Introductory Nuclear Physics Kenneth S. Krane. Cauldrons in the Cosmos C.E.
Rolfs and W.S. Rodney. The Physics of Stars A.C. Phillips. Introductory Nuclear ...
Level III Nuclear Astrophysics
By
Dr. P.H. Regan, Room 29BC04, tel x6783 email:
[email protected]
Lecture Notes Spring Semester 2006 recommended texts, Introductory Nuclear Physics Kenneth S. Krane Cauldrons in the Cosmos C.E. Rolfs and W.S. Rodney The Physics of Stars A.C. Phillips Introductory Nuclear Physics Hodgson, Gadioli and Gadioli Erba
Contents 1 Nuclear Fusion in Stars. 1.1 1.2 1.3
2
Binding Energy Per Nucleon. . . . . . . . . . . . . . . . . . . . . Sources of Energy in Stars. . . . . . . . . . . . . . . . . . . . . . . Hydrogen Burning . . . . . . . . . . . . . . . . . . . . . . . . . .
3 5 5
1.3.1 1.3.2
The Proton-Proton Chain . . . . . . . . . . . . . . . . . . CNO Cycles . . . . . . . . . . . . . . . . . . . . . . . . . .
5 7
1.3.3 1.3.4
Additional Cycles . . . . . . . . . . . . . . . . . . . . . . . The Hot CNO Cycle and the Rapid Proton Process . . . .
11 12
4
He Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Triple-α Process . . . . . . . . . . . . . . . . . . . . . 1.4.2 Heavier Element α-Burning . . . . . . . . . . . . . . . . .
15 16 18
1.5
1.4.3 Neutron Production in (α,n) Reactions. . . . . . . . . . . . The s and r-Neutron Processes. . . . . . . . . . . . . . . . . . . .
18 18
1.6
Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2 Nuclear Reactions. 2.1 Cross-Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22 22
1.4
2.2
Stellar Reaction Rates. . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Mean Lifetimes . . . . . . . . . . . . . . . . . . . . . . . .
23 24
2.2.2 2.2.3 2.2.4
Maxwell-Boltzmann Velocity Distribution . . . . . . . . . Differential Cross-Sections. . . . . . . . . . . . . . . . . . . Inverse Nuclear Reactions . . . . . . . . . . . . . . . . . .
25 28 28
2.3
2.2.5 Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . Radiative Capture Reactions. . . . . . . . . . . . . . . . . . . . .
31 32
2.4
2.3.1 Photodisintegration . . . . . . . . . . . . . . . . . . . . . . Determination of Stellar Reaction Rates. . . . . . . . . . . . . . .
33 33
i
2.4.1 2.4.2
Neutron Induced Non-Resonant Reactions . . . . . . . . . Charged Particle Induced Non-Resonant Reactions. . . . .
33 35
Types of Nuclear Reaction . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Compound Nucleus Reactions. . . . . . . . . . . . . . . . .
39 39
2.6
2.5.2 Direct Reactions . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Angular Momentum Transfer in Stripping Reactions. . . . Resonance Reactions . . . . . . . . . . . . . . . . . . . . . . . . .
41 44 46
2.7
2.6.1 Subthreshold Resonances . . . . . . . . . . . . . . . . . . . Nuclear Fusion Revisited: The Effect of Nuclear Structure . . . .
49 51
2.7.1 2.7.2
The p-p Chain . . . . . . . . . . . . . . . . . . . . . . . . The CNO Cycles . . . . . . . . . . . . . . . . . . . . . . .
51 54
2.7.3 Helium Buring Processes. . . . . . . . . . . . . . . . . . . Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56 60
2.5
2.8
3 Experimental Nuclear Astrophysics 3.1
3.2
62
Accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Cyclotron Accelerators . . . . . . . . . . . . . . . . . . . . 3.1.2 Tandem Van de Graaff Accelerators . . . . . . . . . . . . .
63 63 64
3.1.3 Linear Accelerators . . . . . . . . . . . . . . . . . . . . . . Radioactive Ion Beam production . . . . . . . . . . . . . . . . . .
64 65
ii
List of Tables 1.1
Examples of binding energies for various nuclei. . . . . . . . . . .
3
2.1
Examples of cross-sections for different reactions at lab energies of 2 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
iii
List of Figures 1.1 1.2
Schematic of the Coulomb barrier. . . . . . . . . . . . . . . . . . . The binding energy per nucleon as a function of mass number (see Krane p67). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3 1.4
Summary of the three p-p chains. . . . . . . . . . . . . . . . . . . (a) Electron capture of 7 Be and (b) positron decay of 8 B. . . . . .
6 8
1.5 1.6
Reactions in the simple CN cycle. . . . . . . . . . . . . . . . . . . Breakout from the simple simple CN cycle into the CNO Bi-cycle.
9 10
1.7 1.8 1.9
Higher order parts of the CNO cycles. . . . . . . . . . . . . . . . . The hot or β-limited CNO cycle. . . . . . . . . . . . . . . . . . . The predicted rp-process at medium T and ρ (from Champagne
11 12
and Wiescher). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 The predicted rp-process at high T and ρ (from Champagne and
13
Wiescher.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 The NeNa and MgAl cycles (see Rolfs p375). . . . . . . . . . . . .
14 15
1.12 Schematic of the triple-α process in the formation of 12 6 C. . . . . . 1.13 Predicted paths of the s and r-processes (Krane p779). . . . . . .
17 20
2.1 2.2
Schematic of the Maxwell-Boltzmann energy distribution of a gas a temperature T (see Rolfs p143). . . . . . . . . . . . . . . . . . . Schematic of the measurement of nuclear cross-section with a de-
2.3
tector which covers only a fraction of the full solid angle. . . . . . Schematic of compound nucleus formation, showing the entrance
2.4
2
26 28
channel (1 and 2) and the exit channel (3 and 4). In the inverse process, these are reversed. . . . . . . . . . . . . . . . . . . . . . .
29
Schematic of a non-resonant neutron reaction cross-section with energy following the v1n form. . . . . . . . . . . . . . . . . . . . .
34
iv
2.5 2.6 2.7 2.8
Schematic of the Coulomb barrier experienced by a charged particle reaction with a nucleus. . . . . . . . . . . . . . . . . . . . . . .
35
Dependence on cross-section and S(E) for the reaction 3 He(α, γ)7 Be (see Rolfs p157). . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Dependence of reaction cross-section for charged particle reactions as a function of energy (see Rolfs p159). . . . . . . . . . . . . . . Reaction products from two compound nucleus reactions showing the independence of cross-sections for the exit channel on the entrance channel (see Krane p417). . . . . . . . . . . . . . . . . . .
38
40
2.9 A direct capture reaction. . . . . . . . . . . . . . . . . . . . . . . 2.10 Schematic of the geometry of direct reactions (see Krane p470,
42
Rolfs p171). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Angular distribution measurements for the 90 Zr(d,p)91 Zr reaction
43
(see Krane p422). . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Schematic of a resonant capture reaction. . . . . . . . . . . . . . . 2.13 Matching of nuclear and particle wavefunctions for certain energies
45 46
giving rise to resonant behaviour (see Krane p425). . . . . . . . . 2.14 Helium burning sequence of (α, γ) radiative capture reactions (Rolfs
48
p411). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15 Schematic of a subthreshold resonance (see Rolfs p186). . . . . . .
49 50
2.16 The d(p, γ)3 He reaction (Rolfs p357). . . . . . . . . . . . . . . . . 2.17 The 3 He(α, γ)7 Be reaction (Rolfs p361). . . . . . . . . . . . . . . . 2.18 The 7 Be(p,γ)8 B reaction (Rolfs p363). . . . . . . . . . . . . . . .
52 53 54
2.19 The CNO cycle reactions (Rolfs p367). . . . . . . . . . . . . . . . 2.20 The 14 N(p,γ)15 O reaction (Rolfs p368). . . . . . . . . . . . . . . .
55 56
2.21 The 15 N(p,γ)16 O reaction (Rolfs p370). . . . . . . . . . . . . . . . 2.22 The triple-α reaction to form 12 C (see Rolfs p388). . . . . . . . . .
57 58
2.23 Helium burning upto
20 10 Ne
(Rolfs p411). . . . . . . . . . . . . . . .
1
59
Chapter 1 Nuclear Fusion in Stars. (Krane p65-67, Rolfs p134) Stars produce their energy by nuclear reactions. The positive charges of two nuclei will try to oppose the nuclei getting too close together by forming a Coulomb Barrier between them. Classically, for two nuclei to fuse, their relative kinetic energy must be greater than this Coulomb barrier.
Coulomb Barrier Ec
Potential (V)
KE=projectile energy
Rc
Distance (r)
electro-mag repulsion
-Vo nuclear part
Figure 1.1: Schematic of the Coulomb barrier. In stars, this kinetic energy of the nuclei is generated by the gravitational contraction of the star. In its normal phase, this gravitational contraction is balanced by a gas pressure. The kinetic energy of the nuclei comes from the 2
energy excess given off in the nuclear fusion reactions which occur in the star. If two nuclei fuse together, their total mass/energy will either be greater or less than the mass/energy of the two constituent nuclei. A particular nucleus is described in terms of its number of protons (Z), neutrons (N) and the total number of nucleons (A=N+Z). The total mass energy of a nucleus is less than the sum of the masses of the constituent protons and neutrons due to the binding energy of the system. This binding energy can thus be calculated by BE = ∆E = ∆MN c2 = (MN − ZMp − NMn )c2
(1.0.1)
where MN = nuclear mass, Mp = proton mass, Mn = neutron mass and c is the speed of light in vaccuo. The quantity ∆E is thus the energy required to separate the bound nucleus into it constituent nucleons.
1.1
Binding Energy Per Nucleon.
Rolfs p135 The binding energy per nucleon gives a good guide as to whether energy will be be released in a nuclear fusion reaction. Fe has the highest EAB (=8.79 MeV/A). For two nuclei A1 and A2 , generally, if A1 + A2 56, it 56
costs energy to fuse the nuclei. Nucleus 2 1H 4 2 He 12 6 C 40 20 Ca 56 26 Fe 238 92 U
EB (MeV) 2.22 28.30 92.16 342.05 492.26 1801.70
EB A
MeV/A 1.11 7.07 7.68 8.55 8.79 7.57
Table 1.1: Examples of binding energies for various nuclei. The Q-value for reaction if given by the total mass energy before a reaction minus the total mass energy after the reaction. Symbolically, a nuclear reaction may be written as 3
Figure 1.2: The binding energy per nucleon as a function of mass number (see Krane p67).
A+B →C +D
or
B(A, C)D
where A=projectile nucleus, B=target nucleus and C and D are the residual nuclei. A and B are described as the entrance channel and C and D are known as the exit channel. The Q-value for a reaction is given by Q = (MA + MB − MC − MD )c2
(1.1.2)
The atomic masses Mx are usually found in tabular form in terms of the mass excess, ∆, where ∆ = (M − AMu )c2
(1.1.3)
where M=atomic mass, A=mass number, Mu =1 atomic mass unit (AMU)=931.5 MeV/c2 . For Q > 0 the reaction is said to be exothermic, for Q < 0 it is said to be endothermic.
4
1.2
Sources of Energy in Stars.
Fusion reactions which take place in stars and produce energy include: (a) p-p chain ; (b) CNO and other hydrogen burning cycles; (c) Helium burning (triple-α process; (d) fusion of alpha conjugate nuclei. These fusion processes are responsible for the creation of many elements with A56 it costs energy to fuse 2 nuclei together.
1.3
Hydrogen Burning
Krane p534, Rolfs p328 In main sequence stars (such as the sun), the main energy source is thought to come from converting four protons (hydrogen nuclei) into an α particle (42 He nucleus). This can be achieved in a number of ways, as discussed below.
1.3.1
The Proton-Proton Chain
Rolfs p352-6, Krane p535 The basis of the proton-proton chain is to turn four protons in a 42 He nucleus (and two electrons and two neutrinos). 4p →42 He + 2e+ + 2ν
(1.3.4)
Because the p + p di-proton (2 He) system is unbound, the first step in this reaction is thought to be the three body reaction, p + p →21 H + e+ + ν
(1.3.5)
which has a total Q-value of 1.44 MeV. Note that this reaction proceeds via the weak interaction and therefore has a very low probability (or cross-section, see late). It relies on the uncertainty principle (∆E∆t = h ¯ ) which means that occasionally, two protons can be quasi-bound in a resonant, di-proton state for long enough for the weak interaction to occur. Note that this (arguably the most important nuclear reaction in astrophysics) has never been observed experimentally. Following the formation of the deuteron, the following, proton capture reaction is very likely to occur. 5
2 1H
+11 H →32 He + γ
(1.3.6)
which has a Q-value of +5.49 MeV. Note that while the reaction d+d→4 He is possible, it is less likely due to the relatively small number of deutrons present compared to the large ‘sea’ of protons. It is estimated that in the sun there are about 1 deuteron formed for every 1018 protons. Once the 32 He nucleus is formed, it cannot capture another proton since the compound system 43 Li is unbound, ie, 3 2 He
+11 H →43 Li∗ →32 He +11 H u=neutrino e=electron
p+p -> d + e + u d+p -> 3He 86%
(1.3.7)
3He(a,g)7Be
14% 14%
3He(3He,2p)4He
7Be(e,u)7Li 7Li(p,4He)4He
Chain I
Chain II
0.02% 7Be(p,g)8B 8B->e ++ u + 8Be 8Be -> a + a Chain III
Figure 1.3: Summary of the three p-p chains. However, other possible fusion reactions involving the 32 He nucleus include fusion with a second 32 He nucleus to form a 42 He nucleus (α-particle) and two protons, 3 2 He
+32 He →42 He + 2p + γ
Q = 12.86MeV
(1.3.8)
or fusing with an alpha-particle to form 74 Be as below. 3 2 He
+42 He →73 Be
(1.3.9)
The 74 Be can then be used to form two α particles using either of the following reactions,
6
7 3 Be
7 3 Li
+ e− →73 Li + ν
(1.3.10)
+ p →42 He +42 He
(1.3.11)
7 4 Be
(1.3.12)
or
8 5B
+ p →85 B
→84 Be∗ + e+ + ν 8 4 Be
NB.
