Consider an electron in the $n=3$ orbit of a hydrogen-like atom with atomic number $Z$. At absolute temperature $T$, a neutron having thermal energy $k_{\mathrm{B}} T$ has the same de Broglie wavelength as that of this electron. If this temperature is given by $T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_{\mathrm{N}} k_{\mathrm{B}}}$, (where $h$ is the Planck's constant, $k_B$ is the Boltzmann constant, $m_{\mathrm{N}}$ is the mass of the neutron and $a_0$ is the first Bohr radius of hydrogen atom) then the value of $\alpha$ is ______.

A Hydrogen-like atom has atomic number $Z$. Photons emitted in the electronic transitions from level $n=4$ to level $n=3$ in these atoms are used to perform photoelectric effect experiment on a target metal. The maximum kinetic energy of the photoelectrons generated is $1.95 \mathrm{eV}$. If the photoelectric threshold wavelength for the target metal is $310 \mathrm{~nm}$, the value of $Z$ is _________.

[Given: $h c=1240 \mathrm{eV}-\mathrm{nm}$ and $R h c=13.6 \mathrm{eV}$, where $R$ is the Rydberg constant, $h$ is the Planck's constant and $c$ is the speed of light in vacuum]

In a photoemission experiment, the maximum kinetic energies of photoelectrons from metals P, Q and R are E_P, E_Q and E_R, respectively, and they are related by E_P = 2E_Q = 2E_R. In this experiment, the same source of monochromatic light is used for metals P and Q while a different source of monochromatic light is used for the metal R. The work functions for metals P, Q and R are 4.0 eV, 4.5 eV and 5.5 eV, respectively. The energy of the incident photon used for metal R, in eV, is ___________.

A perfectly reflecting mirror of mass M mounted on a spring constitutes a spring-mass system of angular frequency $$\Omega $$ such that $${{4\pi M\Omega } \over h} = {10^{24}}{m^{ - 2}}$$ with h as Planck's constant. N photons of wavelength $$\lambda $$ = 8$$\pi $$ $$ \times $$ 10^{$$ - $$6} m strike the mirror simultaneously at normal incidence such that the mirror gets displaced by 1 $$\mu $$m. If the value of N is x $$ \times $$ 10¹², then the value of x is ................ [Consider the spring as massless]