5 Dielectric Properties Lecture 4

K Asokan

    Learning Outcomes:

 

From this module students may get to know about the following:

  1. The derivation and interpretation of Clausius-Mosotti Relation
  2. You will learn about relation between dielectric constant and refractive index .
  3. Detailed study of some Problems related toElectronic polarizability of nonpolar gases, Electronic polarizability of a van der Waals solid, Relative permittivity of ionic crystals, Dielectric constant of water (a dipolar liquid) and Electronic Ploarizability of covalent solids.

    4.1 Clausius-Mosotti Relation:

 

In this lecture we will relate the dielectric constant of an insulator to the polarisability of atoms comprising it. The dipole moment of single atom is proportional to the local field. i.e., dipole moment  , where  is the polarisability of the atoms. If there are N atoms per unit volume, the electric moment per unit volume which is called Polarization is given by

 

This is Clausis-Mosotti relation which relates the macroscopic dielectric constant with the microscopic polarisabilities.

 

multiply both sides of equation (5) by the molar volume, one gets

 

 

but we know that

 

This equcation is called the debyeequcation and it forms the basis for the method of determining permanent dipole moment. Dielectric constants are measured at different temperatures and a graph is drawn between

 

Where the slope b is given by

Thus  can be found out using the above equation (7). In any case if the permanent dipoles moment is zero, the dielectric constant, like the polarization is independent of temperatures. in such case the straight line is parallel to the x- axis. The accuracy of measurement of the dipole moment by means of the above method is determined by the precision with which the slope is determined.

 

 

4.2 Relation between dielectric constant and the refractive index:

 

Here the idea is that there exists a definite relation between dielectric constant and the refractive index comes from the propagation of electromagnetic waves through a medium. The electromagnetic waves as developed by Maxwell by the idea that electromagnetic induction consists of varying electric and magnetic fields with time. The electric and magnetic vectors in such waves are perpendicular to each other and also perpendicular to the direction of propagation. Maxwell has shown that the velocity of propagation of such waves for an unbounded medium is given by

 

where  µ  is  the  magnetic  permeability  of  the  medium  and    is  the  absolute  permittivity.  The permeability for nonmagnetic media is . Hence the velocity in such a medium is.

 

The refractive index of the medium is given by

 

4.3 Problems:

 

4.3.1 Electronic polarizability of nonpolar gases:

 

The electronic polarizability of the Ar atom is 1.710-40 F-m2. What is the static dielectric constant of Ar gas at 1 atmosphere at room temperature (300 K)?

 

Ans:

 

To calculate we need the number of Ar atoms per unit volume, N. If P is the pressure, V is the volume and is the total number of atoms, and then the ideal gas law is

 

if we use Clausius- Mosostti relation here then we get  , which is almost same. The dielectric constant of most gases is small for one major reason. The number of atoms or molecules per unit volume N is very small compared with the number of atoms or molecules in the liquid and solid states. Generally the dielectric constant of most non-polar gases (including air) can be takes as 1, the same as vacuum except at very high pressures.

 

 

4.3.2 Electronic polarizability of a van der Waals solid:

 

The electronic polarizability of the Ar atom is 1.7. What is the static dielectric constant of solid Ar (an FCC crystal below 84 K) if its density is 1.8 g cm3?

 

Ans:

 

To   calculate  we   need   the   number   of   Ar   atoms   per   unit   volume    N   from   the   density   d. If is the relative atomic mass of Ar and NAis Avogadro’s number then we have

The two values are different by about 7 percent.the reason is explained in the above equation.

 

4.3.3 Relative permittivity of ionic crystals:

 

Consider a CsCl crystal which has the CsCl unit cell crystal structure (one Cs+-Cl-pair per unit cell) with a lattice parameter (a) of 0.412 nm. The electronic polarizability of Cs+ and Cl- ions are 3.3510-40 F m2, and 3.4010-40 F m2 respectively, and the mean ionic polarizability per ion pair is 6 10-40 F m2. What is

 

the low frequency dielectric constant and that at optical frequencies?

