Calculating Madelung Constant Using Evjen Method

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In summary, the conversation is discussing the use of the Evjen method to calculate the Madelung constant for an infinitely long series of alternately charged ions. The method involves looking at neutral structures, with the first structure consisting of one whole ion and two neighbors cut in half, and the second structure consisting of all other ions. The resulting Madelung constant for the first structure is 1, and for the second structure it is equal to 2(0.5 - 1/2 + 1/3 - ...). The overall Madelung constant, A_n, is equal to the sum of the first and second structure constants. The aim is to show that the deviation of A_n from the actual value A is less
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Petar Mali
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Homework Statement


Applying the method of Evjen calculate Madelungovu constant of infinitely long series of alternately opposite charged ions. Show that summarize by the Evjen cells gives the value Madelung constant [tex]A_n[/tex] whose deviation from actual value [tex]A[/tex] is less than [tex]\frac{1}{n^2}[/tex]

Homework Equations


Madelung constant Madelung constant for the infinite number of ions alternately changing signs

[tex]A=2(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...)=2ln2[/tex]

The Attempt at a Solution


Evjen method is method in which we look in neutral structures. So I think that first neutral structure is one whole ion and two neighbours cut in half. And second neutral structure is all other ions. Is it than Madelung constant for first neutral structure

[tex](A)_I=2\cdot 0,5=1[/tex]

and for second neutral structure

[tex](A)_{II}=2(0,5-\frac{1}{2}+\frac{1}{3}-...)[/tex]

And

[tex]A_n=(A)_I+(A)_{II}[/tex]

But where I have [tex]n[/tex] in here?

Homework Statement


Homework Equations


The Attempt at a Solution

 
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Any idea?
 

1. How is the Madelung constant calculated using the Evjen method?

The Madelung constant is calculated by summing the contributions of the lattice points surrounding the central ion, taking into account their charges and distances from the central ion. The formula for the Evjen method is: M = ∑(qi/r), where q is the charge of the lattice point and r is the distance from the central ion.

2. What is the significance of the Madelung constant in materials science?

The Madelung constant is an important parameter in materials science as it determines the electrostatic interactions between ions in a crystal lattice. It affects the physical and chemical properties of materials, such as their melting point, conductivity, and stability.

3. How does the Evjen method differ from other methods for calculating the Madelung constant?

The Evjen method takes into account the polarization of the ions in the crystal lattice, which can affect the overall electrostatic interactions. Other methods, such as the Ewald method, assume that the ions are fixed and do not take polarization into account.

4. Can the Madelung constant be negative?

Yes, the Madelung constant can be negative. This occurs when the lattice points surrounding the central ion have charges of opposite sign, resulting in a cancellation of the electrostatic interactions.

5. What factors can affect the accuracy of the Madelung constant calculated using the Evjen method?

The accuracy of the Madelung constant calculated using the Evjen method can be affected by several factors, such as the number of lattice points considered, the cutoff distance used, and the choice of lattice structure. It is also important to consider the accuracy of the ion charges and distances used in the calculation.

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