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Lecture Notes in Nuclear Structure Physics
B. Alex Brown
November 2005
National Superconducting Cyclotron Laboratory
and Department of Physics and Astronomy
Michigan State University, E. Lansing, MI 48824
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Lecture Notes in Nuclear Structure Physics, Lecture notes of Nuclear Physics
Lecture notes on nuclear structure physics, covering topics such as nuclear masses, charge radii, decay, Fermi gas model, nuclear shell model, and many-body problem for fermions. The notes were written by B. Alex Brown in November 2005 and are based on research conducted at the National Superconducting Cyclotron Laboratory and the Department of Physics and Astronomy at Michigan State University.
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Lecture Notes in Nuclear Structure Physics
B. Alex Brown
November 2005
National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy
Michigan State University, E. Lansing, MI 48824
Contents
1 Nuclear masses
1.1 Masses and binding energies
A basic quantity which can be measured for the ground states of nuclei is the atomic mass M(N, Z) of the neutral atom with atomic mass number A and charge Z. Atomic masses are usually tabulated in terms of the mass excess defined by
∆(N, Z) ≡ M(N, Z) − uA, (1.1)
where u is the Atomic Mass Unit defined by u = M(^12 C)/12 = 931.49386 MeV/c^2. (^1) I will use the data from the 2003 compilation of Audi, Wapstra and Thibault [1].
Fig. (1.1) shows the position on the nuclear chart for these measured masses together with the experimental error. There are 2127 nuclei measured with an accuracy of 0. MeV or better and 101 nuclei measured with an accuracy of greater than 0.2 MeV. For heavy nuclei one observes several chains of nuclei with a constant N − Z value whose masses are obtained from the alpha-decay Q values.
Nuclear binding energy is defined as the energy required to break up a given nucleus into its constituent parts of N neutrons and Z protons. In terms of the atomic masses M(N, Z) the binding energy B(N, Z) 2 is defined by:
B(N, Z) = ZMH c^2 + NMnc^2 − M(N, Z)c^2 , (1.2)
where MH is the mass of the hydrogen atom and Mn is the mass of the neutron.^3 In terms of the mass excess the binding energy is given by:
B(N, Z) = Z∆H c^2 + N∆nc^2 − ∆(N, Z)c^2 , (1.3)
where ∆H c^2 = 7.2890 MeV and ∆nc^2 = 8.0713 MeV.
How do we know that nuclei are made up of protons and neutrons? In the 1920’s when it was observed that nuclei decay by the emission of alpha particles, protons and electrons one tried to make nuclear models out of constituent protons and electrons. (^4) However, after the discovery of the neutron in 1932, it was observed that the
(^1) This and other constants can be found on the website:
http://physics.nist.gov/cuu/Constants/index.html (^2) The binding energy will also be denoted by BE. (^3) This binding energy also includes contribution from the Coulomb interaction between electrons which is approximately given by −1.43x10−^5 Z^2.^39 MeV. On the scale of nuclear binding this can usually be ignored. It is most important for heavy nuclei where, for example, for Z = 120 the electronic contribution is −1.34 MeV. (^4) “Just because barks come out of dogs does not mean that dogs are made of barks.” (Denys Wilkinson).
Nuclei with measured masses
Proton Number
Neutron Number
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160
0 0.
0.40.
0.8>1.0 MeV error
Figure 1: Nuclei with measured masses. The solid line is for N = Z. The dashed lines indicate the locations of the magic numbers 8, 20, 28, 40, 50, 82 and 126
atomic masses can be qualitatively understood by the contribution of the masses of the individual protons and neutrons which make up the nucleus. To emphasize this point the ratio M(N, Z) − [ZMH + NMn] [ZMH + NMn]
is plotted in Fig. (1.2) as a function of mass. The largest deviation is found near A = 60 where the total mass is only about one percent smaller than expected from the sum of nucleon masses. The intrinsic properties of neutrons and protons inside the nucleus are essentially the same as those of the free nucleons. Nuclear properties are a result of these nucleons interacting with each other through the exchange of mesons. At some level we will need to include small admixtures of other baryons.
We are interested in understanding the binding energy as a function of N and Z. The total BE are shown in Fig. (1.3) as a function of A. One observes an overall linear increase with A reaching a maximum value of about 2 GeV for the heaviest nuclei. One can bring out more detail by plotting BE/A as in Figs. (1.4) and (1.5) [Note that Fig. (1.4) is the inverse Fig. (1.2)]. The maximum as a function of A in Fig. (1.4) are shown separately in the bottom of Fig. (1.6). These represent the nuclei which are most stable and the most abundant in nature. An expanded portion for the experimental for light nuclei is shown in Fig. (1.7). These are the nuclei at the
BE/A (MeV)
A
0
2
4
6
8
10
0 50 100 150 200 250 300
Figure 4: BE/A as a function of A.
Measured BE/A
Proton Number
Neutron Number
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160
<6. 7.27.
8.08.
8.8 MeV
Figure 5: BE/A as a function of N and Z.
top of the “ridge-of-stability” in binding energy or those at the bottom of the “valley- of-stability” in mass. For a fixed A value nuclei away from the valley of stability beta decay until they reach the bottom of the valley. Nuclear structure models are used to understand the overall features of these data as well as the deviations from the averge.
