# 1 Nuclearfusion reactions

## Transcript Of 1 Nuclearfusion reactions

1

Nuclear fusion reactions

1.1 Exothermic nuclear

reactions: ﬁssion and

fusion

2

1.2 Fusion reaction

physics

3

1.3 Some important fusion

reactions

10

1.4 Maxwell-averaged

fusion reactivities

14

1.5 Fusion reactivity in very

high density matter

21

1.6 Spin polarization of

reacting nuclei

24

1.7 µ-catalysed fusion

25

1.8 Historical note

27

Most of this book is devoted to the physical principles of energy production by fusion reactions in an inertially conﬁned medium. To begin with, in this chapter we brieﬂy discuss fusion reactions.

We ﬁrst deﬁne fusion cross section and reactivity, and then present and justify qualitatively the standard parametrization of these two important quantities. Next, we consider a few important fusion reactions, and provide expressions, data, and graphs for the evaluation of their cross sections and reactivities. These results will be used in the following chapters to derive the basic requirements for fusion energy production, as well as to study fusion ignition and burn in suitable inertially conﬁned fuels.

In the last part of this chapter, we also brieﬂy discuss how high material density and spin polarization affect fusion reactivities. Finally, we outline the principles of muon-catalysed fusion.

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1.1 Exothermic nuclear reactions: ﬁssion and fusion

1.1 Exothermic nuclear reactions: ﬁssion and fusion

Reaction Q

Nucleus binding energy Q and binding energy Fission vs fusion

According to Einstein’s mass–energy relationship, a nuclear reaction in which the total mass of the ﬁnal products is smaller than that of the reacting nuclei is exothermic, that is, releases an energy

Q=

mi − mf c2

1.1

i

f

proportional to such a mass difference. Here the symbol m denotes mass, the subscripts i and f indicate, respectively, the initial and the ﬁnal products, and c is the speed of light. We can identify exothermic reactions by considering the masses and the binding energies of each of the involved nuclei. The mass m of a nucleus with atomic number Z and mass number A differs from the sum of the masses of the Z protons and A−Z neutrons, which build up the nucleus by a quantity

m = Zmp + (A − Z)mn − m.

1.2

Here mp and mn are the mass of the proton and of the neutron, respectively. For stable nuclei m is positive, and one has to provide an amount of

energy equal to the binding energy

B = mc2

1.3

in order to dissociate the nucleus into its component neutrons and protons. The Q value of a nuclear reaction can then be written as the difference

between the ﬁnal and the initial binding energies of the interacting nuclei:

Q = Bf − Bi .

1.4

f

i

Accurate data on nuclear masses and binding energies have been pub-

lished by Audi and Wapstra (1995). A particularly useful quantity is the average binding energy per nucleon B/A, which is plotted in Fig. 1.1 as a function of the mass number A. We see that B/A, which is zero for A = 1, that is, for the hydrogen nucleus, grows rapidly with A, reaches a broad maximum of 8.7 MeV about A = 56 and then decreases slightly. For the heaviest nuclei B/A ∼= 7.5 MeV. Notice the particularly high value of B/A for 4He nucleus (the α-particle). The symbols D and T indicate, as usual, deuterium and tritium, that is, the hydrogen isotopes

with mass two and three, respectively. According to the above discussion,

exothermic reactions occur when the ﬁnal reaction products have larger B/A than the reacting nuclei. As indicated in Fig. 1.1, this occurs for ﬁssion reactions, in which a heavy nucleus is split into lighter fragments,

and for fusion reactions, in which two light nuclei merge to form a heavier

nucleus.

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Fig. 1.1 Binding energy per nucleon versus mass number A, for the most stable isobars. For A = 3 also the unstable tritium is included, in view of its importance for controlled fusion. Notice that the mass number scale is logarithmic in the range 1–50 and linear in the range 50–250.

B/A (MeV/nucleon)

1.2 Fusion reaction physics

3

10

8

4He

6

Fission

4 T

3He

2

D

Fusion

p

0

1

5

10 20

50 100 150 200 250

Mass number A

1.2 Fusion reaction physics

Cross section

In most fusion reactions two nuclei (X1 and X2) merge to form a heavier nucleus (X3) and a lighter particle (X4). To express this, we shall use either of the equivalent standard notations

X1 + X2 → X3 + X4,

1.5

or

X1(x2, x4)X3.

1.6

Due to conservation of energy and momentum, the energy released by the reaction is distributed among the two fusion products in quantity inversely proportional to their masses.

We indicate the velocities of the reacting nuclei in the laboratory system with v1 and v2, respectively, and their relative velocity with v = v1 − v2. The center-of-mass energy of the system of the reacting nuclei is then

= 21 mrv2,

1.7

where v = |v|, and

mr = m1m2

1.8

m1 + m2

is the reduced mass of the system.

