As implied by its name and defined in Table 6.1, the impulse response, h(t), is the system output, y(t), when its input, x(t), is a very narrow pulse (i. e., a unit pulse of time duration much less than the smallest important time constant of the system). The impulse response is also the Green’s function or weighting function. It is appropriate to discuss the impulse-response function in connection with transfer functions since, as shown in Table 6.1, it is the Fourier transform of the transfer function. To date the impulse-response function has not enjoyed the popularity of the transfer function as an analytical tool. Recently, however, Dorf2 pointed out that since digital computers greatly facilitate time-domain analyses of systems the impulse-response function should become more popular

Table 6.2 gives an example of the impulse response of a system with a single time constant. Evidently excitation by an input pulse, x(t), having a width much less than the system time constant stretches this pulse to a width or duration of the order of the system time constant. A physical interpretation of this is that the output, y(t), “remembers” an input pulse and shows its effect (up to about the time constant) after the input pulse.

FREQUENCY (со), radians/sec

Fig. 6.1—Magnitude and phase of zero-power reactor transfer function vs. frequency for 2 3 3U, 2 3 5 U. and 2 39 Pu. Curves are shown for various values of l, the neutron lifetime,3

Table 6 2—Illustration of Dynamic Functions for a System Having a Single Time Constant, tc, One Input, and One Output Description Frequency-domain expression Time-domain expression Differential equation of the system and its Laplace transform x(s) = (1 + itcco) y(s) x(t) = y(t) + TC Transfer function Impulse response 4 y(u>) 1 G(0J) x(w) 1 + ітсш h(t) = 0 (r < 0) e~t/Tc , h(t) =*-z—<-T> 0) Tc Spectral density of output for an arbitrary input p f Px(u>) РУ(Ш) " 1 + Tgto2 Cross spectral density of input and output (for an arbitrary input) px(m) PXy(to) = -2————- y 1 + ircOJ Autocorrelation function of output for a constant spectral-density (uncorrelated) input C(r) = e-lTl/Tc Cross-correlation function of output with a constant spectral-density input Сху(т) = 0 (r < 0) Cxy(r) = e-T/Tc (r > 0)

Finally, another insight into the nature of the impulse response comes from using it as a weighting function in relating arbitrary input and output time functions

WATTMETER о і о v (

BAND-PASS FILTER OF WIDTH df

ION CHAMBER

R = 1 .

y(t) = /0 h(t’) x(t — t’) dt’

(6 3)

3. A continuum of frequency components make up the randomly fluctuating or nonperiodic part of the signal.

The spectral-density concept applies primarily to the last. Spectral densities associated with the first two contributions are additive with the total of the third

Ptotal = (i)2 + (y + Y * ) + /-Т P<f> df <6 5)

If the signal is a current through a resistor, the three terms are, respectively, the d-c power, the a-c power of discrete frequencies, and the a-c power of random noise The usefulness of the spectral-density concept is m char­acterizing the last term, and hence P(f) is sometimes called a random-noise spectrum.

Fundamental relations associated with spectral density are given in Table 6.3. As also indicated in Table 6.1, the

spectral density may be obtained from Fourier amplitudes or, alternatively, by integration of an autocorrelation function. Ideally the signal duration, T of Table 6.3, would be infinite. In practice, the finite duration of the signal available for spectral analysis is an important experimental limitation (see Sec 6-7).

5- 1.5 Cross Spectral Density

Just as the spectral-density function, P(f), is used to display the relative importance of various frequency com­ponents in a single random signal, the cross spectral density, PXy(f), is used to show the joint importance of these frequency components in two related random signals, x(t) and y(t). Its definition in terms of Fourier amplitudes and in relation to the cross-correlation function is given in Table 6.1. Evidently the cross spectral density is a more general concept, which reduces to the simple spectral density, P(f), for the case x = y.

Figure 6.1 shows conceptually how one might measure the cross spectral density using a wattmeter and filters with switchable phased outputs. With the switches in the positions indicated, a quadrature spectral density is indi­cated by the time-average value of the current from one chamber and the 90° phase-shifted voltage from another, if the filters are switched in phase, the meter shows the cospectral density. In both instances the extent to which the two signals are similar in a frequency band df is being measured.

Unlike the spectral density, P(f), but like the transfer function, the cross spectral density, Pxy(f), requires two numbers at each frequency for its specification. These may be the “со” and “quadrature” spectral values or the amplitude and phase, with the relations

IPXyP = (cross-spectrum amplitude)2

= (cospectrum amplitude)2 + (quadrature spectrum amplitude)2

= (Coxy)2 + (Quxy)2 (6.6)

в = phase angle

quadrature spectrum amplitude
cospectrum amplitude

Q. uxy

Coxy

Because of the similarities in the descriptions of the transfer function and the cross spectrum, it is not surprising to find that these are related, as shown in Fig 6 3. In spite ■Ф

n 1 (t) n 2(t)

Transfer-function relation Y(s) = G(s) [ X(s) + N1 (s)] + N2(s) where (s) and N2(s) are Fourier transforms of n1 and n2

Spectral-density relation P.. = IGI2 [Pv + P_ ] + Prt

у * M2

Cross-spectral-density relation PXy = GPX

Fig. 6.3—Input—output relations of Fourier transforms and spectra in a system having uncorrelated additive noise signals, n, (t) and n2 (t), at its input and output

of additional signals (such as unwanted noise) at the input and output, a simple relation exists the transfer function, G, times the input spectral density, Px, is the cross spectral density, Pxy On the other hand, only when the unwanted signals can be neglected are the input and output spectral densities related by the square of the transfer function.

A quantity called coherence, cxy(f), has been defined to quantitatively assess the extent to which the presence of

Its square is the ratio of IgI2 (the numerator) to the spectral-density ratio (the denominator), the latter being IgI2 plus effects from extraneous signals nj and n2, according to Fig. 6.3. In frequency ranges over which cxy(f) is 1 or nearly so, the input and output can be related with negligible effects from other uncorrelated signals. Conversely, the input and output can be consideied virtually uncorrelated in frequency ranges where N[(f) and/or N2(f) are large enough to cause cxy(f) to be near zero Evidently the ease of making transfer-function mea­surements will be in proportion to how near cxy(f) is to 1.

