Basic reactor physics

APPENDIX

Basic reactor physics

C

C.1 Introduction It is assumed that most readers are familiar with basic reactor theory. But for those lacking that familiarity or those needing a refresher, this appendix provides a brief overview of basic concepts relevant to a study of reactor dynamics. Greater detail may be found in pertinent references [1–4].

C.2 Neutron interactions Interaction of neutrons with matter results in various outcomes depending on the neutron energy and the nature of target material (target nuclei). The interactions that are important in reactor operation are as follows: Elastic collision: This occurs when the neutron shares its energy with the target nucleus without exciting the nucleus. In a collision between a target nucleus and a neutron, the target nucleus recoils and the neutron continues with lower energy. This is analogous to a “billiard ball collision”. The total kinetic energy (KE) is conserved. This is the primary mode of slowing down of neutrons to thermal energies by interactions with light nuclei of the moderator. Inelastic collision: In an inelastic collision the target nucleus becomes excited, emits a gamma ray, and emits a neutron with lower energy than the incident neutron. Radiative capture: In radiative capture the neutron is absorbed by the target nucleus, produces an excited nucleus that becomes stable by emitting gamma rays. Transmutation: In a transmutation reaction, neutron absorption yields new isotopes. For example 10

B + 1 n ! 7 Li + 4 He ðαÞ

(C.1)

Fission: The essential reaction that takes place in a nuclear reactor is the fission reaction, which occurs in certain heavy nuclei. 92U-235 is the only naturally occurring isotope of uranium that has this property for reactions with slow neutrons. The other main isotopes that undergo fission by slow (or thermal) neutrons are 92U-233, 94Pu-239, and 94Pu-241. The natural abundance of the isotopes of uranium (as it is found in nature) is as follows: U-234 ¼ 0:006%

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U-235 ¼ 0:714% U-238 ¼ 99:28%

C.3 Reaction rates and nuclear power generation In this section, some of the interactions between neutrons and atomic nuclei, are reviewed. Since neutrons have no electrical charge, they can enter into nuclear reactions even when their velocities are low. A brief review of neutron cross sections, neutron flux, reaction rates, and power generation follows in this section. In a nuclear reactor, the issue is the fate of fission neutrons. Fission neutrons result in new fissions, non-fission neutron captures, and neutron leakage. The average energy of fission neutrons is around 2 MeV. These fast neutrons interact with the core materials (structure, fuel, moderator, etc.) by absorption and scattering reactions. Collisions resulting in scattering will slow down the neutrons. Neutron cross sections are the basic data used for determining nuclear reaction rates. The microscopic cross section is represented by the symbol, σ. Microscopic cross sections are basically target areas for incident neutrons. The units for cross sections are cm2. Typical values for cross sections are 1022 to 1026 cm2. To simplify specification of cross section values, a new unit, called the barn is used. A barn is defined as 1024 cm2. Early workers, apparently a jocular bunch, said that, to a neutron, a target with area, 1024 square centimeters, is as big as a ‘barn door’. Reactions of importance in nuclear reactors are fission, capture, absorption (fission + capture), elastic scattering and inelastic scattering. Cross sections are energy dependent. Fission and capture cross sections decrease with increasing neutron energy. For many isotopes these cross sections vary as the reciprocal of the neuron velocity at low energies (1/v or as the reciprocal of the square root of the neutron energy). Isotopes that follow the 1/v law are called 1/v absorbers. The total microscopic cross section, σT, available for interaction between a neutron and a target nucleus is σT ¼ σa + σs

(C.2)

σa ¼ microscopic absorption cross section σs ¼ microscopic scattering cross section. These may be further classified as σa ¼ σf + σc ðfission + captureÞ σs ¼ σse + σsi ðelastic + inelasticÞ

For a given concentration of target nuclei, the number of collisions in a given time interval is proportional to the distance traveled by the neutrons in the volume. Some important relationships follow: •

Neutron flux: ϕ ¼ nv (number of neutrons/cm2-s)

