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Chapter: 4 Atomic and nuclear physics
    Section: 4.7 Nuclear fission and fusion, and neutron interactions
        SubSection: 4.7.2 Neutron cross-sections

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4.7.2 Neutron cross-sections

If n(E) dE is the number of neutrons per unit volume of a material having an energy between E and E + dE, then the neutron flux, Φ, is defined as Φ(E) = n(E)υ, where υ is the neutron velocity. The number of neutrons giving a particular type of reaction per unit volume per unit time in a material containing N nuclei per unit volume is ∫ NΦ(E)σR(E) dE, where σR(E) is the nuclear cross-section for the particular type of interaction R. The product R(E) is called the macroscopic cross-section, ΣR, and its reciprocal can be shown to be the mean free path.

After a neutron interacts with a nucleus, charged particles (e.g. a proton or α particle), γ-rays or one or more neutrons may be emitted. In heavy materials fission may take place. These reactions are known as (n, p), (n, α), (n, γ), (n, Xn) [where X has a value typically of 1 to 3] and (n, f) reactions. The (n, γ) reaction is referred to as radiative capture and the corresponding cross-section written as σnγ. The (n, n) interaction indicates neutron scattering though it does not mean that the same neutron is emitted as produced the interaction. If after the interaction the nucleus remains in the ground state the event is referred to as elastic scattering and the cross-section is written σnn. If it is left in an excited state it is referred to as inelastic scattering and is denoted by (n, n') with the cross-section written as σnn′. The fission cross-section is written σnf and the total cross-section for all events is written σnT. The absorption cross-section, σnA, is defined as σnA = σnTσnnσnn'. The transport cross-section (σtr) is defined as σnTnn where b is the average value of cos ψ and ψ is the angle of neutron scatter in a collision.

The energy range of interest in most applications of nuclear energy (i.e. for fission and fusion reactors) is 10−5 eV to 20 MeV. Evaluated cross-section values are available in a number of nuclear data files that have been prepared both locally and internationally—see below. In Figures A, B, C, D, E and F a selection of cross-sections is shown. Certain features may be distinguished.

At low energies absorption cross-sections tend to be proportional to 1/υ (i.e. to E−1/2) where υ is the neutron velocity and for 10B and 6Li this law is good to ±5% up to about 100 keV (see Figures A and C). For light nuclides, scattering is dominated by potential scattering which can be considered to be the deflection of a neutron by the nuclear field without the formation of a compound nucleus, and the cross-section tends to be independent of energy. Departures, however, arise at low energies in crystalline materials (e.g. C and Be) when the neutron wavelength is comparable to the lattice spacing, and in molecular compounds (e.g. H2 and H2O) when the collision energy is comparable to the vibrational quanta of the molecular oscillations (see Figure B).

At intermediate energies the cross-sections show sharp maxima at neutron energies at which the reacting particles have the same energy (including binding energy) as one of resonance levels in the compound nucleus. Light nuclei have larger spacings between these resonances than heavier nuclei (cf. Figures C and D). It is thus more probable to find resonances at low energies in heavy nuclei and such resonances can be seen in Figures A and B for U and Pu.


(Hint: Click the Images to view Larger Images)
Figure A. (n, a), capture and fission cross-sections in thermal region




Figure A. (n, α), capture and fission cross-sections in thermal region.



Figure B. Elastic scattering cross-sections in thermal region




Figure B. Elastic scattering cross-sections in thermal region.

Figure C. Cross-sections of light nuclides in the slowing-down region




Figure C. Cross-sections of light nuclides in the slowing-down region.

For heavy nuclei having an odd number of neutrons (e.g. 235U) fission is possible at all neutron energies and such materials are called fissile (see Figures A, D and E) and are used as fuel in fission reactors. For heavy nuclei having an even number of neutrons (e.g. 238U) fission is only significant above a threshold energy (see Figure F). Since neutron capture in such materials produces a fissile material they are called fertile. See Section 4.7.1 for a further discussion on fission.

All (n, n′) and (n, 2n) reactions have threshold energies. In some light nuclides (n, p) and (n, α) reactions have appreciable cross-sections at all energies but in the heavier nuclides these cross-sections always show an effective threshold (due to the coulomb potential barrier) at an MeV or so, even when the reaction is energetically possible with low energy neutrons (see Figure F).

Because of the strong dependence of cross-section with energy it is usually necessary in practical calculations to make use of group cross-sections which are values averaged over chosen energy intervals and weighted according to the neutron energy spectrum expected in the system, i.e.

Figure D. Cross-sections of heavy nuclides in the slowing-down region




Figure D. Cross-section of a heavy nuclide in the slowing down region.(Above 20 eV the resonances are too numerous to be shown correctly on this scale.)