6- 1.6 Autocorrelation

Table 6.1 shows that the function m the time domain that corresponds to the spectral density is the auto­correlation function C(t). The definition indicates that it is a measure of the amount of correlation existing at a time interval tin a signal x(t). It has its largest values at т = 0 and at other time intervals during which the signal has essentially the same value, it is smallest during time intervals over which signal values are uncorrelated. In the example shown in Table 6 2, the autocorrelation function decreases from 1 to e_1 in a time tc and approaches zero when t is large. Thus tc may be called a correlation time within which signal values are similar and beyond which they are rather unrelated.

Table 6.1 shows that the spectral density, P(f), can be obtained from either the square of the Fourier transform of x(t) or from the transform of its autocorrelation function. Conversely, the autocorrelation function can be obtained by transforming P(f). However, x(t), when random, cannot be reconstructed from either P(f) or C(r).

6- 1.7 Cross Correlation

The concept of cross correlation is more general than that of autocorrelation since the latter is a special case of the former in which the two signals are the same. The cross-correlation function defined in Table 6.1 is an applica­tion to continuous time functions of the digital concept of a correlation coefficient of statisticians if x, and y, are two time series of variable values spaced in time (x, being at the same time as yi+(T/At)) in which the degree of correlation is sought, then

is a measure of this. However, it is customary to define a normalized correlation coefficient in terms of fluctuations from means.

where ax and ay are the standard deviations 0^x and This is +1 or —1 for perfect correlation or anticorrelation, respectively, and is 0 if there is no correlation. The integral expression for cross correlation in Table 6.1 is evidently digitally evaluated in Eq. 6.9.

In the example in Table 6.2, Cxy is zero for t<0 because the output cannot “know” ahead of time what the perfectly random input, x(t), will be. A significant input — output correlation, however, does exist for values of r up to the order of tc, the correlation time of the system. In other more complex systems, the maximum value of Cxy might occur at some time other than zero, in which case a time-lag effect between x and у will have been identified.

Table 6.1 shows the frequency-domain function corre­sponding to Cxy(t) to be the cross spectral density, Pxy(f) These are Founer-transform pairs, and, if one is known, the other can be found from the relations shown.

5- 2 REACTOR APPLICATIONS

6- 2.1 Neutron Kinetics

For a study of the time behavior of reactors, the equations giving the time dependence of the neutron density, N + n (mean value plus deviations therefrom), and the )th group of delayed-neutron precursors, Cj + Cj, are

/-3r=[kd — (3) — 1] (N + n) + £ /Xj(Cj + Cj) + IS (611)

dc. — л r, „ ) . 0Jk<N + n>

_ Aj^j + Cj)+ і

where the sum of the delayed-neutron fractions, |3j, is the total fraction, /3, Xj is the decay constant of a precursor, S is a source, / is the prompt-neutron lifetime, and к is the effective multiplication constant that, when not unity, represents the departure of the reactor from exact criticality,

P=l-f (6.13)

к

being the excess reactivity.

The solution to these equations, under conditions of all variables undergoing small oscillations about their mean values, is the zero-power transfer function, G0, defined as

lG0l = [(amplitude of power oscillation)/(average power)] /(amplitude of reactivity oscillation) (6 14)

and having the phase

Phase angle = 360° X (fraction of a cycle that the power lags behind the reactivity oscillation) (6 15)

The zero-power transfer function can also be regarded as the quotient of the Fourier transforms of the power and reactivity divided by the average power.

Table 6.4 gives explicit formulas for this transfer function in terms of reactor constants and the frequency.

Table 6.4—Forms of the Complex Amplitude of the
Zero-Power-Reactor Transfer Function G0(u>)

Formula for complex Conditions amplitude of G0 (со)

6

1 — no 2] + loo)

Also, G has been tabulated in detail in Ref. 3. In essentially all but subcntical reactor applications, к may be set equal to 1 to further simplify the approximations there At mid­frequencies, where 2X < со < 0 5(3//, a very simple result, G = 1 //3, exists At these frequencies the physical inter­pretation is

Percent power oscillation about its mean

= — X (percentage reactivity amplitude) = reactivity amplitude in cents

where (3 is typically 0.007.

The application of nuclear instrumentation demands an understanding of the behavior of a reactor. Because power density varies with position in the reactor, an average power measurement is needed. Out-of-core detectors are con­sidered to be spatially averaging and are discussed here from this viewpoint. Detectors for measuring spatial variations in nuclear fluxes are discussed in Chap. 3. Out-of-core detectors are reasonably good averaging devices for most reactors if their installation is properly designed, especially in regard to shadowing by movable objects in or around the reactor.

To a limited extent, nuclear instrumentation influences reactor design. For example, it may be necessary to adjust the location of control elements to avoid shadowing effects on radiation sensors. It may be desirable to introduce a window to cause streaming that will ensure an adequate level of radiation for reliable instrumentation response. Although it is always desirable to avoid reactor-vessel penetrations, penetrations are sometimes necessary to ensure an adequate signal. The minimum reactor power level must be determined to ensure measurements at all reactor levels. If the minimum is too low or uncertain, a neutron source must be provided to maintain the minimum level at a measurable value. There must be provisions for renewing or replacing the source.

All these requirements stem from the mandate that the state of the reactor must always be known. In other words, the reactor level and the rate at which the reactor level is changing must always be known and must always be under control. To ensure this knowledge, redundancy is always used to some degree (see Chap. 12). A common mode of redundancy is to make measurements with three separate detectors or channels, each with independent circuitry. The shutdown signal from any one channel must be in coincidence with another signal (i. e., a two-out-of-three coincidence) before shutdown is allowed.

Radiation detectors sample radiation intensity. Initially, the relationship between reactor power level and the sampled radiation intensity is based on design calculations alone. At power levels near full-power operation, the detectors must be calibrated. This is best accomplished by making heat-balance measurements. Subsequent cal­culations, using the calibration, then relate the detector response to the reactor power level. Periodic recalibration is required to take into account changing radiation patterns and spectra, fuel burnup, and changes in detector sen­sitivity.

The great range in reactor power (from watts to hundreds of megawatts) makes it impossible to use one set of detectors and circuits, despite the wide range of the detectors. Research has produced detector and circuit arrangements capable of measuring over a range of 10 decades. Signal-conditioning circuits are the key to success (see Chap. 5).

A single set of detectors can be used to measure only a part of the reactor range and must be complemented by additional sets of detectors. For safety and reliability, a part of the range of the detectors is sacrificed by having them duplicate the measurements of a part of the range of other detectors. This duplication, or overlapping, is needed for a smooth transfer of control and safety functions from one detector set to another. The amount of overlap is typically one to two decades.