1.2.1 Cross section, reactivity, and reaction rate

A most important quantity for the analysis of nuclear reactions is the cross section, which measures the probability per pair of particles for the

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1.2 Fusion reaction physics

Beam-target and centre-of-mass cross section

Averaged reactivity

Volumetric reaction rate

occurrence of the reaction. To be more speciﬁc, let us consider a uniform beam of particles of type ‘1’, with velocity v1, interacting with a target containing particles of type ‘2’ at rest. The cross section σ12(v1) is deﬁned as the number of reactions per target nucleus per unit time when the target

is hit by a unit ﬂux of projectile particles, that is, by one particle per unit

target area per unit time. Actually, the above deﬁnition applies in general to particles with relative velocity v, and is therefore symmetric in the two particles, since we have σ12(v) = σ21(v).

Cross sections can also be expressed in terms of the centre-of-mass energy 1.7, and we have σ12( ) = σ21( ). In most cases, however, the cross sections are measured in experiments in which a beam of particles

with energy 1, measured in the laboratory frame, hits a target at rest. The corresponding beam-target cross-section σ1b2t( 1) is related to the centre-of-mass cross-section σ12( ) by

σ12( ) = σ1b2t ( 1),

1.9

with 1 = · (m1 + m2)/m2. From now on, we shall refer to centre-ofmass cross-sections and omit the indices 1 and 2.

If the target nuclei have density n2 and are at rest or all move with the same velocity, and the relative velocity is the same for all pairs of

projectile–target nuclei, then the probability of reaction of nucleus ‘1’ per unit path is given by the product n2σ (v). The probability of reaction per unit time is obtained by multiplying the probability per unit path times the distance v travelled in the unit time, which gives n2σ (v)v.

Another important quantity is the reactivity, deﬁned as the probability

of reaction per unit time per unit density of target nuclei. In the present simple case, it is just given by the product σ v. In general, target nuclei move, so that the relative velocity v is different for each pair of interacting nuclei. In this case, we compute an averaged reactivity

∞

σ v = σ (v)vf (v) dv,

1.10

0

where f (v) is the distribution function of the relative velocities, normalized in such a way that 0∞ f (v) dv = 1. It is to be observed that when projectile and target particles are of the same species, each reaction is

counted twice.

Both controlled fusion fuels and stellar media are usually mixtures of elements where species ‘1’ and ‘2’, have number densities n1 and n2, respectively. The volumetric reaction rate, that is, the number of reactions

per unit time and per unit volume is then given by

R12 = n1n2 σ v = f1f2 n2 σ v .

1.11

1 + δ12

1 + δ12

Here n is the total nuclei number density and f1 and f2 are the atomic fractions of species ‘1’ and ‘2’, respectively. The Kronecker symbol δij (with δij = 1, if i = j and δij = 0 elsewhere) is introduced to properly take

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The reaction rate is proportional to the square of the density

1.2 Fusion reaction physics

5

into account the case of reactions between like particles. Equation 1.11 shows a very important feature for fusion energy research: the volumetric reaction rate is proportional to the square of the density of the mixture. For future reference, it is also useful to recast it in terms of the mass density ρ of the reacting fuel

f1f2 ρ2

R12 = 1 + δ12 m¯ 2 σ v ,

1.12

where m¯ is the average nuclear mass. Here, the mass density is computed as ρ = j nj mj = nm¯ , where the sum is over all species, and the very small contribution due to the electrons is neglected. We also immediately

see that the speciﬁc reaction rate, that is, the reaction rate per unit mass, is

proportional to the mass density, again indicating the role of the density

of the fuel in achieving efﬁcient release of fusion energy.

1.2.2 Fusion cross section parametrization

In order to fuse, two positively charged nuclei must come into contact, winning the repulsive Coulomb force. Such a situation is made evident by the graph of the radial behaviour of the potential energy of a two nucleon system, shown in Fig. 1.2. The potential is essentially Coulombian and repulsive,

Z1Z2e2

Vc(r) = r ,

1.13

at distances greater than

rn ∼= 1.44 × 10−13(A11/3 + A12/3) cm,

1.14

which is about the sum of the radii of the two nuclei. In the above equations Z1 and Z2 are the atomic numbers, A1 and A2 the mass numbers of the

Fig. 1.2 Potential energy versus distance between two charged nuclei approaching each other with center-of-mass energy . The ﬁgure shows the nuclear well, the Coulomb barrier, and the classical turning point.

Energy

Vb Coulomb potential

e rn

Approaching nucleus

rtp

r

Nuclear well –U0

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1.2 Fusion reaction physics

Coulomb barrier

interacting nuclei, and e is the electron charge. At distances r < rn the two nuclei feel the attractive nuclear force, characterized by a potential well of depth U0 = 30– 40 MeV.