APPENDIX C Basic reactor physics

N ¼ Density of target nuclei (number of nuclei/cm3) Macroscopic cross section: Σ ¼ σN (cm2/cm3 or cm1) Reaction rate: R ¼ Σϕ (number of interactions/cm3-s) Fission reaction rate: Rf ¼ Σf ϕ Rf ¼ [Nσf] [n(t)v] Rf is the number of fission reactions/cm3-s σf is the microscopic fission cross section

• • • • • • •

The reaction rate in a 1/v absorber is given by R¼Nσϕ

or R ¼ Nðc=vÞ nv

where c is a constant. Note that the velocity terms cancel. Therefore, the reaction rate for 1/v absorbers is independent of the neutron energy. Example C.1 The energy of neutrons in an experimental reactor is approximately equal to 0.0253 eV. This corresponds to a speed of about 2200 m/s for neutrons. As an exercise, let us let flux ϕ ¼ 2  1012/(cm2 s). Calculate the neutron density. n ¼ ϕ=v ¼

2  1012 cm2 s1 ¼ 9  106 =cm3 2200  100m s1

In the above example, use the microscopic absorption cross section, σa ¼ 694 b for U-235. The density of target nuclei is N ¼ 0.05  1024/cm3. Then Σa ¼ Nσa ¼ 34 cm1

The reaction rate is given by the following: R ¼ ϕ Σa ¼ 6:8  1013 =cm3 s

R is also the rate at which U-235 nuclei are consumed. Note that the microscopic fission cross section of U-235 at E ¼ 0.025 eV is, σf ¼ 582 b. The capture cross section is σc ¼ 112 b and the absorption cross section is σa ¼ σf + σc ¼ 694. The ratio σf/σa ¼ 582/694 ¼ 0.84. That is, 84% of thermal neutron absorptions in U-235 result in a fission reaction. If an absorber cross section decreases slower with increasing neutron energy (and neutron velocity) than 1/v, then the absorption rate relative to 1/v absorbers increases as neutron energy increases. The reverse is true for absorbers whose cross section decreases faster than 1/v. The energy dependence of low energy absorptions plays an important role in determining the dynamics of power reactors. The power produced in a reactor is given by the following: P ¼ ðNVÞ σf nv F Watt

(C.3)

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where V ¼ Total volume of core. F ¼ Energy produced per fission (3.225  1011 Watt s/fission). If we assume that all the terms on the right-hand side of Eq. (C.3) are constant, except neutron density, n, the reactor power is directly proportional to the neutron density or the neutron flux. P/n/ϕ

(C.4)

This relationship is important because in the neutronics equations we can replace the neutron density with actual reactor power within a multiplication factor.

C.4 Nuclear fission The nuclei of U-235, U-233, Pu-239 and Pu-241 can undergo fission by low-energy neutrons. Such materials are called fissile materials. Isotopes such as U-238 and Th-232 can undergo fission with fast neutrons and are said to be fissionable (as opposed to fissile) and are referred to as fertile materials. Note that U-238 and Th-232 can be converted to Pu-239 and U-233, respectively, by absorption of neutrons and final decay into the respective fissile isotopes. The reactions are as follows 92 U

238

+0 n1 !

92 U

239

!

239 93 Np

#

+1 β0 (C.5)

93 Pu

239

+1 β0

The half-life for U-239 is 23 min and for Np-238 is 2.3 days 232 90 Th

+0 n1 !

233 90 Th

!

91 Pa

233

#

92 U

+1 β0 (C.6)

233

+1 β0

The half-life for Th-233 is 22 min and it decays to Protactinium (Pa-233). Pa-233 has a half-life of 27 days and has a quite large neutron absorption cross section (around 7.5% as large as the U-233 fission cross section). Therefore, Pa-233 is a significant neutron absorber and its presence diminishes potential U-233 production. The residence of Pa-233 in an operating reactor core results in a significant loss in U-233 production. Numerous fission fragments are released during the fission reaction. These are classified according to the percent of fission yield. Fission yield is the percentage of a given isotope atoms in the total of all fission fragments. The reactions are as follows: 235 236 ∗ 1 92 U + 0 n ! 92 U

A2 1 ! A1 z1 F + z2 F + υ0 n + Energy

where υ ¼ number of neutrons produced in the fission reaction (2 or 3).

