# Study of electric and magnetic polarization for various tunneling model in presence of electric and magnetic field separately and their comparison.

INTRODUCTION

In the limit of infinite potential barrier diatomic and off-centered monotomic impurities align themselves along certain preferred crystallographic directions. The related physical properties of the system depend critically on the actual minimum energy orientational configuration of the impurity apart from the certain other parameters such as the height of the potential barrier, the amount of off-center displacement etc. The simplest form of the octahedral potential for the angular motion of diatomic impurity in the solid state matrices was first given by Devonshire (Devonshire 1936) and subsequently modified by Pandey et al (1968). The condition was subscribed later by Bayelar (1972) whose condition was 260< a < 1730 where tan [alpha] = k'/k.

The effect of externally applied electric field on the energy levels, specific heat, absorption spectrum and induced dipole moment of the impurity were studied in detail in the paper of Gomez et al (1967). It was concluded that the thermal average of the system's dipole moment be proportional to A = E ([ex.sub.0])/[DELTA] for weak field (A<<1) and saturate for strong fields (A>>1) (Gomez et al 1967). The possibility of two sets of minimum energy orientational configurations of the impurity in the matrix exhibits a perfect agreement with < 111> off-centered model in the case of KCl - Li+. But for the Na Br - F - system Quingley and Dass (1973) pointed out a deeper minima along <111> direction while the high field polarization experiment by Rollefson (1972) are better fitted by the <110> off-center displacement model.

The electric field polarization for the <110> off centered model and for the model of simultaneous minima along the direction <110> + <111> has been explained by Pandey et al (1986). Further theoretical expression for polarization, dielectric constant etc. for various tunneling models like <100>, <100> + <111> etc. has been given by Tiwari et al (2001, 2001, 2002, 2002, 2003). In the present paper we have attempted to find out the expression for magnetic polarization and susceptibility for the <100>,<110>,<100> + <111> and <110> + <111> models.

Theory

If an external electric field E is applied at an angle ? with the dipole orientation then the average field polarization is given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (1)

Where k=Boltzman constant, T = absolute temperature N = dipole concentration, [[mu].sub.E] =Electric dipole moment.

Taking [x.sup.t]= [[mu].sub.E] E/ kT, the expression for polarization takes the form

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (2)

Applying the electric field in the direction <100> the expression for polarization in electric field is listed in (Tiwari 2001). It is well known phenomenon that every electron in the atom constitute a magnetic dipole if pictured as a tiny spinning sphere of charge. Mostly alkali halides are the paramagnetic substances. Hence to find out the expression for polarization of magnetic dipoles in the presence of magnetic field the valuable Maxwell's equation is given by

[nabla] x B = [mu]J + [mu][epsilon] [delta]B/[delta]T (3)

As for as magnetostatics is concerned where E is constant we still have

[nabla] x B = [mu]J = [mu] [sigma] E (4)

Here [sigma] = Conductivity of the material.

[mu] = Permeability of the medium.

Transforming E with the help of above relation we obtain the magnetic field polarization for various models given below:

Case I: System <100> and magnetic field along <100> direction: For model <100> polarization in presence of magnetic field is given as

M = Nm [e.sup.x] - [e.sup.-x]/4 + [e.sup.x] + [e.sup.-x] (5)

Where M = magnetic polarization

N = Magnetic dipole concentration

m = Magnetic dipole moment

x = m ([nabla] x B)/[mu][sigma][kappa]T, T = absolute temperature

The above relation in the limit x<<1 reduces to

M = Nmx/3

Thus

M/Nm/3 = m([nabla] x B)/[mu][sigma][kappa]T

Let the magnetic field along <100> be

B = [??][B.suy.x] (6)

Then the above equation reduces to

[??] [delta][B.sub.x]/[delta]z - [??] [delta][B.sub.x]/[delta]y = M/([Nm.sup.2]/[mu][sigma][kappa]T) (7)

Solution of equation (7) may be given by Lagrange's partial differential equation method giving the polarization along y and z directions-

[M.sub.y] = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T([c.sub.1] - y) (7a)

[M.sub.z] = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]Tz (7b)

Where [c.sub.1] and [c.sub.2] are determined by boundary condition. The net polarization may be written as

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (8a)

Where F(x,y,z) is the position function.

