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Home , Nuclear force, Nuclear structure

PHY401 - Nuclear and Particle

Monsoon Semester 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 08

Wednesday, September 9, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Contents

1 The Nuclear 1 1.1 -Nucleon Scattering ...... 2 1.2 The Deuteron ...... 4 1.3 Nature of the ...... 4

2 The Structure of Nuclei 6 2.1 The Fermi Gas Model ...... 7

1 The Nuclear Force

The nuclear force is responsible for holding the nucleus together. This is an interaction between colorless . Its range is of the same order of magnitude as the nucleon diameter. In contrast to the situation in , it is not possible to obtain information about the nuclear force by studying the structure of the nucleus. To a first approximation, the nucleus may be viewed as a collection of nucleons in a potential well. The potential must be obtained by analyzing two-body systems such as nucleon-nucleon scat- tering and the - bound state (the deuteron). There are also considerably greater theoretical difficulties in elucidating the connection between the nuclear and the structure of the nucleon than for the atomic case. 2 gs This is primarily a consequence of the strong coupling constant αs = 4π . PHY401 - Nuclear and Monsoon Semester 2020

1.1 Nucleon-Nucleon Scattering

Nucleon-nucleon scattering at low , below the production threshold, is purely elastic. Note that in process the kinetic of a particle is conserved in the center- of- frame, but its direction of propagation is modified (by interaction with other particles and/or potentials). At such energies the scattering may be described by non-relativistic . It is found that the nuclear force depends upon the total and of the two nucleons. (In and particle physics, isospin (I) is a quantum number related to the .)

Scattering phases

Consider a nucleon coming in “from infinity” with E and momentum p which scatters off the potential of another nucleon. The incoming nucleon may be described by a plane wave and the outgoing nucleon as a spherical wave. The cross-section depends upon the phase shift between these two waves. For states with well defined spin and isospin the cross-section of nucleon-nucleon scattering into a solid angle element dΩ is given by the scattering amplitude f(θ) of the reaction

dσ = |f(θ)|2. (1) dΩ

For scattering off a short-ranged potential a partial-wave decomposition is used to describe the scattering amplitude. The scattered waves are expanded in terms with fixed l. In the case of elastic scattering the following relation holds at large distances r from the centre of the scattering ∞ 1 X f(θ) = (2l + 1)eiδl sin δ P (cos θ), (2) k l l l=0 where √ 2π E 1 |p| 2ME k = = = = = (3) λ ~c λ¯ ~ ~ is the is the wave number of the scattered nucleon, δl a phase shift angle, and Pl the angular momentum eigenfunction, an l-th order Legendre polynomial.

The phase shifts δl describe the phase difference between the scattered and unscattered waves. They contain the information about the shape and strength of the potential and the energy dependence of the cross-section. The partial wave decomposition is especially convenient at low energies since only a few terms enter the expansion given in Eq. (2).

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This is because for a potential with range a we have

|p|a l ≤ . (4) ~

The phase shift δ0 of the partial waves with l = 0 (i.e., s waves) is decisive for nuclear binding. From the above equation we see that the s waves dominate proton-proton scattering (potential range about 2 fm) for relative momenta less than 100 MeV/c.

The Legendre polynomial P0 is just 1, i.e., independent of θ.

The phase shifts δ0 can be measured in nucleon-nucleon scattering.

For momenta larger than 400 MeV/c δ0 is negative, below this it is positive. We learn from this that the nuclear force has a repulsive character at short distances and an attractive nature at larger separations. The scattering experiments indicate that the force has a short ranged repulsive and a longer ranged attractive nature. Since the repulsive part of the potential increases rapidly at small r, it is known as the hard core potential. A complete scattering phase analysis leads to the nuclear potential shown in Fig. 1.

100

50

MeV 1 2 3

) 0 r (

V r fm

−50

−100

Figure 1: Sketch of the radial dependence of the nucleon-nucleon potential for l = 0. Note that the spin and isospin dependence of the potential is not shown.

We may obtain a general form of the nucleon-nucleon potential from a consideration of the relevant dynamical quantities. We will, however, neglect the internal structure of the nucleons, which means that this potential will only be valid for nucleon-nucleon bound states and low energy nucleon-nucleon scattering.

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1.2 The Deuteron

The deuteron is the simplest of all the nucleon bound states, that is, the atomic nuclei. It is therefore particularly suitable for studying the nucleon-nucleon interaction. This proton-neutron system is mostly made up of an l = 0 state. If it were a pure l = 0 state then the would be spherically symmetric. The moment would vanish. Deuteron has a non-zero quadrupole moment. The magnetic dipole moment would be just the sum of the proton and neutron magnetic mo- ments. It is slightly different for a deuteron. Current values for the range of the nuclear force, and hence the effective extension of the poten- tial, is a ≈ 1.2 to 1.4 fm. This implies that the depth of the potential is V ≈ 50 MeV. This is much greater than the deuteron B, which is just 2.23 MeV. Binding energy represents the energy required to break up a nucleus into its constituent nucleons ( and ). The nuclear binding in the deuteron is relatively “weak” and the bound state is much more spread out inside the potential well. This means that the average kinetic energy is comparable to the average depth of the potential, and so the binding energy, which is just the sum of the kinetic and potential energies, must be very small. The binding energy of the nucleons in larger nuclei is somewhat greater than that in the deuteron and the nuclear density is accordingly larger. Qualitatively we still have the same situation: a relatively weak effective force is just strong enough to hold nuclei together. The properties of the nuclei we obtain from scattering experiments bear witness to this fact: it tells us that we can describe the nucleus as a degenerate Fermi gas and and also that the nucleons have a greater mobility in . The weak nuclear bond, since the potential energy is comparable in size to the kinetic energy, means that the nucleons are delocalized inside the potential well.

