# Do ionic crystals have total electric dipole moment?

## Main Question or Discussion Point

I have come up with a paradox: Ionic crystals, in which cations and anions form a lattice, seems to have total electric dipole moment!

For example, consider a one dimensional example:
$+ - + - + - ... + - + -$
In the above picture, a $+$ represents a cation and a $-$ represents an anion.

So in calculating the total magnetic dipole moment, with the definition $P=\int x \rho (x) dx$, I pair the ions and each cation anion pair have a dipole moment $-ql$, where $q$ is charge and $l$ is the distance between the cation and anion. Then the total dipole moment is $\frac{n}{2}ql$!

I don't think ionic crystals can have total dipole moment because if so, salt will have electric field around it!

Can anybody explain this to me?

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Dotini
Gold Member
Here's some interesting data possibly related to your question.

Strong Forces at Work in Simple Table Salt
Intense electric fields alter electrons arrangement to produce light during NaCl crystallization
http://www.pnnl.gov/science/highlights/highlight.asp?id=1438

Crystalluminescence – the glow given off when a salt becomes a solid – was a critical “hint” to the team that the conventional wisdom underlying salt formation was incomplete. This led them to ask if intense electric fields actually occur in concentrated aqueous electrolytes and thus could be responsible for driving the electronic processes leading to the emission of blue light. The answer? Yes.

That's dynamics. What I concern is the static case.

Delta2
Homework Helper
Gold Member
Big dipole moment doesnt necessarily imply strong static electric field.

Dipole moment is $p(r)=\int\limits_{V} \rho(r')(r-r')d^3r'$ while for example the solution to the electrostatic potential $\phi(r)=\int\limits_{V} \rho(r')\frac{1}{|r-r'|}d^3r'$. Or put even more simply, from gauss's law the electric field seems to depend on $\int\limits_{V}\rho(r')d^3r'$ rather than the integral in the expression for p(r). So i guess you understand why although $p(r)$ can be big, $\phi(r)$ can be small.

Delta2
Homework Helper
Gold Member
I thought of something else, if we try to put some numbers in the formula $\frac{n}{2}ql$, n would be of the order of avogadro 10^24, q is of 10^(-19)Cb and l of order of 10^(-9)m so dipole moment would be of order of 10^(-4)Cb x m which seems pretty small to me. Also with the reasoning of my previous post i believe in points outside the ionic crystal (where we can make the approximation $\frac{1}{|r-r'|}\approx\frac{1}{r}$ ) the potential $\phi(r)$ of the n/2 dipoles will tend to cancel out because $\phi(r)\approx \frac{1}{r}\int\limits_{V}\rho(r')d^3r'=0$ because the total charge of an ionic crystal is zero.

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So the total dipole moment is not zero? It seems so counter-intuitive to me.

Delta2
Homework Helper
Gold Member
Well hmm, if we take for example a cubic crystal one side of the cube will look like

+-+-+-+-+-...+-
-+-+-+-+-+...-+
....
....
+-+-+-+-+...+-
-+-+-+-+-+...-+
I believe thats how the ionic cubic crystal is formed, that is, below above and next to an anion, always a cation is positioned.

So,we can see that the dipole moment of the first 2 lines is cancelled (n/2 *q*l for the first row, n/2*q*(-l) for the second row) and thus the whole dipole moment of the side will be zero. The fault was that we were thinking in 1-D afterall.

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Dotini
Gold Member
Spontaneous electric fields in solid films
http://en.wikipedia.org/wiki/Spontelectrics
http://www.tandfonline.com/doi/abs/10.1080/0144235X.2013.767109#.VTFVjUt-_8s
http://astrochemistry.hw.ac.uk/docs/talks/JL_Astrosurf2013.pdf [Broken]

Related topics
Apparently it you freeze water into a block of ice under the influence of an electric field, you will have a static crystalline structure with an electric dipole.

Electric charge separation during ice formation
http://pubs.acs.org/doi/abs/10.1021/j100244a027?journalCode=jpchax

Piezoelectricity / Rochelle salt
http://en.wikipedia.org/wiki/Piezoelectricity
http://en.wikipedia.org/wiki/Potassium_sodium_tartrate

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mfb
Mentor
The fault was that we were thinking in 1-D afterall.
I think that is the important point.

Yes a 1D lattice will have a dipole moment, of the order of 1 positive charge at one side and one negative charge at the other side (actually half of that, doesn't matter) - that is negligible for macroscopic objects. And if you add the second dimension, most of those dipole moments cancel for pairs of rows. You can still end up with a small net polarization from a few atoms, but most of the 1D lines cancel each other and the net effect is negligible.

Actually, I am thinking about the definition of local dipole moment. How to define it properly in ionic crystals?

mfb
Mentor
I'm not sure if that is a useful quantity.
You can do something like weighting each atom by its distance with a Gaussian. The width of the Gaussian then defines how local your definition is.

I think local dipole moment is a basic quantity in electromagnetic dynamics. For example: $D=\epsilon E + P$.

mfb
Mentor
Only if you consider volumes so large that you don't have to care about atoms any more. As far as I understand you try to do that here.

The dipole moment (or polarization) for an infinite lattice is best studied using periodic boundary conditions. In such case the dipole moment is uncertain by a quantum (or defined modulo quantum). The topic is not trivial but an illuminating explanation can be found here:

http://www.physics.rutgers.edu/~dhv/pubs/local_preprint/dv_fchap.pdf

DrDu
There are crystals with a macroscopic electric moment, namely the so called ferroelectric substances. However, rocksalt does not belong to this class.

DrDu
I think local dipole moment is a basic quantity in electromagnetic dynamics. For example: $D=\epsilon E + P$.
No, because polarisation is not simply the density of dipole moments, only in special cases.

DrDu
Well hmm, if we take for example a cubic crystal one side of the cube will look like

+-+-+-+-+-...+-
-+-+-+-+-+...-+
....
....
+-+-+-+-+...+-
-+-+-+-+-+...-+
I believe thats how the ionic cubic crystal is formed, that is, below above and next to an anion, always a cation is positioned.

So,we can see that the dipole moment of the first 2 lines is cancelled (n/2 *q*l for the first row, n/2*q*(-l) for the second row) and thus the whole dipole moment of the side will be zero. The fault was that we were thinking in 1-D afterall.
That's the correct line of argument for e.g. rocksalt, but there are substances which behave like your one-dimensional example.

No, because polarisation is not simply the density of dipole moments, only in special cases.
Then what is polarisation?

DrDu
Namely, they (and others) use $P(t)= \int_{-\infty}^t dt' j(t')$, where j is the current density.