What is the nuclear radius of ^40Ca? What energy electrons and protons are required to probe the size of ^40Ca if one wants to “see” at least half the radius? Strategy We use Equation (12.2) to determine the radius of ^40Ca. From the radius we can determine the size of the probe, which subsequently

Question

What is the nuclear radius of [latex]{ }^{40} Ca[/latex]? What energy electrons and protons are required to probe the size of [latex]{ }^{40} Ca[/latex] if one wants to “see” at least half the radius?

Strategy We use Equation (12.2) to determine the radius of [latex]{ }^{40} Ca[/latex]. From the radius we can determine the size of the probe, which subsequently determines the de Broglie wavelength and the momentum p. The total energy and kinetic energy can be found from the momentum.

[latex]R=r_{0} A^{1 / 3}[/latex] (12.2)

Accepted Answer

We determine R from Equation (12.2).

[latex]R=1.2 fm (40)^{1 / 3}=4.1 fm[/latex]

In order to distinguish a distance at least half the radius, we need a de Broglie wavelength of 2.0 fm. If we use the relation for the de Broglie wavelength [latex](\lambda=h / p)[/latex], we have [latex]p=h / \lambda[/latex]. The total energy of the probing particle is

[latex]E^{2}=\left(m c^{2} ight)^{2}+(p c)^{2}=\left(m c^{2} ight)^{2}+\frac{h^{2} c^{2}}{\lambda^{2}}[/latex]

For a wavelength of 2.0 fm, the last term becomes

[latex]\frac{h^{2} c^{2}}{\lambda^{2}}=\left(\frac{1240 MeV \cdot fm }{2.0 fm } ight)^{2}=3.8 imes 10^{5} MeV ^{2}[/latex]

Now if we insert the rest energy [latex]m c^{2}[/latex] of the appropriate particle, we can determine the total energy and required kinetic energy of either the electron or proton.

Electron energy:

[latex]\begin{aligned}E^{2} &=(0.511 MeV )^{2}+3.8 imes 10^{5} MeV ^{2} \E &=620 MeV \K &=E-m c^{2}=620 MeV -0.5 MeV =620 MeV\end{aligned}[/latex]

Proton energy:

[latex]\begin{aligned}E^{2} &=(938.3 MeV )^{2}+3.8 imes 10^{5} MeV ^{2} \E &=1120 MeV \K &=E-m c^{2}=1120 MeV -938 MeV =180 MeV\end{aligned}[/latex]

We have quoted the results to two significant figures because of the uncertainty in the radius R.

Question 12.2: What is the nuclear radius of ^40Ca? What energy electrons a...

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