 

Ans:

 

At high frequencies, that is near optical frequencies, the ionic polarization is too sluggish to allow ionic polarization to contribute to . Thus, relative permittivity at optical frequencies, is given by

 

    4.3.4 Dielectric constant of water (a dipolar liquid)

 

Given the static dielectric constant of water as 80, its density as 1 g cm-3 calculate the permanent dipole moment per water molecule assuming that it is the orientational polarization of individual molecules that gives rise to the dielectric constant. Use both the simple relationship in Equation (3) and also the Clausius-Mossotti equation and compare your results with the permanent dipole moment of the water molecule which is 6.110-30 C m.

 

Solution:

 

Using the expression for orientational polarization we have

 

This is three times greater than the actual permanent dipole moment of H2O ( C m). On the other hand, if we use the Clausius-Mossotti equation we find = 3.1 10-30 C m, which is half the actual permanent dipole moment of H2O. Both are unsatisfactory calculations. The reasons for the differences are two- fold. First is that the individual H2O molecules are not totally free to rotate. In the liquid, H2O molecules cluster together through hydrogen bonding so that the rotation of individualmolecules is then limited by this bonding. Secondly, the local field can neither be totally neglected nor taken as the Lorentz field. A better theory for dipolar liquids is based on the Onsager theory which is beyond the scope of this document. Interestingly, if we usethe actual = 6.1 10-30 C in the Clausius- Mossotti equation, then turns out to be negative, which is nonsense.

 

4.3.5 ELECTRONIC POLARIZABILITY OF COVALENT SOLIDS

 

Consider a pure Si crystal that has  = 11.9.

 

a. What is the electronic polarizability due to valence electrons per Si atom (if one could portion the observed crystal polarization to individual atoms)?

b. Suppose that a Si crystal sample is electroded on opposite faces and has a voltage applied across it. By how much is the local field greater than the applied field?

 

Solution:

 

Given the number of Si atoms, we can apply the Clausius-Mossotti equation to find

 

This is larger, for example, than the electronic polarizability of an isolated Ar atom, which has more electrons. If we were to take the inner electrons in each Si atom as very roughly representing Ne, we would expect their contribution to the overall electronic polarizability to be roughly the same as the Neatom, which is 0.45 x F m2.

 

 

The local field is a factor of 4.63 greater than the applied field.

 

Summary:

 

Dielectric constant of an insulator are related to the polarisability of atoms comprising it also relation between dielectric constant and refractive index were discussed. In addition derivation and interpretation of Clausius-Mosotti Relation were done.

 

References:

  1. Mossotti, O. F. (1850). Mem. di mathem. efisica in Modena. 24 11. p. 49.
  2. Clausius, R. (1879). Die mechanische U’grmetheorie. 2. p. 62.
  3. Rysselberghe, P. V. (January 1932). “Remarks concerning the Clausius–Mossotti Law”. J. Phys. Chem. 36 (4): 1152–1155.doi:10.1021/j150334a007.
  4. Atkins, Peter; de Paula, Julio (2010). “Chapter 17”. Atkins’ Physical Chemistry. Oxford University Press. p. 622-629. ISBN 978-0-19-954337-3.
  5. Hughes, Michael Pycraft (2000). “AC electrokinetics: applications for nanotechnology”. Nanotechnology 11 (2): 124–132. Bibcode: 2000Nanot.. 11..124P. doi:10.1088/0957-4484/11/2/314.
  6. Markov, Konstantin Z. (2000). “Elementary Micromechanics of Heterogeneous Media”. In Konstantin Z. Markov and Luigi Preziosi.’Heterogeneous Media: Modelling and Simulation’ (PDF). Boston: Birkhauser. pp. 1–162. ISBN 978-0-8176-4083-5.
  7. Gimsa, J. (2001). “Characterization of particles and biological cells by AC-electrokinetics”. In A.V. Delgado. Interfacial Electrokinetics and Electrophoresis. New York: Marcel Dekker Inc. pp. 369–400. ISBN 0-8247-0603-X. 