For a general comparison of the cross-section values for the interactions of neutrons in various materials it is often appropriate to consider four energy regions. The highest energy region around 14 MeV which is appropriate to fusion reactors (and for some activation analyses) is considered in section 4.7.4. The other three regions are applicable to fission reactor applications.

At high energies the cross-sections can be averaged over a fission neutron spectrum which can be expressed as Φ(E) = (4/πT3)1/2E1/2 exp(−E/T) where T is typically 1.35 MeV. In the tables the average values have been taken over the energy range 1 keV to 20 MeV, although most of the contribution comes from a much narrower range around 2 MeV.

At intermediate energies the spectrum in a well moderated system is nearly proportional to 1/E and the average cross-section is equal to


=  

(σ(E)/E) dE

(1/E) dE

Figure E. Non-threshold cross-sections in fast region.




Figure E. Non-threshold cross-sections in fast region. (Note different scales on left and right.)

The numerator is referred to as the resonance integral (R.I.) since it gives the cumulative absorption in the various resonances as a neutron slows down from energy E2 to E1. For cross-sections it is customary to quote the resonance integrals rather than the average values, and the tables show values where E1 = 0.5 eV and E2 = 100 keV though the values are insensitive to the upper limit.

At lower energies, when the neutrons are in thermal equilibrium with their surroundings, Φth = E exp(−E/kTn)/(kTn)2, where Tn is the temperature of the surroundings. If the cross-section varies as 1/ν then the average cross-section in a thermal flux can be shown to be equal to σ0(πT0/4Tn)1/2,

Figure F. Threshold reactions in fast region



Figure F. Threshold reactions in fast region. (Curves refer to (n, 2n) reactions unless stated otherwise.)
where σ0 is the cross-section corresponding to an energy kT0. It is customary to refer cross-sections to the value σ0 at a standard neutron velocity of 2200 ms1, which corresponds to a neutron energy kT0 of 0.025 30 eV and a temperature of 293.59 K. Values of σ0 have been given in the Table of Nuclides (section 4.6.1) and values averaged over a thermal spectrum with Tn= T0 are given in the following tables. The departure of these average values from the value of σ0(π/4)1/2 is indicative of the departure of the particular cross-section from the 1/ν characteristic. The integrations were taken over the range 0.0001 eV to 1.0 eV although most of the contribution comes from a narrower range around 0.025 eV.

The table of neutron cross-sections gives properties of nuclides which have been grouped according to their function in nuclear reactors. Cross-sections are given in barns (10−28 m2). For fissionable materials the number of neutrons produced per fission and denoted by ν are given as average values over the three spectral regions.

In systems which are not well moderated, such as fast nuclear reactors, the above division into three energy regions is not appropriate. There are virtually no thermal neutrons and no region at intermediate energies where the flux is proportional to E1. Detailed calculations require the neutron energy region to be split into a large number of groups with appropriate equations and corresponding cross-sections for each group and each nuclide present. For rough estimates and comparisons the average cross-sections table can be used for systems typical of a plutonium–uranium oxide fuelled, liquid metal cooled fast reactor with a fertile blanket.

Neutrons can be detected by both active and passive devices. Active detectors include fission chambers, BF3 ion chambers, proton recoil counters and their response to neutrons is determined by the appropriate cross-section for the particular interaction employed in the device. The table of neutron cross-sections for reactor materials includes materials used in these detectors. The passive detectors contain materials which become radioactive when irradiated in a neutron flux and on removal from the flux the radioactivity can be measured. The activity A(t) (particles s−1) at a time t seconds after the removal of a thin foil from a neutron flux having an energy spectrum Φ(E) in which the foil has been irradiated for T seconds is given by

   

A(t) =

VρNA

 (1 − exp(−λT)) exp(−λt)

σact(E)Φ(E) dE

A

where V, ρ and A are the volume, density and atomic weight of the foil material respectively, NA is Avogadro's constant, λ (s−1) is the decay constant of the radioactive species produced by the irradiation and σact(E) is the activation cross-section of the material. Values of σact averaged over the thermal and fission energy regions are given in the table together with the activation resonance integral. The materials listed either have a predominant resonance or a threshold and may be used to detect neutrons over a narrow energy region. Additional decay properties of the isotopes listed may be found in the Table of Nuclides (section 4.6.1).

