The nuclear fission process utilized in today’s power-producing reactors is initiated by interaction between a neutron and a fissile nucleus, such as 235 U.+ The nucleus then divides into two fragments, with release of an enormous amount of energy and with production of several new neutrons. Under proper conditions, these product neutrons can react with additional 331U atoms and thus give rise to a neutron chain reaction, which continues as long as sufficient 235 U remains to react. Fission of a single nucleus of 23SU is represented pictorially in Fig. 1.1, and a fission chain reaction is shown in Fig. 1.2. To keep the rate of the chain reaction constant, neutrons are allowed to leak from a nuclear reactor or are absorbed in boron, 238 U, or other nonfissionable materials placed in the reactor. A steady chain reaction is depicted in Fig. 1.3.

The fission of 235 U can take place in a number of ways, one of which is shown in Fig. 1.4. The nucleus of 235 U, which contains 92 protons and 143 neutrons, divides into two fragments, plus some extra neutrons, in such a manner that the total number of protons and neutrons in the product nuclei equals the total number in the reactant neutron and 235 U nucleus. In the example of this figure, the fission fragments are 144 Ba, containing 56 protons and 88 neutrons; **Kr, containing 36 protons and 53 neutrons; and three extra neutrons. The fission fragments are unstable and subsequently undergo radioactive decay. In the radioactive decay some of the neutrons of the nucleus are converted into protons, which remain in the nucleus, and into electrons, which fly out as beta radiation. In this example, four neutrons in 144 Ba are successively converted into protons, resulting in 144 Nd as end product, and three neutrons in 89Kr are converted into protons, resulting in 89Y as end product.

The numbers assigned to each reactant or end product represent its mass in atomic mass units (amu). This unit is defined as the ratio of the mass of a neutral atom to one-twelfth the mass of an atom of 12 C. In the present instance the mass of the products is less than that of the reactants[1]:

A fraction 0.210675/235.043915 = 0.0008963 of the mass of the 235 U atom disappears in this fission reaction. This reduction in mass is a measure of the amount of energy released in this fission reaction. The Einstein equation (1.1) expressing the equivalence of energy and mass,

AE = c2Am (1.1)

predicts that when Am kilograms of mass disappears, ДE joules of energy appears in its place. In this relation, c is the velocity of light, 2.997925 X 10s m/s. t The energy released in this fission reaction thus is

(0.0008963) (2.997925 X 108)2 = 8.06 X 1013 J/kg 235 U (1.2)

or 3.46 X 1010 Btu/lb.

Energy changes associated with a single nuclear event are commonly expressed in terms of millions of electron volts (MeV), defined as the amount of energy acquired by an electronic charge (1.602 X 10’19 C) when accelerated through a potential difference of 1,000,000 V. One MeV therefore equals 1.602 X 10"19 X 106 = 1.602 X 10’13 J.

The energy released when one atom of 235 U undergoes fission in the above reaction is

(8.06 X 1013 J/kg)(235.04 g/g-atom)

(1.602 X 10-13 J/MeV)(6.023 X 1023 atoms/g-atom)(1000 g/kg)

+ Fundamental physical constants are listed in App. A. A table of mass and energy equivalents is given in App. B

Ne

Atoms of 235 U may undergo fission in a variety of ways, of which the reaction shown in Fig. 1.4 is only one. The average yield of particles and energy from fission of 235 U in all possible ways is shown in Fig. 1.5. In the primary fission reaction shown at the top of this figure, 23S U splits into two parts, the radioactive fission products, while at the same time giving off several fast neutrons (2.418 on the average) and gamma radiation. One of these neutrons is used to maintain the fission reaction. The remaining neutrons may either be used to bring about other desired nuclear reactions or be lost either through leakage from the reactor or through capture by elements present in the reactor to produce unwanted or waste products.

Following the primary fission reaction, the radioactive fission products undergo radioactive disintegration, yielding beta particles and delayed gamma rays and ending up as stable fission products. Since the radioactive fission products have half-lives ranging from fractions of a

Neutron Uranium-235 Barium-144 Krypton-89

1.008665 amu 235.043915 amu

Captured in shield and reactor

Producing-з to 12 MeV of energy

Used to continue chain reaction

Initial fission reaction

— О +

L

Stable fission products

/

Delayed gamma rays
6 MeV

Later radioactive disintegrations Figure l. S Average yields in fission of 23S U.

second to millions of years, the emission of beta particles and delayed gamma rays takes place over a long period of time after a reactor has been shut down, but at a diminishing rate.