8
Be is unbound
→42 He +42 He
(1.3.13)
(1.3.14)
Note that in all cases, some of the energy released in these reactions will be taken away by the neutrino. In all three branches of the p-p chain, the total energy released is the same (26.73 MeV), but the fraction which is carried away by the neutrinos differs. The effective energies remaining in the chains (Qeff ) are 26.20 MeV (chain I), 25.66 MeV (chain II) and 19.17 MeV (chain III). Note that the weak interactions, 7 4 Be
8 5B
+ e− →73 Li + ν
(1.3.15)
→84 Be+ + e+ + ν
(1.3.16)
have nuclear structure effects which determine the average energy of the neutrinos (ie. the effective energy which is lost in the process). For example, as figure 1.4 shows, the positron decay of 85 B goes mainly to an excited (unbound) resonance state in 84 Be which consequently decays into two α–particles.
1.3.2
CNO Cycles
(Rolfs p365-371, Krane p537) Stellar interiors in main sequence stars are made up of mostly hydrogen and helium. There are however also small amounts of heavier elements such as carbon 7
(a)
0
3/27Be electron capture Q=862 keV
10.4%
89.6%
478
0
1/2gamma decay 3/27Li
(b)
0
2+ 8B
2.9
Q=16.957 MeV
2+
0
0+ 8Be
2 x 4He -0.092 MeV
Figure 1.4: (a) Electron capture of 7 Be and (b) positron decay of 8 B. (Z=6). (This is particularly relevent for the more massive stars). Most of the present stars are thought to be second or third generation states (sometimes called population 1 type stars) made up from hydrogen and helium mixed with heavier nuclei formed when old stars experienced violent supernova. In these type of stars, energy can be released by burning 4 protons into a 42 He nucleus (as in the p-p chains) via proton reactions on heavier elements starting with oxygen. The simplest and most common of these reaction-cycles uses the fusion of a proton with a 12 6 C nucleus to start the Carbon-Nitrogen(-Oxygen) or CN(O) cycle. The simple CN cycle is shown in figure 1.5. Note that the initial
12 6 C
material is not ‘used up’ in the reactions and can 8
(p,g) 13C
14N
(p,g)
e++ u
15O
13N
e++ u (p,g) (p,a) 12C
15N
Figure 1.5: Reactions in the simple CN cycle. thus be used over and over again, thus the name cycle. The CNO Bi-Cycle (Rolfs p369) The simple CN cycle discussed above assumes the reaction, 15 7 N
∗ 12 4 +11 H →16 8 O →6 C +2 He
(1.3.17)
However, about one time in every thousand reactions, instead of the excited nucleus breaking up into an α-particle and 12 C nucleus, it can decay via gamma-emission to the ground state in 16 8 O, thus breaking out of the simple CN
16 8 O
cycle (as shown in figure 1.6 Thus, this second part of the CNO cycle will contribute only a very small 9
(p,g)
(p,a)
13C
17O
14N
(p,g)
e++ u
e++ u
15O
13N
17F
e++ u
(p,g)
(p,g) (p,a)
(p,g)
15N
12C
16O
Figure 1.6: Breakout from the simple simple CN cycle into the CNO Bi-cycle. portion of the overall energy released (since the 16 O decay to its ground state only 1:1000 cycles). However, it does play a very important role in the nucleosynthesis (formation) of 16 O and 17 O. Note that the reaction 17 8 O
+ p → α +14 7 N
(1.3.18)
returns most of the catalytic material back into the orginal CN cycle. The CN cycle burns hyrdogen at a much faster rate than the p-p chain, (because of the delay associated with the p + p → d + e+ + ν reaction). The CNO cycle is however slowed down due to the larger Coulomb barrier associated with the protons fusing with heavier-Z nuclei such a carbon. The CNO cycle is thought to dominate over the p-p chain in being the major form of energy production in larger, more massive stars as they have much higher internal temperatures. This means that the thermal energy of the protons is much higher and they can tunnel through the Coulomb barrier more easily than in lower temperature, smaller stars. The probabilty of penetrating, or rather tunneling through the Coulomb barrier is given by P(pene) where,
EG P(pene) = exp − E 10
12
(1.3.19)
Where EG is the Gamov energy and gives a measurement of the height of the Couomb barrier (EG = (παZA ZB )2 2µc2 ).
1.3.3
Additional Cycles
(Rolfs p371) The 17 O(p,γ)18 9 F reaction can occasionally compete with the
17
O(p,α)14 N re-
action. As figure 1.7 shows, the unstable 18 F9 will decay by β + emission to 18 8 O10 . This 18 8 O
nucleus can undergo a (p,α) reaction to form 15 N, thus returning the catalytic 19 material back to the simple CN cycle. However, the competing 18 8 O(p,γ)9 F reaction can start off a fourth branch of the CNO cycle, which returns to the 16 CNO Bi-cycle via the 19 9 F(p,α)8 O reaction.
(p,g) 13C
(p,a) 14N
(p,g)
e++ u
e++ u
(p,a)
15N
(p,g)
(p,g)
20Ne
e++ u
17F
(p,g)
12C
18F
e++ u
15O
13N
(p,a)
17O
(p,g)
16O
18O
19F
(p,a)
(p,a)
Figure 1.7: Higher order parts of the CNO cycles. Note that in all of these CNO reactions, the initial catalytic material remains present in the cycle and can be re-used over and over again. The reaction rates for the different (p,α) and/or (p,γ) reactions in the cycle depend crucially on the stellar temperature (ie. proton energy). As discussed later these reactions can be studied in the laboratory. 11
1.3.4
The Hot CNO Cycle and the Rapid Proton Process
(Rolfs p373-374) At very high temperatures (ie. particle energies) and particles densities, such as the conditions in very massive stars, supernova or neutron stars, the hydrogen burning will occur at very high temperatures of 108 → 109 K. For these conditions, in the CNO cycle, the lifetimes of some of the beta unstable systems such as 13 7 N
and 15 8 O are long enough (and the proton density high enough) that proton capture can occur on these unstable nuclei before they beta-decay. For example 13 7 N
+ p →14 8 O+γ
(1.3.20)
This high temperature and density limit is known as the hot or β-limited CNO cycle. The rate of burning of 4 protons to an α-particle is limited by the 15 beta-decay lifetimes of the unstable, proton rich 14 8 O and 8 O nuclei. Note that at lower temperatures, the
14 7 N(p,γ)
reaction sets this limit.
(p,g) 13N
(a,p) 14O
17F
(p,g)
(p,g)
(p,g)
14N
12C
18Ne
e++ u
e++ u
(p,a) e++ u
15N
15O
(p,a)
18F
(a,g) "breakout" (p,g)
19Ne
Figure 1.8: The hot or β-limited CNO cycle.
12
For T>5×108 K, the catalytic material from the hot CNO cycle can be lost or break out and form other nuclei. This breakout and the subsequent proton capture of these breakout nuclei is thought to play an important part in the synthesis of heavier elements. The first breakout reaction from the hot CNO cycle (see fig. 1.8) is, 18 9 F
+ p →19 10 Ne + γ
(1.3.21)
Once the material has broken out of the CNO cycle, it can rapidly capture extra protons, making more beta-unstable nuclei. This is known as the rapidproton or rp-process (see fig 1.9) and is analogous to the standard rapid-neutron or r-process for neutron capture. For review articles on the rp-process, see (i) A.E.Champagne and M.Wiescher, Annual Review of Nuclear and Particle Science 42 (1992) p39-76; (ii) K¨appler, Thielemann and Wiescher, Ann. Rev. Nucl. Part. Sci. 48 (1998) p175-251; and (iii) H. Schatz et al. Physics Reports 294 (1998) p167-264).
Figure 1.9: The predicted rp-process at medium T and ρ (from Champagne and Wiescher). At each step along the rp process, the material can either take on an extra proton via proton capture or have to wait for a β-decay if for example the extra 13
Figure 1.10: The predicted rp-process at high T and ρ (from Champagne and Wiescher.) proton will form a compound nucleus which is unbound to proton emission or the beta-lifetimes is so fast that it can compete with the rate of proton capture. Note that unlike the r process, the rate of the rp-process is hindered for heavier nuclei by the increasing Coulomb barrier experienced by the proton trying to fuse with higher Z systems. Thus for proton reactions to occur for heavier nuclei, higher and higher temperatures (and densities) are required (see figs. 1.9 and 1.10). The rp-process can also lead to the population of nuclei (for example
20 10 Ne
and 24 12 Mg), from which heavier hyrdogen burning cycles, analagous to the CNO cycle can be formed. These include the NeNa and MgAl cycles (see fig 1.11). Note that due to the relatively high Coulomb barriers involved in these cycles, they are relatively unimportant as energy sources in stars. The path of the rp process (ie which nuclei are populated) depends crucially on the temperature and density conditions. Note that at very high temperatures and densities (T∼ 1.5 × 109 K, ρ ∼ 106 g/cm3 ), the main reactions upto Argon nuclei 14
(Z=18) are (α,p) reactions, where an α-particle, rather than a proton is captured. These reactions are less likely at low temperatures due to the higher Coulomb barrier experienced for alpha-particles (Z=2) compared to protons (Z=1). The Early rp–Process and Bottleneck Reactions. The breakout of the hot CNO cycle via the
15
O(α, γ)19 Ne reaction can give
rise to the early stages of the rp-process, via the reaction 20 21 11 Na(p, γ)12 Mg.
19
Ne(p, γ)20 11 Na and
Proton capture reactions for nuclei with A >20 close to the line of betastability (N=Z2a) (c) decays to ground state of 12C
7.654 379 keV
Q=7.275 MeV
0+
287 keV
γ
Q=7.367 MeV
4.439
e+e2+
γ 0 3a
8Be+a
0+ 12C
Figure 1.12: Schematic of the triple-α process in the formation of
12 6 C.
As figure 1.12 shows, the second stage of the 12 C formation requires the capture of a third alpha particle by the resonant state in 84 Be, followed by a gamma-decay to the ground state of 12 6 C, ie, 8 12 4 Be(α, γ)6 C
(1.4.23)
Hoyle suggested that in order for the above reaction to proceed at a fast enough rate to account for the natural abundance of 12 C, there had to be resonant state at energies close to the breakup threshold energy of 12 C into an α-particle and a 84 Be resonance. More specifically, this state should have spin/parity 0+ . The existence of such resonant states, greatly increases the nuclear reaction rate (see next chapter). 17
Experimentally, such as state was observed at an excitation of 7.654 MeV in or 287 keV above the 84 Be+α breakup threshold and thus 287+92=379 keV
12 6 C,
above the α + α + α breakup threshold.
1.4.2 Once
Heavier Element α-Burning
12 6 C
has been formed in the triple alpha-process, radiative capture (α, γ)
reactions can occur to form other α-conjugate nuclei. For example formed via the 12 C(α, γ)16 O.
16
O8 can be
In principle, if the stellar temperature is high enough (to tunnel through the increased Coulomb barrier), this α-capture chain can continue, allowing the 24 28 formation of 20 10 Ne, 12 Mg, 14 Si....etc. However, the rates for these reaction are highly senstitive to the nuclear structure of the individual nuclei involved, and as we shall see later, the presence of low-lying resonances plays a vital role.
1.4.3
Neutron Production in (α,n) Reactions.
In second generation stars, helium burning reactions can take place with other, non-alpha-conjugate nuclei which are present, such as 13 C and 22 Ne. These reactions produce neutrons, ie, 13 16 6 C(α, n)8 O
22 25 10 Ne(α, n)12 Mg
and
(1.4.24)
Such reactions are thought to be the source of neutrons which play a role in the formation of heavier elements (above
1.5
56
Fe) via the slow neutron or s-process.
The s and r-Neutron Processes.
Krane p776 As the binding energy per nucleon curve shows, fusion is no longer energetically favoured for compound nuclei with A>56. However, clearly, nuclear species exist above this mass, upto 238 92 U, so the question remains as to how they are formed. It is thought that the production of heavier elements comes primarily from neutron capture reactions on 56 Fe (the heaviest, stable nucleus formed by heavy nucleus fusion reactions). For example, in a large neutron flux, the following reactions could occur,
18
Note that
56,57,58
56
Fe + n →57 Fe + γ
(1.5.25)
57
Fe + n →58 Fe + γ
(1.5.26)
58
Fe + n →59 Fe + γ
(1.5.27)
Fe are all nuclei which are stable againsst radioactive decay,
however, 59 Fe is β-unstable. Thus, whether 59 Fe takes on another neutron depends on (a) the density of the neutron flux and (b) the decay half-life of 59 26 Fe (45 days). If the neutron flux is too small, the 59 Fe will decay to the stable nucleus 59 Co which will subsequently capture a neutron to form the unstable sys59 tem 60 27 Co. However, if neutron density is high enough, 26 Fe will capture another neutron before it decays to form 60 26 Fe. This sequence of neutron captures will continue until an isotope is reached whose decay half-life is short enough for it
to β-decay before another neutron can be captured. The process where the neutron flux is small and there is sufficient time for β-decay to occur before a further neutron is captured is known as the slow or s-process, while the second, faster process, where many neutrons can be captured before β-decay takes place is known as the rapid or r-process. The neutrons for the s-process are thought to come mainly from (α, n) reactions such as 13 C(α, n)16 O and 22 Ne(α, n)25 Mg which occur during the helium burning phases of second and third generation stars or in Red giants. For typical Red-giant internal temperatures of 1→ 2 × 108 K and estimated
neutron densities (N) of ∼1014 m−3 , the reaction rate per target atom (see later) is given by r ≈< σv > N
(1.5.28)
T=2×108 K corresponds to a neutron velocity of around 2×106 ms−1 (En ∼ 20 keV). For typical neutron capture cross-sections (see later) around this energy of 100 mb, this corresponds to a rate of around 2×10−9 per atom per second, or about one capture every 30 years. Thus clearly, for the r-process to occur, much more dense neutron fluxes must be present.
