <n,  where

So, substitution of (3-84) and (3-89) into (3-83) yields t/At

<V2(t)> = ^ F £ At і/2 (t — m At)

. m = — 00 . t/At m

+ 2 2 [<F£At>2 + F

m = — °° n = -°°

x v (t — m At) v (t — n At)

t/At t/At _ , n

JT £ [(F£At)2 + F£2 Ae~a (n At — m At) (At)2 J x v (t.- m At) v (t — n At) . Note that we have dropped the requirement m *,n from the second and third summations; the argument for the validity of this step is given earlier in this section. Now let

 ■* 01 *. ш: ОДІ

 m,

 At

 At a = о

 (F t At)2 + Ft2 Ae"° (aAt ‘ bAt) (At)2

 * E E a = о b = о

 x v (a At), v (b At)

 00 oo r ■ . + ^ (Ft At)2 + Ft2 Ae _

 x v (a At) v (b At) , which becomes, as At — o,

 (3-92)

 = T Ft v2 (x) dxj r

 I (Ft)S (x) V (y) dy

 dx dx dx.

 (3-93)

 + Fe2AQe2Z122
 < V2(t)> c (WH+. ") (U’L+ a

Because of Equation (3-11), the second term in Equation (3-93) is equal to zero; hence 2

the expected value of V (t) is

 + F£A I e‘°’xv(x)

 + Fe2A I eaxv(x)      <r(t)> = Fc v (x) dx ‘o

 2 FeQe2zi22 = S—-

 “h*  substitution of (3-51) into.(3-94) yields

The product A a is  v{v-) Kp2

9-2 _ 2

2 " "О

A < V2(t)> = ev ^ ~ ^ KP2__________  <V2(t)>C і~2r02 (*H + «) (a;L+ a)

And finally, since

a ~ /3/т0 ,         where /3 is the delayed-neutron fraction, then

Equation (3-101) can be rearranged into the canonical form:  ■ ‘ У ‘ ‘ <!I(w)|2> = q2<; 11

7ГТ,

‘ ‘■ . ‘ ‘

(Equation (3-102) has been somewhat simplified by using the assumption that /3 » т’ X ) .

‘ f. ‘ ‘ ■ ■

But the efficiency c , in terms of counts per neutrons lost, is related to the efficiency e in-terms of counts per fission by «!• . .h.

So Equation (3-102) becomes     у

 n Q2 t Kp n - = ————- b- ■ ■ 1,7 ov

 і * JL  Since the low-frequency, break point of the system is at a frequency much greater than X, we will disregard that part of the spectrum below w = Л and write

which is equivalent to Equation (3-104) for u; > Л.

GEAP-4900.

 (3-106) and the ratio of reactor-noise contribution to the mean square voltage at the amplifier output to the "correct" value is given by (3-107)

 or

 A c

 6 „ (и — 1) Кр (Зг108)  Comparison of Equations (3-100) and (3-108) shows them to be equal within a fgctor of Kp.