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Chapter: 4 Atomic and nuclear physics
    Section: 4.7 Nuclear fission and fusion, and neutron interactions
        SubSection: 4.7.4 Nuclear fusion

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4.7.4    Nuclear fusion

Fusion reactions

The first generation of controlled thermonuclear (fusion) reactors (CTR) producing electrical energy from nuclear fusion reactions between light ions will almost certainly exploit D–T reactions occurring in a hot plasma that is confined magnetically, although a successful inertial confinement reactor cannot be ruled out. The D–T reaction is specified because of its high cross-section at low ion kinetic energies and its large energy release. However, even in a plasma containing equimolar quantities of deuterium and tritium there will occur fusion reactions between like as well as unlike ion species and between the energetic charged particle reaction products and the fuel (and impurity) ions. In low power test reactors, the use of radio-frequency wave heating may lead to acceleration of light ions to MeV energies so that nuclear reactions between these accelerated ions and fuel (and impurity) ions may occur; the resulting reaction rates may be significant in comparison with D–D (but not D–T) fusion rates. The important fusion reactions are listed below. Particle energies (in MeV) are quoted for the two-particle exothermic reactions, calculated relativistically for zero energy reactants; the total energy release is given for the other reactions.


Fusion reactions of CTR interest

1a.

D + D

T(1.011) + p(3.022)

1b.

D + D

3He(0.820) + n(2.449)

2.

p + T

 

3He + n − 0.764

3.

D + T

4He (3.561) + n(14.029)

4.

T + T

4He + 2n + 11.332

5.

D + 3He 

4He(3.712) + p(14.641)

6a.

T + 3He

4He + n + p + 12.096

6b.

T + 3He

4He(4.800) + D(9.520)

7

3He + 3He

 4He + 2p + 12.860

Fusion reaction cross-sections

Plasma reactivity calculations require reaction cross-sections for energies well below those at which direct measurements are practicable, so it is necessary to extrapolate downwards using the theoretical formula

   

σ(E) = 

 S(E)

 exp(−R/)

  with  R = π

e2

Z1Z2

 

E

c

where the cross-section is expressed in centre-of-mass units, E = mv2, m = m1m2/(m1 + m2) and v is the relative velocity of the interacting particles which have masses m1 and m2 and charges Z1 and Z2, respectively. The constants e, and c have their usual meaning. S(E) = A exp(−βE) and the parameters A, β and R are given in Table 1. Note that laboratory energies may be used if the substitution E = (m/m1)Elab is made.


Table 1. Low-energy cross-section parametrization

Reaction

A

β

R

 

(barns-keV)

(keV−1)

(keV1/2)

 

 

 

 

D–Dp     .     .     .     .     .     .     .     .     .     .     .     . 

      52.6

 −5.8 × 10 3

31.39

D–Dn     .     .     .     .     .     .     .     .     .     .     .     . 

      52.6

 −5.8 × 10 3

31.39

D–T       .     .     .     .     .     .     .     .     .   

9821

 −2.9 × 10 2

34.37

T–T        .     .     .     .     .     .     .     .     .   

  175

    9.6 × 103

38.41

D–3He   .     .     .     .     .     .     .     .     . 

5666

 −5.1 × 103

68.74

T–3He    .     .     .     .     .     .     .     .     .

2422

   4.5 × 103

76.82

3He–3He      .     .     .     .     .     .     .     .

5500

 −5.6 × 103

153.7   

 

 

 

 

Fig. 1. Fusion reaction cross-sections

The theoretical formula quoted above applies only for energies well below the Coulomb barrier. A more complex formula is required for higher energies. Cross-sections for the primary fusion reactions are plotted in Fig. 1, as a function of projectile energy (Elab). Bosch and Hale (Nuclear Fusion 1992, 32, 611) have recently reviewed (and have provided convenient parametrizations for) the cross-section data for the reactions D(d, n)3He, D(d, p)T, T(d, n)4He and 3He(d, p)4He.





