Madelung constant

Petar Mali

1. The problem statement, all variables and given/known data
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]


2. Relevant 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]



3. 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?
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

Petar Mali

Any idea?