19
In order to generate the vast, dense neutron flux required for the r-process, the flux density needs to be increased by around ten orders of magnitude!! One suggestion is that such a density may occur during a violent supernova. Close to the magic neutron shell at N=50, 82, 126, the β-decay half-lives are so short compared even to the r−process neutron capture reaction times. This gives rise to the predicted linear increase in proton number for these magic neutron numbers (see fig. 1.13) . These are known as waiting points in the r-process.
Figure 1.13: Predicted paths of the s and r-processes (Krane p779). These nuclei subsequently β-decay to form stable elements and this accounts for the observed increase in abundance for stable nuclei with A=80, 130 and 192 (corresponding to the magic neutron numbers at N=50, 82 and 126).
20
1.6
Questions
24 64 1. Calculate the Q-value of the reactions (a) 21 H+21 H→42 He, (b) 40 20 Ca + 12 Mg→32 Ge. (∆(64 Ge)=-54.430 MeV, 1u=931.5 MeV/c2 ) 4 marks.
2. Discuss the difference between the simple CN and hot or β-limited CNO cycles. What reactions set the limit of rate of conversion of 4 protons to an α-particle in each case ? What nuclei are formed in the hot CNO cycle that are not present in the simple CN cycle 6 marks. 3. Using the Saha equation, estimate the ratio of number of 84 Be nuclei to 4 7 8 9 2 He nuclei for a temperatures of 10 , 10 and 10 K. (Assume an α-particle density of 3×105 g/cm3 ) 8 marks. 4. How can nuclei with A>56 be formed in stars 9 marks?
21
Chapter 2 Nuclear Reactions. 2.1
Cross-Section
(Krane p392, Rolfs p137) One may associate the nucleus with a geometrical area which an other nucleus will ‘see’ in a nuclear reaction. The size of this area is related to the probability of a nuclear reaction occuring. A good analogy is the probability of hitting a target increasing as the target gets larger. This ‘area’ is known as a cross-section. Classically, it is simply equal to the geometrical area of the projectile and target nuclei. If the projectile and target nuclei have radii of RP and RT respectively, then the classicaly cross-section, σ can be estimated by σ = π(RP + RT )2
(2.1.1)
For most nuclei of mass A, 1
R ≈ R0 A 3
R0 ≈ 1.3 × 10−15 m
(2.1.2)
Thus for 1 H+1 H, σ ≈ 0.2 × 10−24 cm2 and for 238 U+238 U, σ ≈ 5 × 10−24 cm2 . Since most nuclear cross-sections are of the order of 10−24 cm2 (10−28 m2 ), this unit of area is adopted and is called the barn (b). Nuclear cross-sections are governed not simply by geometric area, but also by quantum mechanical processes. In order to take into account the wave like nature of the nuclei, we can replace (RP + RT ) with the wavenumber, λdb which corresponds to the De Broglie wavelength and is given by
22
λdb =
MP + MT h ¯ 1 MT (2MP El ) 2
(2.1.3)
where El is the energy of the incident particle in the lab frame. The interaction probability between two nuclei can be reduced by the presence of Coulomb and centrifugal potential barriers, which are nuclear charge and angular momentum. These effects, together with the strength of the interaction (force) between the nuclei results in a very large energy dependence on the interaction cross-section, eg. see table 2.1. Reaction N(p,α)12 C 3 He(α,γ)7 Be p(p,e+ ν)d 15
Force strong nuclear electromagnetic weak nuclear
Cross-section 0.5 b 10−6 b ∼ 10−20 b
Table 2.1: Examples of cross-sections for different reactions at lab energies of 2 MeV.
2.2
Stellar Reaction Rates.
(Rolfs p140) Nuclear cross-sections are usually energy (and thus velocity) dependent. Thus σ = σ(v), where v is the relative velocity between the target and projectile nuclei. The reaction rate, r (per cm3 per second) can be written as r = Nx Ny vσ(v)
(2.2.4)
where Nx and Ny are the number of particles of type x and y per cm3 , v is the velocity of the projectile, x (y is a target and thus assumed to be at rest) and σ(v) is the cross-section for a single target nucleus. If we chose x as the projectiles and y as the target nuclei, then the flux of particles of type x, is given by J, where, J = Nx v
(2.2.5)
Then, the effective area (per cm) made by having Nx (/cm3 ) particles, each with cross-section σ(v) is given by F , where
23
F = σ(v)Ny
(2.2.6)
Note then that the reaction rate (per second per cm3 ), r can be written as r = JF = Nx vσ(v)Ny
(cm−3 s−1 )
(2.2.7)
The matter inside stars has a distribution of particle velocities. If the velocity probability function is given by φ(v) it can be normalised such that, Z
∞
0
φ(v)dv = 1
(2.2.8)
where φ(v)dv is the probablity that the relative velocity has a value between v and v + dv. Then we can write, < σv >=
Z
∞
σ(v)φ(v)vdv
0
(2.2.9)
where the quantity < σv > is the reaction rate per particle pair. Now for exothermic reactions (ie Q>0), the integral has limits of v = 0 → ∞. For endothermic reactions (ie Q 1 + δxy
(2.2.10)
Where the Kronecker δxy accounts for the fact that the reaction rate must be divided by two for identical particles (to avoid double counting).
2.2.1
Mean Lifetimes
(Rolfs p141-142) The decay rates of the nuclei involved is an important factor in the calculation of nuclear reaction rates in stars. The mean lifetime of nuclei of type x against destruction by interacting with nuclei of type y is given by dNx dt
!
y
= −λy (x)Nx = − 24
1 Nx τy (x)
(2.2.11)
This gives the change in abundance of nuclei of type x as a consequence of reactions with nuclei of type y. The change in the abundance of type x is related to the total reaction rate, r by !
dNx y = −(1 + δxy )r dt
(2.2.12)
Thus, τy (x) =
1 Ny < σv >
(2.2.13)
ie. the lifetime of the nuclei of type x only depends on the number of ‘destructive’ nuclei (Ny ) and the reaction rate per particle pair. By analogy, τx (y) =
2.2.2
1 Nx < σv >
(2.2.14)
Maxwell-Boltzmann Velocity Distribution
(Rolfs p142) Usually (apart from at very high temperatures), nuclei in stars are nondegenerate (ie they have different energies) and are non-relativistic. The gas of nuclei is in thermodynamic equilibrium and the particle velocities obey a MaxwellBoltzmann distribution (see figure 2.1). Thus, φ(v) = 4πv 2
m 2πkT
3 2
mv 2 exp − 2kt
!
(2.2.15)
where T =temperature, m=nuclear mass, v=particle velocity (E = 12 mv 2 ). E Therefore, φ(v) is proportional to Eexp − kT . At low energies where E > kT , φ(E) α exp − kT Note that the peak of the MB distribution lies at E = kT and thus defines the
‘temperature’ of the gas of particles. Thus if the reacting nuclear particles, x and y, in stars have MB velocity (and thus kinetic energy) distributions, then,
25
Figure 2.1: Schematic of the Maxwell-Boltzmann energy distribution of a gas a temperature T (see Rolfs p143).
φ(vx ) =
4πvx2
mx 2πkT
3
mx vx2 exp − 2kT
!
(2.2.16)
φ(vy ) =
4πvy2
my 2πkT
3
my vy2 exp − 2kT
!
(2.2.17)
2
and 2
Thus the reaction rate per particle pair is the double integral over both velocity distributions, ie. < σv >=
Z
0
∞
Z
0
∞
σ(v)φ(vx )φ(vy )vdvx dvy
(2.2.18)
where v is the relative velocity between the interacting nuclei. The individual particle velcoities vx and vy can be transformed into the centre of mass velocity V and the relative velocity of the two particles, v using the following. By conservation of linear momentum, if M = mx + my , mx vx + my vy = MV
(2.2.19)
v 2 = vx2 + vy2 − 2vx vy cosθ
(2.2.20)
Using the cosine rule,
26
mx my , mx +my
then we can re-write
σ(v)vφ(v)φ(V )dvdV
(2.2.21)
If µ is the reduced mass of ths system and µ = < σv > in terms of v and V . Thus < σv >=
Z
Z
∞
0
∞ 0
where φ(v) = 4πv
2
µ 2πkT
3
µv 2 exp − 2kT
2
!
(2.2.22)
and φ(V ) = 4πV
2
µ 2πkT
3 2
µV 2 exp − 2kT
!
(2.2.23)
Now, since σ(v) does not depend on V , we can separate the variables and write < σv >=
Z
∞
φ(V )dV
0
Z
∞
σ(v)vφ(v)dv
0
(2.2.24)
but by the normalisation condition ∞
Z
φ(V )dV = 1
0
(2.2.25)
Thus by substituting for φ(v), we can write, < σv >=
Z
∞
vσ(v)4πv 2
0
3
µv 2 exp − 2kT
!
dv
(2.2.26)
µv 2 v σ(v)exp − 2kT
!
dv
(2.2.27)
µ 2πkt
2
Re-arranging µ < σv >= 4π 2πkT
3 Z 2
∞
0
3
Writing this in terms of the centre of mass energy, E = 12 µv 2 , then < σv >=
8 πµ
!1
2
1 3
(kT ) 2
Z
0
∞
E σ(E)Eexp − kT
27
dE
(2.2.28)
2.2.3
Differential Cross-Sections.
(Krane p393) Experimentally, when one measures a nuclear reaction cross-section, one actually detects the outgoing reaction products in a detector, which will (usually) only cover a limited amount of the full possible solid angle of 4π steradians (see fig 2.2) .
detector covering solid angle Ω
target
beam, current=I θ,φ
Figure 2.2: Schematic of the measurement of nuclear cross-section with a detector which covers only a fraction of the full solid angle. The fraction of the solid angle which the detector covers (and thus to which Ω the experiment is sensitive) is 4π where Ω is the solid angle occupied by the detector in steradians. In general, the particles from a nuclear reaction are not emitted isotropically,
ie, their intensity distribution has an angular dependence1 . dσ The differential cross-section, dΩ is often measured rather than the straight cross-section, σ. Often, in the literature, the differential cross-section is measured as a function of lab angle, with the further restriction that the outgoing particle has a certain energy (corresponding a to fixed excitation energy in the residual 2 nucleus). This quantity is known as the doubly differential cross-section, dEd bσdΩ .
2.2.4
Inverse Nuclear Reactions
(Rolfs p144-6) 1 This angular dependence can be used to determine the spin and parity of the nuclear states involved in the reaction
28
For low temperatures, nuclear reactions with positive Q-values, (those which release energy) are preferred, ie 1+2→3+4
Q>0
(2.2.29)
However, as the temperare increases, so does the number of nuclei with kinetic energies greater than the Q-value. For these particles, the inverse process can occur (see fig 2.3, ie 3+4→1+2
Q C -> 3+4
8Be(4He,gamma)12C
Figure 2.3: Schematic of compound nucleus formation, showing the entrance channel (1 and 2) and the exit channel (3 and 4). In the inverse process, these are reversed. In order to fully understand and model the evolution and burning of stars and the production of heavier nuclear species, the rates for these inverse processes must be known. 29
Most of these inverse reaction processes form an excited state in some, intermediate, compound system, which subsequently breaks up into the exit channel components. If the reaction 1 + 2 → 3 + 4 proceeds through an excited compound state, the reaction cross-section is given by,
σ12 = πλ212
2J + 1 (1 + δ12 ) × | < 3 + 4|MII |C >< C|MI |1 + 2 > |2 (2J1 + 1)(2J2 + 1) (2.2.31)
The expression can be thought of as the product of the following terms, 1. πλ212 : λ12 is the wavenumber and reflects the geometrical and quantum mechanical character of the cross-section. 2.
2J+1 (2J1 +1)(2J2 +1)
: where J is the angular momentum of the compound nuclear state and J1 and J2 are spins of the two nuclei in the entrance channel. This is a statistical factor which reflects the fact that the probability of a process occuring increases as the number of possible nuclear magnetic substates increases. For a state of spin J there are 2J + 1 final magnetic substates (ml = −J, −J + 1, ....., J − 1, J). The entrance channel can be in
any one of (2J1 + 1)(2J2 + 1) magnetic substates and thus the probability in 1 . Since there are (2J + 1) final states, being in any one pair is (2J1 +1)(2J 2 +1) this leads to the statistical factor in the equation. This term is sometimes given the symbol ω. 3. (1 + δ12 ) : Takes into account that the cross-section will increase by a factor of 2 if the two particles in the entrance channel are indentical (ie. the same species). The cross-section is doubled because the ‘beam’ and ‘target’ nuclei are indistinguishable. 4. | < 3 + 4|MII |C >< C|MI |1 + 2 > |2 : is the quantum mechanical de-
scription of the interaction (force) involved in the reaction process.....ie the physics! < C|MI |1 + 2 > is the matrix element for the transition from the entrance channel to the compound system C. < 3+4|MII |C > is the matrix element for the transition from the compound system to the exit channel. The interaction (force) responsible for the first transition is given by MI 30
while the second is given by MII . (Note that the interactions involved in the entrance and exit channels may be different). Due to the formation of an intermediate, compound state, these reactions are known as two state or resonant processes. By analogy, the cross-section for the inverse process is given by,
σ34 = πλ234
2J + 1 (1 + δ34 ) × | < 1 + 2|MI |C >< C|MII |3 + 4 > |2 (2J3 + 1)(2J4 + 1) (2.2.32)
Note that the matrix elements involved are identical (except for the ordering). This is property of time-reversal invariance. Reactions involving the strong nuclear and electromagnetic (but not the weak nuclear) are time-reversal invariant, thus the two matrix elements are identical and thus if, h ¯2 λik = (2µik Eik )
(2.2.33)
where µik =reduced mass for particles i and j and Eik is the centre of mass energy, then σ12 m3 m4 E34 (2J3 + 1)(2J4 + 1)(1 + δ12 ) = σ34 m1 m2 E12 (2J1 + 1)(2J2 + 1)(1 + δ34 )
(2.2.34)
This is a very useful result since it is often easier experimentally to measure a nuclear reaction cross-section in the reverse direction. By using the above equation, one can directly obtain the reaction cross-section of interest.
