Question

The unit of length convenient on the nuclear scale is a fermi : 1 f = 10¯¹⁵ m. Nuclear sizes obey roughly the following empirical relation : r = rₒ A⅓ where r is the radius of the nucleus, A its mass number, and ro is a constant equal to about, 1.2 f. Show that the rule implies that nuclear mass density is nearly constant for different nuclei. Estimate the mass density of sodium nucleus. Compare it with the average mass density of a sodium atom obtained in Exercise. 2.27.

Answer 1

The radius of the nucleus is given by the relation,

r= r 0 A 1 3 (1)

Where, r is the radius of the nucleus, A is the atomic number and r 0 =1.2 f is a constant.

Volume of nucleus is given as,

V= 4 3 π r 3

Substituting the value of r from equation (1) in the above formula, we get

V= 4 3 π ( r 0 A 1 3 ) 3 = 4 3 π r 0 3 A  m 3

Mass of a nucleus in kg can be written as,

M=A amu =A×1.66× 10 −27  kg

Density of nucleus is given as,

ρ= M V

By substituting the given values in the above expression, we get

ρ= A×1.66× 10 −27 4 3 π r 0 3 A = 3×1.66× 10 −27 4π r 0 3   kgm −3

The above relation shows that nuclear mass depends only on the constant, r 0 . Hence, the nuclear mass densities are nearly constant for different nuclei.

Consider the case of sodium nucleus.

The density of sodium nucleus is given as,

ρ sodium nucleus = 3×1.66× 10 −27 4π ( 1.2× 10 −15 ) 3   = 4.98 21.71 × 10 18 =2.29× 10 17   kgm −3

The average mass density of sodium atom is,

ρ sodium atom =0.7× 10 3   kgm −3 .

Comparing density of sodium nucleus with that of a sodium atom, we get

ρ sodium nucleus ρ sodium atom = 0.3× 10 18 0.7× 10 3 ≈ 10 15

Thus, we find that density of nucleus is typically 1015 times the atomic density of matter.