The most common way of dividing the power-level range is to use three ranges: source, intermediate, and power ranges.6 This nomenclature is used in commercial

practice. Figure 2.2 shows a typical selection of neutron detectors to cover these ranges.

Each range has peculiarities that depend on the radia­tion levels corresponding to that range and on whether or

not a reactor has been operated. Special features, some­times temporary, must be incorporated into the instru­mentation design to ensure reliable performance during the initial period of a reactor when it is “clean and cold.” The same instrumentation must operate when the reactor has accumulated its full burden of radioactivity and at every condition in between.

In the source and intermediate ranges, the reactivity of the reactor is limited by controlling or limiting the rate (period) at which the power can be increased. In the power range, instrumentation must prevent the reactor from exceeding its rated or licensed operating limit.

In the fully shutdown condition, the neutron density to which the detectors are exposed is frequently quite low, in fact, so low that individual neutrons are counted in order to gain information about the reactor status. Counting is also the only way to detect neutrons in the relatively high gamma fields that may be present.

The limits of the source range (or counting range) are determined by permissible counting rates, expressed in counts per second. The low end of the source range is determined by the counting rate needed to achieve a safe condition, as specified in the safety review (see Chap. 12). This minimum counting rate is normally from 1 to 10 counts/sec. The counting rate is established during the preliminary design and is fixed by consideration of the statistical nature of the neutron population and the time interval needed to achieve a measurement of prescribed accuracy. The counter must be located where the flux density is sufficiently large to ensure that this counting rate is achieved. The magnitude of the neutron source is selected to attain (at the detector location) a neutron flux that results in at least the minimum counting rate at all times. The maximum or high end of the source range is determined by the ability of the counter and the associated electronic circuits to resolve the individual counts. If the counting rate is too high, the resolution loss produces a serious error in the signal. Typical maximum counting rates are 0.5 to 1 X 106 counts/sec, and the allowable resolution loss is less than 10%.

The source range presents an adverse situation for the detection of neutrons. The detector used must be carefully selected for its sensitivity to neutrons in the presence of a large gamma background. The condition of few neutrons and many gammas exists immediately following a scram from full power.

The intermediate range overlaps the source range, and its gamma background is not as severe. However, since the neutron flux is high, individual neutrons are no longer resolved, and the signal takes on a direct-current aspect, becoming indistinguishable from the gamma background (also a d-c signal). Here again, it is essential to know the gamma level at the low end of the intermediate range. Through sensor design the gamma contribution at the low end is normally kept below 10% of the neutron signal. Again, the worst condition exists during a start-up immedi­ately following full-power scram. The intermediate range usually extends into and completely overlaps the power range.

The power range covers from 1 to 150% of full power to provide some allowance for small power excursions. In the power range there is normally no great difficulty with interfering radiation. Neutron detectors that are not gamma compensated are satisfactory, but gamma-compensated sensors may be used for uniformity. These are similar to those used in the intermediate range.

(a) Definition. The pH is defined as the logarithm (to the base 10) of the reciprocal of the hydrogen-ion concentration m moles per liter

pH — lot* + . .

H concentration m moles per liter

figure 4.42 shows pH vs. hydrogen-ion concentration. Points indicate the pH values of various common acids and bases.

All water solutions owe their chemical activity to their relative H+ and OH concentrations In water, the equilib­rium product of the H+ and Oil concentrations is a constant 10 14 at 25°C. When concentrations of H+ and OH in pure water at 25°C are equal, the H concentration is 10 7 and, from the definition, the pH is 7.0 Note that the stale of pH values is not linear with concentration A change of one unit in pH represents a 10-fold change in the effective strength of the acid or base

The pH value depends only on the concentration of hydrogen ions actually dissociated in a solution and not on
the total acidity or alkalinity. Therefore, because dissocia­tion of water increases with temperature and pH is a measure of II concentration only (and not the ratio of H+ to OH ), the pH of pure water increases above 7.0 if the temperature is increased above 25°C. There is no simple way to predict the pH of a solution at a desired temperature from a known pH reading at some other temperature.

(b) Measurement Techniques. Chemical Indicators The pH of a sample may be determined by adding a small quantity of an indicator solution to the sample and comparing the color with that of a color standard. When good color standards are available in steps of 0.2 pH unit and observations are made in a comparator, the limit of accuracy is considered to be 0.1 pH unit. Turbid and colored solutions cannot be observed with accuracy, and indicators are not stable in many strongly oxidizing or reducing solutions. Table 4.21 lists some common pH indicators and their range of use.

Potentiometnc pH Measurement. A potentiometric pH-measuring system consists of (1) a pH-responsive elec­trode, such as glass, antimony, quinhydrone, or hydrogen,

(2) a reference electrode, usually calomel or silver—silver chloride, and (3) a potential-measuring device, such as a pH meter, usually some form of vacuum-tube voltmeter figure 4.43 shows a typical potentiometnc system.

Table 4.22 lists the characteristics of six pH-measuring electrodes. Glass electrodes are electrically sensitive to hydrogen-ion concentration The voltage response to hydro­gen-ion concentration is

E = E° — 0 0591 log H+ (at 25°C)

where E° is the voltage of the particular glass electrode at pi I zero

In Figs. 4.44 to 4.47, some pH-measuring meters are illustrated. The feedback type pH meter (Fig. 4.49) has a circuit capable of good performance if matched tubes are employed to minimize drift The electrodes must be checked periodically against standards for asymmetry.

Figure 4.48 shows the theoretical curve for the pH at 25°C vs. the concentration of ammonia. Figure 4.49 gives the temperature correction for the ammonia curve.

HIGH-IMPEDANCE POTENTIAL MEASURING INSTRUMENT CALIBRATED IN pH UNITS CALOMEL

——————— ГТ——- Ш———— V REFERENCE

/ ELECTRODE

PLATINUM WIRE MERCURY

BUFFER

SOLUTION

Figure 4.50 shows the theoretical curve for the pH at 25°C vs. the conductivity of ammonia. Conductivity measurements can be used to monitor the pH of the feedwater or the ammonia concentration in the feedwater (Fig. 4.51).

Limitations and Practical Considerations.

1. Glass electrodes can develop cracks, which allow some diffusion between the inner filling solution and the sample. When diffusion occurs responses are erratic and nonrcproducible.

2. Glass is soluble in strongly alkaline solutions and thus has a shorter service life. Special alkali-resistant electrodes should be used for these applications.

