Chapter 28, Problem 48P

(a)

To determine

The particle with greater tunnelling probability.

Answer to Problem 48P

The particle with greater tunnelling probability is $\text{proton}$.

Explanation of Solution

Write the expression for tunnelling probability.

$P=e^{-2\kappa a}$                                                            (I)

Here, $P$ is the tunnelling probability, $a$ is the barrier width and $\kappa$ is a measure of the barrier height.

Write the expression for $\kappa$ barrier width.

$\kappa =\sqrt{\frac{8{\pi }^{2}m}{h^2}\left(V-E\right)}$                                                            (II)

Here, $m$ is the mass of the particle, $h$ is the Planck’s constant, $V$ is the barrier height and $E$ is the total energy.

$\kappa$ is directly proportional to $\sqrt{m}$ and $P$ is inversely proportional $\kappa$. Thus, an increased mass results in increased $\kappa$ which in turn decreases $P$.

Since the mass of deuteron is greater than that of the proton, the proton has greater tunnelling probability.

(b)

To determine

The ratio of the tunnelling probabilities.

Answer to Problem 48P

The ratio of the tunnelling probabilities is $121$.

Explanation of Solution

Write the expression for tunnelling probability of proton and deuteron.

$\frac{P_p}{P_d}=\frac{e^{-2{\kappa }_{p}a}}{e^{-2{\kappa }_{d}a}}$                                                           (III)

Here, $P_p$ is the tunnelling probability of the proton and $P_d$ is the tunnelling probability of the deuteron.

Write the expression for ${\kappa }_p$ and ${\kappa }_d$.

${\kappa }_p=\sqrt{\frac{8{\pi }^2{m}_p}{h^2}\left(V-E\right)}$                                                           (IV)

Here, $m_p$ is the mass of the proton.

${\kappa }_d=\sqrt{\frac{8{\pi }^2{m}_d}{h^2}\left(V-E\right)}$                                                            (V)

Here, $m_d$ is the mass of the deuteron.

Subtracting (IV) form (V)

${\kappa }_d-{\kappa }_p=\sqrt{\frac{8{\pi }^2}{h^2}\left(V-E\right)\left(m_d-m_p\right)}$                                                     (VI)

The mass of the deuteron is two times the mass of the proton i.e. $m_d=2m_p$

Rearranging (I)

$\frac{P_p}{P_d}=e^{2\left({\kappa }_d-{\kappa }_p\right)a}$                                                          (VII)

Substituting  (VI) in (VII) and using $m_d=2m_p$

$\frac{P_p}{P_d}=e^{2a\sqrt{\frac{8{\pi }^2{m}_p}{h^2}\left(V-E\right)\left(2-1\right)}}$                                                          (VIII)

Conclusion

Substitute $10.0\text{fm}$ for $a$, $3.0\text{ MeV}$ for $E$, $10.0\text{ MeV}$ for $V$, $1.673\times {10}^{-34}\text{ Js}$ for $m_p$ and $6.626\times {10}^{-34}\text{ Js}$ for $h$ in (VIII) to find $P_p/P_d$.

$\begin{array}{c}\frac{P_p}{P_d}=\exp \left(2\left(10.0\text{ fm}\right)\left(\sqrt{2}-1\right)\sqrt{\frac{8{\pi }^2\times 1.673\times {10}^{-27}\text{ kg}}{{\left(6.626\times {10}^{-34}\text{ Js}\right)}^2}\left(10.0\text{ MeV}-3.0\text{ MeV}\right)}\right)\\ =\exp \left(2\left(10.0\times {10}^{-15}\text{ m}\right)\left(\sqrt{2}-1\right)\sqrt{\frac{8{\pi }^2\times 1.673\times {10}^{-27}\text{ kg}}{{\left(6.626\times {10}^{-34}\text{ Js}\right)}^2}\left(10.0-3.0\right)\times {\text{10}}^6\text{ eV}\times \frac{1.602\times {10}^{-19}\text{}\text{Js}}{\text{1 eV}}}\right)\\ =\exp \left(4.8\right)\\ =121\end{array}$

Thus, the ratio of tunnelling probability is $121$.