IB Physics Topic E.1.3 Properties of Nuclei and High-energy Scattering Revision Notes (HL)

IB Physics Topic E.1.3 Properties of Nuclei and High-energy Scattering Revision Notes (HL) | RevisionDojo

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Nuclear Radius and its Relationship to Nucleon Number

The size of a nucleus is determined by its radius, which depends on the number of nucleons (protons and neutrons) it contains.
This relationship is described by the formula: $$R = R_0 A^{\frac{1}{3}}$$ where:
1. $R$ is the nuclear radius,
2. $R_0$ is a constant, approximately $1.2 \times 10^{-15} \, \text{m}$ (or 1.2 femtometers),
3. $A$ is the nucleon number (mass number) of the nucleus.

Why nuclear radius scales with nucleon number

The formula $R = R_0 A^{\frac{1}{3}}$ reflects the fact that nuclei are approximately spherical, and their volume is proportional to the number of nucleons.
Since the volume of a sphere is proportional to the cube of its radius ($V \propto R^3$), the radius must scale with the cube root of the nucleon number.

Example question

Find the radius of a gold nucleus ($A = 197$).

Solution

Using $R_0 = 1.2 \times 10^{-15} \, \text{m}$:

$$R = 1.2 \times 10^{-15} \times 197^{\frac{1}{3}}$$

$$R \approx 7.0 \times 10^{-15} \, \text{m}$$

Thus, the radius of a gold nucleus is approximately $7.0 \, \text{fm}$.

Hint

Use the mass number A in this equation, not the proton number Z.

Nuclear Density: Why Nuclei Are So Dense

Despite their small size, nuclei contain nearly all the mass of an atom.
This results in extraordinarily high densities.
The density of a nucleus can be calculated using the formula for the density of a sphere: $$\rho = \frac{\text{Mass}}{\text{Volume}}$$
For a nucleus:

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Nuclear Radius and its Relationship to Nucleon Number

The size of a nucleus is determined by its radius, which depends on the number of nucleons (protons and neutrons) it contains.
This relationship is described by the formula: $$R = R_0 A^{\frac{1}{3}}$$ where:
1. $R$ is the nuclear radius,
2. $R_0$ is a constant, approximately $1.2 \times 10^{-15} \, \text{m}$ (or 1.2 femtometers),
3. $A$ is the nucleon number (mass number) of the nucleus.

Why nuclear radius scales with nucleon number

The formula $R = R_0 A^{\frac{1}{3}}$ reflects the fact that nuclei are approximately spherical, and their volume is proportional to the number of nucleons.
Since the volume of a sphere is proportional to the cube of its radius ($V \propto R^3$), the radius must scale with the cube root of the nucleon number.

Example question

Find the radius of a gold nucleus ($A = 197$).

Solution

Using $R_0 = 1.2 \times 10^{-15} \, \text{m}$:

$$R = 1.2 \times 10^{-15} \times 197^{\frac{1}{3}}$$

$$R \approx 7.0 \times 10^{-15} \, \text{m}$$

Thus, the radius of a gold nucleus is approximately $7.0 \, \text{fm}$.

Hint

Use the mass number A in this equation, not the proton number Z.

Nuclear Density: Why Nuclei Are So Dense

Despite their small size, nuclei contain nearly all the mass of an atom.
This results in extraordinarily high densities.
The density of a nucleus can be calculated using the formula for the density of a sphere: $$\rho = \frac{\text{Mass}}{\text{Volume}}$$
For a nucleus:

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Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

IB Physics E.1.3 Properties of Nuclei and High-energy Scattering Notes

Topic A - Space, time and motion5 subtopics

Topic B - The particulate nature of matter5 subtopics

Topic C - Wave behaviour5 subtopics

Topic D - Fields4 subtopics

Topic E - Nuclear and quantum physics5 subtopics

Nuclear Radius and its Relationship to Nucleon Number

Why nuclear radius scales with nucleon number

Nuclear Density: Why Nuclei Are So Dense

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Nuclear Radius and its Relationship to Nucleon Number

Why nuclear radius scales with nucleon number

Nuclear Density: Why Nuclei Are So Dense

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