Fusion reaction rates

The reaction rate, r, between two species of ion with densities ni and nj in a plasma is given by r = ninj(1 + δij) 1 σv, where σv is the appropriate average of the fusion cross-section σ over the relative velocities v, and δij is the Kronecker delta function. The relative ion energy distribution in a plasma is customarily taken to be of Maxwellian form,

N(E) = 2π− 1/2θ − 3/2E1/2exp(−E/θ),

where E is the relative energy and θ =kT, so that

σv = (8/π)1/2m−1/2θ−3/2

(E)exp(−E/θ)dE.

Provided the cross-sections have the simple form σ(E) = (A/E) exp(−βE − R/), applicable at low energies, the integration can be performed approximately by saddle-point integration to give

σv = 0.8052 × 10−22 

AR1/3m−1/2θ−2/3

 exp

−3

R2

1/3

(1 + βθ)1/3θ−1/3

(1 + βθ)5/6

4

in m3 s1, where A is in barns, θ in keV and m is the relative mass in a.m.u. Using the values of A, R and β from Table 1, σv values for temperatures well below the Coulomb barrier may be calculated. It is customary to neglect terms involving β.

Fig. 2. Fusion reactivities

For calculations of plasma reactivities at high energies, an accurate representation of σ is required, necessitating a numerical integration of the σv product over energy. Thermal reactivity values for the primary fusion reactions are plotted in Fig. 2. Bosch and Hale (Nuclear Fusion, 1992, 22, 611) have provided convenient parametrizations for the thermal reactivities for the reactions D(d, n)3He, D(d, p)T, T(d, n)4He and 3He(d, p)4He. Harris, Fowler, Caughlan and Zimmermann (Ann. Rev. Astron. and Astrophys., 1983, 21, 165) have presented a comprehensive set of stellar thermonuclear reaction rate formulae applicable over a very wide range of temperatures.


Thermonuclear neutron energy spectra

In a Maxwellian plasma containing two sets of particles at a common temperature θ, the mean energy of the centre-of-mass of the colliding pairs is

C = (m1 + m2)V2 =θ

and the relative mean kinetic energy is

K =

1

m1m2

v2 =

 θ

2

m1 + m2

where V is the centre-of-mass velocity and v is the relative velocity. The mean relative velocity of reacting particles is obtained by weighting with the reactivity σv and Brysk (Plasma Physics, 1973, 15, 611) has shown that there exists a convenient relationship

Kr Kσv/σv = θ2(d/dθ){ln(σvθ3/2)}.

Use of the simple Gamov cross-section weighted by a Maxwellian energy distribution results in

σv = −2/3 exp(−−1/3) which gives Kr = 2/3 + θ.


      The mean energy and energy distribution. of neutrons emitted from D–D and D–T plasmas provide information on the plasma temperature. The average neutron energy is

En = mnV2

mα  (Q + Kr)
(mn + mα)

where the subscripts n and α refer to neutrons and helium ions and Q is the reaction Q-value. The shift in En with temperature is small. The energy distribution of the neutrons is approximately Gaussian in form


f(En) = 

 1

 exp

{EnEn}2

σw 2σ2w

where

   

σw =

   1

4mnEnθ

1/2

mn + mα

The distribution fwhm is 2σw. The width of the Gaussian distribution is a measurable quantity, thus providing a valuable diagnostic technique for determining the temperature, θ, of a Maxwellian plasma. At low ion temperatures, the relationship between F and θ is F = 82.6√θ for D–D neutrons and F = 177.2√θ for D–T neutrons, where F and θ are in keV. The multipliers are weakly temperature-dependent.


Neutron blanket reactions

A reactor based on the D–T reaction will consume appreciable quantities of tritium. In a pure fusion reactor system, this tritium will have to be derived from nuclear reactions between the 14 MeV fusion neutrons and lithium in a breeding blanket surrounding the reaction chamber. The breeding reactions are

n + 6Li → 4He + T + 4.784 MeV

and

n + 7Li → n′ + 4He + T − 2.467 MeV.

