PHYSICS FOR SCIEN & ENGNR W/MOD MAST
4th Edition
ISBN: 9780134112039
Author: GIANCOLI
Publisher: PEARSON
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Chapter 40, Problem 19P
To determine
Find the relationship between
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The angular frequency of vibrations in a one-dimensional monatomic crystal are given by
where
w (k)
=
4K
m
sin
k =
ka
2
2π
λ
is the wavenumber of the vibration pattern in terms of its wavelength >, the spring constant
between neighbouring atoms, and m the mass of each atom.
(i) Show that the frequency does not change if one shifts
2π
k→ k +.
a
(ii) Briefly explain the physical reason for this periodicity.
Atoms vibrate relative to one another in molecules with the bond acting as a spring.
Consider the H – CI bond, where the heavy Cl atom forms a stationary anchor for the very
light H atom. That is, only the H atom moves, vibrating as a simple harmonic oscillator.
(a) Give the equation that describes the allowed vibrational energy levels of the bond.
(b) The force constant kf for the H – Cl bond is 516.3 N m'1. Given the mass of H equal to
1.7 x 1027 kg, determine the difference in energy (separation) between adjacent energy
levels.
(c) Calculate the zero-point energy of this molecular oscillator.
(ii) We build representations of the angular momentum algebra labeled by the quantum
numbers (j, m) of J² and J,. As an exercise, compute the matrix elements
(j — 2, т —
-1|J_|j = 2, m = 0),
(j = 1, m = 1|J+lj = 1, m = 0),
(j = 1, m = –1|J_\j = 2, m = 0).
Chapter 40 Solutions
PHYSICS FOR SCIEN & ENGNR W/MOD MAST
Ch. 40.4 - Determine the three lowest rotational energy...Ch. 40.6 - Prob. 1BECh. 40.6 - Prob. 1CECh. 40.8 - Prob. 1DECh. 40 - What type of bond would you expect for (a) the N2...Ch. 40 - Describe how the molecule CaCl2 could be formed.Ch. 40 - Does the H2 molecule have a permanent dipole...Ch. 40 - Although the molecule H3 is not stable, the ion...Ch. 40 - The energy of a molecule can be divided into four...Ch. 40 - Would you expect the molecule H2+ to be stable? If...
Ch. 40 - Explain why the carbon atom (Z = 6) usually forms...Ch. 40 - Prob. 8QCh. 40 - Prob. 9QCh. 40 - Prob. 10QCh. 40 - Prob. 11QCh. 40 - Prob. 12QCh. 40 - Prob. 13QCh. 40 - Prob. 14QCh. 40 - Prob. 15QCh. 40 - Prob. 16QCh. 40 - Prob. 17QCh. 40 - Prob. 18QCh. 40 - Prob. 19QCh. 40 - Prob. 20QCh. 40 - Prob. 21QCh. 40 - Prob. 22QCh. 40 - Prob. 23QCh. 40 - Prob. 1PCh. 40 - (II) The measured binding energy of KCl is 4.43eV....Ch. 40 - (II) Estimate the binding energy of the H2...Ch. 40 - (II) The equilibrium distance r0 between two atoms...Ch. 40 - Prob. 5PCh. 40 - Prob. 6PCh. 40 - (III) (a) Apply reasoning similar to that in the...Ch. 40 - (I) Show that the quantity 2/I has units of...Ch. 40 - Prob. 9PCh. 40 - Prob. 10PCh. 40 - Prob. 11PCh. 40 - Prob. 12PCh. 40 - Prob. 13PCh. 40 - Prob. 14PCh. 40 - Prob. 15PCh. 40 - Prob. 16PCh. 40 - (II) Calculate the bond length for the NaCl...Ch. 40 - Prob. 18PCh. 40 - Prob. 19PCh. 40 - Prob. 20PCh. 40 - Prob. 21PCh. 40 - Prob. 22PCh. 40 - Prob. 23PCh. 40 - Prob. 24PCh. 40 - Prob. 25PCh. 40 - Prob. 26PCh. 40 - Prob. 27PCh. 40 - Prob. 28PCh. 40 - Prob. 29PCh. 40 - Prob. 30PCh. 40 - Prob. 31PCh. 40 - Prob. 32PCh. 40 - Prob. 33PCh. 40 - Prob. 34PCh. 40 - Prob. 35PCh. 40 - Prob. 36PCh. 40 - Prob. 37PCh. 40 - Prob. 38PCh. 40 - Prob. 39PCh. 40 - Prob. 40PCh. 40 - Prob. 41PCh. 40 - Prob. 42PCh. 40 - Prob. 43PCh. 40 - Prob. 44PCh. 40 - Prob. 45PCh. 40 - Prob. 46PCh. 40 - Prob. 47PCh. 40 - Prob. 48PCh. 40 - Prob. 49PCh. 40 - Prob. 