3. If the glass becomes coated, the response is sluggish.

4. High sodium-ion concentration for extended periods of time results in loss of sensitivity.

5. Avoid temperature transients.

6. New electrodes should be soaked several hours before use to improve stability.

7. Avoid electrical leakage in the high-impedance input circuit by preventing moisture buildup on the glass elec­trode body and lead, blectrical leakage is sometimes caused by the buildup of humidity and dust inside the instrument case.

8. Grounding problems Many pH meters provide for separate grounding of the amplifier chassis and case. The ground of the amplifier is maintained at the glass-electrode potential by connections with feedback circuits.

9. Shorting of the electrodes causes polarization. The pH reading drifts under these conditions.

10. Colloids are sensitive to salt and may precipitate at the liquid junction as the result of the diffusion of the salt-bridge electrolyte or may form a film on the glass-elec­trode bulb. Slurries cause similar trouble.

11. Glass is attacked by soluble silicates and by acid fluorides. Special alkali-resistant electrodes are available.

12.
Radioactivity in sample solutions may result in ion collection in the high-impedance input circuit, which, in turn, may produce error signals.

13. Glass electrodes respond to high concentrations of sodium, potassium, and lithium ions Sodium-ion correc­tions are usually available from the electrode manufacturer. The need for correcting data for high concentrations of other ions should be investigated.

Although on-lme digital-computer analysis of reactor dynamics is possible73 and perhaps will be prevalent in the future, it has been the practice until now to perform digital analysis off-line. As shown in Table 6 11, the digitizing process may be on-line (creating the proper magnetic-tape format for a computer) or off-line The off-line digitizing may be automatic from an f-m tape or semiautomatic, as in the case of manually operated strip — chart readers whose electrical output punches cards In any event the result is that one or more sequences (x,, у,, etc ) of variables at time spacings At are generated m a form suitable for input to a digital computer

The selection of a digitizing interval, At, and of a total duration of the data collection, T, is discussed in Sec 6-7 4 It will suffice here to note that the digitizing interval determines the upper frequency limit, fmax = l/(2At), of the analysis and the total duration is associated with the frequency resolution (1 e, minimum frequency interval between independently determined spectral values) and accuracy of results The quotient, T/At, is the number of digital values per signal and may be 103 to 10s in typical experiments

A number of versatile programs are available to users of the various commercial computers for statistical analysis of large quantities of data Typical of these are the Biomedical Computer Programs,115 a series of 42 programs that are useful not only in biomedical research but also in any field requiring analysis of data for frequency counts, variances, correlations, and related functions Table 6 14 lists the

Ckn(T) = iGlNkg cos (сот — в) (6 36)

iGl and 0 may be determined from as few as two values, Ckn(0) and Скп(яУ2со)

Whether the digital approach discussed here or the continuous-signal approach discussed in previous sections should be used depends on a variety of factors, some of which are mentioned in Table 6 15 The digital approach has been more common in recent years as digitizing costs and computer rental costs per data point decrease and as demands for computer versatility (see Table 6.14) increase

available computer outputs from just one of these 42 programs (BMD-02T) if one has, for example, three related system variables All possible time-correlation functions and their Fourier transforms are computed with x(t) regarded as an input signal Evidently there is sufficient versatility and generality to permit adaption to almost any type of transfer function experiment

Even more versatile than the Biomedical Computer Programs series is the BOMM system of programs 1 1 6 Here the user describes in few-word control statements the step by-step data-handhng operations to be performed on a time series, such as finding the mean, doing a cross correlation, or plotting an answer These control statements call in standard subprograms to the computer that perform all the detailed calculations Thus individualized data — processing needs can be satisfied with BOMM, although more effort is required to list the control statements than in the Biomedical Computer Programs

Although a computer is a virtual necessity for per­forming the required analysis on random fluctuations, it is not necessarily required in the special case of obtaining transfer functions from strip-chart recordings of sinusoidal oscillations where the signal-to-noise ratio is good 15 76 For example, the transfer function of the BORAX-4 reactor1 could be obtained to an accuracy of ±5% by chart reading and simple hand calculations, even though the root-mean-square oscillatory amplitude, lGlNk0/(2)^, ex­cited by k0 sin cot was only about twice the root-mean — square boiling noise In this technique the digitally de­termined (Eq. 6.9) normalized cross-correlation function, Ckn, of the reactivity and the power [which is lGlNk0 sin (cot + в) + noise] is equated to its theoretical expectation

However, for applications requiring many repetitive de terminations of a single function, the special-purpose continuous analyzer is strongly entrenched The consider­ations of Table 6 15 also apply to frequency domain analysis, as discussed in the following sections

As the use of on-line digital computers becomes more prevalent and accepted in reactor operation, on-line digital analysis of noise may be expected to be used competitively with other methods Cohn683’73 has demonstrated the ability to sample noise as often as every 0 5 Rsec and to do on-line correlations with a digital computer. Polarity correlating (ie., replacing the noise amplitude value by +1 or —1 for its fluctuation about an average of zero in computing correlation functions) was found useful in this application.

Typical specifications are summarized in Table 2 2 Since many different types are available, the values in the table have been entered as ranges

Tables 2 3 through 2 7 indicate the variety of neutron and gamma sensors available from one manufacturer The sensors listed are limited to those discussed in this chapter Commercial in-core neutron sensors are described in Chap 3