The total energy released in fission is the sum of the energies associated with the different particles shown in this figure, 196 to 205 MeV. As up to 5 MeV of gamma energy escapes from a typical power reactor and is not utilized, a nominal figure for the energy released in fission is 200 MeV. This corresponds to around 35.2 billion Btu of energy per pound or 0.95 MWd of energy per gram of 235U undergoing fission. In addition, some 235U is consumed without undergoing fission by reacting with neutrons to form 236 U. When this reaction is taken into account, the energy released is around 29 billion Btu per pound, or 0.78 MWd per gram of 233 U consumed. This is about 2 million times the energy released in the combustion of an equivalent mass of coal.

After 135 Xe, the fission product with highest cross section and appreciable yield is 149 Sm, whose cross section for 2200 m/s neutrons is 41,000 b and whose effective cross section in a typical water-cooled reactor is over 70,000 b. In addition, many of the fission-product nuclides that produce 149 Sm by neutron capture or radioactive decay and several of the nuclides produced from 149 Sm by successive neutron captures have high cross sections. Figure 2.17 illustrates the generic relationship between 149 Sm and the principal nuclides that lead to it or are produced from it.

Table 2.15 gives direct fission yields у [B3], effective thermal-neutron absorption cross sections a and half-lives (cf. App. C) for radioactive decay that are used below to evaluate the poisoning ratio for this chain. Effective cross sections were calculated from cross sections for 2200 m/s neutrons and for neutrons of higher energy from cross-section data given by Bennett [B3], applied to the neutron spectrum of a typical pressurized-water reactor.

The set of 11 differential equations that describe the rate of change of each of the 11 nuclides in the 149 Sm fission-product chain, assuming no processing removal, are

(2.124)

(2.125)

(2.126)

(2.127)

(2.128)

(2.129)

(2.130)

(2.131)

(2.132)

(2.133)

(2.134)

y = Direct yield from fission

Figure 2.17 The fission-product chain leading to 149 Sm.

The poisoning ratio for this set of nuclides is

The solution of this set of equations, with zero initial amount of each nuclide, can be written directly by applying Eq. (2.113). To do so, the nuclide chains of Fig. 2.17 are reformulated into an equivalent set of linear chains with constant formation rate of the first

member of each chain and with each subsequent member of a chain formed only by decay or neutron reaction of its single precursor within the chain. From the data in Table 2.15, only four of the nuclides, 147Nd, 149Pm, lslSm, and 152Sm, have finite direct yields from fission. Each of these four nuclides is the first member of a chain formed at a rate

P,=y, Nfof<t> (2.136)

Two of these chains, i. e., those originating from the direct yields of 147Nd and I49Pm, involve chain branching. For the purpose of calculating the amount of the nuclide at which the branched chain converges, and to calculate the amount of the daughters of this nuclide in the chain, the branched chain must be subdivided into a subset of linear chains as illustrated in Sec. 6.1. For example, for the purpose of calculating the amounts of I50Sm, 151 Sm, and 152Sm formed from the chain initiated by the direct yield of 149Pm, this chain is expressed as two subchains:

y6Nfor<t>

149Pm 149Sm 150Sm ^ 151 Sm ^ 152Sm

Ф°(,

and

Уь^ГОf<t>

149 Pm IS0Pm — Ъ — lsoSm 151 Sm ^ 1S2Sm ^

^6 0CT 8 ^-10

Similarly, the chain originating with the direct yield of 147Nd branches at 147Pm, 148mPm, and I49Pm. It is subdivided into two linear chains to calculate the contribution to Л’5, three to calculate the contribution to N6, and six to calculate the contributions to N9, N10, and Nu. In this way the summation Eq. (2.113) can be used to write the solution for this series of chains.