1.3 Nature of the Nuclear Force

We now turn to the task of understanding the strength and the radial dependence of the nuclear force. A sketch of the radial shape of the nucleon-nucleon potential, that has been derived from a huge amount of elastic proton-proton and neutron-proton scattering data, is shown in Fig. 2. It resembles very much the potential between two that is repulsive when the two atoms overlap and attractive at larger distances. Ideally, we would like to derive the nuclear force between nucleons from quantum chromody- namics (QCD), the theory of the strong interaction between , mediated by the exchange of .

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V(r) short range intermediate long range

1 2 3

rfm

Figure 2: Sketch of the nucleon-nucleon potential derived from phase shift analyses of low-energy elastic nucleon-nucleon scattering data.

Such a derivation is, however, not yet possible, one of the reasons being that nucleons are color-neutral. At distances larger than the confinement scale only color-neutral objects can be exchanged between them, i.e., only the exchange of two or more gluons, of -antiquark pairs or of is possible. Let us briefly look at a particular description of the nuclear force: the -exchange model.

The nuclear force in meson-exchange models

Ever since Yukawa in 1935 first postulated the existence of the pion, there have been attempts to describe the inter-nuclear forces in terms of meson exchange. The exchange of mesons with mass m leads to a potential of the form

− mc r e ~ V = κ , (5) r where κ is a constant. This is known as the . Nucleon-nucleon interaction by meson exchange is sketched in Fig. 3. Nucleons can exchange several types of mesons depending on the distance between them. The range of the exchange potential decreases as the meson mass increases. At large distances the exchange of single dominates. At shorter distances massive mesons, such as the σ mesons, come into play. (It has mass 400-550 MeV/c2 and decays dominantly to two pions.) At distances below approximately 0.8 fm the potential is governed by the exchange of the two

5 / 11 PHY401 - Nuclear and Particle Physics Monsoon Semester 2020 lightest vector mesons, ρ and ω, with masses of 780 MeV/c2. For vector particles, one finds a strong repulsive central force. The most recent approach to describe the nucleon-nucleon potential and the nuclear force is an effective field theory based on chiral . This theory has pions, the nucleon, and the ∆ resonance (all point-like) as effective fundamental degrees of freedom rather than quarks and gluons. It is nowadays considered by its advocates as the state-of-the-art method for microscopic calculations of interactions at low energies of and light mesons, of two-nucleon and many- nucleon forces, and many aspects of .

N

N mesons

N

N

Figure 3: At distances between neighbored nucleons larger than 0.5 fm the nuclear force is mediated by the exchange of mesons. At large distances one-pion exchange dominates, the repulsion at small distances is due to the exchange of vector mesons. In the intermediate range the exchange of a scalar object (two-pion exchange) plays an important role.

2 The Structure of Nuclei

Nuclei that are in their ground state or are only slightly excited are examples of degenerate Fermi gas. The nuclear density is determined by the nucleon-nucleon interaction – essentially by the strong repulsion at short distances and the weak attraction between nucleons that are further apart. The nucleons are not localized in the nuclei but rather move around with comparatively large momenta, of the order of 250 MeV/c. This mobility of the nucleons in the nucleus is a consequence of the fact that, as we have seen for the deuteron, the bonds between nucleons in the nucleus are “weak”. The average distance between the nucleons is much larger than the radius of the nucleon hard core. Not only the nuclear density but also the shapes of the nuclei are fixed by the nucleon-nucleon interaction. A nucleus in equilibrium is not always a sphere; it may be ellipsoidal or even more deformed.

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2.1 The Fermi Gas Model

At zero temperature, all will occupy the lowest state in the potential well. See Fig. 4.

EF

Bosons

Figure 4: Bosons and fermions in the potential well.

Bosonic attraction will result in a collapse of the system into a coherent state of high density, the so called Bose-Einstein condensate. Observation of the Bose-Einstein condensate in cold atomic clouds trapped with laser light led to the Nobel Prize in 1997. The total energy of bosonic system at zero temperature is equal to the number of bosons times the energy of the lowest state in the well. Fermions cannot occupy the same state, even at zero temperature. At zero temperature, the first sits in the lowest state in the potential well, each next fermion sits in the lowest free state in the well. Consequently, n fermions occupy the n lowest-energy states in the potential well.