APPENDIX C Basic reactor physics

The most probable fission fragments are Cs-140 (Cesium) and Rb-93 (Rubidium). For example, the reactions involving these fission fragments are given as follows: 235 236 ∗ 1 92 U + 0 n ! 92 U

93 1 !140 55 Cs + 37 Rb + 30 n + Energy ð 200 MeVÞ

(C.7)

The high-speed fission fragments lose energy by interaction with the molecules of the surrounding medium (fuel, structure, moderator, etc.), thus converting kinetic energy (KE) to thermal energy. There is also heating due to gamma radiation and slowing down of neutrons to lower energy levels. An average of υ ¼ 2.43 neutrons are produced per fission induced by thermal neutrons in U-235. Avogadro’s Number (AN): is the number of molecules per gram mole or atoms per gram atom of a substance. One gram-atom is the quantity of substance in grams, numerically equal to its atomic mass. Avogadro’s Number is numerically equal to AN ¼ 6.023  1023 atoms per gram-atom. Example C.2 The number density (N) of U-235 atoms in natural uranium is the number of atoms of U-235 per cm3. The number density is given by N ¼ ðANÞ ρ e=m where ρ ¼ material density, gm/cm3 e ¼ enrichment [U-235/(U-235 + U-238)] m ¼ gram U per gram atom U.

The number density of U-235 in 1% enriched uranium (with a material density of 19.0 g/cm3 is [3]. N235 ¼


   19:0 gramU=cm3  6:023  1023 ðatomsU=gram  atomUÞ atomsU  235  0:01 238 ðgramU=gram  atomUÞ atomU

¼ 4:80  1020 atoms U-235=cm3 :

Some energy equivalents are as follows: 1 eV is the amount of kinetic energy imparted to an electron when accelerated through a potential difference of 1 V. 1 eV ¼ 1.602  1019 J 1 cal ¼ 4.184 J. The absorption of a neutron by the U-235 nucleus has the form 235 236 ∗ 1 92 U + 0 n ! 92 U

(C.8)

This results in extra internal energy of the product, because the sum of masses of the two interacting particles is greater than that of a normal U-236 nucleus (at ground state). This excess energy is sufficient to cause nuclear fission (electrostatic repulsion dominates nuclear attraction).

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Table C.1 Energy distribution for fission induced by thermal neutrons in U-235. Source

Energy (MeV)

Fission product KE Neutron KE Energy in gamma radiation (instantaneous and delayed) Energy in β-decay of fission products • Total (available as thermal energy): Energy not available as heat • Total energy created in one fission reaction:

168 5 11 7 191 11 202

The neutron binding energies of U-235, U-233, and U-239 nuclei (odd number of neutrons) are about I MeV higher than the nuclei of Th-232 and U-238 (even number of neutrons). This additional binding energy is sufficient to exceed the critical energy for fission, with low-energy neutrons. For example: for U-235, critical energy for fission to occur is 5.5 MeV and the binding energy of an extra neutron is 6.6 MeV. The critical and binding energies for U-238 are 5.9 MeV and 4.9 MeV. Out of two to three neutrons released per fission (depending on the fissioning isotope involved), one of these is used to produce the next fission reaction in a steadystate chain reaction. The remaining neutrons are consumed by • • • •

Leakage from the core Capture by non-fuel reactor constituents (such as coolant, moderator and structural materials) Non-fission capture in the fuel (radiative capture) Capture by the fertile nuclei (such as U-238, resonance capture).

The distribution of fission energy in various forms is shown in Table C.1.

C.5 Fast and thermal neutrons Immediately following fission, the neutrons possess high kinetic energy, in the million-eV range (0.1–15 MeV). Most current-generation reactors include a material (called a moderator) whose purpose is to slow down neutrons while capturing few neutrons. Fast neutrons lose their energy due to scattering collisions with various nuclei in the medium (especially in the moderator, see Section C.2), and become slow neutrons (energy <1 eV). It should be noted that tabulated cross sections are usually for monoenergetic neutrons at an energy of 0.0253 eV or a speed of 2200 m/s. This energy corresponds to a temperature of 20 °C or 293 K. Thermal neutrons are those whose kinetic energy reaches equilibrium with the thermal energy of the moderator. Higher moderator temperature means greater thermal motion of moderator atoms and a consequent