Using eqns 1.13 and 1.14 we ﬁnd that the height of the Coulomb barrier

Vb Vc(rn) = A1/Z3 1+ZA2 1/3 MeV 1.15

1

2

is of the order of one million electron-volts (1 MeV). According to classical mechanics, only nuclei with energy exceeding such a value can overcome the barrier and come into contact. Instead, two nuclei with relative energy < Vb can only approach each other up to the classical turning point

Z1Z2e2

rtp =

.

1.16

Fusion reactions rely on tunnelling

Barrier transparency Gamow energy

Quantum mechanics, however, allows for tunnelling a potential barrier of ﬁnite extension, thus making fusion reactions between nuclei with energy smaller than the height of the barrier possible.

A widely used parametrization of fusion reaction cross-sections is

σ ≈ σgeom × T × R,

1.17

where σgeom is a geometrical cross-section, T is the barrier transparency, and R is the probability that nuclei come into contact fuse. The ﬁrst quantity is of the order of the square of the de-Broglie wavelength of the

system:

σ ≈ λ2 = h¯ 2 ∝ 1 ,

1.18

geom

mr v

where h¯ is the reduced Planck constant and mr is the reduced mass 1.8. Concerning the barrier transparency, we shall see that it is often well

approximated by

T ≈ TG = exp(− G/ ),

1.19

which is known as the Gamow factor (after the scientist who ﬁrst computed it), where

G = (π αf Z1Z2)22mrc2 = 986.1Z12Z22Ar keV

1.20

is the Gamow energy, αf = e2/h¯ c = 1/137.04 is the ﬁne-structure constant commonly used in quantum mechanics, and Ar = mr/mp.

Equation 1.19 holds as far as

G, which sets no limitations to the

problems we are interested in. Equations 1.19 and 1.20 show that the

chance of tunnelling decreases rapidly with the atomic number and mass,

thus providing a ﬁrst simple explanation for the fact that fusion reactions

of interest for energy production on earth only involve the lightest nuclei.

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Astrophysical S factor Partial wave expansion

1.2 Fusion reaction physics

7

The reaction characteristics R contains essentially all the nuclear physics of the speciﬁc reaction. It takes substantially different values depending on the nature of the interaction characterizing the reaction. It is largest for reactions due to strong nuclear interactions; it is smaller by several orders of magnitude for electromagnetic nuclear interactions; it is still smaller by as many as 20 orders of magnitude for weak interactions. For most reactions, the variation of R( ) is small compared to the strong variation due to the Gamow factor.

In conclusion, the cross section is often written as

σ ( ) = S( ) exp(− G/ ), 1.21

where the function S( ) is called the astrophysical S factor, which for many important reactions is a weakly varying function of the energy.

An excellent introduction to the computation of fusion cross-sections and thermonuclear reaction rates can be found in the classical textbook on stellar nucleosynthesis by Clayton (1983). Classic references on nuclear physics are Blatt and Weisskopf (1953), Segrè (1964), and Burcham (1973). In the following portion of this section, we outline the evaluation of the fusion cross-section for non-resonant reactions, which justiﬁes the parametrization 1.21. The treatment is simpliﬁed and qualitative, but still rather technical. The reader not interested in such details can skip Section 1.2.3 without loss of the comprehension of the rest of the chapter.

1.2.3 Penetration factors for non-resonant reactions

The total cross-section can be obtained as a sum over partial waves, that is over the contributions of the different terms of an expansion of the particle wave-function in the components of the angular momentum l. We then write

σ (v) = σl(v),

1.22

l

Far from resonances the partial cross-section can be put in the form:

σl(v) ≈ 2π λ2(2l + 1)βlTl,

1.23

where βl is a function taking into account nuclear interactions and Tl is the barrier transmission coefﬁcient. This last factor, deﬁned as the ratio of particles entering the nucleus per unit time to the number of particles incident on the barrier per unit time, can be written as

λ2 −1/2

U0 −1/2

1/2

Tl ≈ Pl 1 + λ2

= Pl 1 +

≈ U0 Pl, 1.24

0

that is, the product of the barrier penetration factor Pl, measuring the probability that nucleus ‘2’ reaches the surface of nucleus ‘1’, and of

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1.2 Fusion reaction physics

a potential discontinuity factor, due to the difference between the wave-

length of the free nucleus and that of the compound nucleus in the nuclear well λ0 = h¯ /(2mrU0)1/2. According to quantum mechanics, the barrier penetration factors Pl are computed by solving the time-independent Schroedinger equation

h¯ 2 ∇2ψ + ( − V )ψ = 0

1.25

2mr

c

for the wavefunction ψ(r) describing the relative motion of the two interacting nucleons in a Coulomb potential extending from r = 0 to

inﬁnity. As usual for problems characterized by a central potential, we separate radial and angular variables, that is, we write ψ(r, θ , φ) = Y (θ , φ)χ (r)/r. We then expand the function χ (r) into angular momentum components, χl(r), each satisfying the equation

