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Section 7: BASIC NUCLEAR CONCEPTS
In this section, we present a basic description of atomic nuclei, the stored energy contained within them, their occurrence and stability
Basic Nuclear Concepts
EARLY DISCOVERIES [see also Section 2] Radioactivity - discovered in 1896 by Henri Becquerel. Types of radiation observed: alpha ( ) rays (^4 He nuclei); beta ( ) rays (electrons) ; gamma ( ) rays (photons) Proposed atomic models : built of positively and negatively charged components.
- Planetary model: Light electrons (-charge) orbiting a massive nucleus (+charge): - 'Plum pudding' model (J. J. Thompson): In this model, electrons are embedded but free to move in an extended region of positive charge filling the entire volume of the atom. Thompson found it difficult to develop this model. For example he could not account for the patterns of discrete wavelengths in light emitted from excited atoms. In the early 1900s, Rutherford and co-workers, by performing experiments scattering particles off gold, confirmed the planetary model with a small, massive nucleus at its centre. The problem of the stability of such an atom was realized early on but not explained until the development of quantum mechanics [see Section 3]. Discovery of the neutron 1932 – The neutron was identified by James Chadwick from observations of the effects of radiation emitted when beryllium is bombarded with alpha particles. This gave the basic nuclear framework (Heisenberg, Majorana and Wigner) that the nucleus consists of nucleons (neutrons and protons) held together by a strong, short-range binding force, with a strength independent of the type of nucleon.
Nuclear size and density Scattering experiments showed that the nuclear radius varies as cube root of the mass number A , i.e. r = r 0 A 1/3^ (5.1) with r 0 1.2 10 -15^ m = 1.2 fm. Example: Carbon, A = 12; we get r nuc = 2.7 10 -15^ m. From Section 1, the radius of a carbon
atom = 1.3 10 -10^ m. Therefore, the ratio: 12 .. 37 101010 50 ,^1000
15 atom
nuc r r.
The nucleus is a tiny object compared with an atom. The nuclear volume (4 r^3 / 3) A, which implies nuclear matter density ~ constant. Also, nuclear mass ~ 99.98% of the atomic mass, from which we deduce that the density of nuclear matter is huge ~ 1018 kg m-3^ 1 cm^3 ≈ 10,000 battleships!
NUCLEAR MASS AND ENERGY Binding energy Nuclear mass < sum of the masses of constituent neutrons and protons. E.g. Deuteron (bound neutron and proton): m d = 2.01355 u < m n + m p = 1.00867 + 1.00728 = 2.01595 u Difference in mass Δ m = m d – ( m n + m p). E = mc^2 (Einstein) enables us to express energies in either mass or energy units. Thus, Δ m c^2 = the binding energy of nucleons in a nucleus, which is negative. Its magnitude is the energy released when the nucleons fuse into a nucleus. Conversely, it is the energy needed to separate the nucleus into its N neutrons and Z protons. Released energy or nuclear binding energy B is given by B(A,Z) = ( Zm p + Nm n -m(A,Z)) c^2 = Δ mc^2 (5.2) where m p, m n and m(A,Z) are masses of the proton, neutron and nucleus (atomic mass A ), respectively. B increases with A – and we usually quote the average binding energy per nucleon ( B/A ). This figure shows the variation of B/A with A :
The constant inter-nucleon separation explains the approximately constant density of nuclear matter. The short range of the nuclear force means a nucleon senses only its nearest neighbours. In a light nucleus, all the nucleons will sense each other – so the binding energy of the nucleon ( B/A ) increases with A. In heavy nuclei, nuclear size > range of nuclear force. A nucleon senses approximately a constant number of neighbours and hence, the nuclear B/A levels off at high A. This feature is referred to as the saturation of the nuclear force.
Coulomb force This is a repulsive force acting between protons inside the nucleus. It is weaker than the nuclear force. At a typical inter-nucleon separation, the Coulomb energy of two protons is about 1/50th^ of the nuclear energy. However, the Coulomb force is long-ranged: A proton feels the electric force due to all the other protons. Thus, while the nuclear B/A levels off as A and the nuclear size increase, the Coulomb energy continues to increase because Z increases approximately as A/ 2. The Coulomb energy is of opposite sign to the nuclear energy, hence B/A gradually falls off at large A from a maximum. at A ≈ 60, as shown in the figure.
NUCLEAR POTENTIAL AND ENERGY LEVELS Nuclear potential Outside the nucleus r > R , (nuclear radius): the nuclear PE experienced by a neutron → 0 in a few fm.
Inside the nucleus, any nucleon feels an average potential due to the interaction with its neighbours.
In addition, a proton feels the repulsive Coulomb potential, which decreases the net interior PE. It also gives rise to a (Coulomb) barrier in the surface region, which means that energy is required to push a proton towards the surface of a nucleus. This energy increases up to the edge of the nucleus where the total PE reaches a peak B as the nuclear attraction begins to take effect. Beyond the peak, the nuclear force dominates and the PE decreases at closer distances to form a potential well.
Nucleon states Nucleons of both types bound in a nucleus are confined in a potential-energy well. These nucleons can occupy only allowed states – like electrons in an atom. However, energies of nuclear states ( millions of eV) are much greater than electron atomic energies (eV).
We can estimate the nucleon kinetic energy, Using the de Broglie relation: p = h/ λ Inside a nucleus, the nucleon wavelength λ ~ nuclear size D ≈ 10 fm.
Therefore, p = h/D and the kinetic energy: 2 2
2 2 2 KE (^22) mc D h c m p (5.4)
where m = nucleon mass.
Substituting values gives KE (^229311238) (MeV)(MeV 100 fm)(fm) 2 8 MeV.
2 2 2 2
2 2 mc D
hc
This energy must be overcome by the nuclear PE to bind a nucleon in a nucleus.
Inside a nucleus, nucleons arrange themselves in shells (like electrons in an atom). They obey the Pauli Exclusion Principle that no two identical nucleons can exist in the same state.
However, at higher A , we find N > Z. The Coulomb energy becomes more and more important at high Z , making the potential well shallower for protons. Their allowed states are shifted upwards in energy compared with the neutron states, which means that the lowest energy is when N > Z.
NUCLEAR INSTABILITY
Either side of the Valley of Stability: For a given A , nuclei for which N and Z lie above or below the line of stability, are not the most stable. Energy is released by changing a neutron into a proton or vice versa. The force, causing such a change, is the weak nuclear force , whose strength is about 10-12^ times that of the strong nuclear force and so it plays no part in holding the nucleus together. It causes a neutron-rich or proton-rich nucleus to decay into a more stable form via β radioactivity – see Section 8.
Beyond the Valley of Stability: There are upper limits to A and Z for which stable nuclei can exist ( A = 209 and Z = 83). Beyond 209 Bi, nuclei may transform into a more stable product by emitting an α particle - via α radioactivity. Naturally occurring thorium and uranium undergo alpha decay, albeit with very long average lifetimes. Very heavy nuclei can also decay by undergoing fission into two approximately equal fragments