APPENDIX C Basic reactor physics

higher energy of neutrons that interact with the moderator atoms. The energy spectrum of moderator atoms, and consequently the energy spectrum of thermalized neutrons is given by the Maxwell-Boltzmann distribution. Fig. C.1 shows distributions at three different temperatures. In a reactor, the absorption and fission cross sections must be corrected for the actual temperature of the moderator. The Maxwell-Boltzmann distribution applies for neutrons in equilibrium with moderator atoms. As shown in elementary reactor physics books, the “effective” cross section for a material with 1/v dependence in a moderator at temperature, T is σ ðT Þ ¼

1 1:128

rffiffiffiffiffiffiffiffi 293 σ ð0:0253 eV Þ T

(C.9)

The average and most probable energies for neutrons in a Maxwell spectrum are as follows: – average neutron energy ¼ 1/2 kT – most probable neutron energy ¼ 3/2 kT. where k is the Boltzmann constant and T is the absolute temperature. Clearly the thermal spectrum shifts to higher energies as moderator temperature increases. This is called spectral hardening. Resonances are spikes in a material’s cross section. See Fig. C.2 for the energydependent U-238 cross section. Strong resonances are apparent. Resonances are important in their effects on steady state and dynamic characteristics of a reactor. Capture and fission cross sections both exhibit resonant behavior.

Fractional neutron density

4

× 10–4 293 K 493 K 593 K

3

2

1

0 0

1000

2000

3000

5000 6000 4000 v, Neutron velocity (m/s)

7000

8000

FIG. C.1 Maxwell-Boltzmann distribution of thermal neutrons for three temperatures.

9000

10000

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APPENDIX C Basic reactor physics

Incident neutron data / ENDF/B-VII.1 / U238 / MT = 102: (z,Y) / Cross section 10000 1000 100

Cross-section (b)

262

10

1

1/d/vntregion regi g rgy on(m 1/v Incide Incident cide ene energy n (meV) meV) V)

0,1

0,01

Incident Inc n ide ident nt energy energy y (m ((meV) eV))

0,001 1E-4

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

0,001

0,01

0,1

1

10

Incident energy (MeV)

FIG. C.2 U-238 neutron cross section as a function of neutron energy ([5], source ENDF, JANIS 4.0]).

Resonances influence the ability of fission neutrons to slow down to thermal energy without capture. Resonance escape probability influences criticality calculations. Resonances also affect reactor dynamics by two mechanisms. The heavy isotopes in the fuel experience Doppler broadening as the fuel temperature increases. Doppler broadening is the reduction of a resonance peak while increasing the energy span of the resonance. See Fig. C.3. Since the cross section is still very large in the broadened energy region, the net effect is an increase in the resonance absorption rate. Neutron capture in fertile materials (U-238 and Th-232) and Pu-240 (produced by captures in U-238 and Pu-239) exceeds fissions due to Doppler broadening in fissile material resonances. Therefore, Doppler broadening causes reactivity to decrease when fuel temperature increases. So, the fuel temperature coefficient of reactivity (reactivity change/fuel temperature change) is always negative. The negative fuel temperature coefficient is very important in ensuring that a reactor responds satisfactorily in a transient. Low energy resonances (principally between 0.1 eV and 1.0 eV) affect the reactivity change following a change in moderator temperature. The thermal energy spectrum hardens (shifts to higher energies). Consequently, higher moderator temperature causes more thermal neutrons to exist at energies where the resonances occur.

APPENDIX C Basic reactor physics

ABSORPTION CROSS SECTION sa

UNBROADENED

DOPPLER BROADENED

E0 ENERGY

FIG. C.3 Doppler broadening of resonance energy as a function of temperature.

