PHY583 - Note 1a – Some Properties of Nuclei

NUCLEAR STRUCTURE

Milestones in development of nuclear physics:

1896 – birth of nuclear physics, Henri Becquerel discovered radioactivity in uranium compounds.

1911- Rutherford, Geiger & Marsden established that most atomic mass is in the nucleus. Reveal the presence of the nuclear force (short-range, predominant 1014 m & zero at great

distances.)

1930 – Observation of nuclear reactions by Cockroft & Walton, using artificially accelerated particles.

1932 – Discovery of neutrons by Chadwick & conclude that neutrons make up about half of the nucleus.

1933 – Discovery of artificial radioactivity by Joliot & Irene Curie.

1938 – Discovery of nuclear fission by Meitner, Hahn & Strassmann.

1942 – Development of the first controlled fission reactor by Fermi & his collaborators.

SOME PROPERTIES OF NUCLEI

All nuclei composed of protons & neutrons (except ordinary hydrogen nucleus, a single proton)

atomic number (Z ), number of protons in the nucleus.mass number (A), number of nucleons in the nucleus.neutron number (N = A – Z) number of neutrons the nucleus.

X chemical symbols for the elementsAE.g. mass number 56, atomic number 26 i.e. 26 protons & 30 neutrons.

Subscript Z can be omitted, because the chemical symbol can always be used to determine Z.

The isotopes of an element have the same Z value but different N & A values. E.g. Isotopes of hydrogen:

The natural abundances of isotopes can differ substantially. Eg. four isotopes of Carbon

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PHY583 - Note 1a – Some Properties of NucleiNatural abundance of 98.9%;

Charge and Mass

Particle ChargeProton e = + 1.6021773 × 1019 CNeutron e = 1.6021773 × 1019 CElectron 0

Atomic mass (the mass of an atom containing a nucleus & Z electrons) can be measured with great precision with the mass spectrometer.

mP 1836 me

Atomic mass unit (u): Masses of atoms are measured with reference to the carbon-12 atom, which is assigned a mass of exactly 12 u.

1 u = 1.660540 × 1027 kg, mP me

1u 931.4943

Size & Structure of Nuclei

First investigated in Rutherford’s scattering experiment where -particles were directed at a thin piece of metal foil.

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PHY583 - Note 1a – Some Properties of NucleiSome particles were deflected backward, through 180o from the incident direction. Those particles were moving towards a nucleus in a head-on collision.

Rutherford employed an energy calculation to determine distance d, at which a particle approaching the nucleus is turned around by Coulomb repulsion.

kinetic energy of the -particle = electrical potential energy of the system (-particle plus target nucleus)

Rutherford found: 7 MeV -particles approach the nuclei to within 3.2 × 1014 m when the foil was made of gold.

For silver foil, d = 2 × 1014 m.

Rutherford concluded: the positive charge in an atom is concentrated in a small sphere with a radius 1014 m, which he called nucleus.

Most nuclei are approximately spherical & have average radius of:

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Distance of closest approach

PHY583 - Note 1a – Some Properties of Nucleir = r0 A 1/3 ……………13.1 Nuclear radius

A is the mass number & r0 is a constant equal to 1.2 × 1015 m.

Assuming the nucleus is a sphere:

implies that all nuclei have nearly the same density.

When nucleons combine to form a nucleus, they combine as though they were tightly packed spheres as in Fig. 13.3.

Current view: neutron is a fundamental particle & not a proton-electron combination.

Electron cannot exist in the nucleus because:

1. The uncertainty principle would require them to possess unrealistically large kinetic energy compared to the energies actually observed in beta-decay.

2. the much larger moment of the electron would dominate & the nuclei would have magnetic moments of the order of e /2me, in disagreement with actual observation.

Important properties of electrons, protons & neutrons are listed in Table. 13.2.

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PHY583 - Note 1a – Some Properties of Nuclei

Nuclear Stability

Nuclei are stable because of the presence of nuclear force (attractive force, with short very range 2 fm, that act between all nuclear particles)

The protons attract each other via the nuclear force, and at the same time repel each other through the Coulomb force.

The attractive nuclear force also acts between pairs of neutrons, and between neutrons & protons.

No. of stable nuclei 260.

Fig. 13.4 N vs Z of some stable nuclei

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PHY583 - Note 1a – Some Properties of Nuclei

Light nuclei are most stable if they have equal numbers of protons & neutrons (N = Z).

Heavy nuclei are more stable if the number of neutrons exceed the number of protons.

As the number of protons increase, the strength of the Coulomb force increase, which tend to break the nucleus apart.

Hence, more number of neutrons are needed to keep the nucleus stable, since neutrons only experience attractive nuclear forces.