Potential for l -th wave

Penetration factors from WKB method

d2

2mr

dr2 χl(r) + h¯ 2 [ − Wl(r)] χl(r) = 0,

1.26

where

Wl(r) = Vc(r) + h¯ 2l(l + 1)

1.27

2mr r 2

takes the role of an effective potential for the lth component. This last equation shows that each angular momentum component sees an effective potential barrier of height increasing with l. We therefore expect the l = 0 component (S-wave) to dominate the cross section, in particular for light elements. An exception will occur for reactions in which the compound

nucleus, formed when the two nuclei come into contact, has forbidden l = 0 levels. This latter case, however, does not occur for any reaction of relevance to controlled fusion.

Once the solution χl(r) of eqn 1.26 is known, the penetration factor for particles with angular momentum l is given by

Pl = χl∗(rn)χl(rn) .

1.28

χl∗ (∞)χl (∞)

Exact computations of the wavefunctions χl(r) are feasible, but involved (Bloch et al. 1951). However, much simpler and yet accurate evaluations of the penetration factors can be performed by means of WKB method (after the initials of Wentzel, Kramers, and Brillouin), discussed in detail in standard books on quantum mechanics (Landau and Lifshitz 1965; Messiah 1999) or mathematical physics (Matthews and Walker 1970). A pedagogical application to the computation of

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1.2 Fusion reaction physics

9

penetration factors is presented by Clayton (1983). Here it sufﬁces to say that application of the method leads to

Pl = Wl(rn) − 1/2 exp(−Gl), 1.29

with the dominant exponential factor given by

Gl = 2 (2mr)1/2

rtp( )

[Wl(r) − ]1/2 dr,

1.30

h¯

rn

where rtp is the turning point distance 1.16. For l = 0, using eqn 1.27 for Wl(r), we get

G0 = 2 G arccos rn − rn 1 − rn .

1.31

π

rtp

rtp

rtp

Since for eqns 1.15 and 1.16, rn/rtp( ) = /Vb, and in the cases of interest Vb, we can expand the right-hand side of eqn 1.31 in powers of

( /Vb), thus obtaining

G

4

1/2 2

3/2

G0 =

1 − π Vb + 3π Vb + · · · .

1.32

In the low energy limit, we have G0 penetration factor becomes

( G/ )1/2, and the S-wave

Vb 1/2

G

P0

exp −

.

1.33

Penetration factors for l > 0 are approximately given by

Vl 1/2 Pl = P0 exp −2l(l + 1) Vb

= P0 exp −7.62l(l + 1)/(Arrnf Z1Z2)1/2 ,

1.34

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10

1.3 Some important fusion reactions

S -wave cross section

where rnf is the nuclear radius in units of 1 fermi = 10−15 cm. Equation 1.34 conﬁrms that angular momentum components with l > 0 have penetration factors much smaller than the l = 0 component. This allows us to keep the S-wave term only in the cross-section expan-

sion 1.22, which leads us to evaluate the barrier transparency and the

cross section as

Vb 1/2

G

T T0 = U0 exp −

1.35

and

h¯ 2

Vb 1/2 exp(−√ G/ )

σ ( ) σl=0( ) π mr βl=0 U0

,

1.36

respectively. Equation 1.36 for the cross section has the same form as the parametrization 1.21, with the term in square brackets corresponding to the astrophysical S-factor.

Another form of eqn 1.35, which will turn useful later, is

Vb 1/2

rtp 1/2

T = U0 exp −π a∗

,

1.37

B

where

aB∗ = h¯ 2/(2mrZ1Z2e2)

1.38

may be looked at as a nuclear Bohr radius.

1.3 Some important fusion reactions

In Table 1.1 we list some fusion reactions of interest to controlled fusion research and to astrophysics. For each reaction the table gives the Q-value, the zero-energy astrophysical factor S(0) and the square root of the Gamow energy G. For the cases in which S( ) is weakly varying these data allow for relatively accurate evaluation of the cross section, using eqn 1.21, with S = S(0).

For some of the main reactions, Table 1.2 gives the measured crosssections at = 10 keV and = 100 keV, as well as the maximum value of the cross-section σmax, and the energy max at which the maximum occurs. Also shown, in parentheses, are theoretical data for the pp and

CC reactions. In the tables and in the following discussion, the reactions

are grouped according to the ﬁeld of interest.

