Question

3.1. Potentials Between Atoms As a model of thermal expansion, we study the distance between two nearest neighbor atoms in an anharmonic potential that looks roughly like this V(2) ...AkpT 10 where r is the distance between the two neighboring atoms. This potential can be expanded around its minimum as V(x) = (x – ro) - (x – 2)*+... (2) where the minimum is at position to and K3 > 0. For small energies, we can truncate the series at the cubic term. (Note we are defining the energy at the bottom of the well to be zero here). A very accurate approximate form for inter-atomic potentials (particularly for inert atoms such as helium or argon) is given by the so-called Lennard-Jones potential V(a) = 4e (@"-03 + where € and o are constants that depend on the particular atoms we are considering. What is the meaning of the exponent 6 in the second term of this expression (i.e., why is the exponent necessarily chosen to be 6). By expanding Eq. 3 around its minimum, and comparing to Eq. 2, calculate the values of the coefficients ro, K, and K3 for the Lennard-Jones potential in terms of the constants e and o. We will need these results in exercise 3.3. below.

Answer 1

Mul() = ve [ ()*-()] +6 The attractive term origin in Lennard Jones potential is fluctuating dipole yer yn dipole interactions The potential emergy between two dipoles (That do That are instantaneous dipole moments not permanent ). Let There me 2 dipoles with dipole moments .bi 4 P2 who a direction upto ad so The electric field due to po now The dipole moment in other atomb is induced due to the electric field i my po sog. Pq ~+KE, + api 73 the energy of interaction so U = - P2. El C any arbitrary const

Y(*) =.ut () )'] ++ we have to expand the above term around a point xo suell that V'(x)= 0 Because the bencurial is minimum Lat *=*: That seperonony it is talviubnom V"(2) = 4€ (-12 ( ) + (6) => webmaster >> 20%=x - so now if we want to expand v(x) about (x=xo) and forms v(x) = į dv (?*- 7 = 0 we need to find out for kq <3 The co officient of (x-xo) 21 X = X . -X.)3 which are v"(x) & V" (X-) respectively *.= 2'6r V"(x) = ut [42x13) - (ex? vi (s) = ve fear is op - (690) A ] - )(a ) (42313) – (6** *2)] = () () - (2623)

and V"'(x) = 46 - (12*13*14) r"_* (** + x 15 v" (39) = us ( 413 ) + (6 394 terra ] to ) (2012-) (123153 ) – (261780) :) -43 on ko) -)- SO bXlYxI8 *3=( ) ( voete) 2512 7.= 2lor