Solid State Physics | ionic crystal bonding | Born & Meyer equation

[heycoa]

Homework Statement

1) Find the expression for the equilibrium interatomic distance as a function of A, n, and α
2) Determine the expression for the binding energy at equilibrium.
3) Calculate the constant n for NaCl, using the data from the Table 1.2 of Omar book and the fact that the measured binding energy is 7.95 3V per molecule (pair of ions). The Madelung constant for NaCl is 1.75.
4)What does the Madelung constant depend on?

Homework Equations

The only equation given to us, the energy: U(R)=-N(m*e²)/(4∏ε₀*R)+N(A/Rⁿ)

The Attempt at a Solution

for 1) I took the derivative of the equation and set that to zero which seems like it would give me the expression at equilibrium however, I do not know where I am supposed to get α. Assuming I did it correctly,
I got: dU/(dR)=A*R^-n-(0.588002*m)/(ε<sub>0[/SUB*R²)=0.
I am not sure if this even answers the question in all honesty.

for 2) Not sure how to go about this one, I think it is just the above (derived) equation equal to some U(R)equilibrium. But am not too sure.

for 3) If I have my numbers correct,
N=6.022*10^23 (Avogadro constant)
ε0=8.854*10^-12
e=1.602*10^-19 (elementary charge)
R=5.63 angstroms (the value in book)
U(R)=7.95
m=1.75 (given Madelung constant)

So I need to solve for n, however I do not know what A is! I think that A is the Born constant but I have no idea how to get that. All I need is to figure out the Born constant then I can solve for n.

for 4) I said that the madelung constant depends on the shape (Dimension) and the geometry of the crystal.

Please help me I am lost!</sub>

 

[mark726]

Hello there, thank you for your post. Let me break down each question and provide some guidance:

1) The equilibrium interatomic distance can be found by setting the derivative of the energy equation to zero, as you have done. However, in order to solve for the equilibrium distance, you will need to use the given values for A, n, and α. These values should have been given to you in the problem or in the textbook. If they are not given, you may need to make some assumptions or use values from a similar problem.

2) The expression for the binding energy at equilibrium can be found by substituting the equilibrium distance value (from part 1) into the energy equation. This will give you the binding energy at equilibrium.

3) To solve for n, you will need to use the given values for A, ε0, e, R, U(R), and m. Again, if any of these values are not given, you may need to make some assumptions or use values from a similar problem. Once you have substituted in all the values, you can solve for n.

4) You are correct in saying that the Madelung constant depends on the shape and geometry of the crystal. It also depends on the arrangement of the atoms or ions in the crystal lattice. In the case of NaCl, the Madelung constant is 1.75 because of the arrangement of the Na+ and Cl- ions in a face-centered cubic lattice.

I hope this helps guide you in the right direction. If you have any further questions, please don't hesitate to ask. Good luck with your calculations!