50PCh. 40 - Prob. 51PCh. 40 - Prob. 52PCh. 40 - Prob. 53PCh. 40 - Prob. 54PCh. 40 - Prob. 55PCh. 40 - Prob. 56PCh. 40 - Prob. 57PCh. 40 - Prob. 58PCh. 40 - Prob. 59PCh. 40 - Prob. 60PCh. 40 - Prob. 61PCh. 40 - Prob. 62GPCh. 40 - Prob. 63GPCh. 40 - Prob. 64GPCh. 40 - Prob. 65GPCh. 40 - Prob. 66GPCh. 40 - Prob. 67GPCh. 40 - Prob. 68GPCh. 40 - Prob. 69GPCh. 40 - Prob. 70GPCh. 40 - Prob. 71GPCh. 40 - Prob. 72GPCh. 40 - Prob. 73GPCh. 40 - Prob. 74GPCh. 40 - Prob. 75GPCh. 40 - Prob. 76GPCh. 40 - Prob. 77GPCh. 40 - Prob. 78GPCh. 40 - Prob. 79GPCh. 40 - Prob. 80GPCh. 40 - Prob. 81GPCh. 40 - Prob. 82GPCh. 40 - Prob. 83GPCh. 40 - Prob. 84GPCh. 40 - Prob. 85GPCh. 40 - Prob. 86GPCh. 40 - Prob. 87GPCh. 40 - Prob. 88GPCh. 40 - Prob. 89GP
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- 2(6) Calculate the fundamental vibrational wavenumber (in cm-1) for HI molecule, if its angular vibrational frequency is 4.394×1014 s-1. Calculate the vibrational energy of the molecule in the ground state and the force constant. Assume the mass is the mass of a proton.arrow_forward(a) Emission of microwave radiation from the J = 1→0 transition of a molecule has been detected at 88.63 GHz from a region of interstellar space in which there is evidence of thermal equilibrium and a temperature of around 20 K. Estimate the frequency and relative intensity of the J = 2 → 1 transition of the same molecule.arrow_forward3. (a) One way of treating the vibrational modes of a linear diatomic solid is to assume that the atoms have the same masses, but the springs on either side of an atom have spring constants K and G, respectively. Show that the dispersion relation of such a lattice is given by (K+G` +G)' - 4KGsin ka M where M is the mass of the ion, G and K the lattice constants, a is the periodic distance between masses and k the lattice wave vector. (i) (ii) Sketch the dispersion relation Discuss what happens when K = G and K >> G. (b) In diatomic (linear) lattice, why do we assume same o and k.arrow_forward
- Check Your Understanding Would the result in Example 6.4 be different if the mass were not 1.0 kg g a tiny mass of 1.0 pg, and the amplitude of vibrations were 0.10 m?arrow_forward3) (i) Draw each one of the planes given below in a separate lattice cell and provide coordinate values at which the plane intercepts the coordinate axes. (a) (112) (b) (321) (c) (222) (ii) Calculate the planar density of (i) (a) for an FCC lattice with a lattice constant of a. (iii) Draw each one of the directions given below in a separate lattice cell and show coordinate values on the axes. (a) [112) (b) [113] (c) [221] (iv) Calculate the linear density of (ii) (a) for a BCC lattice with a lattice constant of a. (v) What is the number of planes in {121} for a cubic crystal structure?arrow_forwardVibrations in the diatomic molecule CO can be approximated as a harmonic oscillator, where the angular frequency w = 6.505 × 1013 Hz and the reduced mass is equal to u = 1.14 × 10-27 kg. Assume the molecule is in its first excited vibrational state. Its vibrational wavefunction can then be written as V1 (æ) = (÷)i /2a xe-T where a = 2α πe If we were to measure the bond length of the molecule, what is the most likely displacement from the equilibrium bond distance in the first excited vibrational state? Give your answer in Angstroms [Note: The equilibrium displacement in the Quantum harmonic oscillator corresponds to x = = 0, ie the coordinate x measures displacement from equilibrium]arrow_forward
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