Nuclear electric quadrupole moment

Introduction

In this particular article Nuclear electric quadrupole moment, we are going to discuss about basics of electric dipole . Charge distribution in nuclides and units of quadrupole moment.

Meaning of electric moments

We know that nuclides have the positive charge. The meaning of electric moments of charges in the nuclide can be understood, if it is assumed that positive charge is continuously distributed throughout the nucleus. To calculate the potential at an external point, the nucleus is divided into infinitesimally small volume elements. Let the volume of one of these elements. Let the volume of one of these elements be dV at a position vector r from the center of the nucleus and the charge density in it be ρ(r) C/m².

∴ The amount of charge inside volume element dq = ρ(r) dV

Let a point A is an external point on the Z-axis at a distance ‘a’ from the volume element dV and let OA = R. The potential at a point A due to charge inside volume dV will be-

dφ = 1/ 4π∈(ρ(r)/a)dV…………..(1)

Where ∈ is a permittivity of the medium.

The total potential at an external point A due to the entire charge of the nucleus.

φ = 1/4π∈ ∫(ρ(r)/a)dV…………..(2)

If the position vector r of dV makes an angle θ with Z-axis, then from Fig.

Theory

a = R – r

or a = √ R² + r² – 2Rr cosθ

or 1/a = 1/R [1+r²/R²-2rcosθ/R]−½…….(3)

Since charge distribution is in a limited volume, thus for external points in the limit R>>r and using binomial expansion, we can write

1/a = 1/R [1-1/2(r²/R²-2rcosθ/R)+3/8(r²/R²-2rcosθ/R)²….]

or 1/a = 1/R [1+rcosθ/R+r²/2R²(3cos²θ-1)+….]…….(4)

Using equation  (4) in equation (2),

φ = 1/4π∈ [1/R∫ρ(r)dV + 1/R²∫ρ(r)rcosθdV+1/2R³∫ρ(r)r² (3cos²θ-1)dV+….]…(5)

In equation (5), if we put

P₀ = ∫ρ(r)dV

p₁ = ∫ρ(r)r cosθdV = ∫ρ(r)zdV

p₂= ∫ρ(r)r²(3cos²-1)dV = ∫ρ(r)(3z²-r²)dV…(6)

then this equation can be written be as,

φ = 1/4π∈(p/R) + 1/4π∈(p/R²) + 1/4π∈(p/2R³)…

In this equation first term is the potential due to a point charge (monopole), second term is a potential due to dipole moment along Z-axis, third term is a potential due to quadrapole moment along Z-axis and so on. therefore p₀, p₁, p₂ are called  monopole, dipole, quadrapole moments respectively.

Nuclear electric dipole moment p is zero for all nuclides in ground state as well as in non-degenerate excited states. This can be explained by law of conservation of parity.

Quadrupole moment and charge distribution in nuclides

The quadrupole moment of a nuclide having charge density ρ(r) is defined classically as

Q = ∫ρ(r)(3z²-r²)dV…(8)

where r is the distance of the volume element dV from the centre of the nucleus and z is the component of its distance along Z-axis. The distribution of charge in the nucleus is due to the protons. If the coordinates of ith proton are (xi ,yi, zi), then

Q = ∑e(3zi²-ri²)  …(9)

where e is the charge of proton.

The quadrupole moment of the nucleus depends on the symmetry of the charges. three possibilities of charge distribution in the nucleus is as follows

In case of spherical symmetry charge distribution,

r² = x² + y² + z²

thus, ‾r² = 3z²‾

In this case quadrupole moment of a nucleus is found to be zero ,i.e. Q=0. Those nuclei, whose quadrupole moment is zero, have spherical symmetric charge distribution is streched along Z-axis, then

3z²‾>r²

and Q>0. Such type of charge distribution is called spheroidal charge distribution. If the nuclear charge distribution is of type as shown in above fig. i.e. charge distribution is stretched perpendicular to Z-axis, then

3z²‾< r²

and Q < 0, such type of charge distribution is called oblate spheroidal charge distribution.

Unit of quadrupole moment of nucleus

In the integral of quadrupole moment expression, total charge of the nucleus is already incorporated. Therefore it has become as a custom to divide the value of e in writing the observed values of quadrupole moemnt and then the unit of quadrupole moment comes out to be equivalent to the unit of area i.e m². Unit m² is very large for quadrupole moment. Therefore, another unit barn is  generally written for the unit of quadrupole moment.

1 barn = 100¯²  m²