(i) What is the concept of effective mass of the electron ? (ii) The energy near a valence band edge is given by E(k) = -1 \times 10^{-26} k^2\text{ ergs}. An electron is removed from the orbital k = 1 \times 10^7 k_x\text{ cm}^{-1}. Find the sign and magnitude of effective mass of the hole.
Give logic diagram, Boolean equation and truth tables of AND, OR and NOT gates. Show that NAND and NOR gates can be utilized as a Universal Gate.
(ii) Recalculate the intrinsic carrier concentration at T = 450\text{ K} by assuming that the effective electron and hole masses in the respective bands decrease by 0\cdot 5\%.
Given that for intrinsic Si at T = 300\text{ K}, the band gap is 1\cdot 1\text{ eV} and the intrinsic carrier concentration is n_i = 1\cdot 5 \times 10^{10}\text{ cm}^{-3}. (i) Calculate the intrinsic carrier concentration at T = 450\text{ K}. Assume that the band gap at T = 450\text{ K} is 1\cdot 08\text{ eV}.
The density of a liquid monovalent metal near absolute zero temperature is given to be 0\cdot 081\text{ g cm}^{-3}. Calculate the Fermi energy \varepsilon_F, the electron velocity v_F at the Fermi surface and the Fermi temperature T_F.
Consider the planes with indices (1\ 0\ 0) and (0\ 0\ 1); the lattice is f.c.c., and the indices refer to the conventional cubic cell. What are the indices of these planes when referred to the primitive axes : \vec{a}_1 = \frac{a}{2}(\hat{x}+\hat{y}), \vec{a}_2 = \frac{a}{2}(\hat{y}+\hat{z}) and \vec{a}_3 = \frac{a}{2}(\hat{z}+\hat{x})?
For the ground state of deuteron, prove that the radius of nucleon is of the order of \sim 2\cdot 15 \times 10^{-13}\text{ cm}.
(ii) \bar{\nu}_e + p^+ \rightarrow n^0 + e^- What are the various conservation laws for elementary particles ? Apply these conservation laws to confirm whether the following reactions are possible or not : (i) \pi^+ + n^0 \rightarrow K^0 + K^+ (ii) \bar{\nu}_e + p^+ \rightarrow n^0 + e^-
(ii) From a study of the radial part of the Schrödinger equation of a deuteron for any angular momentum l, what do we conclude?
What is meant by strength of the interactions of elementary particles ? Classify the different forces on the basis of this strength of interaction.
Write the decay reactions of the neutral pions (\pi^0) and the charged pions (\pi^+, \pi^-).
Does the nucleus possess magnetic moment ? Justify your answer. Define nuclear magneton (\mu_{\text{N}}) and Bohr magneton (\mu_{\text{B}}). Calculate their values.
What were the difficulties faced by the initial theory of \beta-decay? How did Pauli eliminate the difficulties? What were the expected properties of the new particle proposed by Pauli?
Apply the meson theory of nuclear forces to write the interaction between protons, neutrons, proton to neutron and neutron to proton.
Assume that the range of the interaction of nuclear force and of meson is 2 \times 10^{-15}\text{ m}. Estimate the mass of the meson.
A radionuclide (N_1) decays to a radioactive daughter (N_2) that subsequently decays to a third daughter (N_3). If at t = 0, the initial concentrations of N_1, N_2 and N_3 are N_1^0, 0 and 0 respectively, and the decay constants are \lambda_1 (from N_1 \rightarrow N_2) and \lambda_2 (from N_2 \rightarrow N_3), then derive the expression for N_2 at time t. Given that the daughter N_3 is stable.
A sample contains 4\text{ mg} of {}^{210}\text{Bi}. If the half-life of radioactive {}^{210}\text{Bi} is 5\text{ days} and the average energy of the \beta-particles emitted is 0\cdot 34\text{ MeV}, then at what rate does the sample emit energy?
(i) Write semi-empirical mass formula. Calculate the atomic number (Z) of most stable nucleus for given mass number (A) using the above formula. (Use the value of fitted coefficients for Coulomb energy a_3 = 0\cdot 711\text{ MeV} and that for asymmetry energy a_4 = 23\cdot 702\text{ MeV}). (ii) Calculate the Q-value of the following nuclear reaction : {}_4\text{Be}^9 + {}_2\text{He}^4 = {}_6\text{C}^{12} + {}_0\text{n}^1 Given : the mass of neutral atoms of \text{Be}, \text{He} and \text{C} are 9\cdot 015060, 4\cdot 003874 and 12\cdot 003815\text{ amu}, respectively. The mass of neutron is 1\cdot 008986\text{ amu}.
In the case of deuterons, answer the following : (i) What percentage of the time do the nucleons of a deuteron spend within the range of the nuclear forces?
Compare nuclear density of hydrogen ({}_1\text{H}^1) with its atomic density. (Assume the atom to have the radius of its first Bohr orbit). What inference can one get from the above comparison ?