To calculate the growth and decay of these nuclides after reactor shutdown, the assumed equilibrium amounts at the time T of shutdown are calculated as above, using Eq. (2.114). These become the initial amounts N® for application of the batch decay, Eq. (2.18) for time t after shutdown. During shutdown the branching and convergence involving neutron reactions disappear, and we have only four simple linear chains to solve by applying Eq. (2.18).

Alternatively, the differential equations may be solved directly by numerical methods with a digital computer [С2]. Results obtained from the latter approach are shown in Figs. 2.18 to 2.20. Calculations were made for a thermal-neutron flux of 3.5 X 1013 «/(cm2 -s), considered representative of a 1060-MWe pressurized-water reactor similar to one manufactured by Westinghouse for the Donald C. Cook Nuclear Plant [A 1 ].

Figure 2.18 shows the contribution of individual nuclides to the poisoning ratio as a function of time, starting with fresh, unirradiated fuel at time zero. The poisoning ratio of 149Sm builds up very quickly to 0.0113, the fission yield at mass 149, and then increases more gradually because of additional 149 Sm production by neutron capture in nuclides of mass 147 and 148. Other nuclides of this chain that make appreciable contributions to the poisoning ratio include 147Pm, 148mPm, ls0Sm, 151 Sm, and 152Sm. The overall poisoning ratio, the sum of the contributions of individual nuclides, is shown in Fig. 2.19.

Figure 2.20 shows how the poisoning ratio of this chain varies if the reactor is shut down after initial operation for 7300 h for various periods of time Ґ and then operated at a flux of 3.496 X 1013 n/(cm2,s) for additional time T. The behavior shown in this figure is considered representative of this reactor after it has been refueled several times with one-third of the oldest fuel replaced by fresh fuel.

When an aqueous solution of an extractable component is brought into equilibrium with an immiscible solvent for the component and the two phases are then separated, the component will be found distributed between the two phases. This distribution may be characterized by the distribution coefficient D, defined as

£ _ concentration of component in organic phase concentration of component in aqueous phase

for the two phases leaving the equilibrium contactor. The distribution coefficient is a function of the nature of the solvent, the temperature, and the equilibrium compositions of the aqueous and organic phases, but is independent of the amount of either phase.

The fraction of component initially present in the aqueous feed that is extracted in one stage of equilibrium contacting depends on the relative volumes of aqueous and solvent phases. Nomenclature for deriving an equation for the fraction extracted is given in Fig. 4.1.

The material balance on the extractable component may be expressed as

Ey _ ED/F P Fz 1 + ED/F

F Volume E

The ratio of solvent to feed needed for a given fraction extracted is

E_ _ P

F D( 1 — p)

The fraction extracted therefore becomes greater, the greater the ratio E/F of solvent to feed, but an infinite amount of solvent is needed for complete extraction in a single contact.

Solvent leaving this equilibrium contactor is capable of extracting more of the metallic component from additional aqueous feed, because the feed concentration z is greater than the concentration x in the aqueous phase with which this solvent is in equilibrium. It is therefore possible to reduce the amount of solvent needed for a given fraction extracted by using multiple contact between solvents and aqueous phases in a countercurrent cascade, as illustrated in Fig. 4.2. As the number of stages is increased indefinitely, the organic extract approaches equilibrium with the aqueous feed, so that in the limit

У max ~ (4.7)

where D is the distribution coefficient for the feed stage.

For a given ratio E/F, the ymax from an infinite number of contacting stages results in the maximum recovery pmax, i. e.,

. = ЕУтах = §D

Ртах pz p

Alternatively, for a specified recovery p, the minimum ratio (E/F)to achieve a specified recovery p occurs for an infinite number of stages and is given by

Thus with a large number of stages it becomes possible to approach complete extraction,

i. e., p = 1, with a finite amount of solvent. With a finite number of stages, the relative amount E/F of solvent for a given fraction extracted lies between the minimum, given by Eq. (4.9), and the single-stage maximum value given by Eq. (4.6).

Because the solvent (Fig. 4.2) ordinarily is valuable, it is desirable to wash the extracted component out of it and to recycle the solvent for reuse. A flow sheet of this type is shown in

Figure 4.2 Multistage countercurrent solvent extraction. M, mixer; S, settler.