Unless otherwise specified the values in the tables are mainly obtained from the JEF-2 library [which is mainly for fission applications] (Nordborg et al, 1991) and the average values were calculated by C. Nordborg and M. Konieczny of the NEA Data Bank. Other libraries that are available include EFF-2 [fusion applications] (Nordborg et al., 1991), EAF-3 [Fusion and other activation applications] (Kopecky et al. (1991)), ENDF/B-VI [fission and fusion applications] (Dunford, 1991) and JENDL-3 [fission and fusion applications] (Kikuchi et al., 1991).


References

Neutron cross-sections
       S. F. Mughabghab, M. Divadeenam and N. E. Holden (1981) Neutron Cross-sections, Volume 1, Neutron
    
       Resonance Parameters and Thermal Cross-sections, Part A: Z = 160, Academic Press, New York.
       S. F. Mughabghab (1984) Neutron Cross-sections, Volume 1, Neutron Resonance Parameters and Thermal
           Cross-sections, Part B: Z = 61–100, Academic Press, New York.
       V. McLane, C. L. Dunford and P. F. Rose (1988) Neutron Cross-sections, Volume 2, Neutron Cross-section
           Curves, Academic Press, New York.
       J. E. Lynn (1968) Theory of Neutron Resonance Reactions, Clarendon Press, Oxford.
       C. Nordborg, H. Gruppelaar and M. Salvatores (1991) ‘Status of JEF and EFF Projects’, Nuclear Data for
           Science and Technology, Springer-Verlag, Berlin, p. 782.
       C. L. Dunford (1991) ‘Evaluated Nuclear Data File ENDF/B-VI’, Ibid., p. 788.
       Y. Kikuchi and members of the JNDC (1991) ‘Japanese Evaluated Nuclear Data Library—JENDL-3’, 
            Ibid., p. 793.
       J. Kopecky, H. Gruppelaar and R. A. Forrest (1991) ‘European Activation File for Fusion’, Ibid., p. 828.
Data centres
   For USA and Canada
       National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY 11973, USA.
   For other OECD countries (e.g. Western Europe and Japan)
       NEA Data Bank, Le Seine Saint-Germain, 12, boulevard des Iles, F-92130 Issy-les-Moulineaux, France.
    For all other countries
       IAEA Nuclear Data Section, P.O. Box 100, A-1400 Vienna, Austria.
In addition there are also some National Data Centres.


Activation detectors for neutron foil detectors

Nuclide and reaction 

Half-life of 
  principal
  activity

Activation cross-section*

Average over
thermal
spectrum
(b)

Resonance
integral
(b)

Average over
fission
Spectrum
(mb)

 

 

 

 

 

164Dy(n, γ)165Dy#   .     .     .     .     .     .     .

2.33 h   

2207           

328       

98.5

176Lu(n, γ)177Lu#    .     .     .     .     .     .     .

6.71 d   

3048           

918       

130.2  

103Rh(n, γ)104mRh    .     .     .     .     .     .     .

4.34 min

9.15

70.6  

44.4

115In(n, γ)116mIn       .     .     .     .     .     .     .

54.15 min  

146.9     

2586        

84.6

197Au(n, γ)198Au      .     .     .     .     .     .     .

2.70 d   

87.98 

1562        

78.6

152Sm(n, γ)153Sm     .     .     .     .     .     .     .

46.7 h       

183.3      

2977        

91.0

107Ag(n, γ)108Ag     .      .     .     .     .     .     .

2.37 min

33.0    

105.7   

62.9

186W(n, γ)187W   .     .    .     .     .     .     .     .

23.9 h       

33.3    

518.7   

36.9

59Co(n, γ)60Co# .     .     .     .     .     .     .     .

5.27 y   

33.0    

75.5 

4.91

55Mn(n, γ)56Mn   .    .     .     .     .     .     .     .

2.58 h   

11.80  

15.3 

3.31

63Cu(n, γ)64Cu     .    .     .     .     .     .     .     .

12.7 h       

3.99

   4.94

10.76  

23Na(n, γ)24Na#  .    .     .     .     .     .     .     .

15.0 h       

   0.471

     0.310

0.23

115In(n, n′)115mIn       .     .    .     .     .     .     .

4.49 h   

0    

      0.211a

204.6a 

31P(n, p)31Si       .     .     .     .     .     .     .     .

2.62 h   

0    

0  

36.5

32S(n, p)32P        .     .     .     .     .     .     .     .

14.2 d      

0    

0  

69.2

58Ni(n, p)58Co#       .     .     .     .     .     .     .

70.8 d     

0    

0  

106.5  

27Al(n, p)27Mg         .     .     .     .     .     .     .

  9.46 min

0    

0  

     4.32

56Fe(n, p)56Mn         .     .     .     .     .     .     .

2.58 h 

0    

0  

     1.60

27Al(n, α)24Na# .     .     .     .     .     .     .      .

