A common approximation to the interatomic forces in a material
is the Lennard-Jones potential between neighboring atoms in a
solid:
U(r) = A/r12 - B/r6
This can be used to find various information about a solid
material.
Assume that for the lattice of a particular alloy of copper, the
Lennard-Jones constants are:
A = 2.39 ⨯ 10-133 J·m12
B = 3.19 ⨯ 10-76 J·m6
Find the following:
(a) The equilibrium distance between neighboring atoms in the copper lattice (this is called the lattice constant a).
Answer: 0.000000000338 m
(b) Model this as an approximate potential around the equilibrium position
Ueff(r) = (1/2)k(r-a)2 + C
(i) What is the effective spring stiffness k?
(ii) What is the energy minimum C?
(c) Find the resonance frequency of a copper atom in a linear chain of atoms? [Assume only nearest neighbor interactions]
(d) Find the vibration amplitude at the temperature T = -2⁰C.
Assume that the oscillator has one degree of freedom.
[ Use kB = 1.38⨯10-23 J/K/atom as Boltzmann's constant. ]
A common approximation to the interatomic forces in a material is the Lennard-Jones potential between neighboring...
2. The interaction between neutral atoms can be approximated with the Lennard- Jones potential U(r)-4e where r is the interatomic distance, and E and σ are constants. a) If U is measured in Joules and r is measured in nanometers (nm), what are the units of є and σ. b) What is the interatomic distance at which the potential is zero?. c) Find the value of r for which U is a minimum d) What is the minimum value of...
1. a) A simple cubic crystal is formed of atoms which interact with a Lennard-Jones potential with a bond energy of 0.3 eV, and nearest neighbor distance -0.4 nm. Find A and B (note that r r is a minimum in U) b) What is the force between near est neighbors when rro? c) What is the force between nearest neighbors when r-1.01 ro? This is a small force, but on a VERY small object. d) What is the spring...
The Lennard-Jones potential energy Ulj(x) is important in chemical bonding theory and is given by 1012 ool ULJ 1 1 1 ULJ = 4€ | (x12 26 I 0 E 012 where e and o are constants and ř=r/o. (a) Sketch a graph of this potential energy. (b) Use IDL to obtain a careful graph of ULJ(x)/€ versus x/0. (c) Locate any equilibrium points, (d) determine which ones are stable, and (e) find the frequency of small oscillations about about...
The potential energy of a diatomic molecule (a two-atom system like H, or 02) is given by А B E, (1) 76 where r is the separation of the two atoms of the molecule and A and B are positive constants. Eq. 1 is also known as Lennard-Jones potential. (a) Derive the force acting between the two atoms as a function of their distance r. [4 marks] (b) Find the system's equilibrium separation. [4 marks] (c) For a total mechanical...
#2 This is a solid state physics problem. ( I use this book : kittel introduction to soild state physocs ) thank you ! 5. [Graphene] Consider graphene, a hexagonal lattice of carbon atoms as shown in the figure. The distance between neighboring carbon atoms is a0.143 nm (a) B A (a) Write down the unit lattice vectors ā, and a and unit reciprocal lattice vectors b, and b,. [5] a2 (b) The two sites A and B are not...
EENG 245 Physical electronics HW 1 1) The NaCl crystal is cubic, and can be described as follows. Na atoms sit at the corners and faces of a cube, and Cl atoms sit in between two Na atoms. This means that a Clatom is found half-way along each of the cube edges, and there is a Cl in the center of the cube. (We could also have described the lattice by interchanging Na and Cl in the description above.) Another...