A large and continuously updated database on fusion reactions, quot-

ing original references for all included data, has been produced and is

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Nuclear fusion reactions

1.1 Exothermic nuclear

reactions: ﬁssion and

fusion

2

1.2 Fusion reaction

physics

3

1.3 Some important fusion

reactions

10

1.4 Maxwell-averaged

fusion reactivities

14

1.5 Fusion reactivity in very

high density matter

21

1.6 Spin polarization of

reacting nuclei

24

1.7 µ-catalysed fusion

25

1.8 Historical note

27

Most of this book is devoted to the physical principles of energy production by fusion reactions in an inertially conﬁned medium. To begin with, in this chapter we brieﬂy discuss fusion reactions.

We ﬁrst deﬁne fusion cross section and reactivity, and then present and justify qualitatively the standard parametrization of these two important quantities. Next, we consider a few important fusion reactions, and provide expressions, data, and graphs for the evaluation of their cross sections and reactivities. These results will be used in the following chapters to derive the basic requirements for fusion energy production, as well as to study fusion ignition and burn in suitable inertially conﬁned fuels.

In the last part of this chapter, we also brieﬂy discuss how high material density and spin polarization affect fusion reactivities. Finally, we outline the principles of muon-catalysed fusion.

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1.1 Exothermic nuclear reactions: ﬁssion and fusion

1.1 Exothermic nuclear reactions: ﬁssion and fusion

Reaction Q

Nucleus binding energy Q and binding energy Fission vs fusion

According to Einstein’s mass–energy relationship, a nuclear reaction in which the total mass of the ﬁnal products is smaller than that of the reacting nuclei is exothermic, that is, releases an energy

Q=

mi − mf c2

1.1

i

f

proportional to such a mass difference. Here the symbol m denotes mass, the subscripts i and f indicate, respectively, the initial and the ﬁnal products, and c is the speed of light. We can identify exothermic reactions by considering the masses and the binding energies of each of the involved nuclei. The mass m of a nucleus with atomic number Z and mass number A differs from the sum of the masses of the Z protons and A−Z neutrons, which build up the nucleus by a quantity

m = Zmp + (A − Z)mn − m.

1.2

Here mp and mn are the mass of the proton and of the neutron, respectively. For stable nuclei m is positive, and one has to provide an amount of

energy equal to the binding energy

B = mc2

1.3

in order to dissociate the nucleus into its component neutrons and protons. The Q value of a nuclear reaction can then be written as the difference

between the ﬁnal and the initial binding energies of the interacting nuclei:

Q = Bf − Bi .

1.4

f

i

Accurate data on nuclear masses and binding energies have been pub-

lished by Audi and Wapstra (1995). A particularly useful quantity is the average binding energy per nucleon B/A, which is plotted in Fig. 1.1 as a function of the mass number A. We see that B/A, which is zero for A = 1, that is, for the hydrogen nucleus, grows rapidly with A, reaches a broad maximum of 8.7 MeV about A = 56 and then decreases slightly. For the heaviest nuclei B/A ∼= 7.5 MeV. Notice the particularly high value of B/A for 4He nucleus (the α-particle). The symbols D and T indicate, as usual, deuterium and tritium, that is, the hydrogen isotopes

with mass two and three, respectively. According to the above discussion,

exothermic reactions occur when the ﬁnal reaction products have larger B/A than the reacting nuclei. As indicated in Fig. 1.1, this occurs for ﬁssion reactions, in which a heavy nucleus is split into lighter fragments,

and for fusion reactions, in which two light nuclei merge to form a heavier

nucleus.

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Fig. 1.1 Binding energy per nucleon versus mass number A, for the most stable isobars. For A = 3 also the unstable tritium is included, in view of its importance for controlled fusion. Notice that the mass number scale is logarithmic in the range 1–50 and linear in the range 50–250.

B/A (MeV/nucleon)

1.2 Fusion reaction physics

3

10

8

4He

6

Fission

4 T

3He

2

D

Fusion

p

0

1

5

10 20

50 100 150 200 250

Mass number A

1.2 Fusion reaction physics

Cross section

In most fusion reactions two nuclei (X1 and X2) merge to form a heavier nucleus (X3) and a lighter particle (X4). To express this, we shall use either of the equivalent standard notations

X1 + X2 → X3 + X4,

1.5

or

X1(x2, x4)X3.

1.6

Due to conservation of energy and momentum, the energy released by the reaction is distributed among the two fusion products in quantity inversely proportional to their masses.

We indicate the velocities of the reacting nuclei in the laboratory system with v1 and v2, respectively, and their relative velocity with v = v1 − v2. The center-of-mass energy of the system of the reacting nuclei is then

= 21 mrv2,

1.7

where v = |v|, and

mr = m1m2

1.8

m1 + m2

is the reduced mass of the system.