The energy of the highest occupied state is called the Fermi energy, EF . See Fig. 4. At zero temperature all states up to the Fermi energy are occupied, while all states above the Fermi energy are free. The total energy of the system is equal to the sum of energies of the occupied states. According to the spin-statistic theorem, particles with integer spins are bosons, and particles with half-integer spins are fermions. For example He-4 is a (this is true for with even neutron number), He-3 is a fermion (true for isotopes with odd neutron number). The Fermi gas model defines properties of a system of non-interacting fermions in an infinite potential well. This model predicts gross properties of various quantum-mechanical systems, for example elec- trons in metals, or nucleons in nuclei. It assumes that all fermions occupy the lowest energy states available to them up to the Fermi energy, and that there is no excitations across the Fermi energy at zero temperature.

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In nuclei, the Fermi gas model assumes that the protons and the neutrons fill two separate potential wells, independently. See Fig. 5. However, this model assumes a common Fermi energy for the protons and neutrons in stable nuclei. If Fermi energy for protons and neutrons are different, then the β decay transforms one type of nucleons into the other until the common Fermi energy (that is stability) is reached. The protons and neutrons that together build up the nucleus are viewed in the Fermi gas model as comprising two independent systems of nucleons. As spin-1/2 particles they naturally obey Fermi-Dirac statistics. It is assumed that the nucleons, inside those constraints imposed by the Pauli principle, can move freely inside the entire nuclear volume. The potential that every nucleon feels is a superposition of the potentials of the other nucleons. We can assume that this potential has the shape of a well, i.e., that it is constant inside the nucleus and stops sharply at its edge. See Fig. 5.

Proton potential

Protons Neutrons B

Neutron potential p n EF EF

Figure 5: Sketch of the proton and neutron potentials and states in the Fermi gas model.

Let us denote the dimensions of the well along the x, y, z coordinates as Lx, Ly and Lz. Then the energies are 2 2 2 2 2 ! π ~ nx ny nz Enx,ny,nz = 2 + 2 + 2 . (6) 2m Lx Ly Lz The momentum components of the particles are

nx px = ~kx = ~π , (7) Lx ny py = ~ky = ~π , (8) Ly nz pz = ~kz = ~π . (9) Lz

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For simplicity we assume

Lx = Ly = Lz = L. (10)

This gives π2 2 E = ~ n2 + n2 + n2 . (11) nx,ny,nz 2mL2 x y z The corresponding Fermi momentum is

p2 p E = F =⇒ p = 2mE . (12) F 2m F F

We count the number of occupied states up to the Fermi momentum. We get π2 2 p2 + p2 + p2 < p2 =⇒ ~ (n2 + n2 + n2) < p2 . (13) x y z F L2 x y z F Thus we have 2 2 2 2 2 pF L nx + ny + nz < . (14) π2~2 For each combination of (nx, ny, nz) we have a single spacial state.

The quantum numbers (nx, ny, nz) change by one. A change by a unit in each direction defines a cube of a unit volume. Thus we can state that there exists one spacial state per unit volume in the space defined by the quantum numbers (nx, ny, nz).

The volume in the (nx, ny, nz) space is limited by the condition

2 2 2 2 2 pF L 2 nx + ny + nz < = R . (15) π2~2 That is p L R = F . (16) π~ This condition defines a sphere with radius R in the (nx, ny, nz) space. See Fig. 6. The volume of the sphere is the number of states up to the Fermi momentum since there is one state per unit volume.

Since nx > 0, ny > 0 and nz > 0 we have to take only 1/8 of the sphere cut by positive x, y, z axes. Nucleons have spins (up or down). Thus we have two states per unit volume as defined by two orientation of spins. With these corrections we can calculate the number of occupied states. We have 1 4 1 p L3 n = 2 × × πR3 = π F . (17) 8 3 3 π~

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Unoccupied states ny

Fermi surface rn

Occupied states nx

nz

Figure 6: Occupied and unoccupied states in the Fermi gas model.

The Fermi momentum is

√3 1 √ p = 3π2 3 n F ~ L r √3 n = 3π2 3 ~ L3 r √3 n = 3π2 3 . (18) ~ V

We have calculated the Fermi momentum pF in terms of the density of states n/V . Denoting A as the number of nucleons and Z as the number of protons, we have the volume

4 4 V = π(r A1/3)3 = πr3A, (19) 3 0 3 0 where r0 is an experimentally measured number, r0 = 1.2 fm. Thus we have the Fermi momentum r √3 n p = 3π2 3 F ~ V r r π Z = ~ 3 32 3 . (20) r0 4 A

The Fermi momentum for protons is

r r p ~ 3 2 π 3 Z pF = 3 . (21) r0 4 A

The Fermi momentum for neutrons is r r n ~ 3 2 π 3 (A − Z) pF = 3 . (22) r0 4 A

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Assuming A/Z ∼ 2 we get an estimate for the Fermi momentum

p n pF = pF ≈ 250 MeV/c. (23)

This corresponds to the speed

vF pF c 250 βF = ≈ 2 ≈ = 0.27 (24) c mpc 940

References

[1] B. Povh, K. Rith, C. Scholz and F. Zetsche, Particles and Nuclei: An Introduction to the Physical Concepts, 6th edition, Springer (2008).

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