Table 2.3—Uncompensated Ionization Chambers* Nominal dimensions Neutron- sensitive material Thermal- neutron sensitivity, amp/nvt Gamma sensitivity, amp/(R/hr) Max. oper. thermal — neutron flux, nvt Typical oper. voltage, volts (d-c) Min signal resistance, ohms Signal capacitance, pF Max. oper. temp., °F Detector insulator І Length Sensitive, in. Overall, in. Detector O. D., in. 2 3 5 u 1 4 х 10 1 3 4 2 x 10’1 1 1 4 X 10’0 300-1000 109 150 300 ai2o3 6 n1/, 2 1 °B 4 4 х КГ1 4 4 5 x 10’1 1 5 0 x 10′ 0 200-1000 10′ 1 170 300 ai2o3 7 135/, 3 IJSU 2 6 х 1ЄГ1 4 3 0 x10 1 1 8 5 x 10‘ 0 300-1000 10’ 140 300 ai2o3 6 11 y2 2 ”SU 3 0 х10 1 4 4 2 x10 1 1 6 0 X 10′ 0 300-1000 109 150 300 ai2o3 6 11‘/2 2 2 3 5 u 1 4 х10 1 4 42×10" 1 4 x 10′ 1 300-1000 109 150 300 ai2o3 6 u1/. 2 1 °B 4 4 х10 1 4 4 5 x 10" 1 5 0 x 10‘ 0 200-1000 10′ 0 170 575 ai2o3 7 3 2 3 3 и 40 х 10 1 4 40 x 10" 1 2 7 X 10′ 0 200 1000 10" 160 575 ai2o3 7 135/s 3 2 3 5 u 1 4 х 10" 3 4 2 x 10 1 ‘ 1 4 X 10’0 300-1000 109 150 300 ai2o3 6 11’/, 2 10 в 4 4 х10 1 4 4 5 x 10" ‘ 50×10’° 200-1000 10‘ 1 170 300 ai2o3 7 “7. 3 10 в 1 5 х10 14 3 5 x 10 1 2 5 0 x 10 1 0 300 800 10′ 3 110 175 Rex 5‘/2 10’/2 3‘/2 2 3 5 U 2 8 х 10" 4 40 x 10" ‘ 5 0 x 10′ 0 300 1000 109 170 500 ai2o3 7 135/8 3 2 3 5 и 5 1 х 10 1 4 5 0 x 10" 1 2 7 x 10’0 300-1000 107 283 850 ai2o3 10 157. 17. 2 3 5 и 40 х 10 1 4 4 0 x 10 1 1 2 7 x 10’0 300-1000 107 150 700 ai2o3 7 135/8 3 2 3 5 и 1 4 х 10 1 3 4 2×10» 1 4 X 10‘ 0 300-1000 10* 1000 390 ai2o3 6 276 3 . ов 1 2 х 10 1 4 3 0 x 10 1 2 1 0 x 10’0 200 1000 10′ 2 1880 175 ai2o3 10 15/s 1 10 в 3 0 х 1 O’1 3 1 8 x 10" 0 2 5 x 10′ 0 300-1100 10′ 3 1850 175 Rex 108 1133/8 3‘/2 *Courtesy Westwghouse Electric Corp. +Nv is expressed in neutrons cm 2 sec 1 tAl2 Oj is a high alumina content ceramic Rex is a cross linked styrene

VJ 00

Table 2.4—Compensated Ionization Chambers* Thermal- Uncomp. Max. oper. Typical Min. Max. Nominal dimensions Length Sensitive, Overall, neutron sensitivity, gamma sensitivity, thermal- neutron oper. voltage, signal resistance, Signal capacitance, oper. temp., Insulation type$ Detector O. D., amp/nvt amp/(R/hr) flux, nv+ volts (d-c) ohms pF "F Detector Conn. in. in. in.

4 4 x 10 1 4 2 3 x10 1 1 2 5 xlO10 300-1000 10’4 275 175 Rex Rex 14 2 37, 3’/. 4 4 x 10" 4 2 3×10" 2 5 x 10′ 0 300-1000 10′ 4 275 175 Rex Rex 14 24’/я з1/. 4 4 x 10 1 4 2 5 x 10Г1 1 2 5 x 1010 300-1000 10′ 2 315 575 ai2o3 ai2o3 14 2il 3’/. 4 4 x 10’4 2 3 x 10" 1 2 5 x 10′ 0 300-1000 10′ 3 275 175 Rex Rex 14 2 Я ЗУ, 1 5 x 10Г14 3 5 x 10 1 2 2 5 x 10′ 0 300-800 10′ 3 130 175 Rex Rex 5’/, ю У, ЗУ, 1 5 x10 1 4 3 5 x 10" 2 25×10’" 300-800 10′ 3 135 175 Rex Rex 5’/2 ю ■/, з1/, 4 4 x10’4 2 3×10" 25×10’" 300-1000 10′ 3 290 400 ai2o3 ai2o3 14 !9’/8 з1/. 1 0 x 10 1 5 1 5 x10 1 3 1 5 x 10‘2 25-250 101 1 155 660 ai2o3 ai2o3 2У16 75/>6 3 4 4 x10 ’ 4 2 3 x 10 1 1 2 5×10" 300-1500 10′ 3 290 300 ai2o3 ai2o3 14 19’/, ЗУ.

Table 2.5—Fission Counters’ Thermal — neutron sensitivity, (counts/sec)nv + Max. oper thermal — neutron flux, nv + Typical oper. voltage, volts (d-c) Min. signal resistance, ohms Signal capacitance, pF Max. oper. temp., °F Insulator type;;: Detector Conn Nominal dimensions Length Detector O. D., in Sensitive, in. Overall, in 0 7 14x10s 200-800 109 150 300 ai2o3 ai2o3 6 ny 2 0 2 5 0 x 105 200-800 10“ 140 300 ai2o3 ai2o3 6 11 у 2 0 14 7 0 x 10s 200-800 109 150 300 ai2o3 ai2o3 6 ny 2 1 25 x 10 3 10×10* 250 500 109 55 575 ai2o3 Rex % 5 35/1 6 0 210 0 52 2 0 x 10s 200-800 109 150 300 ai2o3 ai2o3 6 11У 2 0 07 1 4 x 10‘ 200-800 109 150 300 ai2o3 ai2o3 6 11У 2 0 7 1 4 x 10s 200-800 109 160 575 ai2o3 ai2o3 7 13s/8 3 0 7 1 4x10s 200 800 109 150 300 ai2o3 ai2o3 6 ny 2 0 7 1 4 x 105 200 800 107 150 700 Al2 O3 ai2o3 6 11У 2 0 1 1 0 x 10" 300 800 109 30 575 ai2o3 ai2o3 4% ?7. 1 0 14 7 0 x 10s 200 800 109 170 500 ai2o3 ai2o3 7 13 % 3 0 5 2 0 x 10 s 200 800 10 7 283 850 ai2o3 ai2o3 10 15 7. 1% 0 25 4 0 x 105 200 800 107 150 700 ai2o3 ai2o3 7 3 3 % 3 5 x 10 3 2 0 X 108 350 650 1010 40 750 ai2o3 1 5 2 0 7 14 x10s 75 109 160 500 ai2o3 — 7 14 3 0 35 2 8 x 10s 200 800 109 150 300 ai2o3 ai2o3 6 ny 2 § lOx 10′ 200 800 109 150 300 ai2o3 ai2o3 6 11У 2 0 7 14×10’ 200-800 108 1000 390 ai2o3 ai2 o3 6 276 3 2 2 x 10 4 5 0 x 108 250 800 5 x 108 2 500 alo3 — 5/1(, 1’/,. 0 220 1 5 x 10 3 1 0 x 108 300- 500 108 260 500 ai2o3 A1,03 7, 243 0 210 1 x 10-5 10×10” 100-200 109 125 250 I h Rex ■Уза 6з5/,6 0 090 0 18 6 0 x 10s 200-800 109 45 575 ai2o, ai2o3 «7. 12 1 0 5 2 0 x 10s 200-800 108 150 390 a 12 0 3 ai2o3 6 ny 2 0 35 3 0 x 10s 200 800 108 90 390 ai2o3 ai2o3 3 "У 2 *Courtesy Westtnghouse Electric Corp. + Nv is expressed in neutrons cm 2 sec 1 $A1203 is a high alumina-content ceramic Rex is a cross linked stvrene §Sensitive material is 238 U Sensitivity to ^ 1 5 MeV neutrons = 10 3 to thermal neutrons = 1 4 x 10 4

Chi SO

‘Courtesy Westinghouse Electric Corp.

