Finding nearest neighbour equilibrium distance

In summary, for the given problem, the energy per ion in CsCl can be calculated using the formula (αe 2 /(4πε0)) + 8Ae -(R/ρ), where α is the Madelung constant, A = 5.64 x 103 eV, and ρ = 0.34 Å. The nearest neighbor equilibrium distance for a simple cubic lattice is equal to the lattice parameter, a, and can be determined by taking the derivative of the energy equation and setting it equal to zero.
  • #1
Cocoleia
295
4

Homework Statement


The energy per ion in for CsCl is nearly – (αe 2 /(4πε0)) + 8Ae -(R/ρ) , where α is the Madelung constant and A = 5.64 x 103 eV and ρ = 0.34 Å. Calculate the nearest neighbour equilibrium distance.

Homework Equations


alpha = 2 ln 2

The Attempt at a Solution


I think that CsCl is a simple cubic structure
I found online

For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a. Therefore, for a simple cubic lattice there are six (6) nearest neighbors for any given lattice point.

so then my answer would be 0.34 A ? Is this correct ?
 
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  • #2
The first line of the problem statement appears incomplete. I don't see any definition of ## R ## and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.
 
  • #3
Charles Link said:
The first line of the problem statement appears incomplete. I don't see any definition of ## R ## and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.
That's the entire problem copied and pasted from the assignment.

I'm not sure what you mean by what do I take the derivative of ?
 
  • #4
Cocoleia said:
That's the entire problem copied and pasted from the assignment.

I'm not sure what you mean by what do I take the derivative of ?
The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that ## \frac{dV}{dR}=0 ##. ## \\ ## Consider for example a mass on a spring in a gravitational field.: ## U=\frac{1}{2}kx^2+mgx ## . Taking derivative and setting equal to zero: ## kx+mg=0 ## ==>> ## x_{equilibrium}=-\frac{mg}{k} ##, which the spring constant equation also tells you the forces are balanced there.
 
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  • #5
Charles Link said:
The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that ## \frac{dV}{dR}=0 ##. ## \\ ## Consider for example a mass on a spring in a gravitational field.: ## U=\frac{1}{2}kx^2+mgx ## . Taking derivative and setting equal to zero: ## kx+mg=0 ## ==>> ## x_{equilibrium}=-\frac{mg}{k} ##, which the spring constant equation also tells you the forces are balanced there.
Ok so it is not enough to say
For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a.

I would have to derive the energy equation that is given to me and set =0
 
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Related to Finding nearest neighbour equilibrium distance

1. What is a nearest neighbour equilibrium distance?

A nearest neighbour equilibrium distance is the shortest distance between two points where the forces of attraction and repulsion between them are balanced. This distance is considered to be an important measure in various fields of science, such as physics, chemistry, and biology.

2. How is nearest neighbour equilibrium distance calculated?

The calculation of nearest neighbour equilibrium distance involves finding the point at which the potential energy of the system is at a minimum. This can be done by using mathematical equations, computational methods, or experimental techniques depending on the specific system being studied.

3. What factors affect the nearest neighbour equilibrium distance?

The factors that affect the nearest neighbour equilibrium distance include the strength of the forces between the two points, the mass of the particles involved, and the medium in which they are interacting. Additionally, the shape and arrangement of the particles can also influence the equilibrium distance.

4. How is nearest neighbour equilibrium distance used in research?

Nearest neighbour equilibrium distance is used in various research fields to understand the physical and chemical properties of different systems. It can help predict the behavior and stability of molecules, the structure of crystals, and the arrangement of atoms in a material. It is also used in the design and development of new materials with specific properties.

5. Can nearest neighbour equilibrium distance be altered?

Yes, the nearest neighbour equilibrium distance can be altered by changing the parameters that affect it, such as the strength of forces or the properties of the medium. Researchers can also manipulate the equilibrium distance by introducing external forces or altering the shape and arrangement of particles. However, in some systems, the equilibrium distance is a fixed characteristic that cannot be changed.

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