Because of parasitic neutron capture and penetration losses, the number of tritium atoms produced per 14 MeV neutron may fall below unity. In this case a neutron multiplier such as beryllium will be needed:

n + 9Be → 8Be* + 2n −1.665 MeV → 2 4He + 2n −1.573 MeV.

Fig. 3. Neutron cross-sections for tritium breeding

Lead, with a threshold for (n, 2n) reactions at about 8 MeV, may be employed instead. The cross-sections for these reactions are displayed in Fig. 3.

      Reaction and scattering cross-sections needed for neutron transport and moderation considerations are discussed in sections 4.7.2 and 4.7.3.

Neutron activation reactions

Since the D–D and D–T plasmas are intense sources of neutrons it will be possible to employ activation techniques for plasma diagnostic and for reactor dosimetric purposes. Standard reactions which may be employed for this purpose are listed in Table 2, ordered by approximate threshold energies. Most of the cross-sections are taken from the DOSCROS84 Dosimetry File (Zijp, Nolthenius and Verhaag (1984) ECN Petten, The Netherlands, Report ECN-160). Related information on these activation reactions may be found in the Nuclear Data Guide for Reactor Neutron Metrology (Baard, Zijp and Nolthenius (1989) Kluwer Academic Publishers, ISBN 0-7923-0486-1).

The structural materials of which the vacuum vessel and breeding blanket can be constructed will become so highly radioactive that direct access to these inner regions of the fusion reactor will not be possible for maintenance purposes. Further, when the reactor reaches the end of its operational lifetime the problem of disposal of the highly radioactive structural materials will have to be considered. This activation problem can be minimized by appropriate choice of structural material. The most important radionuclides produced in each of the chemical elements most likely to be employed in a structural material are listed in Table 3. The Handbook of Fusion Activation Data (Parts 1 and 2) prepared by Forty, Forrest, Compton and Rayner (AEA FUS 180 (1992) and 232 (1993), AEA Technology, Fusion, Culham, Abingdon, UK) contains extensive information for elements from hydrogen to bismuth.

The induced radioactivity R remaining after time T due to a particular reaction channel is given to first order by

 

R(Bq · kg−1) = (1.32 × 10−9) · 

F

 · · · 

exp(−ln 2 · T/T1/2)

A

T1/2

for an irradiation of duration brief compared with the decay time T and radionuclide half-life T1/2. F is the fractional abundance of the target nuclide in the irradiated element, A is the nuclear number, is the energy-averaged neutron reaction cross-section in barns (10 28 m2), is the neutron fluence (n · m2) and the times are in years.

The above formula takes only first generation reaction products into account and ignores the burnup of target and daughter nuclides. For orientation purposes, can be taken to be the reaction cross-section averaged over the energy spectrum of neutrons emitted from a D–T plasma at 20 keV ion temperature as provided in Table 4 for the significant particle-emitting reactions. More detailed cross-section information can be obtained from The European Activation File EAF-3 with Neutron Activation and Transmutation Cross-sections, by Kopecky, van der Kamp, Gruppelaar and Nierop (1992) ECN Petten Report ECN-C-92-058.

The surface gamma dose-rate from a thick slab of material can be estimated from



 

D(Gy . hr1) = 5.76 × 1010(μa/μm)(B/2)SV

where SV is the rate of gamma radiation energy emission (in MeV kg 1 s 1), μa is the energy absorption coefficient in air, μm is the mass absorption coefficient of the element and B is the photon buildup factor, all appropriate to the gamma radiation energy Eγ (MeV). The absorption coefficients may be obtained from Storm and Israel (1970) Nuclear Data Tables, A7, 565. Typically, B ≈ 2 and (μa/μm) ≈ Eγ/2, which leads to the simplified form

D(Gy · hr 1) = 2.88 × 1010f REγ2

where f is the number of quanta Eγ emitted per decay and R is the specific activity. If several gamma emissions occur, then D must be summed over all of them.