+ Nv is expressed in neutrons cm"2 sec 1

*A1303 is a high-alumina-content ceramic, Rex is a cross linked styrene §Oval case 7l s/ in and 3^ 6 in

REFERENCES

1 S Glasstone and M C Edlund, 7be Elements of Nuclear Reactor Theory, D Van Nostrand Company, Inc, Princeton, N J, 1955.

2 Reactor Physics Constants, USAEC Report ANL-5800 (2nd Rev ), Argonne National Laboratory, Superintendent of Docu­ments, U S Government Printing Office, Washington, D C, 1963

3 J A Crowther, Ions, Electrons and Ionizing Radiations, 8th ed, Edward Arnold, Ltd, London, 1949

4 D R Bates (Ed ), Atomic and Molecular Processes, Academic Press, Inc, New York, 1962

5 В В Rossi and H H Staub, Ionization Chambers and Counters, Experimental Techniques, McGraw-Hill Book Company, Inc, New York, 1949

6 USA Standard Glossary of Terms in Nuclear Science and Technology, USAS N 1 1—1967, United States of America Standards Institute, New York 1967

7 W Abson and F Wade, Nuclear Reactor Control Ionization

Chambers, Proc Inst llec Eng (London), 103B(22) 590

(1956)

8 M L Awcock, U2 3 5 Coated Ionization Chamber, Type 1Z-400, Canadian Report AECL-805, pp 44-45, August 1959

9 L Colli and V Facchini, Drift Velocity of Electrons m Argon, Rev Set Instrum, 23: 39 (1952)

10 V Facchini and A Malvicmi, Argon—Nitrogen Fillings Make Ion Chambers Insensitive to 02 Contamination, Nucleonics, 13(4) 36 (1955)

11 W M Trenholme, Effects of Reactor Exposure on Boron Lined and BF3 Proportional Counters,//?/[3] Trans Nucl Sci, NS-6(4)

1 (1959)

12 J L Kaufman, High Current Saturation Characteristics of the ORNL Compensated Ionization Chamber (Q1045), USAEC Report CF-60 5-104, Oak Ridge National Laboratory, May 25, 1960

13 D P Roux, Parallel-Plate Multisection Ionization Chambers for High-Performance Reactors, USAEC Report ORNL 3929, Oak Ridge National Laboratory, April 1966

14 E В Hubbard, Compensated Ion Chamber, in Proceedings of the 1959 Biannual National Nuclear Instrumentation Sym posium, Idaho Falls, Idaho, June 24—26, 1959, ISA Vol 2, pp 99 106, Instrument Society of America

15 W H Todt, A Gamma Compensated Neutron Ionization Chamber Detector for the NERVA Reactor, ILLI (Inst Elec Electron Eng) Trans Nucl Sci, NS-15(1) 9 1 (1968)

16 H S McCreary, Jr, and R T Bayard, A Neutron Sensitive Ionization Chamber with Electrically Adjusted Gamma Com pensation, Rev Set Instrum, 25. 161 (1954)

17 W Baer and R T Bayard, A High Sensitivity Fission Counter, Rev Set Instrum, 24: 138 (1953)

18 W Abson, P G Salmon, and S Pyrah, The Design, Performance and Use of Fission Counters, Proc Inst Elec Eng (London) 105B(22) 349 (1958)

9 S A Korff, Proportional Counters, Nucleonics, 6(6) 5 (1950)

0 S A Korff, Proportional Counters, II, Nucleonics, 7(5) 46

(1950)

21 W Abson, P G Salmon, and S Pyrah, Boron Trifluoride Proportional Counters, Proc Inst Elec Eng (London), 105B(22) 357(1958)

22 N M Gralenski and J E Schroeder, Description and Analysis of a Sensitive BF3 Filled Uncompensated Ionization Chamber USAEC Report DC-60 11 73, General Electric Company, Nov 10, 1960

23 R В Mendell and S A Korff, Plateau Slopes and Pulse Characteristics of Large, High-Pressure BF3 Counters, Rev Set Instrum, 30: 442 (1959)

24 J W Htlborn, Self Powered Neutron Detectors for Reactor Flux Monitoring, Nucleonics, 22(2) 64(1964)

25 J Moteff, Neutron Flux and Spectrum Measurement with Radioactivants, Nucleonics, 20(12) 56 (1962)

26 W C Judd, Continuous Flux Monitoring of a High-Flux Facility with A41, in Reactor Technology Report No 14, USAEC Report KAPL-2000-11, p II1-1, Knolls Atomic Power Labora tory,1960

27 V Adjacic, M Kurepa, and В Lalovic, Semiconductor Measures Fluxes in Operating Core, Nucleonics, 20(2) 47 (1962)

28 R Babcock, Radiation Damage in SiC, IELE Trans Nucl Set, NS-12(6) 43-47 (1965)

29 R V Babcock and H C Chang, SiC Neutron Detectors for High Temperature Operation, in Neutron Dosimetry Symposium Proceedings, Harwell, Eng, December 1962, pp 613 622, Inter national Atomic Energy Agency, Vienna, 1963 (STI/PUB/69)

30 R R Ferber and G N Hamilton, Silicon Carbide High Temperature Neutron Detectors for Reactor Instrumentation, Nucl Appl, 2. June 1966

(a) High-Voltage Power Supplies. Solid-state power supplies with excellent characteristics are available for
powering the nuclear detectors Solid-state power supplies reduce the package size, the heat generated, and the maintenance time Many power supplies are capable of supplying power to two or more detectors connected in parallel When power supplies are used to provide for two or more detectors, the detectors must be isolated from ground. Also, note that on a common-failure basis the units are not independent and cannot be used as redundant units in a shutdown circuit. It is desirable that the loss of one power supply will not cause a scram.