    References and Suggestive Readings

  1. T. Honegger, K. Berton, E. Picard et D. Peyrade. Determination of Clausius–Mossotti factors and surface capacitances for colloidal particles. Appl. Phys. Lett., vol. 98, no. 18, page 181906, 2011.
  2. Feynman, R. P., Leighton, R. B.; Sands, M (1989). Feynman Lectures on Physics. Vol. 2, chap. 32 (Refractive Index of Dense Materials), sec. 3: Addison Wesley. ISBN 0-201-50064-7.
  3. Kittel, Charles (1995). Introduction to Solid State Physics (8th ed.). Wiley. ISBN 0-471-41526-X.
  4. Solid  State  Physics:  An  Introduction,  by  Philip  Hofmann  (2nd  edition  2015,  ISBN-10: 3527412824, ISBN-13: 978-3527412822, Wiley-VCH Berlin.

    Web Links

  1. http://physics.stackexchange.com/questions/59422/polarizability-and-the-clausius-mossotti-relation
  2. http://farside.ph.utexas.edu/teaching/jk1/lectures/node45.html
  3. http://iopscience.iop.org/article/10.1088/0143-0807/4/3/003/pdf
  4. http://www.gitam.edu/eresource/Engg_Phys/semester_2/dielec/clau_moss.htm

    Additional Topics to be studied

 

I.     Richard Feynman on the Clausius–Mossotti equation.

II.   Magnetic analogue of Clausius-Mossotti equation.

 

Brief historical survey of clausius mossotti equation

 

In this section we briefly discuss the historical development of the Clausius-Mossotti relation and its magnetic analogue which unfortunately lacks a different name for itself. The scientific figures responsible the development of his equation are S.D. Poisson (France), M. Faraday (England), J.C. Maxwell (Scotland), O.F. Mossotti (Italy), R. Clausius (Germany) and H.A. Lorentz (The Netherlands). We discuss their contributions chronologically. The history of this equation begins in 1824, when Poisson presented his book at a meeting of the French Academy in which he had carried out a detail mathematical analysis of the problem of magnetic induction. In his classic text on Electricity and Magnetism, Maxwell mentions that “the mathematical theory of magnetic induction was first given by Poisson…” In order to explain the phenomenon of magnetic induction Poisson hypothesized that an imaginary ‘magnetic matter’ or ‘magnetic fluid’ is confined to certain molecules of the magnetic substance. That molecule is magnetized in which the two opposite kinds of magnetic matter, which are present precisely in equal quantity, are separated towards opposite poles of the molecule. He called such molecules ‘magnetic elements’ of the substance and examined the particular case in which these elements are spherical and are uniformly distributed throughout the substance. He calculated the ratio ‘K’ called Poisson’s Magnetic Coefficient – the ratio of the volume of magnetic elements to the whole volume of the substance. This turned out to be [μr -1 ]/[μr +2], the factor that appears in the expression for magnetic polarizibility. Maxwell ruled out the validity of such a hypothesis by using the experimental works of Thalen. However, he concludes, “… the value of Poisson’s mathematical investigation remains unimpaired, as they don’t rest on his hypothesis.” Later on, to explain this phenomenon of magnetic induction Ampere hypothesized that the magnetism of a molecule is due to an electric current that already exists in it which constantly circulates in some closed path within the molecules of the magnet, and must not flow from one molecule to another. These are the two alternative pictures of a magnetic dipole. Thus, using the hypothesis of Poisson and Ampere, and the magnetic analogue of ClausiusMossotti equation, the problem of magnetic induction was completely resolved. Note all this happened about half a century before the development of the Clausius-Mossotti equation for dielectrics. After a few years of Poisson’s formulation, Faraday for the first time applied Poisson’s idea to dielectrics. It was Mossotti who studied the problem in greater detail and presented it in his memoirs. He introduced the ‘cavity method’ which he later developed in his second book. Meanwhile, Clausius was also studying the same problem [19]. For the first time, he explicitly wrote the formula of what is now famous as the ClausiusMossotti equation, as called by Lorentz. It may be noted that all of them attacked the same problem using different approaches. Coming back to our derivation, the approach that is followed in this paper (use of local field or Lorentz field) significantly departs from that used by Poisson yet resembles the one used by H.A. Lorentz and L.V. Lorenz in their derivation of the Lorentz-Lorenz equation (used in optics). So it’s evident that historically, Poisson’s equation plays a more fundamental role as compared to that by the Clausius-Mossotti equation.

 

The Clausius-Mossotti formula gives the atomic polarizability in terms of the dielectric constant for a linear dielectric. We can check the formula for a few gases. The formula is