2.2.5
Reaction Rates
(Rolfs p 146) Recalling the result that the reaction rate per particle pair in the entrance channel can be given by
< σv >12 =
8 πµ12
!1 2
1 3
(kT ) 2
Z
∞
0
E12 dE12 σ12 (E)E12 exp − kT
and similarly for the exit channel, 31
(2.2.35)
8 πµ34
< σv >34 =
!1
1
2
3
(kT ) 2
Z
0
∞
E34 dE34 σ34 (E)E34 exp − kT
(2.2.36)
By conservation of energy, E34 = E12 + Q
(2.2.37)
and thus we can write the ratio of reaction rates per particle pair as
< σv >34 (2J1 + 1)(2J2 + 1)(1 + δ34 ) = < σv >12 (2J4 + 1)(2J3 + 1)(1 + δ12 )
µ12 µ34
!3
2
exp −
Q kT
(2.2.38)
Q This ratio is dominated by the exponential term, exp − kT . This ratio is thus a rather small number except at very high temperatures. (eg. for Q=8 MeV, 34 ≈10−4 ). T=1010 K, 12 The total reaction rate is the difference between the entrance and exit channel
rates, ie. r12 =‘production’ rate and r34 =‘decay’ rate. r = r12 − r34 =
N1 N2 N3 N4 < σv >12 − < σv >34 1 + δ12 1 + δ34
(2.2.39)
By substitution,
< σv >12 N3 N4 (2J1 + 1)(2J2 + 1) r= N1 N2 − 1 + δ12 (2J3 + 1)(2J4 + 1)
µ12 µ34
!3 2
Q exp − (2.2.40) kT
For eqilibrium conditions, ie rate of production=rate of decay r = 0
2.3
Radiative Capture Reactions.
(Rolfs p147) If one the of ‘particles’ in the exit channel is electromagnetic radiation (ie a γ-ray), the reaction is known as a radiative capture reaction. For example 12
∗ 13 C + p →13 N+γ 7 N →
Q = 1.95MeV
(2.3.41)
These types of reactions are very common in stellar reactions and in general (apart from for very neutron deficient nuclei and A>56) they have positive 32
Q-values and can thus contribute to the energy production of the star. These reactions also play a vital role in the production of heavy elements (in, for example, the rp process).
2.3.1
Photodisintegration
(Rolfs p147) At very high stellar temperatures, the thermal photons in the hot, high density core of the star have high enough energies to give rise to the inverse reaction to radiative capture, known as photodisintegration. For example, 13
∗ 12 N + γ →13 C+p 7 N →
Q = −1.95MeV
(2.3.42)
Photodisintegration is thought to play an important role in the core break up preceeding supernova for very heavy stars.
2.4
Determination of Stellar Reaction Rates.
(Rolfs p150) As shown previously, the reaction rate per particle pair is given by, < σv >=
8 πµ
!1
1
2
3
(kT ) 2
Z
0
∞
E σ(E)Eexp − kT
dE
(2.4.43)
Note that this expression gives the reaction rate at a fixed temperature value, T , and this will change as the star evolves. In order to solve this expression analytically, one must have a knowledge of the variation of cross-section, σ as a function of energy (ie. σ(E)). This depends very heavily on the type of reaction involved. Nuclear reactions in stars can be broadly divided into two groups, (a) resonant and (b) non-resonant.
2.4.1
Neutron Induced Non-Resonant Reactions
(Rolfs p150-153) Neutron induced reactions occur predominantly in the high density stellar core. Note that the decay lifetime of the free neutron is too short for such reactions to occur in the relatively low-density proto-star. Common examples
33
of neutron producing reactions are reactions can also occur.
13
C(α,n)16 O and
22
Ne(α,n)25 Mg. The inverse
For neutrons with thermal energies (E ∼ kT ), and relative angular momentum l=0 (known as an ‘S’-wave neutron), the neutron reaction cross-section,
cross-section σn
σn (En ) is proportional to the inverse of the neutron velocity ( v1n ) (see fig. 2.4).
σn α 1/Vn
neutron velocity (Vn) Figure 2.4: Schematic of a non-resonant neutron reaction cross-section with energy following the v1n form. A reaction whose reaction cross-section follows this
1 vn
relation is described
as a non-resonant reaction. This is the general term for a reaction where the cross-section varies slowly with energy. If σn α v1n , then < σn vn > has constant value. Therefore, in principle only a single measurement of the cross-section in the thermal energy (E ∼ 30 keV, T ∼ 3 × 108 K) range is required to obtain the reaction rate. However, in practice more energies are measured in order to ensure the distribution follows a v1n pattern (and to make sure there are no low-lying resonances which can
dramatically increase the reaction rate). For higher energy neutrons, larger partial waves (l >0) can contribute to the reaction cross-section, thereby modifying the simple v1n law. In these cases, σv is 1 written as an expansion in terms of velocity (or E 2 ). Then
34
σv = S(0) +
dS(0) 1
d(E 2 )
1
E2 +
d2 S(0) 1
d(E 2 )2
E + ....
(2.4.44) 1
where the parameter S(0) and its differentials with respect to E 2 are experimentally determined quantities.
2.4.2
Charged Particle Induced Non-Resonant Reactions.
(Rolfs p153-162) Positively charged nuclei repel each other due to their mutual electromagnetic interaction. Typically, energies corresponding to greater than 107K are required for fusion to occur in stars.
Coulomb Barrier Ec
Potential (V)
KE=projectile energy
Rc
Distance (r)
electro-mag repulsion
-Vo nuclear part
Figure 2.5: Schematic of the Coulomb barrier experienced by a charged particle reaction with a nucleus. If all the nuclei in a star fused when the ”Coulomb Barrier” was overcome, the star would burn instantaneously, in a ‘flash’. We know however that fusion occurs gradually and the rate increases with temperature (energy). These effects can be accounted for by understanding that it is possible for charged particles to quantum mechanically tunnel throught the Coulomb barrier and fuse. The probability for tunneling through the Coulomb barrier can be approximated by P where, 35
EG P = exp(−2πη) = exp − E
12
(2.4.45)
where η is known as the Sommerfeld parameter and EG is known as the Gamow energy. For E in keV and µ=the reduced mass in amu, 1
µ 2 2πη = 31.29Z1 Z2 (2.4.46) E The effect of the energy dependence on the probablity of tunneling through
the Coulomb barrier means that σ(E) α exp (−2πη)
(2.4.47)
The geometrical part of the cross-section depends on the De Broglie wavelength, ie 1 (2.4.48) E Thus by combing these relations, we can see that for charged particle induced σ(E) α πλ2db α
reactions, 1 exp(−2πη)S(E) (2.4.49) E where S(E) is known as the astrophysical ‘S’-factor, as defined by this equation. σ(E) =
As shown in figure 2.6, for non-resonant reactions, S(E) is varies smoothly with energy. By susbstituting the expression for σ(E) into the general equation for the reaction rate per particle pair, we have,
< σv >=
8 πµ
!1 2
1 (kT )
3 2
Z
0
∞
S(E)exp −
E EG − kT E
12
(2.4.50)
dE
where EG = b2 =the Gamow energy=0.978(Z1 Z2 )2 µ and µ in amu.
1
E is small at Note that exp EEG 2 is small at low energies, while exp − kT higher energies. As figure 2.7 shows, the product of these two terms gives a value for the integral which peaks at an energy E0 where,
36
Figure 2.6: Dependence on cross-section and S(E) for the reaction 3 He(α, γ)7 Be (see Rolfs p157).
E0 =
bkT 2
!2
3
1
= 1.22(Z12Z22 µT 2 ) 3 keV
(2.4.51)
where T is the temperature in 106 K and µ is the reduced mass in amu. E0 is known as the effective burning energy. The width of the peak about this effective burning energy (∆) can be obtained using the approximation to a Gaussian function given by, b E − 1 exp − kT E2
!
≈ Imax exp −
where Imax is proportional to < σv > and Imax
3E0 = exp − kT
37
E − E0 ∆ 2
!2
(2.4.52)
(2.4.53)
Figure 2.7: Dependence of reaction cross-section for charged particle reactions as a function of energy (see Rolfs p159). Differentiating equation 2.4.52 twice, one can obtain an expression for the width ∆, where 4
∆=
3
1 2
1
(E0 kT ) 2 =
4 1 2
3 2
1 1 3
5
(2.4.54)
EG6 (kT ) 6
Recalling
< σv >=
1 πµ
!1
1
2
(kT )
3 2
Z
0
∞
E EG S(E)exp − − kT E
12
(2.4.55)
dE
an approximation to the integral is
Z
0
∞
EG E − exp − kT E
21
dE
3E0 = Imax ∆ = 1 (E0 kT ) exp − kT 32 4
38
1 2
(2.4.56)
Nuclear reactions occur mainly in the energy region straddled by the energy window defined by ∆ (2.4.57) 2 In general, this energy is too low to measure the reaction cross-section directly in the laboratory (although the corresponding temperatures can be extremely high!) One typically measures S(E) over a range of available lab energies and E = E0 ±
then extrapolates down to the region around E0 . Using the approximation given in equation 2.4.56, the non-resonant reaction rate per particle pair can be estimated by, < σv >=
2 µ
!1
∆
2
− 3 S(E0 )exp
(kT ) 2
In terms of dimensionless parameter τ =
3E0 , kT
3E0 kT
(2.4.58)
the reaction rate per particle
pair can be calculated approximately by, 1 τ 2 e−τ S(E) cm3 s−1 µZ1 Z2 where S(E) is in units of keVb (b-barns) and µ is in amu. < σv >= 7.2 × 10−19
2.5
(2.4.59)
Types of Nuclear Reaction
Krane p416 Broadly speaking, nuclear reactions can be thought of as three types: (a) compound nucleus reactions, such as fusion-evaporation; (b) direct reactions, such as pick-up and stripping reactions and direct capture; and (c) resonance reactions. These will now be discussed in turn.
2.5.1
Compound Nucleus Reactions.
If a nucleus hits an another (target) nucleus with a small impact parameter, ie close to a head on collision, it will have a high probability of entering the target nucleus and scattering off one of the target nucleons with the loss of some energy. The beam nucleus can then scatter off other nucleons in the target in quick succession, dispersing its energy among the target nucleons until all of its kinetic energy has been shared between the nucleons of the fused, compound system. 39
Figure 2.8: Reaction products from two compound nucleus reactions showing the independence of cross-sections for the exit channel on the entrance channel (see Krane p417).
40
The energy is dissipated among all the nucleons in a statistical distribution with a small probability that any one will have enough energy to escape from the compound system. This is analogous to molecules evaporating from the surface of a hot liquid. The intermediate state, after fusion but before evaporation, is known as a compound nucleus. Symbolically, a + b → c∗ → d + e
(2.5.60)
Note that the same compound nucleus can be formed in a number of different ways, eg. 12 6 C
∗ +42 He →16 8 O
(2.5.61)
and 16 ∗ p +15 7 N →8 O
(2.5.62)
As figure 2.8 illustrates, in a compound reaction, the relative probability of decay into any specific set of final products is independent of the constituents used to form the compound system. That is, the compound nucleus has no ‘memory’ of how it was formed.
2.5.2
Direct Reactions
(Krane p419-20, Rolfs p168-169) In a direct reaction, the incident nucleus interacts with the surface of the target nucleus. Direct processes only involve upto a few valence nucleons and typically occur at energies higher than compound reactions (since as the beam energy increases, the De-Broglie wavelength decreases and the impact parameter increases). A good example of a direct reaction relevent to astrophysical measurements is the process of direct capture (see figure 2.9). In a direct capture (DC) reaction, the incoming particle with energy E (which can be thought of quantum mechanically as a plane wave) goes directly to a standing wave, corresponding to the population of a specific nuclear orbital. The
41
beam (plane wave)
compound system in final state
1 0 0 1 1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
gamma-ray emitted
target
1 0 0 1 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
Energy of projectile Q-value gamma-spectrum is CONTINUOUS with increasing projectile energy
γ γ gamma γ emission to populate final state(s)
E3 E2 E1 discrete gammas decay from excited states
Figure 2.9: A direct capture reaction. energy difference between the incoming particle energy, the Q-value of the reaction and the final excitation energy of the state to which the particle is captured to (Ei ) is emitted in the form of a gamma-ray of energy Eγ where
Eγ = KEbeam − KErecoil + Q − Ei = Ebeam −
AB Ebeam + Q − Ei (2.5.63) AB + AT
Note then that the energy of the emitted photon will increase linearly with the beam energy. This process is analagous to the continuous energies of X-rays emitted in the atomic Bremsstrahlung process. Direct processes are so called because they involved a direct transition from the initial to the final state, without passing through some intermediate state (such as a compound nucleus). Other examples of direct processes include stripping and pickup reactions such a the (p,d) reaction.