Fig. 4.3. Here the group of solvent extraction stages used to extract the desired component from feed has been designated as the extracting section, and the group used to wash this component back into the aqueous phase as the stripping section. In practice, each section may be either batteries of mixers and separators as shown in Fig. 4.2, in which phases are alternatively mixed and separated, or countercurrent columns, in which the two phases flow past one another in continuous contact and continuously exchange material.

A flow sheet similar to Fig. 4.3 could be used to extract uranium from sulfuric acid-leach solutions with organic amines, as illustrated in Fig. 5.9.

When more than one component of the feed is extractable, the flow sheet of Fig. 4.3 is not capable of producing any one component in pure form, because the organic phase leaving the extracting section will carry some of every component with it. To separate the most extractable component in relatively pure form, it is necessary to add an additional scrubbing section, as shown at the top of Fig. 4.4. The purpose of this scrubbing section is to scrub all but the most extractable component from the organic phase leaving the extracting section. This section functions somewhat like the enriching section of a fractional distillation column and provides partial reflux to wash back the components not wanted in the product. An operation of this kind, in which two or more extractable components are separated by distribution between two counterflowing solvents, is called fractional extraction.

Separation of two components by fractional extraction is possible when the distribution coefficient of one component Df differs from that of the other component Dj.

The concentrations of component і in organic phase yt and aqueous phase x,- leaving a stage are related by [9]

and a similar equation holds for component /:

Уі = Dixi

The ratio of concentrations in the organic phase is related to the ratio of concentrations in the aqueous phase by

Уі _ Dixi

Уі Dixi

Thus, separation is possible when DtjDj Ф 1. The ratio of distribution coefficients is a measure of the ease or difficulty of a separation by fractional extraction and is known as the separation factor a:

_ У‘/х‘ = 2l Уііхі Di

A flow sheet like Fig. 4.4 has been used to separate uranium from neutron-absorbing impurities (Chap. 5), and zirconium from hafnium (Chap. 7), by fractional extraction of an aqueous nitrate solution with an organic solution of TBP in kerosene.

For every additional extractable component to be separated in pure form, two additional sections are required, one for scrubbing and the other for stripping. As an example, Fig. 4.5 shows the flow sheet used in the Purex process to extract pure uranium and pure plutonium from fission products and to separate them from each other by fractional extraction between aqueous phases and TBP in kerosene.

Figure 4.5 Fractional extraction of mixture of plutonium, uranium, and fission products. Solid line, aqueous; broken line, organic.

Here, sections C and D have been added to the flow sheet of Fig. 4.4 in order to separate the uranium and plutonium present in the extract leaving section В by fractional extraction.

The probability that a radioactive nucleus will decay in a given time is a constant, independent of temperature, pressure, or the decay of other neighboring nuclei. The disintegrations of individual nuclei are statistically independent events and are subject to random fluctuations. In a large number of nuclei, however, the fluctuations average out, and the fraction that decays in unit time is a constant and is numerically equal to the probability that a single nuclei will decay in that time. This rate of radioactive decay is known as the decay constant X, with dimensions of reciprocal time.

Because the number of nuclei that decay in unit time is proportional to the number present, radioactive decay is a first-order reaction. If N is the number of nuclei present at time t, and if N changes with time only because of radioactive decay, then

This integrates to

N = N(2.2)

where № is the number of nuclei present at time zero. Thus, of № nuclei originally present, №e~remain at time t. The number with lives between t and t + dt is

-dN = Ш°е-М dt

The mean life т is the reciprocal of the decay constant, as may be seen from

(2.4)

It is customary to describe the specific rate of radioactive decay by the half-life f, which is the length of time required for half of the nuclei originally present to decay. The relation between the half-life and the decay constant is found from

The curie (Ci) is a unit frequently used as a measure of the amount of radioactive material. It is defined as the amount of radioactive material that will produce 3.7 X 1010 disintegra­tions^. This is approximately the number of disintegrations per second in 1 g of radium. A more up-to-date unit is the Becquerel, which is the amount of radioactive material that produces one disintegration pet second.