C.6 Relation between specific power and neutron flux The following equation defines the relation between neutron flux and reactor power: P ¼ F Vf Nf σ f φ

(C.10)

where F ¼ conversion from fission rate to power (3.225  1011 Watt s/fission) Vf ¼ fuel volume Nf ¼ fuel atoms per unit volume P ¼ reactor power. Note that (Vf Nf) is simply the number of fissile atoms in the reactor. Using the atomic weight of the fissile material and Avogadro’s number gives the following: Vf Nf ¼ 6:023  1023 mf =Mf ðatoms of fissile material in the reactorÞ

where mf¼ mass of fissile material in the reactor Mf ¼ atomic weight of fissile material. The effective fission cross section is given by σ f ðT Þ ¼

1 1:128

rffiffiffiffiffiffiffiffi 293 σ f ð293Þ T

or σ f ðT Þ ¼

15:17 σ f ð293Þ pffiffiffi T

(C.11)

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APPENDIX C Basic reactor physics

For example, for a moderator temperature of 300 °C, the effective fission cross section of U-235 is 348 b (compared to σf (20 °C) of 549 b). The result is   P=mf ¼ 2:948  1010 σf ð293Þ x T0:5 =Mf x ϕ

For a fuel enrichment of ε, the specific power is usually expressed as power/mass of total uranium   P=mU ¼ 2:948 x 1010 x ε x σf ð293Þ x T0:5 =Mf x ϕ

or



ϕ ¼ ðP=mU Þ= 2:948 x 1010 x ε x σf ð293Þ x T0:5 =Mf

where mU ¼ mass of total uranium in the reactor. For example, for fuel with an enrichment of 3% and a specific power of 30 kw/kg of U, the flux is 3.47  1013 neutrons/(cm2 s).

C.7 Neutron lifetime and generation time The total neutron lifetime, l, is given by l ¼ ls + lth

(C.12)

ls ¼ slowing down time ( 107 s). lth ¼ thermal lifetime ( 104 s). Slowing down time is the time a neutron spends in slowing from fission to thermal energies. Thermal lifetime is the time a neutron spends diffusing at thermal energies before absorption by a fissile nucleus. A quantity called the neutron generation time is also used in the reactor kinetics equations. Neutron lifetime and generation time are equal in a critical reactor. The two quantities have slightly different (but inconsequential) definitions for an unsteady state reactor. The only significant impact is in determining the form of the kinetics equations (see Chapter 3). The neutron lifetime (or generation time) is around 104–105 s for light water reactors (PWRs, BWRs).

C.8 Multiplication factor and reactivity In a nuclear power reactor, the fission reaction is maintained (regulated) to give a desired power level. In a steady-state reactor, the number of neutrons from one generation to the next generation (an interval of one generation time) remains constant.

APPENDIX C Basic reactor physics

Define the multiplication factor (gain) k¼


 Number of neutrons generated in the present generation np Number of neutrons generated in the previous generation ðnÞ

(C.13)

If k ¼ 1, the chain reaction is sustained, and the reactor is said to be critical. If k < 1, the number of neutrons from one generation to the next decreases. Such a reactor is said to be sub-critical. If k > 1, the number of neutrons from one generation to the next increases without bound, and such a reactor is said to be super-critical. In summary: • • •

k ¼ 1, critical reactor k < 1, sub-critical reactor k > 1, super-critical reactor

C.9 Computing effective multiplication factor The following factors determine the magnitude of the multiplication factor k: 1. Thermal fission factor, η: The factor, η, is defined as the number of fast neutrons produced per thermal neutron absorption in the fuel. That is: η¼ν

σfuel f σfuel a

2. ν ¼ number of neutrons produced per fission. A typical value for η is around 1.65 for a U-235-fueled thermal reactor. 3. Thermal utilization factor, f: The factor, f, is defined as the number of neutron absorptions in the fuel per total number of neutron absorptions. That is: f¼

Σfuel a Σtotal a

A typical value for f is around 0.71 for a U-235-fueled thermal reactor. 4. Resonance escape probability, p: The factor, p, is equal to the number of neutrons that reach thermal energy per fast neutron born. It accounts for neutron losses in resonances during slowing down. A typical value for p is around 0.87 for a U-235-fueled thermal reactor. 5. Fast fission factor, ε: This factor is defined as the total number of neutrons from both thermal and fast fissions per number of neutrons from thermal fissions. A typical value for ε is around 1.02 for a U-235-fueled thermal reactor Using the above four factors we define the effective multiplication of an infinite size core as k∞ ¼ η f p ε

