Calculate Poisson's ratio of NaCl crystal

[ukamle]

hello to all
here is a problem i designed myself. i don't know whether it can be solved.

Question:

Consider NaCl lattice of edge length 'x' m. Let Na+ carry +1 C and each Cl- carry -1 charge. Calculate Poisson's ratio of NaCl crystal. Assume no heat loss when deformation takes place.


Other questions (easy):

1) Calculate the locus of a point which moves such that the sum of reciprocals of potentials due to two charges separated 'd' distance apart is constant.


I will post other questions later. I will try to come up with a soln. for the Poisson's ratio problem.

 

[maverick280857]

I think the "easy" problem is not so easy. You can of course set up an equation but I don't know if interpreting the locus will be easy. It will be a 3 dimensional surface sure. Perhaps if we consider the center of the line joining the two charges as the pole and the line joining them the initial line, then we could work this out in cylindrical polar coordinates.

I think what you mean is

$$\frac{1}{V_{1}} + \frac{1}{V_{2}} = k$$

k is a constant

 

[maverick280857]

I think the "easy" problem is not so easy. You can of course set up an equation but I don't know if interpreting the locus will be easy. It will be a 3 dimensional surface sure. Perhaps if we consider the center of the line joining the two charges as the pole and the line joining them the initial line, then we could work this out in cylindrical polar coordinates.

I think what you mean is

$$\frac{1}{V_{1}} + \frac{1}{V_{2}} = k$$
(k is a constant).

Now let the charges A and B be at (0, 0, -r) and (0, 0, +r) where 2r = d. Then we can write

$$V_{A}(x,y,z) = \frac{q_{1}}{4\pi\epsilon_{0}\sqrt{x^2 + y^2 + (z+r)^2}}$$

$$V_{B}(x,y,z) = \frac{q_{2}}{4\pi\epsilon_{0}\sqrt{x^2 + y^2 + (z-r)^2}}$$

We could now impose the required constraint on the sum of reciprocals and simplify to get some locus...but in 3 dimensions I don't see how it can be easily interpreted. Perhaps I haven't worked on it much (as usual)...so if I get an idea...I'll post it here.

As far as the Poisson Ratio is concerned, I tend to think that you have to bring in the notion of an electromagnetic wave mediating the force and causing an equivalent elastic interaction somehow, due to the structure of the atom. Frankly, my explanation sounds useless to me as I cannot back it up with equations.

What you are looking for is an accurate theoretical description of the interaction in electromagnetic terms so that you can find an equivalent Poisson's Ratio (considering a small perturbation of course).

EDIT: I get it now...the reciprocal permits an easier interpretation because the messy radicals come to the numerator...