The maximum in the binding energy per nucleon occurs for 58 Fe. 58 Fe represents the most bound (lowest energy) state for nucleons. Thus fusion of two light nuclei with a combined mass of A < 58 usually results in energy release. The fusion of deuterium and tritium is the main reaction being investigated for controlled fusion reactors. Other fusion processes are important for solar energy and for the creation of elements up to A = 58 in stellar environments. The falloff in binding energy per nucleon above A = 58 implies that most of these nuclei can spontaneously decay into lighter products. The most common of these decay processes are alpha decay, where a (^4) He is emitted, and fission, where the nucleus breaks up into two roughly equal mass
fragments. The fission products are usually accompanied by neutrons. Intermediate decay modes, where light fragments such as 14 C are emitted, are also possible and have also been observed, but their decay rate relative to alpha decay is extremely small. Although most heavy nuclei have a positive Q value for spontaneous decay, many of them have lifetimes on the order of the age of the universe and thus exist in nature, due to the hindrance of tunneling through the Coulomb barrier.
1.2 Q values and separation energies
In this section we consider energy conservation for nuclear transformations that in- clude, for example, the fussion of two nuclei a and b into the combined system c
[N, Z]a + [N, Z]b → [N, Z]c (1.4)
or the decay of nucleus a into two other nuclei b and c
[N, Z]a → [N, Z]b + [N, Z]c (1.5)
In general we consider the combinations
∑
i
[N, Z]i →
∑
f
[N, Z]f (1.6)
where N and Z are conserved.
∑
i
Ni =
∑
f
Nf and
∑
i
Zi =
∑
f
Zf (1.7)
BE/A (MeV)
A
5
6
7
8
9
10
0 10 20 30 40 50 60 70
Figure 7: An expanded portion of the experimental values for BE/A (points connected by a line). The liquid-drop model is shown by the dashed line.
This process is characterized by the Q value:
Q =
∑
i
M(Ni, Zi)c^2 −
∑
f
M(Nf , Zf )c^2 =
∑
f
B(Nf , Zf ) −
∑
i
B(Ni, Zi). (1.8)
Spontaneous decay involves a single initial nuclear state and is allowed if Q > 0. In the decay, energy is released in the form of the kinetic energy of the final products. Reactions involving two initial nuclei and are endothermic (a net loss of energy) if Q < 0; the reactions are exothermic (a net release of energy) if Q > 0.
We can consider the Q values associated with the removal of one or two nucleons from a nucleus. These are conventionally defined in terms of the one-nucleon and two-nucleon separation energies, S:
Sn = −Qn = B(N, Z) − B(N − 1 , Z), (1.9)
Sp = −Qp = B(N, Z) − B(N, Z − 1), (1.10) S 2 n = −Q 2 n = B(N, Z) − B(N − 2 , Z), (1.11)
and S 2 p = −Q 2 p = B(N, Z) − B(N, Z − 2). (1.12)
Nuclear drip lines
Proton Number
Neutron Number
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160
0 0.
0.40.
0.8>1.0 MeV error
HF1 inside the drip linesAW outside the drip lines
Figure 8: Nuclei in the Audi-Wapstra-Thibault mass compilation together with the HF1 prediction for nuclei out to the neutron and proton drip lines. Observed nuclei outside the drip lines are indicated.
A negative value for one of these quantities means that the nucleus can spontaneously decay by the emission of neutrons or protons. The boundary between positive and negative values of S is called the drip line. Nuclei inside the drip lines are stable to the spontaneous emission of nucleons, whereas those outside the drip line can spontaneously decay by emission of one and/or two nucleons. The neutron drip line represents a sharp boundary between those nuclei just inside drip line that beta decay with a lifetime on the order of ms, and those outside the drip line that have lifetimes on the order of 10−^20 s associated with the strong interaction decay widths of MeV. For heavy nuclei, the Coulomb barrier can greatly hinder the proton decay, and the the lifetimes for the one or two proton decay of nuclei just outside the proton-drip line can be comparable to or even longer than the lifetimes for beta decay. Fig. (1.8) shows the nuclear chart for nuclei whose masses have been measured together with those that are predicted to lie within the neutron and proton drip lines from a microscopic nuclear model called HF1 [2]. An expanded version of this figure for light nuclei is shown in Fig. (1.9). HF1 is a Hartree-Fock(*) 5 calculation where on the order of 10 parameters of the Hamiltonian are fitted to the experimental binding energies. These figures also show nuclei with measured masses which lie outside the drip lines. For light nuclei, the masses given in the Audi-Wapstra-Thibault compilation are usually
(^5) The (*) indicates an idea or topic that will be covered latter in the book.