(C.14)

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APPENDIX C Basic reactor physics

The above formula is referred to as the four-factor formula. This does not take into account the probability of neutron leakage from finite size cores. In order to include this effect, two more factors are added to the above calculation [1–4]. 6. Fast non-leakage probability, PFnl: The fast non-leakage factor is the number of fast neutrons do not leak out of the core during slowing down to thermal neutron per fast neutron produced. A typical value for PFnl0.97 is around 0.97 for a U-235-fueled thermal reactor. 7. Thermal non-leakage probability, PTnl: The thermal non-leakage factor is the number thermal neutrons do not leak out of the core per thermal neutron produced. A typical value for PTnl is around 0.99 for a U-235-fueled thermal reactor. 8. Effective multiplication factor, keff: With definition of non-leakage probabilities, we can now calculate the effective multiplication factor as keff ¼ k∞ PFnl PTnl ¼ η f p ε PFnl PTnl

(C.15)

The formula in Eq. (C.15) is generally referred to as the six-factor formula. The reader can find more details in Ref. [1, 2].

C.10 Neutron transport and diffusion The most complete description of the spatial distribution of neutrons in a reactor is given by neutron transport theory. Transport theory defines a reactor in terms of seven independent variables: three position coordinates, two direction vectors, energy and time. The transport theory equation is called the Boltzmann equation. Computer codes have been developed for transport theory analysis, but they suffer from complexity, long computing time, and difficulty in determining detailed finemesh parameters needed for implementation. Most reactor studies treat neutron motion as a diffusion process – that is, neutrons tend to diffuse from regions of high neutron density to regions of low neutron density. Diffusion theory ignores the direction dependence of the neutrons.

Exercises C.1. The neutron flux in a certain reactor is 2  1013 neutrons/(cm2-sec). If the neutrons have a mean velocity of 3100 m/s, calculate the neutron density. Indicate the units. C.2. The neutron flux in a commercial pressurized water reactor (PWR) is 2  1013/ (cm2-s). The macroscopic cross section for fission is 30 cm2/cm3. Calculate the rate of fission reactions per cm3. Indicate the units. Simplify your answer.

APPENDIX C Basic reactor physics

C.3. The neutron flux in a high flux reactor is approximately 1015 neutrons/cm2-sec. The microscopic neutron cross section for fission in U-235 for this reactor is 500  1024 cm2. The density of U-235 target nuclei is 1024 nuclei/cm3. (a) Calculate the rate at which fission reactions take place in this reactor. Simplify your answer and indicate the units. (b) If the energy generated per fission is approximately 200 MeV, calculate the total power produced in the reactor per cm3. Indicate the units. 1 MeV ¼ 1.60  1013 J. C.4. Match the following neutron energies (E) with thermal neutrons, fast neutrons, and neutrons in the resonance energy range. E: 0.1 MeV - 15 MeV _________________. E < 1 eV _________________. E: 1 eV – 0.1 MeV _________________. C.5. The energy of neutrons in a light water reactor is approximately equal to 0.0253 eV. This corresponds to a speed of about 2200 m/s for neutrons at 20 deg. C. (a) If the neutron flux is 2  1012 neutrons/(cm2-sec) calculate the neutron density. Indicate the units. 1 m ¼ 100 cm. (b) If the microscopic absorption cross section is σa ¼ 694  1024 cm2 for U-235 and the density of target nuclei of U-235 is N ¼ 0.048  1024/cm3, calculate the total reaction rate. Indicate units. (c) What is the rate at which U-235 nuclei are consumed in this reactor?

References [1] J.J. Duderstadt, L.J. Hamilton, Nuclear Reactor Analysis, John Wiley, New York, 1976. [2] J.K. Shultis, R.E. Faw, Fundamentals of Nuclear Science and Engineering, second ed, CRC Press, Boca Raton, FL, 2007. [3] A.R. Foster, R.L. Wright, Basic Nuclear Engineering, Allyn and Bacon, Boston, 1983. [4] J.R. Lamarsh, Introduction to Nuclear Reactor Theory, Addison-Wesley, Reading, MA, 1983. [5] https://www.nuclear-power.net/glossary/, 2018.

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