So, the magnetic susceptibility is

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (8b)

Case II:- System <110> and magnetic field along <100> direction : For model <110> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

This gives the net polarization in the limit x <<1 as

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (9a)

and [chi] = [absolute value of [delta]M/[delta][B.sub.x]]

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (9b)

Case III:- System <100> + <111>and magnetic field along <100> direction : For model <100> + <111> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

With, [x.sub.1] = [m.sub.1] ([nabla] x B)/[mu][sigma][kappa]T, [x.sub.2] = [m.sub.2] ([nabla] x B)/[mu][sigma][kappa]T

In the limit x <<1 above equation reduces to

M = N([m.sub.1.sup.2] + [m.sub.2.sup.2])([c.sub.2] - [B.sub.x])/6[mu][sigma][kappa]T F(x, y, z) (10a)

and

[chi] = N([m.sub.1.sup.2] + [m.sub.2.sup.2])/6[mu][sigma][kappa]T F(x, y, z) (10b)

Case IV:- System <110> + <111>and magnetic field along <100> direction : For model <110> + <111> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

In the limit x<<1 above equation reduces to

M = N([3m.sub.1.sup.2] + [2m.sub.2.sup.2])([c.sub.2] - [B.sub.x])/15[mu][sigma][kappa]T F(x, y, z) (11a)

and

[chi] = N([3m.sub.1.sup.2] + [2m.sub.2.sup.2])/15[mu][sigma][kappa]T F(x, y, z) (11b)

RESULTS AND DISCUSSION

The polarization and susceptibility for the model <100>, <110>, <100> + <111> and <110> + <111> in presence of magnetic field is obtained and given by equations (Tiwar and Singh 2001, Tiwari 2001, 2002, 2003). With suitable choosed co-ordinates the expression for susceptibility for each model is found to resemble with the Langevin's theory and satisfies Curie's inverse temperature law. In the availability of general solution a better result for magnetic susceptibility is expected in further theoretical investigation.

REFERENCES

[1.] Bayelar, H.U.1972. Phys. Status Solidi 53 (B) : 419.

[2.] Devonshire, A.F. 1936.Proc. R. Soc. London, Ser. 153 (A) : 601

[3.] Gomez, M, Bowen S.P. & Krumhansl, Z.A. 1967. Phys. Rev.153 : 1009.

[4.] Mitra, D.N., Agrawal, V.K. & Pandey, G.K.Solid State Com. 8: 1645

[5.] Pandey, G. K., Pandey, K.L., Massey, M.& Raj Kumar, 1986.Phy. Rev. (B), 34 (2), 1277.

[6.] Quingley R. J.& Dass, T.P. 1973. Phys. Rev. 7 (B): 4004.

[7.] Rollefson, R. J. 1972. Phys. Rev. (B): 3235.

[8.] Tiwari, R.K. & Singh P.N.,2001. Proc. Nat. Acad. Sci. India, 71 (A) I.

[9.] Tiwari, Raj Kumar et al 2001. J. of Physical Soci. of Japan, Vol.70, No.1 :173-174 Jan.2001.

[10.] Tiwari, Raj Kumar et al. 2002.Journal of Ultra Science 14(3): 545.

[11.] Tiwari, Raj Kumar et al. 2002.Proc. of Nat. Conf. on Tech., Management in Rural area, p.xxv,Oct. 2002.

[12.] Tiwari, Raj Kumar et al. 2003.Bulletin of Pure and applied Sciences, Vol. 22D (No 2) 115-119, 2003.

Raj Kumar Tiwari *,Mukesh Upadhaya, S.R.Shukla, D.N. Pandey & Ambikesh Tripathi

* Department of Physics & Electronics,Dr. R.M.L. Avadh University, Faizabad-224001, India

* Address for correspondence, E-mail: mupadhyaya@rediffmail.com

In the limit of infinite potential barrier diatomic and off-centered monotomic impurities align themselves along certain preferred crystallographic directions. The related physical properties of the system depend critically on the actual minimum energy orientational configuration of the impurity apart from the certain other parameters such as the height of the potential barrier, the amount of off-center displacement etc. The simplest form of the octahedral potential for the angular motion of diatomic impurity in the solid state matrices was first given by Devonshire (Devonshire 1936) and subsequently modified by Pandey et al (1968). The condition was subscribed later by Bayelar (1972) whose condition was 260< a < 1730 where tan [alpha] = k'/k.