Eventually, the repulsive forces between protons cannot be compensated for by the addition of more protons . This occurs when Z = 83. Elements that contain more than 83 protons do not have stable nuclei.

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PHY583 - Note 1a – Some Properties of Nuclei

Most stable nuclei have even values of A. Certain values of Z and N correspond to unusually high stability in nuclei. These values of N & Z are called magic numbers.

Z or N = 2, 8, 20, 28, 50, 82, 126

The unusual stability of nuclei with progressively larger magic number suggests a shell structure for the nucleus that is similar to atomic shell structure.

Nuclear Spin & Magnetic Moment

An electron has an intrinsic angular momentum associated with its spin.

A nucleus, like an electron, has an intrinsic angular momentum that arises from relativistic properties.

Magnitude of nuclear angular momentum = I is a quantum number called nuclear spin, & may be an integer or a half-integer.

Nuclear angular momentum is the total angular momentum of all the nucleons (orbital + spin angular momentum of each nucleon)

The maximum component of the angular momentum projected along the z-axis is I .

Fig. 13.5 illustrates the possible orientations of the nuclear angular momentum & its projections along

the z-axis for the case I = .

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PHY583 - Note 1a – Some Properties of Nuclei

Nuclear angular momentum has a nuclear magnetic moment associated with it.The magnetic moment of a nucleus is measured in terms of the nuclear magneton

A unit of nuclear magneton is defined as: ............13.3

mp = mass of proton

This definition is analogous to spin magnetic moment of a free electron .

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PHY583 - Note 1a – Some Properties of Nuclei

by a factor of 2000 because of the large difference between the masses of proton &

electron i.e. .

There is no general theory of nuclear magnetism that explains this value.

Even neutron has a magnetic moment: showing that the uncharged

neutron has an internal charge distribution. The sign shows that this moment is opposite to the spin angular momentum of the neutron.

Nuclear Magnetic Resonance & Magnetic Resonance Imaging

Nuclear magnetic moment (as well as electronic magnetic moments) precess when placed in an external magnetic field.

The frequency at which they precess, called the Larmor precessional frequency, L, is directly proportional to the magnetic field.

This precession is drawn in Fig. 13.6a.

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PHY583 - Note 1a – Some Properties of Nuclei

e.g. For a proton in a 1-T magnetic field: L = 42.577 MHz

The potential energy of a magnetic dipole moment in an external magnetic field B = B.

When is parallel (as closely as quantum physics allow) to B, the potential energy of the dipole moment in the field is minimum, Emin.

When is as antiparallel as possible, the potential energy has its maximum value, Emax.

Fig. 13.6b shows these two energy states for a nucleus with a spin of .

It is possible to observe transitions between these two spin states through a technique called nuclear magnetic resonance (NMR).

A constant magnetic field B (as in Fig. 13.6a) is introduced to align magnetic moments, along with a second weak, oscillating magnetic field oriented perpendicular to B.

When the frequency of the oscillating field is adjusted to match the Larmor precessional frequency (L), a torque acting on the precessing moments causes them to “flip” between the two spin states.

These transitions result in a net absorption of energy by the spin system, an absorption that can be detected electronically.

A sketch of the apparatus used in NMR is given in Fig. 13.7

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PHY583 - Note 1a – Some Properties of Nuclei

The absorbed energy is supplied by the generator producing the oscillating magnetic field.

NMR and a related technique called electron spin resonance are extremely important methods of studying nuclear and atomic systems & how these systems interact with their surroundings.

Fig. 13.8 shows a typical NMR spectrum

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PHY583 - Note 1a – Some Properties of Nuclei

An NMR spectrum of .

Magnetic resonance imaging (MRI) is a widely used diagnostic technique based on NMR.

Because about 2/3 of the atoms in the human body are hydrogen, which gives a strong NMR signal, MRI works very well for viewing internal tissues.

In MRI the patient is placed inside a large solenoid that supplies a spatially varying magnetic field.

Because of the gradient in the magnetic field, protons in different parts of the body precess at different frequencies, and so the resonance signal can provide information on the positions of the protons.

A computer is used to analyse the position information to provide data for the construction of a final image. Fig. 13.9 shows an MRI scan of a human brain.

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PHY583 - Note 1a – Some Properties of Nuclei

Another advantage of MRI over other imaging techniques: It causes minimal damage to cellular structures.

Photons associated with the radio frequency signal used in MRI have energies of 107 eV.

Because molecular bond strengths are much greater ( 1eV), the radiofrequency radiation causes little cellular damage.

In comparison, x rays & gamma rays have energies ranging from 104 to 106 eV and can cause considerable cellular damage.

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