Because the number of disintegrations per second in 1 g-atom is 7N, where N is Avogadro’s

number, 6.02252 X 1023 atoms/g-atom,^ the number of curies per gram of a nuclide of atomic weight M and decay constant X is

(2.7)

Rieck [Rl] has used the computer codes LEOPARD [Bl] and SIMULATE [FI] to predict the power distribution in the fuel and poison arrangement shown in Fig. 3.19 for the first fuel cycle for this reactor, and the amount of thermal energy produced by each assembly up to the time when the reactor ceases to be critical with all soluble boron removed from the cooling water. Figure 3.20 is a horizontal cross section of one-quarter of the core of this reactor. Each square represents one fuel assembly. The core arrangement has 90° rotational symmetry, about the central assembly 1AA at the upper left of the figure.

The first row of symbols in each square is the serial number of the assembly. The first symbol is the fuel lot number: lot 1 contains 2.25 w/o 235U; lot 2 contains 2.8 w/o 235 U and boron burnable poison; and lot 3 contains 3.3 w/o 235 U and burnable poison. The second

О Zircoloy cladding containing U02 pellets (see detail) (204) ® Zircoloy guide for control rods, water filled (20)

® Zircoloy instrument thimble, empty (I)

symbol is the letter designating the row in which the assembly is placed when initially charged to the reactor. The third symbol is the letter designating the column in which the assembly is placed. The second row of symbols, here a dash (-), gives the bumup of the assembly at the start of the cycle, here zero. The third row gives the bumup at the end of the cycle when the reactivity has dropped to zero. The fourth row gives the power of the assembly relative to the core average. It is a requirement of fuel management in this reactor that the power of every assembly relative to the core average be kept below 1.S8, to prevent the water leaving each assembly from reaching the boiling point at 155 bar. In this first cycle assembly power is controlled by the use of burnable poison and the placement of individual assemblies in the modified scatter pattern shown in Figs. 3.19 and 3.20.

The total thermal energy produced in the first cycle is evaluated by multiplying the bumup increment of each assembly in megawatt-days per metric ton by the mass of uranium in that assembly in metric tons and summing over all assemblies in the reactor, taking into account the total number of assemblies in positions equivalent to those shown in Fig. 3.20. For example, there are four assemblies in the BB position, four in BC, two in AB, and one in AA. The total thermal energy produced in the first cycle thus evaluated is 1341.1 GWd, or 32,188 X 104 kWh.

Table 3.2 gives the local power at 12 axial positions 1 ft apart in six selected assemblies relative to the average reactor power, at the beginning and end of the first cycle. Another requirement of fuel management in this reactor is that the ratio of local to average power at all points not exceed 2.33 to keep the linear power below 16 kW/ft. Table 3.2 shows that the maximum relative power of 1.68 at the beginning of the cycle (in EE) is well below this limit, and that the maximum relative power at the end of the cycle is even lower.

The cross section о has dimensions of length squared (cm2) as is required to make Eq. (2.42) dimensionally consistent. Fundamentally, it is the fraction of the reacting nuclei consumed by the nuclear reaction per unit time per unit flux.

Cross sections for reactions with neutrons vary from a lower detectable limit of around 1 X 10’28 cm2 to a maximum of 2.65 X 10"18 cm2, which has been observed for I3sXe. To avoid using such large negative exponents, cross sections are usually expressed in units of 10"24 cm2, called bams (b). For instance, the xenon cross section is 2.65 X 106 b. The millibarn (mb) is 1СГ27 cm2.

There is a different cross section for every different reaction of a nuclide with neutrons. Examples of cross sections for low-energy neutrons moving at a speed of 2200 m/s are given in Table 2.6.

The sum of the cross sections for all reactions in which a neutron is absorbed is called the absorption cross section, denoted by oa. In the examples of Table 2.6,

aa235U = 680.8 b

aa14N = 1.88 b

The neutron speed, or kinetic energy, is specified in the listing of neutron cross sections in Table 2.6 because the cross section generally varies with neutron speed, in many cases very strongly. Curves for the variation in capture or absorption cross sections with neutron energy for many nuclides are given in BNL-325 [Ml]. A table of the published values of cross sections for neutron-absorption reactions, for 2200 m/s neutrons, is given in App. C. For most of the nuclides the absorption cross sections for low-energy neutrons vary nearly as the reciprocal of the neutron speed v.