15.0 h    

0    

0  

     0.90

63Cu(n, 2n)62Cu       .     .     .     .     .     .     .

   9.74 min

0    

0  

     0.16

 

 

 

 

 


Neutron cross-sections

 

Average over thermal spectrum

Slowing-down region resonance integrals

Average over fission spectrum

 

σnn (b)

σnγ (b)

σnf (b)

σnn (b)

σnγ (b)

σnf (b)

σnn (b)

σnγ (b)

σnf (b)

σnn′ (b)

Fissile materials

 

 

 

 

 

 

 

 

 

 

 

233U

  12.19

  42.20

468.20

2.495

141.90

 134.16

   751.71

2.498

4.229

0.063

1.841

1.132

2.596

235U

  15.98

  86.70

504.81

2.433

152.82

 131.97

   271.53

2.438

4.409

0.095

1.219

1.917

2.583

  239Pu

    7.90

274.32

699.34

2.882

155.87

 184.06

   289.36

2.876

4.566

0.065

1.800

1.369

3.091

  241Pu

  12.19

 334.11

936.65

2.946

148.68

 169.13

   570.66

2.933

5.170

0.226

1.626

1.048

3.151

 

 

 

 

Fertile materials

 

 

 

 

 

 

 

 

 

 

 

238U

    9.37

     2.414

1.05 × 10−5

2.489

319.06

277.70

        2.16 × 10−3

2.490

4.825

0.070

0.300

2.598

2.598

  240Pu

    1.39

   262.65

6.13 × 10−2

2.784

913.76

   8448.7  

    3.74

2.785

4.996

0.095

1.349

1.418

3.013

  232Th

  11.84

    6.533

   

187.36

   84.97

 

 

4.823

0.102

0.071

2.241

2.093

234U

  12.24

    90.45

  0.407

2.352

227.20

 659.19

       0.618

2.353

5.340

0.177

1.215

1.793

2.559

236U

    8.08

     4.582

  0.042

2.317

250.57

 346.02

     4.38

2.318

5.403

0.177

0.582

1.941

2.517

  238Pu

  19.46

   463.10

14.743

2.895

197.72

 142.92

   22.71

2.896

4.635

0.077

1.968

1.143

3.121

  233Pa

    8.41

     35.99

 

 

132.65

 854.43

 

 

4.503

0.187

0.460

2.168

2.471

  237Np

  14.58

  159.15

1.57 × 10−2

2.534

155.90

 657.90

       0.207

2.535

4.576

0.196

1.290

1.515

2.773

  241Am

  10.99

  551.30

     2.924   

3.330

146.18

  1443.8  

     9.77

3.331

4.771

0.322

1.323

1.258

3.573

   242mAm

  14.26

1706.90

    6686.1      

3.210

144.34

261.47

      1630.1  

3.211

4.504

0.080

1.843

1.324

3.414

243Am

    7.09

   67.73

4.42 × 10−2 

3.062

175.89

    1811.4     

             1.194      

3.063

5.020

0.250

1.100

1.324

3.334

 

 

 

 

  σnA (b)

 

 

Cladding and structural materials

 

 

 

 

 

 

 

 

 

Be

  6.328

6.74 × 10−3

 

 

  72.87

  0.004

 

 

2.693

0.0362

 

0.120

 

Al

  1.450

    0.189

 

 

  23.33

  0.129

 

 

3.095

0.0056

 

0.279

 

Si

  2.101

    0.152

 

 

  22.94

  0.082

 

 

3.186

0.0132

 

0.216

 

Cr

  3.418

    2.724

 

 

  66.63

  1.539

 

 

3.261

0.0049

 

0.476

 

Mn

  1.767

  11.794

 

 

601.69

15.297

 

 

2.722

0.0047

 

0.976

 

Fe

11.330

    2.292

 

 

118.89

  1.344

 

 

2.613

0.0100

 

0.612

 

Co

  6.005

  32.981

 

 

760.88

75.449

 

 

3.007

0.0066

 

0.726

 

Ni

17.742

    3.933

 

 

219.85

  2.073

 

 

3.068

0.0864

 

0.446

 

Cu

  8.680

    3.360

 

 

124.24

   4.453

 

 

2.896

0.0264

 

0.726

 

Zr

  6.490

    0.163

 

 

  95.07

   0.910

 

 

5.366

0.0066

 

0.579

 

 Nb

  6.069

    1.015

 

 

  82.30

   9.380

 

 

4.426

0.0271

 

1.298

 

  Mo

  5.557

    2.277

 

 

 113.24

 24.380

 

 

4.719

0.0231

 

