Nuclear Radius

Introduction

The nucleus of an atom consists of protons and neutrons, which are collectively known as nucleons. The size of the nucleus is an important property in nuclear physics, and its radius is determined by the number of nucleons it contains. The nuclear radius increases with the number of nucleons but follows a predictable pattern that can be mathematically described.

Mathematical Formula

The radius $R$ of a nucleus with atomic mass number $A$ is estimated by the following empirical formula:

$R=R$₀ A ⅓

Where $R$₀ is the nuclear radius constant, which has a value of approximately 1.25 femtometers (fm). The atomic mass number $A$ represents the total number of protons and neutrons in the nucleus of the atom. For example, for a carbon-12 nucleus, $A=\; 12$, and for a uranium-238 nucleus, $A=\; 238$.

Radius in Terms of Atomic Mass Number

According to the formula, the nuclear radius increases with the atomic mass number but not linearly. Instead, the radius grows as the cube root of the mass number. For light nuclei, such as hydrogen, the radius is relatively small, while for heavy nuclei, such as uranium, the radius increases significantly.

The nuclear radius for a proton (A = 1) is approximately 1.25 fm, and for larger atoms, it follows the cube-root rule. For example:

• For $A=\; 4$ (helium nucleus), the nuclear radius is approximately 2.52 fm.
• For $A=\; 12$ (carbon nucleus), the nuclear radius is approximately 3.74 fm.
• For $A=\; 238$ (uranium nucleus), the nuclear radius is approximately 7.75 fm.

Range of Nuclear Radii

In nature, atomic mass numbers range from 1 (for hydrogen) to approximately 250 (for very heavy elements such as uranium). Consequently, the nuclear radius for typical nuclei ranges from:

• 1 fm for a single proton (A = 1, such as hydrogen),
• ~2.5 fm for helium (A = 4),
• ~7.75 fm for uranium (A = 238),
• ~8 fm for heavy nuclei (with A > 200).

Relation to Nuclear Forces

The size of the nucleus is also related to the range of the nuclear force, which is responsible for holding protons and neutrons together. The nuclear force is short-range, meaning it only acts over distances of about 1-2 fm. This is why the nucleus has a size on the order of a few femtometers, and why the formula for nuclear radius relies on the atomic mass number raised to the one-third power.

Graphical Representation

In practice, the relationship between nuclear radius and atomic mass number can be represented graphically as a curve where the nuclear radius increases slowly as the cube root of the mass number.

The following graph (which can be added externally using tools like MathJax) will show how nuclear radius increases with the atomic mass number. A straight line fitted to the data points will reveal a clear trend following the cube root law.

Formula Application Example

Let's calculate the nuclear radius for an atom of uranium-238:

Given: $A=\; 238$ and $R$₀ = 1.25 fm , we use the formula:

$R=\mathrm{1.25}x{\mathrm{238}}^{⅓}$

The nuclear radius for uranium-238 is calculated as:

Atomic mass number 𝐴=238 Nuclear radius constant 𝑅0=1.25fm
Formula for Nuclear Radius: 𝑅=𝑅0×𝐴 ^1/3
R=R0 ×A ^1/3
Step-by-Step Calculation: 𝑅=1.25×238^1/3
Now, calculate 238 ^1/3 :
238^1/3≈6.2 Thus, the nuclear radius 𝑅is:
𝑅=1.25×6.2≈7.75fm