42
0 projectile 1 0 1 0 1 b
11111 00000 00 11 00000 11111 00 11 00000 11111 00 11 00000 11111 00 11 00000 11111 00000 11111 00000 11111
target
0=s 1=p 2=d 3=f 4=g
11 00 00 11
0 1 2 3 4
Pb
θ Pa P R
Figure 2.10: Schematic of the geometry of direct reactions (see Krane p470, Rolfs p171).
43
2.5.3
Angular Momentum Transfer in Stripping Reactions.
(Krane p420, Rolfs p171) As figure 2.10 shows, for an incoming particle of linear momentum pA and an outgoing particle of linear momentum pB , by conservation of momentum, the residual nucleus must recoil with momentum p where pA − pB = p
(2.5.64)
In a direct process, to first order, it is assumed that the reaction takes place on the nuclear surface, at a radius R. Thus the transferred nucleon will go into an orbital with angular momentum l, where l = Rp
(2.5.65)
Using the cosine rule
p2 = p2A + p2B − 2pA pB cosθ = (pA − pB )2 + 2pA pB (1 − cosθ)
(2.5.66)
If we know the energy and direction of the incident and outgoing particles we p2 can calculate pA and pB , (since E = 2m ) and thus obtain a value for p to infer a 1
value for l. (R = R0 A 3 for nuclei where R0 ≈1.3 fm). Thus, substituting and re-arranging the above equations, we can show that using this semi-classical argument,
l=
2c2 pA pB 2sin2 θ2 ¯ 2 c2 h R2
1 2
(2.5.67)
Since l is quantised, the outgoing particles should generally lie at larger angles for higher values of l (ie transfer of higher angular momentum) (see figure 2.11). Note that in this simple, semi-classical approach, the spins of the initial and final particles have been ignored as well as interference effects. Both of these will effect the fine structure of the angular distributions. Using the angular distribution to deduce the angular momentum transfer in a reaction can also be used to infer the parity change involved using the parity selection rule,
44
Figure 2.11: Angular distribution measurements for the (see Krane p422).
(−1)l = (−1)π
90
Zr(d,p)91 Zr reaction
(2.5.68)
Note that when a reaction has two components (eg. a direct and a compound reaction such as the (d,p) type reactions) the component can be separated using the angular distributions of the emitted particles. Due to the thermalisation time and associated longer time scale, the emitted particles in a compound nucleus reaction are emitted with a virtually isotropic distribution.
45
2.6
Resonance Reactions
(see Rolfs p173, Krane p428) There is a type of reaction in which a excited state in the compound system is formed in the entrance channel. This state will then decay by particle or γemission. Such a process only occurs when the energy of the entrance channel (Q + ER ) matches the energy of the state in the compound system. ie when ER = Ex − Q beam and target nuclei 1 0 0 1
1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
compound system in resonant state
projectile energy Er
(2.6.69)
gamma-emission to bound states
unbound resonance state
break-up energy (Q-value) gamma e+e- decays to bound states
Figure 2.12: Schematic of a resonant capture reaction. In the formation of compound nuclei, the system is created at an excitation energy where the density of states is very high. Although there are discrete quantum states, there are so many that they lie so close to each other that they form an effective structureless continuum. Each nuclear state which is unstable against decay has a certain energy width, Γ which is related to the decay mean life of the state τ by the uncertainty principle such that Γτ = h ¯ 46
(2.6.70)
The states populated in compound nucleus reactions tend to have energy spacings much less than the intrinsic width of the states, giving rise to the energy continuum. By contrast, the bound states populated by direct reactions have longer meanlifes (since they are bound against particle emission and thus this can not add to the total width of the state Γtot = Γγ + Γα + ....). For a state lifetime of 10−12 seconds, the corresponding energy width is approximately 10−3 eV (1 meV) which is much less than the typical level energy spacing in this region. Between these two extremes lies the resonance region, which corresponds to discrete states in the compound nucleus energy region. These types of states are very important in nuclear astrophysics as they tend to have very large reaction cross-sections (compared to the direct capture process) and are responsible for the formation of many elements. If the wavefunctions of the particle to be captured and the nuclear potential are well matched (ie at certain energies) then the probability for penetration of the potential becomes easier (see fig 2.13). Only at certain energies do the phases and the wavefunctions match, these correspond to the resonance regions. The width and shape of the resonance state is given by the Breit-Wigner formula, such that the nuclear reaction cross-section,
σ(E) = πλ2db
(2J + 1) Γa Γb (1 + δ12 ) 2 (2J1 + 1)(2J2 + 1) (E − ER )2 + Γ
(2.6.71)
2
where ER =Ex − Q, Γ=width of resonant state and E is the beam energy (for a fixed target). Equation 2.6.71 is the Breit-Wigner formula for a single level resonance. It is only valid for isolated resonances where the separation of nuclear levels is large compared to their widths. Angular momentum and parity selection rules (as well as energy) determine whether or not a resonant state is formed in a given reaction channel. By conservation of angular momentum, J1 + J2 + l = J 47
(2.6.72)
Figure 2.13: Matching of nuclear and particle wavefunctions for certain energies giving rise to resonant behaviour (see Krane p425). where J1 and J2 are the angular momentum (spin) of particles in the entrance channel, J is spin of the resonant state and l is the relative spin between the particles in the entrance channel. For spinless particles (even-even nuclei in their ground states) J1 = J2 = 0 and thus l = J. Parity conservation gives, (−1)l π(J1 )π(J2 ) = π(J)
(2.6.73)
Thus for spinless particles, (−1)l = π(J). In the case of spinless particles, the parity of the resonant state is determined by the orbital angular momentum of the channel. Thus is (−1)l is not equal to π(J), the state cannot be formed since the resonant state would have unnatural 48
parity. A good example of this selection rule is in the 16 O(α, γ)20 10 Ne reaction. The effect of resonances in increasing the nuclear reaction rate at stellar temperatures are highlighted in the (α, γ) capture reactions (see figure 2.14).
Figure 2.14: Helium burning sequence of (α, γ) radiative capture reactions (Rolfs p411). The strength of a resonance is defined by the quantity ωγ where, ωγ =
Γa Γb (2J + 1) (1 + δ12 ) (2J1 + 1)(2J2 + 1) Γ
(2.6.74)
where Γa and Γb are the partial widths.
2.6.1
Subthreshold Resonances
(Rolfs p185) If there is an excited state at energy Er in the compound system where Er is less than the Q-value for the decay into the reaction channel (ie. less than the
49
break up threshold), then the resonant state energy ER = Er − Q is negative (see fig 2.15). Such as state is called a subthreshold resonance.
Figure 2.15: Schematic of a subthreshold resonance (see Rolfs p186). However, because all excited nuclear states can decay (and thus have a decay lifetime), this excited, resonant state has a finite width Γ. In the case of reactions which decay by γ emission only, the state lifetimes are relatively long (∼ 10−12 s) and thus the widths are rather narrow. However, if particle decay (such as proton or α emission) can compete with γ-decay (which occurs at the particle decay threshold energy) the associated width of the resonant state can become much larger, of the order of 1 MeV. Thus the high-energy tail of a subthreshold resonance may extend to the energy regime above the particle decay threshold. Thus it is possible to have large cross-section for fusion reactions at energies below the threshold due to the tail of a subthreshold resonance. A good example of this effect is the 14 N(p,γ)15 O reaction which proceeds strongly via a –22 keV subthreshold resonance. 50
2.7
Nuclear Fusion Revisited: The Effect of Nuclear Structure
As we have seen, the presence of low-lying resonant states at excitation energies comparable to stellar temperatures can have a large effect on the reaction crosssection. These are important factors in the calculation of relative production ratios of the elements in nucleosynthesis. In this section, we will look more closely at specific nuclear decay schemes and see their effect on the production/decay cross-sections for different types of reaction.
2.7.1
The p-p Chain
(Rolfs p357-68) The d(p, γ)3He Reaction This has the lowest Coulomb barrier of all fusion reactions in the p-p chain (except the weak p+p→ d) and measurements of the cross-section have been allowed in the laboratory at energies as low as 16 keV (see fig 2.16 and 2.17.) (Note that the Gamow window for T=1.5×107 K for this reaction is E0 =6.3±3.3 keV). The 3 He(α, γ)7Be Reaction This reaction (which is important for the modelling of the solar neutrino problem from the 74 Be→73 Li+β + + ν decay) also appears to go only by direct capture (see figure ??). Note however, that unlike the d(p, γ)3 He reaction, the direct capture has two components, one to the ground state and a second to the excited state at 429 − keV which has spin/parity 12 . Note that the measured S(E) factor as measured experimentally in the range between 160 and 2500 keV is again in good agreement with the direct capture model. The 7 Be(p,γ)8 B Reaction The p-p chain can terminate with 8 B beta-decaying to an excited resonance in the unbound system 84 Be, which subsequently breaks up into two α-particles (see figure 2.18).
51
Figure 2.16: The d(p, γ)3 He reaction (Rolfs p357).
52
Figure 2.17: The 3 He(α, γ)7 Be reaction (Rolfs p361).
53
Figure 2.18: The 7 Be(p,γ)8 B reaction (Rolfs p363). Experimental measurements of the gamma-rays associated with direct capture are very difficult for this reaction due to background from the 478 keV gammaray from decay in the 73 Li daughter (note 73 Be is unstable against β decay). As figure 2.18 shows, it is clear however, that the resonance at ER =640 keV in the 7 Be+p system causes a sharp increase in the cross-section around temperatures corresponding to this energy.
2.7.2
The CNO Cycles
(Rolfs p366-69) The
12
C(p,γ)13 N and
13
C(p,γ)14 N Reactions
The 12 C(p,γ)13 N and 13 C(p,γ)14 N reactions in the CN cycle have been studied in the laboratory at energies comparable to thermal energies in heavy stars. The reactions rates appear to be dominated by the low-energy tails of very wide resonances at ER =457 keV and 555 keV in the 12 C+p and 13 C+p systems 54
Figure 2.19: The CNO cycle reactions (Rolfs p367). respetively. The
14
N(p,γ)15 O Reaction.
The experimental data on the
14
N(p,γ)15 O reactions show the effect of the reso-
nance at ER =278 keV on the cross-section (see figure 2.20). Note however that the experimental data on this reaction does not currently extend below 100 keV, and thus the effect of the subthreshold resonance at ER =−22 keV, corresponding to the bound state at 7271 keV in the 15 O system is not yet fully accounted for in the rate predictions for this reaction. The
15
N(p,γ)16 O Reaction.
This is the reaction which allows break-out from the simple CN cycles into the CNO ‘Bi-Cycle’ via the formation of 16 O in its ground state. The ratio of measured S(E) factors for the 15 N(p,γ)16 O reaction compared to the 15 N(p,α)12 C reaction is about 1 to 1000, ie. one 16 O nucleus is created for every 1000 12 C. As figure 2.21 shows, the 15 N(p,γ)16 O cross-section is dominated at low energies by two resonance capture reactions to 1− resonant states in the 15
N+p system (ie excited, unbound states in
55
16
O).
Figure 2.20: The
2.7.3
14
N(p,γ)15 O reaction (Rolfs p368).
Helium Buring Processes.
The Triple-α Process→12 C The triple-α process requires 2 resonances for unbound states. The α + α system is unbound to breakup into the 2 α particles by 92 keV. This is equivalent
8 4 Be
to there being a (ground state) at 92 keV in the α + α system. The width of this 0+ , resonance (which is effectively the ground state of the unbound 84 Be is 6.8 eV. The 8 Be+α reaction can then form 12 C in a resonant state corresponding to an excitation energy of 7.654 MeV above the 12 C ground state. This resonance lies 287 keV above the breakup threshold into the 84 Be+α system and 287+92=379 keV above the threshold for breakup back into three α-particles (see figure 2.22). The 0+ resonance will decay mainly by α-breakup but a small portion of its partial width will decay via either (a) gamma-emission to the 2+ excited state
56
Figure 2.21: The at 4439 keV in
12
15
N(p,γ)16 O reaction (Rolfs p370).
C or (b) by e+ e− decay directly to the 0+ ground state. (Note
that 0+ → 0+ decays can not proceed by single gamma-ray emission). The
12
C(α, γ)16 O and Heavier He Reactions
(Rolfs p395) The 12 C(α, γ)16 O reaction is one of the most important astrophysical reactions as its rate determines the mass distribution of the heavier elements. There is much uncertainty about the cross-section at energies corresponding to helium burning temperatures (∼300 keV), however, it is thought that the 12 C(α, γ)16 O reaction proceeds mainly through the low energy tails of subthreshold resonances at ER =– 45 keV and –245 keV respectively (see figure 2.23). These resonances correspond to excited states at Ex =7.12 and 6.92 MeV in the 16 O nucleus. The ratio of the contributions from fusion into these two resonant states can be measured by observing the energy (and angular distributions) of the emitted gamma-rays from the decays of the resonances. The state at 7.12 keV has a 57
Figure 2.22: The triple-α reaction to form
12
C (see Rolfs p388).
spin/parity of 1− and decays to the ground state by a 7.12 MeV, E1 (‘electric dipole’) gamma decay, while the state at 6.92 MeV is of a 2+ nature and decays to the ground state by an E2, or electric quadrupole transition. The gamma-ray angular distributions for E1 and E2 decays are quite different and can be readily separated. Current data on the helium burning reactions extends down to approximately 900 keV and a large degree of extrapolation is required to determine a value for the S(E) factor at realistic helium burning energies.
58
Figure 2.23: Helium burning upto
59
20 10 Ne
(Rolfs p411).
