Download PHY583 - Note 2a - Properties of Nuclei

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  1014 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 elements
A
E.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
Natural abundance of
98.9%;
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PHY583 - Note 1a – Some Properties of Nuclei
Charge and Mass
Particle
Proton
Neutron
Electron
Charge
e = + 1.6021773 × 1019 C
e = 1.6021773 × 1019 C
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 × 1027 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.
Some particles were deflected backward, through 180o from the incident direction. Those particles
were moving towards a nucleus in a head-on collision.
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PHY583 - Note 1a – Some Properties of Nuclei
Rutherford employed an energy calculation to determine distance d, at which a particle approaching
the nucleus is turned around by Coulomb repulsion.
Distance of closest approach
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 × 1014 m when the foil was
made of gold.
For silver foil, d = 2 × 1014 m.
Rutherford concluded: the positive charge in an atom is concentrated in a small sphere with a
radius  1014 m, which he called nucleus.
Most nuclei are approximately spherical & have average radius of:
r = r0 A 1/3
……………13.1 Nuclear radius
A is the mass number & r0 is a constant equal to 1.2 × 1015 m.
Assuming the nucleus is a sphere:
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PHY583 - Note 1a – Some Properties of Nuclei
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:
neutron has an internal charge distribution. The
showing that the uncharged
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  107 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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