1.2.1 Cross section, reactivity, and reaction rate

A most important quantity for the analysis of nuclear reactions is the cross section, which measures the probability per pair of particles for the

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4

1.2 Fusion reaction physics

Beam-target and centre-of-mass cross section

Averaged reactivity

Volumetric reaction rate

occurrence of the reaction. To be more speciﬁc, let us consider a uniform beam of particles of type ‘1’, with velocity v1, interacting with a target containing particles of type ‘2’ at rest. The cross section σ12(v1) is deﬁned as the number of reactions per target nucleus per unit time when the target

is hit by a unit ﬂux of projectile particles, that is, by one particle per unit

target area per unit time. Actually, the above deﬁnition applies in general to particles with relative velocity v, and is therefore symmetric in the two particles, since we have σ12(v) = σ21(v).

Cross sections can also be expressed in terms of the centre-of-mass energy 1.7, and we have σ12( ) = σ21( ). In most cases, however, the cross sections are measured in experiments in which a beam of particles

with energy 1, measured in the laboratory frame, hits a target at rest. The corresponding beam-target cross-section σ1b2t( 1) is related to the centre-of-mass cross-section σ12( ) by

σ12( ) = σ1b2t ( 1),

1.9

with 1 = · (m1 + m2)/m2. From now on, we shall refer to centre-ofmass cross-sections and omit the indices 1 and 2.

If the target nuclei have density n2 and are at rest or all move with the same velocity, and the relative velocity is the same for all pairs of

projectile–target nuclei, then the probability of reaction of nucleus ‘1’ per unit path is given by the product n2σ (v). The probability of reaction per unit time is obtained by multiplying the probability per unit path times the distance v travelled in the unit time, which gives n2σ (v)v.

Another important quantity is the reactivity, deﬁned as the probability

of reaction per unit time per unit density of target nuclei. In the present simple case, it is just given by the product σ v. In general, target nuclei move, so that the relative velocity v is different for each pair of interacting nuclei. In this case, we compute an averaged reactivity

∞

σ v = σ (v)vf (v) dv,

1.10

0

where f (v) is the distribution function of the relative velocities, normalized in such a way that 0∞ f (v) dv = 1. It is to be observed that when projectile and target particles are of the same species, each reaction is

counted twice.

Both controlled fusion fuels and stellar media are usually mixtures of elements where species ‘1’ and ‘2’, have number densities n1 and n2, respectively. The volumetric reaction rate, that is, the number of reactions

per unit time and per unit volume is then given by

R12 = n1n2 σ v = f1f2 n2 σ v .

1.11

1 + δ12

1 + δ12

Here n is the total nuclei number density and f1 and f2 are the atomic fractions of species ‘1’ and ‘2’, respectively. The Kronecker symbol δij (with δij = 1, if i = j and δij = 0 elsewhere) is introduced to properly take

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The reaction rate is proportional to the square of the density

1.2 Fusion reaction physics

5

into account the case of reactions between like particles. Equation 1.11 shows a very important feature for fusion energy research: the volumetric reaction rate is proportional to the square of the density of the mixture. For future reference, it is also useful to recast it in terms of the mass density ρ of the reacting fuel

f1f2 ρ2

R12 = 1 + δ12 m¯ 2 σ v ,

1.12

where m¯ is the average nuclear mass. Here, the mass density is computed as ρ = j nj mj = nm¯ , where the sum is over all species, and the very small contribution due to the electrons is neglected. We also immediately

see that the speciﬁc reaction rate, that is, the reaction rate per unit mass, is

proportional to the mass density, again indicating the role of the density

of the fuel in achieving efﬁcient release of fusion energy.

1.2.2 Fusion cross section parametrization

In order to fuse, two positively charged nuclei must come into contact, winning the repulsive Coulomb force. Such a situation is made evident by the graph of the radial behaviour of the potential energy of a two nucleon system, shown in Fig. 1.2. The potential is essentially Coulombian and repulsive,

Z1Z2e2

Vc(r) = r ,

1.13

at distances greater than

rn ∼= 1.44 × 10−13(A11/3 + A12/3) cm,

1.14

which is about the sum of the radii of the two nuclei. In the above equations Z1 and Z2 are the atomic numbers, A1 and A2 the mass numbers of the

Fig. 1.2 Potential energy versus distance between two charged nuclei approaching each other with center-of-mass energy . The ﬁgure shows the nuclear well, the Coulomb barrier, and the classical turning point.

Energy

Vb Coulomb potential

e rn

Approaching nucleus

rtp

r

Nuclear well –U0

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6

1.2 Fusion reaction physics

Coulomb barrier

interacting nuclei, and e is the electron charge. At distances r < rn the two nuclei feel the attractive nuclear force, characterized by a potential well of depth U0 = 30– 40 MeV.

