# Why does dielectric constant of crystal fluctuates in case of Frenkel defect in a crystal?

So I was studying Frenkel defect in crystal and came across this line that said

The closeness of like charges tends to increase the dielectric constant of the crystal.

I'm not able to understand it properly because in a dielectric an number of dipoles align themselves in a direction parallel to electrical field and thus they create a counter field to reduce the electric field that created or aligned it. Now, if we think like that here, dipole moment is charge multiplied by distance, and distance has definitely reduced between a positive and negative ion here, so dipole moment should decrease and consequently dielectric constant should decrease but why is the opposite happening here.

• closeness of like charges. A dipole is based on separation of unlike charges... – Buck Thorn Mar 3 '19 at 19:01
• Yeah one of the reasons why I'm not able to understand this line.The paragraph below the line are my thoughts on the matter.I'd love to be corrected if there are some potential correction or mistake in my thoughts. – Abner Alfred Thompson Mar 4 '19 at 5:38
• No reason that you can't have a dipole consisting of like point charges. The value will vary with where you put the origin, bit it's still a dipole. – Ian Bush Aug 2 '19 at 12:50

The explanation provided in the text you quote is of the "hand waving" sort. Disrupting the crystal structure via Frenkel defects will have a rather complicated effect. That the effect is complicated is illustrated (in a rather rudimentary fashion) in the wikipedia, from which I derived the following figure:

Clearly the dislodged cations are shifted into closer proximity of other cations, but also of anions, and the orientational effects are not simple.

So the text you quote does not provide an entirely satisfactory explanation, I agree.

Now consider your line of argument:

Now, if we think like that here, dipole moment is charge multiplied by distance, and distance has definitely reduced between a positive and negative ion here, so dipole moment should decrease and consequently dielectric constant should decrease but why is the opposite happening here.

The scenario you describe in the paragraph makes sense. You can write the dipole moment $$p$$ for two point charges of opposite polarity $$q$$ and separated by distance $$d$$ as

$$p = qd$$

Note that, as the wikipedia explains, the value of $$p$$ is independent of the choice of reference point, provided the overall charge of the system is zero.

Obviously decreasing $$d$$ decreases $$p$$.

The problem is that this does not describe the situation layed out in the text you quote:

The closeness of like charges tends to increase the dielectric constant of the crystal.

Consider what it means to increase the closeness of like charges. Presumably if the crystal remains constant in volume and overall charge, it requires that opposite charges move apart.

But this is still a hand-waving explanation. You need to attempt some calculations with a model to get closer to a convincing explanation.