Table 2. Fusion neutron dosimetry threshold reactions

Reaction 

Half-life 

Cross-section (barns) 

Threshold 
(MeV)

2.47 MeV*

14.05 MeV*

 

 

 

 

 

93Nb(n, n′)93mNb  .     .     .     .     .

16.1 yr   

0.294

0.036

0.1

103Rh(n, n′)103mRh .     .     .     .     .

56.1 min

0.920

0.304

0.1

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

   4.486 h

0.326

0.066

0.3

238U(n, f)F.P.        .     .     .     .     .

       —

0.540

1.125

0.8

47Ti(n, p)47Sc  .     .     .     .     .     .

3.35 d

0.031

0.119

1.0

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

70.8 d    

0.100

0.423

1.1

232Th(n, f)F.P.      .     .     .     .     .

       —

0.116

0.347

1.2

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

14.29 d   

0.080

0.250

1.5

54Fe(n, p)54Mn     .     .     .     .     .

312.1 d       

0.061

0.368

1.5

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

157.3 min    

0.041

0.092

1.5

64Zn(n, p)64Cu     .     .     .     .     .

12.7 h     

0.021

0.173

1.8

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

    9.458 min

0.077

2.7

46Ti(n, p)46Sc      .     .      .     .     .

83.81 d   

0.259

3.0

63Cu(n, α60)Co    .     .     .     .      .

5.27 yr

0.044

3.5

60Ni(n, p)60Co     .     .     .     .     .

5.27 yr

0.126

3.8

28Si(n, p)28Al       .     .     .     .     .

     2.244 min

0.268

4.0

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

 2.577 h

0.110

4.2

48Ti(n, p)48Sc      .     .     .     .     .

43.7 h    

0.064

4.5

52Cr(n, p)52V      .     .     .     .     .

   3.75 min

0.105

5.0

59Co(n, α)56Mn   .     .     .     .     .

  2.577 h

0.029

5.0

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

15.00 h  

0.124

5.4

24Mg(n, p)24Na   .     .     .     .     .

15.00 h  

0.196

6.0

232Th(n, 2n)231Th      .     .     .     .

25.52 h 

1.538

6.5

197Au(n, 2n)196Au     .     .     .     .

   6.183 d

2.201

8.1

127I(n, 2n)126I     .     .     .     .     .

13.02 d 

1.614

9.3

65Cu(n, 2n)64Cu       .     .     .     .

12.7 h   

0.858

10.1  

55Mn(n, 2n)54Mn      .     .     .     .

312.1 d    

0.720

10.4  

59Co(n, 2n)58Co  .     .     .     .     .

70.8 d  

0.755

10.7  

19F(n, 2n)18F  .     .     .     .     .     .

109.71 min

0.041

11.0  

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

    9.74 min

0.477

11.1  

64Zn(n, 2n)63Zn    .     .     .     .     .

38.5 min

0.174

12.2  

58Ni(n, 2n)57Ni     .     .     .     .     .

35.9 h   

0.025

12.5  

197Au(n, 3n)195 Au  .    .     .    .     .

183 d        

15.0  

 

 

 

 

 

* The cross-sections are given as group averages from 2.40 to 2.55 and from 14.0 to 14.1 MeV.

 
Table 3. Important contributing radionuclides, by half-life*

Element

1 min < T1/2    
           < 1 day    

1 day–30 days

30 days–5 years 

5–100 years    

500 years < T1/2

C    .   .   .   .   .

    —

    —

    —

    3H

    10Be

Mg  .   .   .   .   .

    24Na

    —

    22Na

    3H

    —

Al    .   .   .   .   .

    24Na, 27Mg

    —

    —

    3H

    26Al

Si    .   .   .   .   .

    24Na, 27Mg,

    —

    22Na

    3H

    26Al

 

    28Al, 31Si

 

 

 

 

Ti    .    .    .    .

    —

    47Sc, 48Sc

    46Sc, 45Ca

    3H, 42Ar

    41Ca

V    .    .    .    .

    51Ti, 52V

    48Sc, 51Cr

    46Sc, 49V

    3H

    41Ca

Cr   .    .    .    .