Using eqns 1.13 and 1.14 we ﬁnd that the height of the Coulomb barrier

Vb Vc(rn) = A1/Z3 1+ZA2 1/3 MeV 1.15

1

2

is of the order of one million electron-volts (1 MeV). According to classical mechanics, only nuclei with energy exceeding such a value can overcome the barrier and come into contact. Instead, two nuclei with relative energy < Vb can only approach each other up to the classical turning point

Z1Z2e2

rtp =

.

1.16

Fusion reactions rely on tunnelling

Barrier transparency Gamow energy

Quantum mechanics, however, allows for tunnelling a potential barrier of ﬁnite extension, thus making fusion reactions between nuclei with energy smaller than the height of the barrier possible.

A widely used parametrization of fusion reaction cross-sections is

σ ≈ σgeom × T × R,

1.17

where σgeom is a geometrical cross-section, T is the barrier transparency, and R is the probability that nuclei come into contact fuse. The ﬁrst quantity is of the order of the square of the de-Broglie wavelength of the

system:

σ ≈ λ2 = h¯ 2 ∝ 1 ,

1.18

geom

mr v

where h¯ is the reduced Planck constant and mr is the reduced mass 1.8. Concerning the barrier transparency, we shall see that it is often well

approximated by

T ≈ TG = exp(− G/ ),

1.19

which is known as the Gamow factor (after the scientist who ﬁrst computed it), where

G = (π αf Z1Z2)22mrc2 = 986.1Z12Z22Ar keV

1.20

is the Gamow energy, αf = e2/h¯ c = 1/137.04 is the ﬁne-structure constant commonly used in quantum mechanics, and Ar = mr/mp.

Equation 1.19 holds as far as

G, which sets no limitations to the

problems we are interested in. Equations 1.19 and 1.20 show that the

chance of tunnelling decreases rapidly with the atomic number and mass,

thus providing a ﬁrst simple explanation for the fact that fusion reactions

of interest for energy production on earth only involve the lightest nuclei.

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Astrophysical S factor Partial wave expansion

1.2 Fusion reaction physics

7

The reaction characteristics R contains essentially all the nuclear physics of the speciﬁc reaction. It takes substantially different values depending on the nature of the interaction characterizing the reaction. It is largest for reactions due to strong nuclear interactions; it is smaller by several orders of magnitude for electromagnetic nuclear interactions; it is still smaller by as many as 20 orders of magnitude for weak interactions. For most reactions, the variation of R( ) is small compared to the strong variation due to the Gamow factor.

In conclusion, the cross section is often written as

σ ( ) = S( ) exp(− G/ ), 1.21

where the function S( ) is called the astrophysical S factor, which for many important reactions is a weakly varying function of the energy.

An excellent introduction to the computation of fusion cross-sections and thermonuclear reaction rates can be found in the classical textbook on stellar nucleosynthesis by Clayton (1983). Classic references on nuclear physics are Blatt and Weisskopf (1953), Segrè (1964), and Burcham (1973). In the following portion of this section, we outline the evaluation of the fusion cross-section for non-resonant reactions, which justiﬁes the parametrization 1.21. The treatment is simpliﬁed and qualitative, but still rather technical. The reader not interested in such details can skip Section 1.2.3 without loss of the comprehension of the rest of the chapter.

1.2.3 Penetration factors for non-resonant reactions

The total cross-section can be obtained as a sum over partial waves, that is over the contributions of the different terms of an expansion of the particle wave-function in the components of the angular momentum l. We then write

σ (v) = σl(v),

1.22

l

Far from resonances the partial cross-section can be put in the form:

σl(v) ≈ 2π λ2(2l + 1)βlTl,

1.23

where βl is a function taking into account nuclear interactions and Tl is the barrier transmission coefﬁcient. This last factor, deﬁned as the ratio of particles entering the nucleus per unit time to the number of particles incident on the barrier per unit time, can be written as

λ2 −1/2

U0 −1/2

1/2

Tl ≈ Pl 1 + λ2

= Pl 1 +

≈ U0 Pl, 1.24

0

that is, the product of the barrier penetration factor Pl, measuring the probability that nucleus ‘2’ reaches the surface of nucleus ‘1’, and of

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8

1.2 Fusion reaction physics

a potential discontinuity factor, due to the difference between the wave-

length of the free nucleus and that of the compound nucleus in the nuclear well λ0 = h¯ /(2mrU0)1/2. According to quantum mechanics, the barrier penetration factors Pl are computed by solving the time-independent Schroedinger equation

h¯ 2 ∇2ψ + ( − V )ψ = 0

1.25

2mr

c

for the wavefunction ψ(r) describing the relative motion of the two interacting nucleons in a Coulomb potential extending from r = 0 to

inﬁnity. As usual for problems characterized by a central potential, we separate radial and angular variables, that is, we write ψ(r, θ , φ) = Y (θ , φ)χ (r)/r. We then expand the function χ (r) into angular momentum components, χl(r), each satisfying the equation

