SIZE OF THE NUCLEUS

We know that in an alpha-particle scattering experiment, the distance of closest approach to a gold nucleus of an alpha particle of kinetic energy \fn_cm \large 5.5 Mev is about \fn_cm \large 4\times10^{-14}m. Since the +ve charge is confined to the nucleus, the actual size of the nucleus has to be less than \fn_cm \large 4\times10^{-14}m .

If we use alpha- particles of higher energies than \fn_cm \large 5.5 Mev , the size of the nucleus will be smaller and at some point, nuclear forces will begin to be affected and differ from Rutherford’s calculation, because Rutherford’s calculation is totally based on coulomb’s repulsive force.

By performing scattering experiments in which fast electron, instead of alpha-particles, are projectiles that bombardments, the size of nuclei of various elements have been accurately measured.

It has been found experimentally that the volume of a nucleus is directly proportional to its mass number A.

If R is the radius of the nucleus and assumed spherical

than      \fn_cm \large \frac{4}{3}\pi R^3\propto A

\fn_cm \large R\propto A^{\frac{1}{3}}

\fn_cm \large \left [ R=R_0 A^{\frac{1}{3}} \right ]

\fn_cm \large R_0=1.2\times10^{-15}m=1.2fm\;\;\;\;\;\;\;\;\;fm\rightarrow femto\;meter

i.e we can say that atomic nuclei of different elements have different in size.

NUCLEAR DENSITY

Let      \fn_cm \large m\rightarrow average mass of nucleon( mass of either proton or neutron)

\fn_cm \large R\rightarrow Radius of nucleus

∴ mass of nucleus= \fn_cm \large mA   where \fn_cm \large A is mass number

Volume of nucleus= \fn_cm \large \frac{4}{3}\pi R^3

\fn_cm \large =\frac{4}{3}\pi (r_0 A^\frac{1}{3})^3

\fn_cm \large =\frac{4}{3}\pi R_0 ^3A

∴ The density of the nucleus is given by

\fn_cm \large \rho =\frac{mA}{\frac{4}{3}\pi R_0 ^3 A}

\fn_cm \large \left [ \rho =\frac{3m}{4\pi R_0 ^3} \right ]\;\;(constant)

We know that

\fn_cm \large m=1.66\times10^{-27}kg\;\;\;\;and \;\;\;\;R_0-1.2\times10^{-15}m

we get    \fn_cm \large \left [ \rho =2.98\times10^{-17}kg/m^3 \right ]

Hence the matter of nucleus is very dense and which is constant for all nucleus.

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