The high-voltage power supplies must have a voltage monitoring circuit or an output voltage proportional to the high-voltage output which can be used to monitor the high voltage The high-voltage monitor is used to give an alarm when the high voltage drops below a predetermined value Figure 5.13 shows a high-voltage monitor unit with a variable set-point alarm. The 10-turn potentiometer dial reading corresponds to the trip voltage

The high-voltage power supplies should have short- circuit protection. Protection against voltage surges at the input is also desirable.

(b) Low-Voltage Power Supplies. Power supplies used in the counting equipment must have excellent regulation and stability. The low-voltage supplies should have internal short-circuit protection to prevent supply damage due to overcurrent demands from the unit. Because the counting equipment is of solid-state design, it is important that the
voltage supplied to the printed circuit boards be limited below a value that would cause component damage.

Power supplies with built-m protection are commer­cially available and should be specified when ordering the nuclear equipment. An inexpensive method of providing overvoltage protection is shown in Fig. 5.14. The Zener diodes prevent the voltage from exceeding a given value. Excessive current drawn from the supply through the Zener diodes will cause the fuse to open before circuit damage has occurred.

Control rods are arranged in symmetrical patterns within the core structure and around the core periphery The total absorption capability in all the rods largely determines the shutdown reactivity of the core Shutdown reactivity is defined as the increase in reactivity required to bring the reactor to critical from a fully shutdown condition

Safety rods are specifically designed to effect rapid shutdown or scram should a hazardous reactor or plant condition occur These rods normally contain poison material and are withdrawn to a maximum degree before start-up and are kept in that position during power operation They are designed for rapid release and accelera­tion into the reactor Separate safety rods are seldom used in the present generation of power reactors, but their scram function is combined into the shim rods (Vol 2, Chap 12)

Shim rods comprise the greatest number of rods and control the greatest amount of reactivity Shim rods are used to remove shutdown reactivity during start up and to offset the effects of temperature, xenon, samarium, and fuel depletion during power operation Controlled shim-rod motion is very slow during power operation and may be only a few inches or tenths of inches per day Shim rods are usually arranged in groups or banks, only one of which can be moved at a time In any group, one or several rods can be moved at the same time, at the option of the operator

The regulating rod, similar in design to a shim rod, is used for fine control of reactor power level Small changes of reactivity are needed Regulating-rod motion may be

manually controlled, but usually the regulating rod is driven by a servomechanism in a feedback control loop that finely regulates power or flux To satisfy the performance requirements of automatic control, the drive mechanism for the regulating rod has more stringent performance require ments than the drives for shim or safety rods

Some of the neutrons participating in the chain reaction are emitted at various times after the fission event When certain fission product nuclides decay by emitting beta particles, the resultant nuclides are unstable, and each of the nuclides emits a neutron immediately after the beta decay T he rate of neutron emission therefore is the same as the rate of beta decay of these “precursor” nuclides, і e, a rate that decreases exponentially with time

I ables 1 1 and 1 2 list the half lives and decay constants [X = (In 2)/T^ = 0 693/T^l of the delayed neutron emitters resulting from the fissioning of 2 3 3 U, 2 3 5 U, and 2 3 9 Pu by thermal neutrons and by fast neutrons, respectively The tables also list the absolute yields of delayed neutrons (number of delayed neutrons per fission emitted by each precursor type) and the relative abundances of the delayed neutrons (number of delayed neutrons emitted by each precursor tvpe divided by the total number of delayed neutrons emitted in a fission process)
fraction can be expressed as the sum of the delayed-neutron fractions for each group of the delayed neutron emitters

m

0 = E ft (15)

1=1

where m is the number of delayed-neutron groups

Table 1 3 lists the values of ft for the fission of 2 3 3 U,

2 3 5 U, and 239Pu by thermal and fast neutrons The table

also lists v, the number of prompt neutrons per fission Values of ft are obtained by multiplying the relative

abundance values in Tables 1 1 and 1 2 by the values of /3 in Table 1 3

The average energy of the delayed neutrons is not the same as the average energy of the prompt neutrons Thus, in any chain-reacting system, the effectiveness of the delayed neutrons in propagating the nuclear fission chain reaction differs from that of the prompt neutrons The factor (3 used in f-q. 1 4 does not take this into account

since (3 is a simple ratio of numbers of neutrons To take the neutron energies into account, replace /3 with

7(3 = effective delayed neutron fraction

= (number of fissions caused by delated neutrons)/ (number of fissions caused by delayed plus prompt neutrons) (1.6)

where 7 is the delayed-neutron effectiveness The value of 7 depends on the chain-reacting system and is gen­erally slightly greater than 1 For the power reactors discussed in this book, it is a good approximation to assume 7=1. Likewise, the delayed-neutron effectiveness for the individual delayed-neutron groups can be assumed to be 1, і e., 7 = 7i = 1.

The basic kinetic equations for a nuclear fission chain reaction in which delayed neutrons are taken into account are obtained by writing the rate of change of the neutron density (n = neutrons/cm3) as a sum of two terms

dn

— = (rate of change of prompt-neutron density)

+ (rate of change of delayed neutron density) m

= 7fk(l -0) — 1] + E A, C, (1 7)

‘ 1=1

where C, 1S the density (number/cm1) of delayed-neutron emitters of the uh group and is the decay constant (fraction decaying/sec) of the іth delayed-neutron emittei group The number of groups, m, is 6 (see 1 ables 1 1 and 1.2). The first term of f q 1 7 is obtained by substituting the multiplication factor for prompt neutrons (Eq. 1.4) for к in Eq. 1 1

The density of each of the delayed neutron-emitting groups, C,, is obtained from the equation

dC

-jjA = (rate of production of і th group of delayed- neutron emitters) — (rate of decay of і th group of delayed-neutron emitters)

kftn

/

The first term is the multiplication factor for delayed neutrons (Eq 1 4) divided by the time interval between generations, or the prompt-neutron lifetime, / If there is a source of neutrons present other than the fissionable isotopes and the fission products that emit neutrons, then a source term must be added to the right-hand side of f q 1.8

Equations 1 7 and 1 8 are the basic neutron kinetics equations They are important in the design of the instrumentation and control systems for nuclear power reactors. In Chap 6 the equations are used to show how transfer-function measurements can yield useful informa­
tion on power-reactor behavior In Chap 7 the equations are shown to be basic to the design of reactor control sy stems

The measurement of temperatures in gas-cooled reac­tors requires certain specialized sensors, e. g., sensors based
on the transmission of acoustic energy through gases. For a complete discussion, see Vol. 2, Chap. 18, Sec. 18-2.2.