Potential for l -th wave

Penetration factors from WKB method

d2

2mr

dr2 χl(r) + h¯ 2 [ − Wl(r)] χl(r) = 0,

1.26

where

Wl(r) = Vc(r) + h¯ 2l(l + 1)

1.27

2mr r 2

takes the role of an effective potential for the lth component. This last equation shows that each angular momentum component sees an effective potential barrier of height increasing with l. We therefore expect the l = 0 component (S-wave) to dominate the cross section, in particular for light elements. An exception will occur for reactions in which the compound

nucleus, formed when the two nuclei come into contact, has forbidden l = 0 levels. This latter case, however, does not occur for any reaction of relevance to controlled fusion.

Once the solution χl(r) of eqn 1.26 is known, the penetration factor for particles with angular momentum l is given by

Pl = χl∗(rn)χl(rn) .

1.28

χl∗ (∞)χl (∞)

Exact computations of the wavefunctions χl(r) are feasible, but involved (Bloch et al. 1951). However, much simpler and yet accurate evaluations of the penetration factors can be performed by means of WKB method (after the initials of Wentzel, Kramers, and Brillouin), discussed in detail in standard books on quantum mechanics (Landau and Lifshitz 1965; Messiah 1999) or mathematical physics (Matthews and Walker 1970). A pedagogical application to the computation of

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1.2 Fusion reaction physics

9

penetration factors is presented by Clayton (1983). Here it sufﬁces to say that application of the method leads to

Pl = Wl(rn) − 1/2 exp(−Gl), 1.29

with the dominant exponential factor given by

Gl = 2 (2mr)1/2

rtp( )

[Wl(r) − ]1/2 dr,

1.30

h¯

rn

where rtp is the turning point distance 1.16. For l = 0, using eqn 1.27 for Wl(r), we get

G0 = 2 G arccos rn − rn 1 − rn .

1.31

π

rtp

rtp

rtp

Since for eqns 1.15 and 1.16, rn/rtp( ) = /Vb, and in the cases of interest Vb, we can expand the right-hand side of eqn 1.31 in powers of

( /Vb), thus obtaining

G

4

1/2 2

3/2

G0 =

1 − π Vb + 3π Vb + · · · .

1.32

In the low energy limit, we have G0 penetration factor becomes

( G/ )1/2, and the S-wave

Vb 1/2

G

P0

exp −

.

1.33

Penetration factors for l > 0 are approximately given by

Vl 1/2 Pl = P0 exp −2l(l + 1) Vb

= P0 exp −7.62l(l + 1)/(Arrnf Z1Z2)1/2 ,

1.34

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10

1.3 Some important fusion reactions

S -wave cross section

where rnf is the nuclear radius in units of 1 fermi = 10−15 cm. Equation 1.34 conﬁrms that angular momentum components with l > 0 have penetration factors much smaller than the l = 0 component. This allows us to keep the S-wave term only in the cross-section expan-

sion 1.22, which leads us to evaluate the barrier transparency and the

cross section as

Vb 1/2

G

T T0 = U0 exp −

1.35

and

h¯ 2

Vb 1/2 exp(−√ G/ )

σ ( ) σl=0( ) π mr βl=0 U0

,

1.36

respectively. Equation 1.36 for the cross section has the same form as the parametrization 1.21, with the term in square brackets corresponding to the astrophysical S-factor.

Another form of eqn 1.35, which will turn useful later, is

Vb 1/2

rtp 1/2

T = U0 exp −π a∗

,

1.37

B

where

aB∗ = h¯ 2/(2mrZ1Z2e2)

1.38

may be looked at as a nuclear Bohr radius.

1.3 Some important fusion reactions

In Table 1.1 we list some fusion reactions of interest to controlled fusion research and to astrophysics. For each reaction the table gives the Q-value, the zero-energy astrophysical factor S(0) and the square root of the Gamow energy G. For the cases in which S( ) is weakly varying these data allow for relatively accurate evaluation of the cross section, using eqn 1.21, with S = S(0).

For some of the main reactions, Table 1.2 gives the measured crosssections at = 10 keV and = 100 keV, as well as the maximum value of the cross-section σmax, and the energy max at which the maximum occurs. Also shown, in parentheses, are theoretical data for the pp and

CC reactions. In the tables and in the following discussion, the reactions

are grouped according to the ﬁeld of interest.

A large and continuously updated database on fusion reactions, quot-

ing original references for all included data, has been produced and is

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