Nuclear Engineering

Shape of the Nucleus: Electric Moments and magnetic Moment

The shape of a nucleus is an important field of study. Some nuclei are almost spherical, whereas some are deformed. Determination of degree of deformation of nucleus both theoretically and experimentally is a complicated topic. An initial idea of deformation can be obtained from electric quadrupole moment of a nucleus. The electric quadrupole moment measures the departure of a nucleus from spherical symmetry. Here we adopt a simple way to determine electric quadrupole moment. Let us consider that the charge is situated at the point S/ (Fig. m1.5) having Cartesian co-ordinates (x,y,z).




FIGURE m1.5 Coordinate system for evaluation of potential

The potential at a point S on the Z-axis due to this charge is




where

where r is the distance of the charge from the origin and is given by r = (x ² + y ² + z ²)^1/2 and cosθ = z/r defines the angle between a and r.






In eqn. (m1.6), the coefficient of 1/a is known as monopole strength, the coefficient of 1/a² is the z-component of the dipole moment, the coefficient of 1/a³ is the z-component of the quadrupole moment. Similarly, considering higher order terms (h.o.t.) one get higher moments like octupole moments, etc. The first term in equation (m1.6) is the ordinary Coulomb potential.,
The net quadrupole moment of the nucleus is:


Putting cos θ = z/r, we get



From electrostatics, it is well known that the electric field due to a uniformly charged shell at all points outside of it is the same as if all the charge were concentrated at the centre of the shell, i.e the centre of the nucleus here. Therefore, from equation (m1.8) one can easily find out that a nucleus with a spherical symmetric charge distribution has no electric quadrupole moment or higher electric moments. If a uniformly charged ellipsoid is considered than it has both a net charge and a quadrupole moment, but its dipole moment is zero. Its charge distribution can be well approximated, as shown in Fig. m1. 6, by a spherical charge distribution plus an elementary quadrupole.¹




	FIGURE m1.6 A uniformly positively charged distribution of ellipsoidal shape (at the left), which is electrically equivalent to a uniform positively charged sphere plus extra positive and negative charges as shown at the right.


From a detail quantum mechanical analysis, one can show that for a spherical distribution of charges Q = 0, and Q > 0 and Q < 0 are for prolate (cigar-like shape) and oblate distribution (discus-shape) (Fig. m1.7) respectively. ²




	FIGURE m1.7 Pictorial representation of spherical, prolate and oblate shape nucleus


Nucleus — a tiny magnet

In the above section we have seen that nucleus has a charge distribution and various electrical moments. Apart from charge, nucleus also has a spin,* which essentially makes nucleus to behave like a tiny magnet with well defined magnetic moment. Every nucleon inside a nucleus has a magnetic moment due to its spin motion. The magnetic moment can be calculated considering a classical orbital motion of a particle, one can write the angular momentum³



where and are radius of the orbit and linear momentum of the particle respectively. If the current produced by the motion of the charge is equal to i then the magnetic moment (μ) associated with it is


Whereis a unit vector vertical to the circular motion, in the rotation sense going with a positive current. For a proton or electron combining the above equations one can write

In vector notation

Quantum mechanically one can write the relation between the operators for the above circular motion¹

and

the z component of the magnetic moment. From the quantum mechanical consideration,² Eqn. (m1.14) can be written as


The unit is known as Bohr (for electron) magneton or nuclear (if m is the nucleon mass) magneton. Therefore,

For intrinsic spin an analogous relationship can be written as

where g_s is the gyromagnetic ratio. For electron the gyromagnetic ratio turns out to be almost -2 and for protons and neutrons 5.5855 and -3.8263 respectively¹. The individual moments can be combined and the total moment can be written as

where g_j is the nuclear gyromagnetic ratio. The magnetic moments of electron proton and neutron are


and

respectively. The numerical factors appear in case of nucleons is due to the fact that neutrons and protons have electric charge distribution i.e. of charged meson clouds surrounding them.


———————————————————————————————————————– *‘Spin’ is an intrinsic property of elementary particles, like charge, mass etc. You will understand this only after reading Dirac’s theory of quantum mechanics, where special relativity and Schrodinger’s equation is unified. ———————————————————————————————————————–

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Shape of the Nucleus: Electric Moments and magnetic Moment

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