4- 3 PRESSURE SENSING AND TRANSMITTING

4- 3.1 Sensors

This section deals with elastic sensing elements that respond to a system pressure change and, in so doing, generate a measurable physical quantity, such as position or

•Courtesy Pall Trinity Micro Corp Values from ASMF Boiler and Pressure Vessel Code, Sec. VIII—Pressure Vessels, 1971

tASME Spec. Min Tensile = 45,000 psi $ASME Code, case 1325 (special ruling)

§ASME Code, case 132 3 (special ruling) f ASME Code, case 1321 (special ruling)

mechanical or electrical force. Each sensor is a differential element, and atmospheric pressure is constantly applied in opposition to the system pressure. To sense the absolute pressure, you must apply a second element (e. g., a calibrated spring) in opposition or place that part of the sensor that is normally at ambient (atmospheric) pressure within an evacuated containment.

(a) Materials. In out-of-core pressure sensors, materials coming in contact with the measured fluid must be noncorrosive, must not otherwise deteriorate, and must not contain elements that may become dangerously radioactive by accidental exposure to neutrons. The objective is a device capable of continuous, dependable pressure sensing over an extended period of time. Sensor materials contact­ing the measured fluid should be compatible with the fluid This is the same problem that is involved in choosing thermowell materials. Stainless steels, type 304 or better, are frequently used. Sometimes Inconel is used. Teflon materials for seals and О-rings are avoided as are any components containing cobalt. If the most highly desirable materials are not available at the sensor, diaphragm seals described in Sec. 4-3.4(b) are used.

(b) Basic Types. Elastic metal sensors, available in a variety of forms, consist of slack and rigid diaphragms, multiple or stacked diaphragms, corrugated bellows, and the Bourdon tube in a variety of forms, from single-turn and torsion-bar to helical and spiral multiple-turn designs.

Each manufacturer has his own series of ranges for the various designs based upon the sizing of components and the required performance of a complete linkage system or other device that depends on this initiating element for its successful operation. Table 4.16 gives some typical ranges, and Sec. 4-3 6 gives a sample set of performance specifica­tions.

Strain gages consist of a fine wire or an array of fine wires usually bonded into an assembly for mechanical strength. Under an applied stress the array of fine wires is stretched, this results in an increase in its electrical resistance. If this array is incorporated in a suitable arrangement, the resistance change can be made directly proportional to the imposed pressure. Close temperature control must be maintained by comparing the strain wire with unstressed wire (or compensation), electrical shielding of the sensor wire is also important. In some designs the strain wire may be mounted (bonded) on a Bourdon tube, bellows, or mechanical structure, such as a beam or ring [11] In Fig. 4.17 the strain gage is in the form of a short tube sealed by a diaphragm. Note the variable resistance is applied as a leg of a conventional Wheatstone measuring bridge Because the fractional change in strain-gage resis­tance is very small, electrical amplification and signal conditioning are usually required before use in readout and action modules.

Piezoelectric sensors are similar to strain-gage sensors with a crystal used for stress sensing instead of a wire. The crystal responds to a pressure change (usually expressed by a force in a predetermined direction with respect to the crystal axes) by generating a small electrical potential difference. The latter depends on the magnitude of the imposed stress and on the crystal properties Again, temperature control or compensation, as well as amplifica­tion and signal conditioning of the output, are essential.

Silicon wafer piezoelectric sensors now available are capable of sensing range spans from 0 to 6 psig to 0 to 1500 psig. Output is 10 to 50 mA d-c. Features include a range span adjustment of 4 1 for a given diaphragm and

the capability of elevating the range span (zero suppression) to the maximum pressure range for the unit. Thus a 0- to 1500-psi range device would be expected to calibrate for 1100-to 1500-psig range input for a 10- to 50-mA d-c output.

Historically the rod-oscillator measurement of the zero-power transfer function is one of the oldest reactor — dynamics measurements,4 dating back to CP-2 Since then it has been repeated many times on many reactors with the techniques detailed below

Basically, a sinusoidal reactivity excitation provided by a rotating or reciprocating control rod causes a power oscillation that is detected by ion chambers. The purpose of these measurements may be any of the following

l. To determine j3/1 for a particular reactor by fitting the measurement to a formula in Table 6.4.

2. To verify experimental techniques on a known transfer function.

3 To compare reactivity effects of small samples by the amplitude of the resulting power oscillation.

A considerable number of methods in addition to the rod oscillator have proved useful in obtaining dynamics information, і e., in determining quantities closely related to the reactor transfer function. These methods are listed and classified in Table 6.5. Methods having no reactor excitation by external equipment depend on the random fluctuations or noise in the neutron population, as detected by counters or ion chambers, to provide information about

reactor characteristics The following incentives for applying such methods differ slightly from those for the rod oscillator

1. To determine specific reactor parameters, such as (3// at critical, the subcntical reactivity, or the absolute reactor power.

2 To verify experimental techniques on a known system

3 To investigate spatial neutron effects

Experiments on the dynamic behavior of zero-power reactors have been numerous and also somewhat repetitious owing to similarities among the reactors and kinds of equipment used. The particular reactors studied along with the classes of information obtained are given in Table 6.6. It will suffice here to point out that the quantities being measured are the constant parameters appearing in the neutron kinetics equations and the transfer function. (The numerous cases in which samples have been oscillated to make reactivity measurements are omitted since the emphasis here is on transfer functions.)

Section 6-3, Methods of Measurement, summarizes the features of the various methods and how the results in Table 6.6 are obtained

Although neutron detectors have been used in virtually all the zero-power dynamics studies to date, there has been some theoretical732 and experimental2 3,69a work in­volving gamma detectors. Cerenkov detectors and liquid scintillators have been found suitable for monitoring the fluctuations of prompt gammas from fission. The emission of these gammas, like the neutrons, exhibits statistical fluctuations that depend on the reactor’s zero-power transfer function. Since gammas travel farther than neu­trons in a reactor, the prompt gammas offer a means of using peripheral detectors to monitor deep into the core of a large reactor In one novel application693 two widely separated gamma detectors were pointed in collimated fashion at various fuel elements, and relative local power values were obtained by cross correlation.