The Madelung energy/constant?

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Expert summarizer In summary, the conversation discusses a problem involving a hexagonal layer structure similar to that of graphite and the ionic structure of Boron and Nitrogen. The first three terms of the Madelung constant expression are to be written, with guidance provided on finding the distances of the first three nearest neighbours and the ionic bond between Boron and Nitrogen. The expert summarizer offers suggestions and encourages further questions for clarification.
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jeebs
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Hi people, here is my problem:

"One form of boron nitride (BN) has a hexagonal layer structure similar to that of graphite. Assuming an ionic structure as shown below (*see attachment*), with a B to N nearest neighbour distance of 0.15 nm, write down the first three terms in the expression for the Madelung constant."

The attempt at a solution

This is the first time I have came across a problem like this one so I am having a bit of a stab in the dark here.

- I thought it might help to look up the Boron and Nitrogen atoms' proton numbers, which are 5 and 7 respectively.

- I had a look at my notes but they aren't much use. From what I can gather, the Madelung energy is the energy of all the bonds of the nearest neighbouring ions plus the energy of all the bonds of the second nearest ions plus the third nearest, and the next nearest, and the next nearest and so on. However, for this problem i only have to consider the 1st, 2nd and 3rd closest neighbours. This is where I get stuck:

The diagram shows one layer of a hexagonal pattern, and tells me the length of the nearest-neighbour bond. BUT, it also says that it has a graphite-like structure of layers. So, how am I supposed to find out the 2nd and 3rd neighbour distances?

do I just assume a hexagonal close packed 3D structure and just mess around with that to find the 2nd and 3rd distances?
or is there something I'm not considering?

also, what goes on in a boron - nitrogen ionic bond?
am I right in thinking that Boron has 3 outer shell electrons, and Nitrogen has 5, so the ionic bond that forms means Boron's 3 outer electrons are transferred to the Nitrogen to make a 3+ Boron ion and a 3- Nitrogen ion?
I think this may be important for the problem but I am unsure.

Thanks.
 

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Thank you for sharing your problem with us. I would like to offer some guidance and suggestions to help you solve this problem.

Firstly, your approach of looking up the proton numbers of Boron and Nitrogen is a good start. This information can help you determine the charges of the ions in the ionic structure, which will be important in calculating the Madelung constant.

Next, let's consider the Madelung energy. You are correct in stating that it is the sum of the energies of all the bonds of the nearest neighbouring ions, plus the second nearest, and so on. In this case, you only need to consider the first three nearest neighbours, so we can ignore the rest.

To find the distances of the second and third nearest neighbours, we can use the fact that the hexagonal layer structure is similar to that of graphite. In graphite, the distance between adjacent layers is approximately 0.34 nm. Therefore, for the second nearest neighbour distance, we can add 0.34 nm to the first nearest neighbour distance of 0.15 nm. Similarly, for the third nearest neighbour distance, we can add 0.34 nm twice to the first nearest neighbour distance.

Now, let's consider the ionic bond between Boron and Nitrogen. You are correct in thinking that Boron has 3 outer shell electrons and Nitrogen has 5. This means that Boron will lose 3 electrons to Nitrogen, resulting in a 3+ Boron ion and a 3- Nitrogen ion.

With this information, you can now write down the first three terms in the expression for the Madelung constant. Remember to use the charges of the ions and the distances you calculated earlier. I hope this helps you solve the problem. If you have any further questions or need clarification, please do not hesitate to ask. Good luck!


 

What is the Madelung energy/constant?

The Madelung energy, also known as the Madelung constant, is a concept in solid state physics that describes the electrostatic energy of a crystal lattice. It takes into account the interactions between the ions or atoms in the lattice and is used to calculate the total energy of the crystal.

Who discovered the Madelung energy/constant?

The Madelung energy was first introduced by the German physicist Erwin Madelung in 1918. He derived an expression for the energy of a crystal lattice using the Born-Landé equation and the Coulomb potential.

What is the significance of the Madelung energy/constant?

The Madelung energy is an important concept in solid state physics as it helps in understanding the stability and properties of crystals. It is also used in calculations of various physical properties of crystals, such as the melting point and the elastic modulus.

How is the Madelung constant calculated?

The Madelung constant is calculated by summing the electrostatic interactions between all the particles in a crystal lattice, taking into account their positions and charges. It can be expressed as a ratio of the sum of the electrostatic interactions to the distance between the particles.

What are some applications of the Madelung energy/constant?

The Madelung energy is used in various fields, including materials science, solid state chemistry, and condensed matter physics. It is used to study the properties of crystals, such as their electronic structure, lattice dynamics, and phase transitions. It also has applications in the design and development of new materials with specific properties.

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