The effect of externally applied electric field on the energy levels, specific heat, absorption spectrum and induced dipole moment of the impurity were studied in detail in the paper of Gomez et al (1967). It was concluded that the thermal average of the system's dipole moment be proportional to A = E ([ex.sub.0])/[DELTA] for weak field (A<<1) and saturate for strong fields (A>>1) (Gomez et al 1967). The possibility of two sets of minimum energy orientational configurations of the impurity in the matrix exhibits a perfect agreement with < 111> off-centered model in the case of KCl - Li+. But for the Na Br - F - system Quingley and Dass (1973) pointed out a deeper minima along <111> direction while the high field polarization experiment by Rollefson (1972) are better fitted by the <110> off-center displacement model.

The electric field polarization for the <110> off centered model and for the model of simultaneous minima along the direction <110> + <111> has been explained by Pandey et al (1986). Further theoretical expression for polarization, dielectric constant etc. for various tunneling models like <100>, <100> + <111> etc. has been given by Tiwari et al (2001, 2001, 2002, 2002, 2003). In the present paper we have attempted to find out the expression for magnetic polarization and susceptibility for the <100>,<110>,<100> + <111> and <110> + <111> models.

Theory

If an external electric field E is applied at an angle ? with the dipole orientation then the average field polarization is given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (1)

Where k=Boltzman constant, T = absolute temperature N = dipole concentration, [[mu].sub.E] =Electric dipole moment.

Taking [x.sup.t]= [[mu].sub.E] E/ kT, the expression for polarization takes the form

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (2)

Applying the electric field in the direction <100> the expression for polarization in electric field is listed in (Tiwari 2001). It is well known phenomenon that every electron in the atom constitute a magnetic dipole if pictured as a tiny spinning sphere of charge. Mostly alkali halides are the paramagnetic substances. Hence to find out the expression for polarization of magnetic dipoles in the presence of magnetic field the valuable Maxwell's equation is given by

[nabla] x B = [mu]J + [mu][epsilon] [delta]B/[delta]T (3)

As for as magnetostatics is concerned where E is constant we still have

[nabla] x B = [mu]J = [mu] [sigma] E (4)

Here [sigma] = Conductivity of the material.

[mu] = Permeability of the medium.

Transforming E with the help of above relation we obtain the magnetic field polarization for various models given below:

Case I: System <100> and magnetic field along <100> direction: For model <100> polarization in presence of magnetic field is given as

M = Nm [e.sup.x] - [e.sup.-x]/4 + [e.sup.x] + [e.sup.-x] (5)

Where M = magnetic polarization

N = Magnetic dipole concentration

m = Magnetic dipole moment

x = m ([nabla] x B)/[mu][sigma][kappa]T, T = absolute temperature

The above relation in the limit x<<1 reduces to

M = Nmx/3

Thus

M/Nm/3 = m([nabla] x B)/[mu][sigma][kappa]T

Let the magnetic field along <100> be

B = [??][B.suy.x] (6)

Then the above equation reduces to

[??] [delta][B.sub.x]/[delta]z - [??] [delta][B.sub.x]/[delta]y = M/([Nm.sup.2]/[mu][sigma][kappa]T) (7)

Solution of equation (7) may be given by Lagrange's partial differential equation method giving the polarization along y and z directions-

[M.sub.y] = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T([c.sub.1] - y) (7a)

[M.sub.z] = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]Tz (7b)

Where [c.sub.1] and [c.sub.2] are determined by boundary condition. The net polarization may be written as

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (8a)

Where F(x,y,z) is the position function.