2.8
Questions
1. (a) Calculate the classical cross-section of (i) a proton on 12 C, (ii) 12 C on 12 C. (b) Estimate the De Broglie wavelength of a proton fired at a 12 C target at an energy of (i) 100 keV and (ii) 1 MeV 4 marks. 2. Calculate the ratio of reaction rates per particle pair for the reaction 7 Li(p,α)α at T=109 and 1010 K. The ground state spins of the proton and 7 Li nuclei + − are 12 and 32 respectively. (Hint you‘ll need to calculate the Q-value for the reaction and use the Boltzmann constant) 6 marks. 3. Derive the relationship between the reaction rate per particle pair, < σv > and the total reaction rate per unit volume per unit time 4 marks. 4. Calculate the probability for α-particle tunneling and fusing with a nucleus at a centre of mass energy of 0.1, 1 and 10 MeV 5 marks.
16
O
5. Calculate the effective burning energy of two alpha-particles in a stellar gas of temperature (a) 107 , (b) 5 ×107 and 108 K 5 marks. 6. Calculate the energy of the emitted photon in the direct capture reaction − 3 He(α, γ)7 Be to the 12 excited state at 429 keV in 7 Be. Assume the 3 He is at rest and the α-particle has an energy of 300 keV 4 marks. 7. (a) What is the energy width of a state which gamma-decays with a lifetime of 5×10−12 secs. (b) If this state can now α decay with a partial width of 10 eV, what is the new decay lifetime of the state. (c) The width of the ground state resonance in 84 Be is 6.8 eV, what is the lifetime of 84 Be, why does this resonance not have a γ-decay branch 4 marks. 8. Explain the terms resonance and subthreshold resonance. Give an example of an important subthreshold resonance in a nuclear astrophysics reaction 4 marks. 9. Discuss the effects of nuclear structure in the reactions involved in helium burning from the formation of 12 C upto 20 10 Ne. Comment on the effect of subthreshold and un-natural parity resonant states 6 marks.
60
10. Discuss the design and operation of the following types of ion accelerator (a) tandem Van de Graaff (b) cyclotron (c) linear accelerator. Discuss the possible advantages/disadvantages of using a particular machine (eg. acceleration of noble gas beams, ease in changing beam energy, cost) 8 marks.
61
Chapter 3 Experimental Nuclear Astrophysics In order to fuse nuclei together, they must first have enough relative kinetic energy to overcome the Coulomb repulsion between beam and target. An accelerator is a device for giving energy to a beam of particles and directing them toward a specific target. Depending on the range of energies which can be produced, accelerators are classified as low energy (10-100 MeV), medium energy (100-1000 MeV), and high energy (>1000 MeV) instruments. Heavy ion accelerators are made up of several components. These include the ion source for ionizing the beam of particles that is to be accelerated. Depending on the application and chemistry constraints, the ions may be positive or negative. In an Electron Cyclotron Resonance (ECR) source or positive ion injector, a gas of atoms is ionized by subjecting it to an electric discharge and then extracting the positively charged ions by acceleration toward a negative electrode at a potential of the order of 10 kV. Positive ions are used in cyclotrons and linear accelerators. In Tandem Van de Graaff accelerators, the negative ions are produced in an ion source by passing a beam of positive ions through a neutral gas which has relatively loosely bound electrons (usually Cesium). The positive ions capture these electrons to become negative ions. These negative ions are accelerated towards a positive central voltage and then enter a thin carbon foil or gas stripper where they lose their electrons and become positive ions.
62
3.1
Accelerators
There are different types of accelerator, the design of which depends on the energy and intensity of the beams needed for doing the experiments. Cyclotrons, linear accelerators and tandems are the main accelerators used in the study of nuclear structures and nuclear astrophysics.
3.1.1
Cyclotron Accelerators
The cyclotron accelerator is a circular device in which a pulse of particles makes many cycles through the device, increasing its energy in each turn until the beam comes out from the accelerator. The bending of the beam into a circular orbit is caused by the Lorentz force (qvB), which provides the centripetal acceleration of the circular motion, mv 2 (3.1.1) F = qvB = r where q is the charge of the particle, m and v its mass and velocity, and r its instantaneous radius. The period for completing one circular orbit is, t=
2πr 2πm = v qB
(3.1.2)
and its frequency is, qB 1 = (3.1.3) t 2πm which is called the cyclotron resonance frequency of a cyclotron. Note that the period of a particle through the cyclotron is independent of ν=
the radius of the orbit. This is because as the particle spirals to larger radii, its energy also increases and hence the gradual increase in its orbit is compensated by the increasing speed. The maximum velocity of the particle is at the largest radius (R) where it emerges from the cyclotron, qBR m This corresponds to a kinetic energy T, given by vmax =
63
(3.1.4)
1 2 q 2 B 2 R2 T = mvmax = (3.1.5) 2 2m The accelerating capability of a cyclotron is defined in terms of a parameter (k) which is related to the kinetic energy T of the particle by, AT (3.1.6) Z2 where (Z) is the atomic number and (A) the mass number of the particle. The k=
parameter (k) corresponds to the kinetic energy to which protons (A=1, Z=1) would be accelerated. Heavier ions are accelerated to kinetic energies equal to kz 2 . A
3.1.2
Tandem Van de Graaff Accelerators
A Tandem Van de Graaff is a multistage accelerator which by charge-exchange can produce energetic ion beams above the Coulomb barrier for compound reactions. First a beam of negative ions (q = -1) is accelerated out of its source toward a high voltage terminal in the centre of a pressurized (10-20 atmospheres) tank. The tank contains an insulating stable gas like SF6 to prevent breakdown nd sparking between the machine components (such as the terminal) which are at high voltage, and the tank. Then the negative ions pass through a thin carbon foil at the central terminal where they are stripped of some or all of their electrons depending on the atomic number of the ion (higher Z atoms have more tightly bound electrons which are more difficult to strip off) and become positively charged. The stripped positive ions are accelerated away from the central terminal and after passing through electric and magnetic fields for focusing, bending, and selecting the desired energy the beam can be used directly to bombard a target or injected into a post accelerator for further acceleration.
3.1.3
Linear Accelerators
In a linear accelerator (linac), particles move in a straight line and receive many accelerations by passing through a series of electrodes separated by gaps and alternately connected to opposite poles of an ac voltage source (V=V◦ cosωt). The distance (L) between gaps must be such that an ion enters an electrode when it is at a negative potential and leaves it when it has a positive potential, 64
hence
vT (3.1.7) 2 where v is the velocity of ion in the electrode and T is the period of the oscillating voltage. Nonrelativistically, after crossing n gaps, the velocity is, L=
2nqV◦ v= m
12
(3.1.8)
where q and m are charge and mass of the ions. Hence the length of nth. electrode is, nqV◦ Ln = 2m
21
(3.1.9)
T 1
Therefore, the length of electrodes must increase as n 2 . Relativistically, as v→c, the distances between gaps are approximately constant.
3.2
Radioactive Ion Beam production
Currently, there are two main ways to produce intense beams of radioactive, heavy ions at energies useful for nuclear physics studies. These are the Projectile Fragmentation method (PF) and the Isotope Separator On-Line method (ISOL). Much effort is presently being expended in the construction of radioactive beam facilities in Europe, North America and in Japan. In the PF method, an energetic (>50 MeV/u), stable heavy-ion beam bombards a thin target. Those projectiles which are involved in peripheral reactions with the target nuclei can lead to fragmentation of the projectile nucleus. The fragmentation products of interest can then be selected on-line from the wide range of reaction products using electromagnetic devices. This leads to a wide range of very energetic and reasonably intense beams of short-lived radioactive species. The advantages of this technique are fast separation, no chemical selectivity, relatively simple production targets and beams that do not require further acceleration. The drawbacks are nonoptimum production energies which are normally above the desired energies for many experiments, poor beam emittance and limited primary beam intensity. The four major PF laboratories in operation are GANIL in France, GSI in Germany, NSCL in U.S.A. and RIKEN in Japan.
65
The second approach (ISOL) involves the coupling of a primary radioisotope production accelerator to an isotope separator that is itself coupled to a postacceleration system. In this method, typically a high energy light ion beam is directed onto a thick, high-temperature target which allows the reaction products to diffuse, into an ion source. The radioactive ions are extracted and mass separated before being injected into a second accelerator. The major ISOL laboratory currently in operation are at Louvain-la-Neuve in Belgium and ISOLDE at CERN in Switzerland. The two methods have their own advantages and disadvantages. The ISOL method has the advantage of producing higher intensity beams with better beam quality than the PF method, but at a lower beam energy. The main drawback to the ISOL method is that the diffusion/desorption and ionization processes are strongly element (chemical)-dependent and slower than the PF method. In many cases, significant decay losses can occur in this release stage.
66
Further Reading. Useful web sites. 1. RIA http://www.phy.anl.gov/ria/ 2. Ernst Rehm’s lecture notes (in powerpoint format), http://www.orau.gov/ria/notes.htm 3. nucleosynthesis http://ultraman.ssl.berkeley.edu/nucleosynthesis.html Review Articles. 1. Sythensis of Elements in Stars, E.M. Burbidge, G. Burbidge, W.A. Fowler and F. Hoyle, Rev. Mod. Phys. 29 (1957) p15 2. Experimental and Theoretical nuclear astrophysics: The quest for the orgin of the elements, Rev. Mod. Phys. 56 (1984) p149 3. On the Origin of Light Elements, H. Reeves, Rev. Mod. Phys. 66 vol 1. (1994) p193 4. Radiative Capture Reaction in Nuclear Astrophysics, C. Rolfs and C.A. Barnes, Ann. Rev. Nuc. Part. Sci. 40 (1990) p45 5. Explosive Hydrogen Buring, A.E. Champagne and M. Wiesher, Ann. Rev. Nucl. Part. Sci. 42 (1992) p39 6. Nuclei at the Limits of Particle Stability, A.C. Mueller and B.M. Sherrill, Ann. Rev. Nucl. Part. Sci. 43 (1993) p529 7. Current Quests in Nuclear Astrophysics and Experimental Approaches, F. K¨appler, F.K. Thielmann, M. Wiescher, Ann. Rev. Nucl. Part. Sci. 48 (1998) p175 8. rp-Process Nucleosynthesis at Extreme Temperature and Density Conditions, H. Schatz et al. Phys. Rep. 294 (1998) p16 9. Nuclear Astrophysics Measurements with Radioactive Beams, M. Smith and E. Rehm Ann. Rev. Nucl. Part. Sci. 51 (2001) p91 General Articles. 1. Nuclear Astrophysics: A New Era, M. Wiescher, P.H. Regan and A. Aprahamian, Physics World, 15, No.2 (2002) p33 67
3NA Exam 1997 1. (a) Calculate the Q-value for the reaction 4
He+4 He→8 Be
given that ∆(4 He)=+2.425 MeV and ∆(8 Be)=+4.942 MeV. (You may assume that 1u=931.5 MeV/c2 = 1.66×10−27 Kg. 5 marks. (b) The Saha equation may be written as
N12
N1 N2 = 1 + δ12
2π µkT
!3
2
h ¯ 3 ωexp −
ER kT
Explain the meaning of the terms N12 , N1 , N2 , δ12 , µ, ω and ER in this expression. 4 marks (c) Estimate the ratio of
8 Be 4 He
in a concentration of 4 He atoms at a temper-
ature of 108 K with a density of 3×108 g/cm3 . (k=1.38×10−23 JK−1 ) 5 marks (d) Explain briefly the way
12
C is formed in the triple-α process and disuss
the effect of resonant states in the rates of α-buring reactions and 16 O(α, γ)20 Ne. 6 marks.
12
C(α, γ)16 O
2. (a) Describe the reactions in the simple CN cycle at low stellar temperatures and densities 6 marks. (b) How is this cycle modifed in the high temperature and density limit ? 4 marks (c) Describe how nuclei with A>56 can be created in stars. 10 marks 3. (a) Given the expression < σv >34 (2J1 + 1)(2J2 + 1) = < σv >12 (2J3 + 1)(2J4 + 1)
1 + δ34 1 + δ12
!
µ12 µ34
!3
2
exp −
Q kT
12 , J3 , J4 , δ34 , µ12 , Q and T and briefly explain the meaning of the terms 34 explain the significance of this expression in the experimental determination
of reaction rates. 6 marks. (b) For the reaction, 68
1
H + 7 Li →8 Be∗ →4 He + 4 He
Calculate the ratio of reaction rates per particle pair at temperatures of (i) + 109 and (ii) 1010 K given that the ground state/parities of 1 H and 7 Li are 12 and 32 respectively. You may assume M(1 H)=1.007825u, M(7 Li)=7.016003u, M(4 He)=4.002603u, 1u=931.5MeV/c2 and k=8.62×10−5 eV/K= 1.38×10−23 −
JK−1 8 marks (c) Explain the terms inverse reaction. Under what circumstances are inverse reactions most likely to occur? 2 marks (d) What is the energy width, Γ in eV for a state which only decays via gamma-ray emission with a mean-lifetime of 5×10−12 seconds? 2 marks (e) If this same state can now also decay via α-emission with a partial, α-decay width of 10 eV, what is the new, total decay lifetime of the state? 2 marks.
3NA Exam 1998 1. (a) Draw a careful sketch to illustrate the variation of the binding energy per nucleon with nucleon number, labelling both axes and any salient points on the curve 4 marks. (b) Calculate the binding energy and binding energy per nucleon of the fol208 lowing nuclei (i) 42 He, (ii) 56 26 Fe and (iii) 82 Pb. Comment on the values you obtain. Note (∆(4 He)=+2.425 MeV, ∆(56 Fe)=–60.601 MeV, ∆(208 Pb)=– 21.764 MeV, 1u=931.5 MeV/c2 , m(1 H)=1.007825u and mn =939.573 MeV/c2 and ∆=mass excess. 4 marks. (c) Explain the terms proton and neutron ‘drip lines’. What processes can allow neutron deficient nuclei to exist even though they lie beyond the proton drip line? 2 marks. (d) Locate the above nuclei on the curve of binding energy per nucleon and explain why nuclear fusion is energetically unfavourable for A¿56. 4 marks.
