Atoms in a solid - Calculating Distances and Cohesive Energy

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TFM
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Homework Statement



Consider two different two-dimensional arrangements, (a) and (b), of four atoms (just four!) defined as follows:
(a) in which the centres of the four atoms form a square of side [tex] a_0 [/tex]
(b) in which the centres of the atoms form an equilateral diamond shape with angles 60° and 120°, and with the length of the side being [tex]b_0[/tex]
If the inter-atomic interaction potential is of the Lennard-Jones form [tex]U(r) = 4\epsilon [(\frac{\sigma}{r})^{12} - (\frac{\sigma}{r})^6][/tex]
and you neglect next nearest neighbours, calculate:
(i) the nearest neighbour distances [tex]a_0[/tex] and [tex]b_0[/tex], respectively
(ii) the cohesive energy per atom of each arrangement

Hence deduce which of the two arrangements would be favoured energetically at very low temperature. Would taking the next nearest neighbours change this conclusion?


Homework Equations



Lennerd-Jones Equation given

The Attempt at a Solution



Hi,

I am not quite sure what to do for this question.

For the first part, I am thinking that the atoms are joined together, so a0 would be one atomic diameter, for B0), it would require some trigonometry to get the length

Does this make sense?

Thanks,

TFM
 

Answers and Replies

  • #2
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What do you mean be "one atomic diameter"?

What is special about the potential energy of a system in static equilibrium?

Work out the potential energy of systems (a) and (b) as a function of distance between adjacent atoms, d. What value of d will correspond to equilibrium?
 
  • #3
TFM
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What do you mean be "one atomic diameter"?

What is special about the potential energy of a system in static equilibrium?

Work out the potential energy of systems (a) and (b) as a function of distance between adjacent atoms, d. What value of d will correspond to equilibrium?

Well For the square, i am assuming the atoms are touching so if a0 is the distance form the center of each atom, it would equal the length of one atom.

The potential energy of a system in static equilibrium is a minima, thus = 0 when you differentiate.
 
  • #4
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I'm not sure if you've solved the problem yet, but I'll just make sure you're on the right path.

There is no such thing as the "length of one atom" (other than an order of magnitude estimate), for a couple of reasons. Firstly atoms kind of spread out everywhere and we can't do much more than say "68% of the time an atom will be at most this large". But that only applies to isolated atoms; when we put other atoms around (like we do here) the size and shape of the atom will change.

Even if the atoms had definite size the distance between them wouldn't necessarily depend on their size (think of planets; the distance between planets and the sun doesn't depend on the radius of the planet). So how can we find the distance between the atoms?

"The potential energy of a system in static equilibrium is a minima, thus = 0 when you differentiate."

That hits the nail on the head! All you need to do is find the distance corresponding to minimum potential energy.
 

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