Derivation in Time Domain

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



= 2 ^n v ft ■ 11 At) • ‘


n = -°°




t/At • ’ ■

У] . Vm Vn V(t — mAt) V (t — IlAt) .


m = — ^

n =-<*■•


since the expectation of the sum of a number of random variables is equal to the sum of their expectations.

Подпись: 1 x F c At image267 Подпись: (3-84)

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


Подпись: m n Подпись: 1 xP (count in m) x 1 x Подпись: PA(count, in n)+ P^ (count in n) Подпись: (3-85)

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

Подпись: where P (count in m) PA (count in n) PQ(count in ri) 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.

Подпись:de Hoffmann^ that

Подпись:image276P (count in m) (count in n) +

Подпись:= (F€ At)2 + Ffc2 vlv-1) c-a (tg — 4) (At)2

. 2o Tf2


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.

Подпись: ft At Подпись: Ft At Подпись: 1)KP2 : Є At 2F2(1-KP)T0 • Подпись: (3-87)

P (count in m) ^Рд (count in n) + Pf, (count in n) J


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


image283 Подпись: (3-88)

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]

Подпись: (3-89)for m <n,

image286 Подпись: (3-90)


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 .

Подпись: c2 A e ~a (mAt ■ nAt) (At)2t/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

Подпись: (3-91)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:









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 _<v (bAt ‘ aAt^ (At.)2

a = о b = a


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




<V2(t)> = T Ft v2 (x) dx
j r


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












+ 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£[2]A I e‘°’xv(x)

+ Fe2A I eaxv(x)

Подпись: /;

Подпись: v (t) Подпись: QeZ12 image297 image298
Подпись: (3-94)

<r(t)> = Fc v (x) dx ‘o

2 FeQe2zi22

<v (t)> = S—-


Подпись: (jJ 4. CL! H + L Подпись: (3-95)

substitution of (3-51) into.(3-94) yields

The product A a is

Подпись: (3-97)Подпись: A оv{v-) Kp2

9-2 _ 2

2 " "О

A < V2(t)> = ev ^ ~ ^ KP2__________

Подпись:Подпись: (3-99)<V2(t)>C і~2r02 (*H + «) (a;L+ a)

And finally, since

a ~ /3/т0 ,

image306 image307

Подпись: T,'Подпись: To +Подпись:

Подпись: Q = charge transferred per neutron detected. t . ’ e = detector efficiency in counts/(neutron absorbed in the reactor in whatever manner), n = total number of neutrons in the reactor, T = neutron lifetime. ' M '
Подпись: ,.2 x 2 Ct/ + A.
Подпись: в = p-
Подпись: = equivalent one-group delayed-neutron-emitter's decay constant.

where /3 is the delayed-neutron fraction, then

Equation (3-101) can be rearranged into the canonical form:

Подпись: 1 +image316■ ‘ У ‘ ‘

Подпись: (3-102)<!I(w)|2> = q2<; 11


‘ ‘■ . ‘ ‘

(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

Подпись: (3-103)«!• . .h.

So Equation (3-102) becomes

Подпись: екрл^ image320 Подпись: (3-104)
Подпись: о Q tKp n <| і(ш)!^> = — 4— ' 77 Г о и'
Подпись: 1 +


n Q2 t Kp n

<! 1(<*:)Г>- = ————- b-

■ ■ 1,7 ov

і * JL

image324 Подпись: (3-105)

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.







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









6 „ (и — 1) Кр







Comparison of Equations (3-100) and (3-108) shows them to be equal within a fgctor of Kp.