69
(e) State the possible astrophysical processes which may form (i) nuclei along the proton drip line for A>70 and (ii) neutron rich nuclei approaching the drip line for A>56. 2 marks. (f) Explain the initial reactions involved in the slow neutron capture (‘S’) process starting from a 56 Fe core and forming 59 Co. What is thought to be the source of the neutrons involved? What does this reveal about the types of stars in which slow neutron capture occurs? 4 marks. 2. (a) Briefly discuss the operation of the following types of ion accelerator, (i) Tandem Van de Graaff (ii) cyclotron and (iii) linac. In each case, discuss how the energy of the produced beam is obtained and whether the final beam is AC or DC in nature. 12 marks. (b) A cyclotron has a 2m diameter magnet with a magnetic flux density, B of 1T. Calculate the maximum energy (in MeV) of a beam of fully stripped 12 6 C ions that may be extracted from the cyclotron. Note, −27 Kg=931.5 MeV/c2 and 1e=1.6times10−19 C. 4 marks. 1u=1.66×10 (c) State two methods of production of radioactive ion beams for use in nuclear astrophysics studies. Why are such beams so useful in nuclear astrophysics studies? 4 marks. 3. (a) The Saha equation may be written as N12
N1 N2 = 1 + δ12
(
2π µkT
)3 2
ER h ¯ ωexp − kT 3
Explain the meaning of the terms N12 , N1 , N2 , δ12 , µ, ω, and ER in this expression. 4 marks. 8
Be (b) Estimate the ratio of 4 He in a concentration of 4 He atoms at a temperature of 109 K with a density if 4×108 kg/m3 . 5 marks.
(c) Explain the terms (i) subthreshold resonances, (ii) thermal resonance, (iii) ‘unnatural’ parity state. Give examples pertinent to 4 He burning reactions. 6 marks. (d) Calculate the energy of the photon emitted in the direct capture re+ action 3 He(alpha,γ)7 Be to the 12 excited state at 429 keV in 7 Be. Assume that the 3 He is at rest and the α particle has a kinetic energy of 70
300 keV. Note Q-value for α + α →7 Be = –92keV, k=1.38×10−23 J/K, h ¯ =1.05×10−34 Js, mu =1.66×10−27 Kg=931.5 MeV/c2 , M(4 He)=4.002603u, M(3 He)=3.016029u, M(7 Be)=7.016928u. 5 marks.
3NA Exam 1999 1. (a) Given that the classical expression for a cross-section, σ, can be expressed by σ = π(RP + RT )2 estimate the classical geometric cross-section for a reaction with a beam of protons incident on a 12 C nucleus. The nuclear radius, R, can be expressed as R ≈ 1.3 × 10−15 m. 4 marks. (b) The deBroglie wavelength, λDB , for the particles in this reaction can be expressed by λDB =
MT + MP h ¯ q MT 2Mp El
where MP is the projectile mass, MT is the target mass, EL is the projectile energy in the laboratory frame and h ¯ =1.055×10−34 Js. Hence, re-estimate the cross-section for the reaction in part (a) for a beam or protons at an −27 kg and energy of 5 MeV onto a 12 6 C target at reast. Note 1u≈ 1.66 × 10 1eV=1.6×10−19 J. 4 marks.
(c) The Sommerfeld parameter, η, can be calculated using the expression 2πη = 31.29Z1 Z2
r
µ E
where µ is the reduced mass of the system in amu and E is the energy if keV of the collision in the centre of mass frame. Using the value of the cross-section calculated in section (b), estimate the 13 value of the astrophysical ‘S’-factor for the compound reaction 12 6 C+p→7 N at a proton energy in the centre of mass frame of 5 Mev. 4 marks. 71
(d) Sketch the variation of cross-section with particle energy for the fusion of two charged nuclei in a gas of particles with velocities which follow a Maxwell-Boltzmann type distribution. Explain the origin of the ‘Gamow Peak’ using this diagram. 6 marks. (e) Given that 1
Eo = 1.2(Z12 Z22 µT62 ) 3 keV where Eo is the Gamow peak energy, calculate the effective burning energy 13 8 of the 12 6 C+p→7 N reaction at a temperature of T=10 K. Note, µ is the reduced mass in amu. 2 marks. 2. (a) List with the aid of a diagram, the reactions involved in the Simple CN cycle at low stellar temperatures and densities. Extend this to include the simple CNO Bi-cycle. 6 marks. (b) How is this cycle modified in the high temperature and density limit. what nuclei are formed in the hot CNO cycle that are not present in the Simple CN cycle. 4 marks. (c) Describe briefly what is meant by a radiative capture reaction and two examples of such a reaction in the hot CNO cycle. 3 marks. (d) Given the expression, < σv >34 (2J1 + 1)(2J2 + 1) = < σv >12 (2J3 + 1)(2J4 + 1)
1 + δ34 1 + δ12
!
µ12 µ34
!3
2
exp −
Q kT
calculate the ‘ratio of reaction rates per particle pair’ for the 7 Li(p,α)α reaction at a temperature of 1010 K. The ground state spin/parities of the + − proton and 7 Li nucleus are 12 and 32 respectively and the Q-value for the reaction is 17.35 MeV. Note k = 1.38 × 10−23 JK−1 = 8.62 × 10−5 evK−1 . 7 marks 3. (a) Explain the term ‘direct reaction’ as used in nuclear physics. 3 marks. (b) Calculate the energy of the photon emitted in the direct capture reaction 1+ 12 13 13 6 C(p,γ)7 N to the excited 2 state in 7 N with an excitation energy of 2.366
72
MeV, for a proton of energy 1 MeV. Note M(12 C=12u, M(1 H)=1.007825u, 1u=931.5MeV/c2 and M(13 N)=13.005739u. 5 marks. (c) Using a simple semi-classical geometric model, the angular momentum transferred ina direct reaction can be approximated using the following expression, θ h ¯ 2 l2 = R2 PA PB (2sin2 ( )) 2 with the aid of a simple diagram, explain the meaning of all the terms in this expression. 4 marks. (d) How are the spin and parity of a compound resonant stat related in the most general case to the relative motion and spins and parities of the two nuclei in the extrance channel? 4 marks (e) How are these relations simplified in the case of two even-even nuclei in the entrance channel ? 2 marks (f)
20 10 Ne
has an excited state at an excitation energy of 4.97 MeV which
has spin/parity 2− . Why can this state not be directly populated in the 16 O+4 He→20 Ne reaction ? 2 marks. 3NA Exam 2000 1. (a) The Saha equation may be written as, N12
N1 N2 = 1 + δ12
(
2π µkT
)3 2
h ¯ 3 ωexp −
ER kT
Explain the meaning of the terms N12 , N1 , N2 , δ12 , µ and ω in this expression. 4 marks. 8
Be in a concentration of 4 He atoms at a temper(b) Estimate the ratio of 4 He ature of 2×109 K with a density if 5×108 kg/m3 . 5 marks.
(c) Sketch the variation of fusion rate as a function of particle energy for the fusion of two charged nuclei in a gas of particles which follow a MaxwellBoltznmann type distribution. On the same sketch, show the variation of probability of barrier penetration as a function of energy. Using these sketches, explain the orgin of the ‘Gamow peak’. 6 marks. 73
(d) Give the expressions, 1
Eo = 1.2(Z12 Z22 µT62 ) 3 keV and 1 4 ∆ = √ (Eo kT ) 2 3
calculate the energy and width of Gamow peak for an ensemble of 4 He nuclei at a temperature of T = 109 K 5 marks. Note Q-value α + α →8 Be= −92 keV k = 1.384 × 10−23 JK−1 h ¯ = 1.05 × 10−34 Js mu = 931.5 MeV/c2 ≈ 1.66 × 10−27 kg T6 = temperature in units of 106 K 2. (a) List with the aid of a diagram the reactions involved in the Simple CN cycle at low stellar temperatures and densities. Extend this to include the CNO Bi-cycle. 6 marks. (b) How iis the cycle modified in the high temperature and density limit? What nuclei are formed in the ‘hot’ CNO cycle which are not present in the Simple CNO Bi-cycle. 4 marks. (c) State the possible astrophysical processes which may form, (i) nuclei along the proton drip line for A < 70 and (ii) neutron-rich nuclei approaching the neutron drip-line for A > 56. 2 marks, (d) Explain the initial reactions involved in the slow neutron capture (‘S’) process starting from a 56 26 Fe core. What is thought to be the source of the neutrons involved in these reactions? What does this reveal about the types of stars in which slow neutron capture occurs? 5 marks. (e) State three pieces of evidnece for the formation of elements via the rapid (‘r’) neutron capture process. 3 marks.
74
3. (a) Given that the classical expression for a cross-section, σ, can be expressed by σ = π(Rp + RT )2 estimate the classical, geometric cross-section (in barns) for the fusion of two 1 16 O nuclei. The nuclear radius, R can be estimated by R ≈ 1.3×10−15 A 3 m. 4 marks
(b) State the main difference between the operation of single ended and ‘tandem’ Van de Graaff accelerators. 2 marks. (c) If the laboratory fram energy required to overcome the Coulomb barrier between two nuclei, Ec can be estimated by the expression, Ec (MeV ) =
1.44Z1 Z2 AT + AB R1 + R2 AT
where R1 and R2 are the nuclear radii in 4 fm, Z1 and Z2 are the atomic numbers of the nuclei and AT and AB are the mass numbers of the target and beam nuclei respectively, show that the Coloumb barrier energy in the lab frame for a beam of 26 MeV. 4 marks.
12 6 C
ions onto a fixed,
16 8 O
target is approximately
(d) What terminal voltage would be required using a tandem Van de Graaff to accelerate fully stripped 12 C ions such that they would just have enough energy to fuse with
16
O nuclei at the target position? 4 marks.
(e) What magnetic flux density, B, would be required to accelerate fully stripped 12 C ions to the Coulomb barrier energy om 16 8 O, assuming a cyclotron accelerator with a 2 m diameter? 4 marks. (f) What cyclotron frequency does this correspond to ? 2 marks. 3NA Exam 2001 1. (a) Nuclear reaction rates in stars are related by the expression < σv >34 (2J1 + 1)(2J2 + 1) = < σv >12 (2J3 + 1)(2J4 + 1) 75
1 + δ34 1 + δ12
!
µ12 µ34
!3
2
exp −
Q kT
12 Briefly explain the meaning of the terms , J3 , J4 , δ34 , µ12 , Q and T 34 and explain the significance of this expression in the experimental determi-
nation of reaction rates. 6 marks. (b) For the reaction 15 7 N
∗ 12 4 + 1 H→16 8 O →6 C + 2 He
calculate the ratio of reaction rates per particle pair at temperatures of (i) 109 K and (ii)1010 K, given tha the ground state spins and parities of +
N and 1 H are 12 and 12 respectively. The atomic masses of 1 H, 15 N, 12 C and 4 He are 1.007825u, 15.000109u, 12.000000u and 4.002603u respectively.
15
−
1u=931.5MeV/c2 . k=1.38x10−23 JK−1 =8.62x10−5 eVK−1 . 8 marks. (c) Explain the term ‘inverse reaction’. Under what circumstances are inverse reactions most likely to occur. 2 marks. (d) Calculate the energy of the photon emitted in the direct capture reaction 15 N+1 H to the ground state of 16 O. Note 15 N→16 O Q-value =12.126 MeV and M(16 O)=15.994915u. Assume that the proton has an energy of 2 MeV. 4 marks.
15
N nucleus is at rest and the
2. (a) Given that the classical expression for the cross-section, σ, can be expressed by σ = π(Rp + RT )2 estimate thet classical, geometric cross-section in barns of the fusion of an alpha particle on a 12 C nucleus. The nuclear radius, R can be expressed as R = 1.3x10−15 m and 1b=10−28 m2 . 4 marks. (b) The De Broglie wavelength, λdb for the particles in this reaction can be expressed by λdb =
Mp + MT h ¯ 1 MT (Mp El) 2
where Mp is the projectile mass, MT is the target nass, El is the projectile energy in the lab frame and h ¯ =1.055x10−34 Js. Restimate the cross-section for the reaction in part (a) for a laboratory alpha-particle beam energy of 5 MeV. (Note 1u=1.66x10−27 Kg and 1eV=1.6x10−19 J.) 4 marks. 76
(c) If the laboratory frame energy required to overcome the Coulomb barrier between the two nuclei, Ec can be estimated by the expression, Ec (MeV ) =
1.44Z1Z2 AT + AB . (R1 + R2 ) AT
where R1 and R2 are the nuclear radii in f m, Z1 and Z2 are the atomic numbers of the nuclei and AT and AB are the mass numbers of the target and beam respectively, sgow that the Coulomb barrier in the lab frame for alpha particles on the fixed
12
C target is approximately 4 MeV. 4 marks.