    52V

    51Cr

    49V, 54Mn

    3H

    53Mn

Mn  .    .    .    .

    56Mn

    —

    54Mn

    3H

    53Mn

Fe    .    .    .    .

    56Mn

    —

    54Mn, 55Fe

    3H

    53Mn

Co   .    .    .    .

    58mCo, 60mCo

    —

    59Fe, 58Co

    60Co, 63Ni

    60Fe, 59Ni

Ni    .    .    .    .

    58mCo, 60mCo

    —

    55Fe, 57Co,

    60Co, 63Ni

    59Ni

 

 

 

   58Co

 

 

Cu   .    .    .    .

    62Cu, 64Cu

    —

    65Zn

    60Co, 63Ni

    60Fe, 59Ni

Zn   .    .    .    .

    64Cu, 63Zn

    —

    65Zn

    60Co, 63Ni

    60Fe, 59Ni

Zr   .    .    .    .

    97Zr, 97Nb

    90Y, 89Zr

    88Y, 95Zr,

    90Sr

    93mNb, 93Zr

 

 

 

    95Nb

 

 

Nb  .    .    .    .

    94mNb

    92mNb

    95Nb

    93mNb

    94Nb, 93Zr

Mo  .   .    .    .

    99mTc

    92mNb, 99Mo

    95Nb

    93mNb

    91Nb, 93Mo,

 

 

 

 

 

    94Nb, 99Tc,

 

 

 

 

 

    98Tc

W   .    .    .   .

    187W, 188Re

    186Re

    181W, 185W

    193Pt

    182Hf, 186mRe

 

 

 

 

 

 

* Note the lack of important nuclides with 100 years < T1/2 < 500 years.

Table 4. Reaction cross-section (in barns), averaged for 14 MeV fusion neutrons

Target element

Reaction type

 

(n, 2n)

(n, α)

(n, p)

(n, n′p)

(n, n′α)

 

 

 

 

 

 

C        .    .    .    .    .    .    .    .    .    .

0.003

0.082

0.000

0.000

0.258

Mg     .    .    .    .    .    .    .    .    .    .

0.034

0.087

0.161

0.178

0.062

Al       .    .    .    .    .    .    .    .    .    .

0.016

0.121

0.078

0.337

0.018

Si       .    .    .    .    .    .    .    .    .    .

0.010

0.108

0.220

0.387

0.075

Ti       .    .    .    .    .    .    .    .    .    .

0.280

0.034

0.082

0.054

0.000

V       .    .    .    .    .    .    .    .    .    .

0.461

0.017

0.032

0.063

0.003

Cr      .    .    .    .    .    .    .    .    .    .

0.325

0.039

0.083

0.107

0.000

Mn     .    .    .    .    .    .    .    .    .    .

0.774

0.031

0.060

0.024

0.000

Fe      .    .    .    .    .    .    .    .    .    .

0.355

0.044

0.128

0.070

0.001

Co     .    .    .    .    .    .    .    .    .    .

0.698

0.029

0.073

0.051

0.001

Ni      .    .    .    .    .    .    .    .    .    .

0.153

0.090

0.325

0.406

0.003

Cu     .    .    .    .    .    .    .    .    .    .

0.581

0.035

0.048

0.149

0.008

Zn      .    .    .    .    .    .    .    .    .    .

0.371

0.018

0.102

0.206

0.016

Zr      .    .    .    .    .    .    .    .    .    .

0.872

0.012

0.031

0.036

0.001

Nb     .    .    .    .    .    .    .    .    .    .

0.459

0.009

0.040

0.011

0.002

Mo    .    .    .    .    .    .    .    .    .    .

0.956

0.016

0.036

0.123

0.000

W      .    .    .    .    .    .    .    .    .    .

2.073

0.001

0.002

0.001

0.000

Pb     .    .    .    .    .    .    .    .    .    .

1.971

0.001

0.001

0.000

0.000

 

 

 

 

 

 

O.N.Jarvis

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