So, the magnetic susceptibility is

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.] (8b)

Case II:- System <110> and magnetic field along <100> direction : For model <110> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

This gives the net polarization in the limit x <<1 as

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (9a)

and [chi] = [absolute value of [delta]M/[delta][B.sub.x]]

M = [Nm.sup.2]([c.sub.2] - [B.sub.x])/3[mu][sigma][kappa]T F(x, y, z,) (9b)

Case III:- System <100> + <111>and magnetic field along <100> direction : For model <100> + <111> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

With, [x.sub.1] = [m.sub.1] ([nabla] x B)/[mu][sigma][kappa]T, [x.sub.2] = [m.sub.2] ([nabla] x B)/[mu][sigma][kappa]T

In the limit x <<1 above equation reduces to

M = N([m.sub.1.sup.2] + [m.sub.2.sup.2])([c.sub.2] - [B.sub.x])/6[mu][sigma][kappa]T F(x, y, z) (10a)

and

[chi] = N([m.sub.1.sup.2] + [m.sub.2.sup.2])/6[mu][sigma][kappa]T F(x, y, z) (10b)

Case IV:- System <110> + <111>and magnetic field along <100> direction : For model <110> + <111> the polarization in magnetic field is worked out and given by

[MATHEMATICAL EXPRSSION NOT REPRODUCIBLE IN ASCII.]

In the limit x<<1 above equation reduces to

M = N([3m.sub.1.sup.2] + [2m.sub.2.sup.2])([c.sub.2] - [B.sub.x])/15[mu][sigma][kappa]T F(x, y, z) (11a)

and

[chi] = N([3m.sub.1.sup.2] + [2m.sub.2.sup.2])/15[mu][sigma][kappa]T F(x, y, z) (11b)

RESULTS AND DISCUSSION

The polarization and susceptibility for the model <100>, <110>, <100> + <111> and <110> + <111> in presence of magnetic field is obtained and given by equations (Tiwar and Singh 2001, Tiwari 2001, 2002, 2003). With suitable choosed co-ordinates the expression for susceptibility for each model is found to resemble with the Langevin's theory and satisfies Curie's inverse temperature law. In the availability of general solution a better result for magnetic susceptibility is expected in further theoretical investigation.

REFERENCES

[1.] Bayelar, H.U.1972. Phys. Status Solidi 53 (B) : 419.

[2.] Devonshire, A.F. 1936.Proc. R. Soc. London, Ser. 153 (A) : 601

[3.] Gomez, M, Bowen S.P. & Krumhansl, Z.A. 1967. Phys. Rev.153 : 1009.

[4.] Mitra, D.N., Agrawal, V.K. & Pandey, G.K.Solid State Com. 8: 1645

[5.] Pandey, G. K., Pandey, K.L., Massey, M.& Raj Kumar, 1986.Phy. Rev. (B), 34 (2), 1277.

[6.] Quingley R. J.& Dass, T.P. 1973. Phys. Rev. 7 (B): 4004.

[7.] Rollefson, R. J. 1972. Phys. Rev. (B): 3235.

[8.] Tiwari, R.K. & Singh P.N.,2001. Proc. Nat. Acad. Sci. India, 71 (A) I.

[9.] Tiwari, Raj Kumar et al 2001. J. of Physical Soci. of Japan, Vol.70, No.1 :173-174 Jan.2001.

[10.] Tiwari, Raj Kumar et al. 2002.Journal of Ultra Science 14(3): 545.

[11.] Tiwari, Raj Kumar et al. 2002.Proc. of Nat. Conf. on Tech., Management in Rural area, p.xxv,Oct. 2002.

[12.] Tiwari, Raj Kumar et al. 2003.Bulletin of Pure and applied Sciences, Vol. 22D (No 2) 115-119, 2003.

Raj Kumar Tiwari *,Mukesh Upadhaya, S.R.Shukla, D.N. Pandey & Ambikesh Tripathi

* Department of Physics & Electronics,Dr. R.M.L. Avadh University, Faizabad-224001, India

* Address for correspondence, E-mail: mupadhyaya@rediffmail.com

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Author: | Tiwari, Raj Kumar; Upadhaya, Mukesh; Shukla, S.R.; Pandey, D.N.; Tripathi, Ambikesh |
---|---|

Publication: | Bulletin of Pure & Applied Sciences-Physics |

Date: | Jan 1, 2006 |

Words: | 1391 |

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