2.6 Neutron Speeds in Reactors

Neutrons in a nuclear reactor have velocities, and energies, distributed over a wide range. Neutrons are bom from the fission reaction at an average energy of about 2 MeV

(и = 1.955 X 107 m/s). To maintain a steady-state nuclear chain reaction it is necessary that the rate constant for the neutron fission reaction be sufficiently high so that neutron production will compete favorably with processes that consume neutrons. In addition to neutron absorption, neutrons are consumed by diffusing to the outer surface of the reactor and escaping to the surroundings. The diffusion of neutrons through matter is similar to the diffusion of gas molecules, and the average rate of loss of neutrons of speed v from a volume element in a reactor due to diffusion, or “leakage,” can be expressed as

where n is the average concentration of neutrons of speed v throughout the reactor. The rate constant К і varies as the surface-volume ratio of the reactor and is usually affected but little by neutron speed. From Eq. (2.43) it follows that neutron consumption by leakage increases with neutron speed. On the other hand, the cross sections for fission decrease markedly as neutron speed increases. Unless fissionable fuel in a highly concentrated form is available, it is then necessary to reduce the neutron speed to obtain the proper balance between neutron production and consumption. This is done by designing the reactor to contain sufficient atoms of low atomic weight, such as hydrogen, deuterium, beryllium, or carbon. The fast neutrons from fission undergo elastic collisions with these light nuclei, called moderators, and soon reach thermal equilibrium with the surrounding medium. In a thermal reactor enough moderator material is present so that the neutrons will be quickly degraded to thermal energies, and most of the fissions occur with the thermal neutrons.

A fast reactor is one in which no moderator is present and most of the fissions occur with neutrons of energies near the energies at which they were bom. To overcome the high probability of neutron consumption by leakage in fast reactors, a high concentration of fissionable material is required, as may be obtained by fueling the reactor with plutonium or with uranium highly enriched in 235 U.

Equations (3.44) through (3.80) have been used to calculate the effect of irradiation in this PWR on the composition of fuel initially containing 3.2 w/o 235 U, with results given in Table 3.15. Nuclide concentrations TV,- from these equations have been converted to weight fractions w,-by

where Mj is the atomic weight of nuclide i, and N°s and TVj8 are the atom concentrations of 235U and 238U, respectively, in fresh fuel.

Weight percents from this table are plotted against burnup as the points in Fig. 3.29. These points are to be compared with the lines, taken from Fig. 3.3, representing weight percents calculated by the more accurate point-depletion computer code CELL [B2]. Agreement is excellent for 233 U and 236 U, fair for 239 Pu, but poor for the higher plutonium isotopes at high bumup. This is because the effective absorption cross section of 240Pu decreases as its

concentration increases, as a result of its very high absorption cross section at resonance energies, an effect that the equations of this chapter cannot take into account.

In addition to 235 U, two other isotopes can be used as fuel in nuclear fission reactors. These are plutonium-239, 239Pu, produced by absorption of neutrons in 238 U; and 233U, produced by absorption of neutrons in natural thorium. The reactions by which these isotopes are made are as follows:

+’n-* 239U -* 239Np + e~

Neutron 4

“Fu + e-

Beta particles

232 Th + ln -> 233 Th -*• 233 Pa + e-

4

233 U + e"

Properties of these three fissile fuel nuclides are listed in Table 1.1.

The number of neutrons produced per neutron absorbed by fissile material is less than the number of neutrons produced per fission because some of the neutrons absorbed produce the higher isotopes 236 U, ^Pu, or 234 U rather than causing fission.

Tabic 1.1 Nuclear fuels

The fact that the number of neutrons produced per neutron absorbed exceeds 1.0 for each fuel indicates that each will support a nuclear chain reaction. Neutrons in excess of the one needed to sustain the nuclear chain reaction may be used to produce new and valuable isotopes, for example, to produce 239Pu from 238 U or 233 U from thorium by the reactions cited earlier.