1.109

 

Ta

  6.160

  18.795

 

 

 234.24

739.013

 

 

3.942

0.1053

 

2.569

 

  4.997

  16.077

 

 

1118.37 

361.481

 

 

4.561

0.0431

 

2.331

 

 Pb

 11.221

    0.158

 

 

136.58

    0.095

 

 

5.668

0.0023

 

0.686

 

 

 

σnA (b)

 

 

 

σnA (b)

 

 

 

 

 

 

 

Components and coolants

 

 

 

 

 

 

 

 

 

 

H

28.966

0.294

 

 

240.23

      0.149

 

 

3.99

3.96 ×10−5

 

 

 

D

  4.160

4.48 × 10−4

 

 

  41.28

2.28 × 10−4

 

 

2.54

6.95 ×10−6

 

 

 

  He

  0.849

6.46 × 10−3

 

 

    9.27

3.25 × 10−3

 

 

3.66

1.15 ×10−6

 

 

 

C

  4.935

2.99 × 10−3

 

 

  57.51

1.53 × 10−3

 

 

2.37

1.52 × 10−6

 

0.012

 

N

10.290

1.673

 

 

106.89

       0.848

 

 

1.87

0.123

 

0.009

 

  Na

  3.090

0.417

 

 

122.68

       0.310

 

 

2.62

2.73 × 10−3

 

0.526

 

K

  2.204

1.912

 

 

  26.60

       1.231

 

 

2.54

0.104

 

0.107

 

 

 

 

 

σnA(b)

 

 

Absorbers and poisons

 

 

 

 

 

 

 

 

 

Li

  1.097

      62.53

 

 

12.46

    31.71

 

 

1.63

2.47 × 10−2

 

0.174

 

B

  4.499

  677.57

 

 

52.70

 342.65

 

 

2.35

9.15 × 102

 

0.032

 

Cd

10.237

     2918.3    

 

 

85.12

    71.88

 

 

4.34

4.80 × 10−2

 

1.288

 

       135Xe

383920.0

2720200.0

 

 

5212.05   

7654.47

 

 

5.04

7.63 × 10 4

 

0.780

 

Hf

 8.560

  92.31

 

 

773.33 

1989.03

 

 

5.06

 0.131

 

1.795

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



Average neutron cross-sections in a fast reactor core and blanket

Material 

Core

Blanket

 

 σtr (b)

σnγ (b)

σnf (b)

σtr (b)

σnγ (b)

σnf (b)

 

 

 

 

 

 

 

 

 

10B     .   .   .   .   .   .   .

4.8

2.3*

 

 

  6.0

 3.6* 

 

 

C    .  .   .   .   .   .   .   .

3.6

  0.1 m

 

 

  3.9

    0.05 m

 

 

O    .   .  .   .   .   .   .   .  

3.5

  0.6 m

 

 

  3.6

  0.3 m

 

 

Na     .   .   .   .   .   .   .

3.7

  1.4 m

 

 

  3.9

  1.9 m

 

 

Cr      .   .   .   .   .   .   .

4.2

   0.014

 

 

  4.7

  0.019

 

 

Fe      .   .   .   .   .   .   .

3.8

  8.6 m

 

 

  4.5

10 m   

 

 

Ni      .   .   .   .   .   .   .

8.2

  0.021

 

 

10.6

  0.026

 

 

Mo    .   .   .   .   .   .   .

7.2

0.11

 

 

  7.6

0.16

 

 

232Th     .   .   .   .   .   .

9.2

0.36

    8.9 m

2.30

10.7

0.58

4.5 m  

2.28

233U      .   .   .   .   .   .

8.8

0.22

2.6

2.52

10.5

0.32

3.2     

2.51

235U .    .   .   .   .   .   .

9.7

0.49

1.8

2.45

11.6

0.74

2.3     

2.43

238U .    .   .   .   .   .   .

9.4

0.25

    0.041

2.73

10.3

0.34

0.021 

2.71

239Pu .   .   .   .   .   .   .

9.5

0.40

1.7

2.93

11.2

0.73

2.0     

2.90

240Pu     .   .   .   .   .   .

9.4

0.42

  0.37

3.06

10.8

0.67

0.25   

3.02

241Pu     .   .   .   .   .   .

10.1  

0.40

2.3

2.98

12.0

0.63

3.0     

2.95

242Pu     .   .   .   .   .   .

9.7

0.29

  0.29

3.02

11.8

0.51

0.18   

2.99

Fission product pair

12.7  

0.39

 

 

14.0

0.65

 

 

 

 

 

 

 

 

 

 

 

M.G. Sowerby/R.A. Forrest

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