Measured one-neutron separation energies
Proton Number
Neutron Number
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160
02
46
8
12 MeV separation energy 10
Figure 10: Measured one-neutron separation energies.
obtained from the observation of resonances in nuclear reactions. As such, it would be better to call these quantities resonance energies rather than masses or binding energies. In addition, there are several light nuclei not shown in Fig. (1.9) whose non-observation in experimental studies implies that they lie outside of the drip lines [3]. For light nuclei the proton and neutron drip lines are experimentally established only up to about A = 24. For heavier nuclei the proton-drip line is observed in a few regions from the observation of proton decays with relatively long lifetimes. The HF1 model gives a prediction for the neutron and proton drip lines. Between A = 40 and A = 200 nearly half of the nuclei expected to be inside the drip lines have not yet been observed in experiments. The properties for many of these unobserved nuclei are critical for the understanding of nuclear models as well as the astrophysical processes in element production. New generations of accelerators such are RIA (the Rare Isotope Accelerator project in the US) are being planned to produce and study these nuclei.
The systematics of the one-neutron separation energies, Sn, are shown in Fig. (1.10) for experiment and in Fig. (1.11) for the HF1 model. As one moves from the proton to the neutron drip lines, the one-neutron separation energies decrease. This decrease is not smooth but shows odd-even oscillations associated with the two-body pairing nature of the strong interaction between neutrons(*).
HF1 one-neutron separation energies
Proton Number
Neutron Number
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140 160
02
46
8
12 MeV separation energy 10
Figure 11: HF1 results for the one-neutron separation energies.
The Sn values vs neutron number for the even-even nuclei with N > Z are shown in Fig. (1.12). Just after the magic numbers 28, 50, 82 and 126 there is sudden decrease in the separation energy due to the fact that neutrons go into valence shells which are loosely bound compared to those which have just been filled at the magic numbers(*).
This jump at the magic numbers can be emphasized by taking the differences in one-neutron separation energy:
∆Sn = B(N, Z) − B(N − 1 , Z) − [B(N + 1, Z) − B(N, Z)]
= 2B(N, Z) − B(N − 1 , Z) − B(N + 1, Z). (1.13)
The values of ∆Sn for the even-even nuclei (N, Z) are shown in the bottom panel of Fig. (1.12). One observes clear peaks at the magic numbers 50, 82 and 126. The magic numbers 8, 20 and 28 also appear as peaks, but the peak for 20 goes away for Z = 10 and 12. The value of peak height is related to the size of the shell gaps(*). Similar results are found for the proton separation energies.
1.3 The liquid-drop model
Nuclei are bound due to the overall attractive strong interactions between nucleons. The strong interaction arises from the exchange of mesons. The interactions are short ranged and occur mainly between neighboring nucleons. In addition, the nuclear interaction saturates, resulting in a nearly constant interior nucleon density and a surface radius approximately equal to 1. 2 A^1 /^3. The analogy of this situation with a droplet of liquid, results in the liquid-drop model for the nuclear binding energies in which the binding energy is expressed in the form:
B(N, Z) = α 1 A − α 2 A^2 /^3 − α 3
Z^2
A^1 /^3
− α 4
(N − Z)^2
A
The four terms on the right-hand side are referred to as the volume, surface, Coulomb and symmetry energy terms, respectively. The first term represents the nearest neigh- bor attractive interaction between nucleons, and the second term represents the cor- rection due to the fact that the nucleons on the surface only interact with those in the interior. The third term is due to the Coulomb repulsion between protons. The fourth term called the symmetry energy arises because the proton-neutron strong interaction is on the average more attractive than the proton-proton or neutron-neutron strong interactions and because the total kinetic energy is minimized when N = Z. The constant α 3 =0.697 MeV is fixed by the Coulomb interaction and the nuclear size. Typical values of the liquid-drop constants that reproduce the average trends in the experimental data are given by α 1 = 15.49 MeV, α 2 = 17.23 MeV and α 4 = 22. MeV.
The comparison of the liquid-drop model for the most stable nuclei is shown in Fig. (1.6). The total is shown with by the line (D). Combinations of individual terms are also shown: the volume term (A), the volume plus surface terms (B), the volume plus surface plus Coulomb terms (C). The term “symmetric nuclear matter” refers to extrapolation of properties of nuclei with N = Z and with no Coulomb interaction to infinite sized nuclei where the surface can be ignored. Thus the symmetric nuclear matter extrapolation of the liquid-drop model used for this example gives BE/A= −15.5 MeV.
The size of the symmetry energy term determines how the binding energy de- creases as we move away from stability. To isolate this term we can examine the chain of nuclei for a fixed A value as a function of Z. Experimental values B(N, Z)/A vs Z are shown in Fig. (1.13) for A = 100. The form of this curve for A = 100 is similar to those for other A values. The binding energy has a maximum at Zmax = 44. With- out the Coulomb interaction between protons, the maximum would occur at N = Z due to the symmetry energy (the α 4 term). When the Coulomb interaction between protons (the α 3 term) is added, the peak is shifted to more neutron-rich nuclei. The
mass excess (MeV/c^2) for A=
Z
35 40 45 50
BE (MeV) for A=
820
830
840
850
860
870
Figure 13: Binding energy per nucleon, BE/A, and the mass excess, ∆, for A = 100 as a function of Z. The points connected by a line are the experimental values and the liquid-drop model for the binding energy is shown by the dashed line.