When the number of neutrons produced per neutron absorbed in fissile material is greater than 2.0, it is theoretically possible to generate fissile material at a faster rate than it is consumed. One neutron is used to maintain the chain reaction, and the second neutron is used to produce a new atom of fissile material to replace the atom that is consumed by the first neutron. This process is known as breeding. The reactions taking place in breeding 239Pu from 238 U are shown in Fig. 1.6. 238 U is the only material consumed over all; 239Pu is produced from 238 U and then consumed in fission.

Fission of 239Pu

Second neutron is captured by 238U to produce 239U

Later, decays radioactively to form 25^“u

+ ©

Atom of239Puto replace atom consumed in fission

In thermal reactors fueled with plutonium, the number of neutrons produced per neutron absorbed is less than 2.0 and breeding is impossible. For 233 U, on the other hand, this number is substantially greater than 2.0, and breeding is practicable in a thermal reactor. In fast reactors, the number of neutrons produced per neutron absorbed is close to the total number of neutrons produced per Fission, so that breeding is possible with both 333 U and plutonium. Breeding as here defmed is not possible with 235 U, because there is no naturally occurring isotope from which 235 U can be produced.

A fast reactor is one in which the average speed of neutrons is near that which they have at the moment of fission, around 15 million m/s. At these high speeds the probability of a neutron’s being absorbed by a fissionable atom is low, and the neutron-absorption cross section, which is a measure of this probability, is small.

A thermal reactor is one in which the neutrons have been slowed down until they are in thermal equilibrium with reactor materials; in a typical power reactor, thermal neutrons have speeds around 3000 m/s. At these lower speeds, the neutron-absorption cross sections are much larger than for fast neutrons.

The critical mass of fissile material required to maintain the fission process is roughly inversely proportional to the neutron-absorption cross section. Thus the critical mass is lowest for plutonium in thermal reactors, larger for the uranium isotopes in thermal reactors, and much greater in fast reactors. For this reason, as well as others, thermal reactors are the preferred type except when breeding with plutonium is an objective; then a fast reactor must be used.

6.3 Properties of Laplace Transforms

The Laplace transform f(t) of a function f(t) is defined as

It is a function of the transform variable s. The Laplace transform of a derivative is obtained by integration by parts:

The inverse transform of a function L(s) of j is a function of the variable of which L(s) is the Laplace transform. For example, functions of the variable t have been transformed in Eqs. (2.137), (2.138), and (2.139). It can be seen that the inverse transform of 1/s is e~°, or unity.

These simple properties of the Laplace transform make it a very convenient tool for solving systems of first-order linear differential equations, such as the equations for growth and decay of nuclides in radioactive disintegrations and neutron irradiation. They permit these differential equations to be treated as if they were systems of simple transformed linear equations without derivatives.

Control of values of distribution coefficients is one of the most important factors in achieving successful separations by solvent extraction. In simple extraction, without fractionation (as in Figs. 4.1 and 4.2), a high value of the distribution coefficient is desirable because the volume of solvent required is then small. In simple stripping, without fractionation (as in the stripping section of Fig. 4.3), a low value of the distribution coefficient is desired because the volume of stripping solution is then small and the product concentration high. In fractional extraction, when a separation is to be made between two extractable components, the ratio of their distribution coefficients should differ from unity by as much as possible. In addition, in this case, it is desirable that the geometric mean of the distribution coefficients not depart greatly from unity, because the optimum flow ratio of aqueous to organic phases is given approxi­mately by

(f) t=V^I (4.14)

and most fractional extraction contactors operate best when this flow ratio is near unity.

The principal factors that affect the numerical value of the distribution coefficient are

1. Element being extracted

2. Oxidation-reduction potential of aqueous phase

3. Nature of solvent

4. Concentration of complexing agent

5. Concentration of salting agent

6. Hydrogen ion concentration in aqueous phase

4.1 Element Being Extracted

One of the reasons that solvent extraction is so successful in separating and purifying certain elements is that the distribution coefficients of different elements between certain solvents and aqueous solutions differ enormously. Table 4.2 lists distribution coefficients observed by Furman [F2] between diethyl ether and aqueous nitrate solutions. The much higher distribu­tion coefficient for uranium is the reason for the successful use of diethyl ether in purifying uranium.