Problem set 48

1. State the following decay mode in the category of allowed, forbidden and Fermi or Gammow-Teller transition
   
1. Fermi transition and allowed
2. Fermi transition and second forbidden
3. G-T transition and first forbidden
4. G-T transition and allowed
Selection rules for allowed transitions in $\beta$-decay are: \begin{equation*} \left.\begin{aligned} &\Delta I=0\\ &{\scriptstyle I_i=0\rightarrow I_f=0\quad \text{allowed}}\\ &\Delta\pi=0 \end{aligned}\right\}\text{Fermi rule } \end{equation*} \begin{equation*} \left.\begin{aligned} & \Delta I=0,\pm1\\ & {\scriptstyle I_i=0\rightarrow I_f=0\quad \text{forbidden}}\\ &\Delta\pi=0 \end{aligned}\right\}\text{G.T. rule} \end{equation*} where $I$ is nuclear spin and $\pi$ is parity.

Using these selection rule we can see that the transition is Fermi transition and allowed

Hence, answer is (A)

2. State the following decay mode in the category of allowed, forbidden and Fermi or Gammow-Teller transition
   
1. Fermi transition and allowed
2. Fermi transition and second forbidden
3. G-T transition and allowed
4. G-T transition and third forbidden
Using transitions we can see that the transition is G-T transition and allowed

Hence, answer is (C)

3. The short wavelength cut-off of the continuous X-ray spectrum from a nickel target is 0.0825 nm. The voltage required to be applied to the X-ray tube must be:
1. 0.15 kV
2. 1.5 kV
3. 15 kV
4. 150 kV
According to Duane–Hunt law $$\lambda_{min}=\frac{1239.8}{V \text{ in }kV}pm$$ $$V \text{ in }kV=\frac{1239.8}{\lambda_{min}\text{ in }pm}$$ $$V \text{ in }kV=\frac{1239.8}{82.5}=15\:kV$$ Hence, answer is (C)

4. If the Planck's constant were to be zero, then the total energy contained in a box filled with radiation of all frequencies at temperature T would be (where $k$ is Boltzmann constant and $T\ne0$)
1. zero
2. infinite
3. $\frac{3}{2}kT$
4. $kT$
According to Stefan-Boltzmann law the the energy flux $\Phi$ emitted from a blackbody at temperature $T$, is $\Phi = \sigma T^4$, where $\sigma=\frac{2\pi^5k^4}{15h^3c^2}$. AS $\sigma\propto\frac{1}{h^3}$, energy will be infinite when $h=0$

Hence, answer is (B)

5. The number of electrons per unit volume in metallic potassium is $1.33\times 10^{28}\: atoms/m^3$ and if each potassium atom donates one electron to the electron gas, its Fermi energy in eV is :
1. 4.72
2. 3.23
3. 2.05
4. 1.85
$$E_F=\frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{\frac{2}{3}}$$ \begin{align*}&E_F=\frac{h^2}{8m}\left(\frac{3n}{\pi}\right)^{\frac{2}{3}}\\ &={\textstyle \frac{\left(6.62607004 \times 10^{ −34}\right)^2}{8\times 9.109\times10^{−31}}\left(\frac{3\times1.33\times 10^{28}}{3.14}\right)^{\frac{2}{3}}J}\\ &=3.28446\times10^{-19}J\\ &=\frac{3.28446\times10^{-19}}{1.6\times10^{-19}}\\ &=2.05 eV \end{align*} Hence, answer is (C)