(d) State two methods of production of radioactive ion beams. Giving an example of a reaction from the hot CNO cycle, explain why such beams are so useful in nuclear astrophysics studies. 4 marks. (e) Given that the Sommerfeld parameter, ν can be calculated using the expression µ 1 2πν = 31.29Z1 Z2 ( ) 2 E where µ is the reduced mass of the system in AMU and E is the energy of the collision in the centre of mass frame in keV, calculate the probability of tunnelling through the Coulomb barrier for an alpha-particle on a 12 C target with a centre of mass collision energy of 0.2 MeV. 4 Marks. 3. (a) Using a simple, semi-classical geometric model, the angular momentum transferred in a direct reaction can be approximated using the following expression, θ h ¯ 2 l2 = R2 PA PB {2sin2 } 2 with the aid of a simple diagram, explain the meaning of all the terms in this expression. 6 marks. (b) How are the spin and parity of a compound resonant state related in the most general case to the relative motion and spins and parities of the two nuclei in the entrance channel. 4 marks. (c) Explain briefly with the aid of an energy level diagram, the way in which 12
C is formed in the triple-alpha process. 5 marks. 77
(d) With the aid of energy level diagrams, discuss the effect of resonant states on the rates of the alpha-burning reactions 12 C(α, γ)16 O and 16 O(α, γ)20Ne, including examples of subthreshold resonances and parity forbidden states. 5 marks. 3NA Exam 2003 1. (a) Explain the term ’direct capture reaction’ as used in nuclear astrophyics 3 marks (b) Calculate the energy of the photon emitted in the direct capture re− action 3 He(α,γ)7 Be to the 12 excited state at 429 keV in 7 Be. Assume that the 3 He is at rest and that the α particle has a kinetic energy of 300 keV. ( M(4 He=4.002603u, M(3 He)=3.016029u, M(7 Be)=7.01692u and 1u=931.5MeV/c2 ). (c) Using a simple, semi-classical geometric model, the angular momentum transferred in a direct reaction can be approximated using the following expression, θ h ¯ 2 l2 = R2 PA PB {2sin2 } 2 with the aid of a simple diagram, explain the meaning of all the terms in this expression. 5 marks. (d) How are the spin and parity of a compound state related in the most general case to the relative motion and spins and parities of two nuclei in the entrance channel ? 4 marks (e)
20
Ne has an excited state at excitation energy of 4.97 MeV, which has
a spin/parity of 2− . Why can this state not be directly populated in the 16 O+4 He→20 Ne reaction ? 2 marks 2. (a) Calculate the Q-value for the reaction 4 He+4 He→8 Be given that the mass excesses of 4 He and 8 Be are +2.425 MeV and +4.942 MeV respectively. Comment on the significance of the results with respect to the stability of 8 Be. 3 marks (b) With the aid of a sketch, draw the three main branches of the p-p chain. Include all the relevant reactions and give estimates of the relative probabilities for the each branch. 6 marks 78
(c) The Saha Equation may be written as
N12
N1 N2 = 1 + δ12
(
2π µkT
)3 2
h ¯ 3 ωexp −
ER kT
Explain the meaning of the terms N12 , µ and ER in this expression. 4 marks. (d) Estimate the ratio of
8 Be 4 He
in a concentration of 4 He atoms at a temper-
ature of 1×109 K with a density if 109 kg/m3 . 7 marks. k=1.38x10−23 J/K ,h ¯ =1.05x10−34 Js, 1u=931.5MeV/c2 and 1eV=1.6x10−19 J 3. (a) List the reactions in the simple CN cycle at low stellar temperatures and densities (include on the proton capture branches) 4 marks. (b) How is the cycle modified in the high temperature and density limit ? 4 marks (c) Explain the initial reactions including any relevant decay lifetimes involved in the slow-neutron capture process, starting from a 56 Fe core and forming 59 Co. What reactions are thought to be the source of the neutrons involved ? What does this reveal abouy the types of stars in which slow neutron capture exists ? 6 marks (d) State three pieces of evidence for the formation of elements via the rapid neutron capture process. 3 marks (e) State three possible proposed astrophysical sites for the rapid proton capture process. 3 marks 4. (a) Sketch the variation of cross-section with particle energy for the fusion of two charged nuclei in a gas of particles with velocities which follow a Maxwell-Boltzmann distribution. Draw also the probabilities of fusing through the Coulomb barrier on the same sketch. Explain the significance of the GAMOW WINDOW on this diagram. 7 marks (b) Given the expression 1
Eo = 1.2(Z12 Z22 µT62 ) 3 keV
79
where Eo is the Gamow peak energy, calculate the effective burning energy 14 7 of the 13 6 C+p→7 N reaction at a temperature of T=5x10 K. Note, µ is the reduced mass in amu. 5 marks. (c) The Sommerfeld parameter, ν can be calculated using the expression µ 1 2πν = 31.29Z1 Z2 ( ) 2 E where µ is the reduced mass of the system in amu and E is the centre of mass collision energy in MeV. Assuming a reaction cross-section of 100mb, calculate the value of: (a) the probability of penetrating the Coulomb barrier and (b) the astrophysical S(E) factor, 14 for the reaction 13 6 C+p→7 N at proton energy in the centre of mass frame of 2 MeV. 8 marks
3NA Exam 2004 1. (a) Nuclear reactions rates in stars are related by the expression, < σv >34 (2J1 + 1)(2J2 + 1) = < σv >12 (2J3 + 1)(2J4 + 1)
1 + δ34 1 + δ12
Briefly explain the meaning of the terms, marks
!
12 , 34
µ12 µ34
!3
2
Q exp − kT
J3 , J4 , δ34 , µ12 , and T . 6
(b)Explain the significance of this expression in the experimental determination of reaction rates. 2 marks. (c) For the nuclear reaction 1 H+73 Li→ 4 He+4 He, calculate the ratio of reaction rares per particle pair at temperatures of (i) T=108 K and (ii) T=3×109 K. The ground-state spins anre parities of teh 1 H and 7 Li are f rac12+ and 32 respectively. You may also assume that m(1 H)=1.007825u, m(7 Li)=7.016003u, m(4 He)=4.002603u, 1u=931.5MeV/c2 and k=8.62×10−5eVK−1 −
and k=1.38×10−23 JK−1 . 8 marks. (d) What is the energy width, Γ, in electron volts for a particle bound nuclear state which decays only via electromagnetic decay with a mean decay lifetime, τ of 5×10−12 . 2 marks. 80
(e) If the same state can now also decay via an alpha-particle emission process with a partial width of Γα =10eV, what is the new, total decay mean-lfetime of this state ? (Note, 1 eV=1.6×10−19 eV, h=6.64×10−34 Js. 2. (a) List the reactions of the simple CN cycle at low stellar temperature and density (include only the proton capture branches). 4 marks (b) How is this cycle modifies in the high-temperature and high-densith limit. 4 marks. (c) (i) Write down the initial nuclear reactions, including any relevant decay lifetimes involved in the slow (’s’) neutron capture process beginning from a 56
Fe seed nucleus and finally forming 59 Co. (ii) What reactions are thought to be the source of the neutrons involved ? (iii) What does this reveal about the types of stars in which slow neutron-capture is thought to take place ? 9 marks (d) State three pieces of proposed evidence of the formation of heavy elements with A>56 via the rapid (’r’) neutron capture process 3 marks. 3. (a) Using a simple sketch, explain why energy is released in eth fusion of nuclei with a compound mass of A∼56, but not for heavier nuclei 4 marks (b) Sketch the Segre Chart of the nuclides, approximately including and labelling the following features: (1) the valley of stability; (ii) the proton and neutron drip lines; (iii) the N=Z line; and (iv) any magic numbers relevant to nucleosynthesis paths. Indicate on the same sketch the approximate paths and limits of the S, r and rp processes for nucleosynthesis, highlightling any prominent features. 10 marks (c) Explain briefly, with the aid of an energy level diagram the way in which C is though to be formed by the triple-α process. Give the names of the first state in 12 C which is populated in this process and show its possible
12
decay modes. 6 marks 4. (a) Given that the classical expression for the cross-section, σ, can be expressed by σ = π(Rp + RT )2 81
estimate thet classical, geometric cross-section in barns of the fusion of a protons on a 12 C nucleus. The nuclear radius, R can be expressed as R = 1.3x10−15 m and 1b=10−28 m2 . 5 marks. (b) The De Broglie wavelength, λdb for the particles in this reaction can be expressed by λdb =
h ¯ Mp + MT 1 MT (Mp El) 2
where Mp is the projectile mass, MT is the target nass, El is the projectile energy in the lab frame and h ¯ =1.055x10−34 Js. Restimate the cross-section for the reaction in part (a) for a laboratory alpha-particle beam energy of 5 MeV incident on a 12 C nucleus at rest. (Note 1u=1.66x10−27 Kg and 1eV=1.6x10−19 J.) 5 marks. (c) Sketch the variation of cross-section with particle energy of the fusion of two charged nuclei in a gas of particles with velocities which follow a Maxwell-Boltzmann distribution. Explain the origin of the ’Gamow Peak’ using this diagram. 6 marks. (d) The Sommerfeld parameter, ν can be calculated using the expression µ 1 2πν = 31.29Z1 Z2 ( ) 2 E where µ is the reduced mass of the system in AMU and E is the energy of the collision in the centre of mass frame in keV. Given that the total reaction cross-section, σ is 200mb, calculate the astrophysics S(E) factor in units of MeV.b for the reaction 12 C+4 He→16 O at a centre of mass collision energy of 10 MeV. 4 marks. 3NA Exam 2005 1. (a) List the reactions in the three main branches of the proton-proton reaction chain, giving estimates for the relative branching ratios for each part of the chain. 8 marks (b) Explain the term ’radiative capture reaction’ as used in nuclear physics. 2 marks 82
(c) Calculate the energy of the photon emitted in the direct capture reaction − 3 He(α,γ)7 Be to the excited 12 state in 7 Be with an excitation energy of 429 keV. Assume that the 3 He nucleus is at rest and that the α particle has a kinetic energy of 100 keV in the laboratory frame. Note: M(4 He=4.002603u, M(3 He)=3.016029u, 1u=931.5MeV/c2 and M(7 Be)=7.0169289u. 6 marks. (d) Calculate the Q value for the reaction 2 H+2 H→4 He given that he masses of the deuteron and alpha particle are M(2 H)=2.014102u and M(4 He)=4.002603u. (e) Why is the reaction stated in part (d) thought to play a negligible role in the production of 4 He in the p-p chain ? 2 marks 2. (a) i. Write down the initial nuclear reactions, including any relevant decay lifetimes involved in the slow (’s’) neutron capture process beginning from a
56
Fe seed nucleus and finally forming
59
Co.
ii. What nuclear reactions are thought to be the source of the neutrons involved ? 6 marks (b) Sketch the expected form of the neutron capture cross-section for l=0 (s-wave) neutrons as a function of neutron energy for non-resonant neutron capture. What does the nature of this curve imply about the reaction rate per particle pair, < σn vn > as a function of neutron energy for non-resonant neutron capture ? 6 marks (c) The reaction rate per target atom, r, can be described by the expression r ≈< σv > N. If for typical red giant stars the neutrons in the core have kinetic energies of 20 keV and typical neutron capture cross-sections at this energy are of the order of 100mb, assuming a neutron density N = 1014 neutrons per cubic metre, estimate the typical time between successive
neutron captures on the same seed nucleus that one would expect for a medium sized nucleus in the core of a red giant star. Comment on the consequences of your result in terms of the s-process path along the Segr´e chart. Note 1u=1.66×10−27 Kg, 1 eV=1.6×10−19 J and 1 barn = 10−28 M2 . 8 marks 3. (a) Using a simple sketch, explain why energy is generally released in the fusion of nuclei with a resulting compound of A>56, but not for heavier nuclei ? 4 marks 83
(b) Sketch the Segre Chart of the Nuclides, indicating and labelling the following features: (i) the valley of stability; (ii) the proton and neutron drip lines; (iii) the N=Z line; and any magic numbers relevant to nucleosynthesis paths. Indicate on the same sketch the approximate paths and limits of the s-, r- and rp- processes for nucleosynthesis, highlighting any prominent features. 8 marks (c) The energy (in MeV) in the laboratory frame energy required to overcome the Coulomb barrier between the two nuclei, Ec can be estimated by the expression, Ec (MeV ) =
1.44Z1Z2 AT + AB . (R1 + R2 ) AT
where R1 and R2 are the nuclear radii in fermis (10−15 m), Z1 and Z2 are the atomic numbers of the nuclei and AT and AB are the mass numbers of the target and beam respectively, estimate the Coulomb barrier in the lab frame for alpha particles on the fixed 12 C (Z=6) target. Note that the size of the nuclear radius can be approximated using the expression R = 1.3A−1/3 . 4 marks. (d) What terminal voltage would be required to accelerate singly ionized He ions to the energy needed in section (c) using a single-ended Van de Graaff accelerator ? 2 marks 4
(e) How is this voltage modified for fully stripped ions in a tandem Van de Graaff accelerator? 2 marks 4. (a) Sketch the variation of cross-section with particle energy for the fusion of two charged nuclei in a gas of particles with velocities which follow a Maxwell-Boltzmann distribution. Explain the origin of the ’Gamow Peak’ using this diagram. 5 marks (b) Explain the terms ’resonance reaction’ and ’subthreshold resonance’ in terms of nuclear reactions. Give an example of an important sub-threshold resonance in the CNO hydrogen burning cycle. 5 marks (c) Using a simple geometric model, the angular momentum transferred in a direct reaction can be approximated with the expression, 84
θ h ¯ 2 l2 = R2 PA PB {2sin2 } 2 with the aid of a simple diagram, explain the meaning of all the terms in this expression. 3 marks. (d) In the 90 Zr(d,p)91 Zr direct transfer reaction, if the l=2 transfer reaction protons in the exit channel are scattered in such a way that their angular distributions are peaked at an angle of approximately 20◦ in the centre of mass frame, at what scattering angle would you expect the l=4 transfers to states of similar excitation energy to be peaked ? 3 marks (e) If the ground state spin/parity of the deuteron and proton are 1+ and 1+ respectively, what spin/parity values would one expect to populate in 2 the reaction outlined in section (d) for l=2 transfer ? 4 marks.
85
