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MAGNETIC MATERIALS

MAGNETIC MATERIALS. DKR-JIITN-PH611-MAT-SCI-2011. PARAMAGNETISM. Paramagnetism occurs in those substances where the individual atoms, ions or molecules posses a permanent magnetic dipole moments. The permanent magnetic moment results from the following contributions:.

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MAGNETIC MATERIALS

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  1. MAGNETIC MATERIALS DKR-JIITN-PH611-MAT-SCI-2011

  2. PARAMAGNETISM Paramagnetism occurs in those substances where the individual atoms, ions or molecules posses a permanent magnetic dipole moments. The permanent magnetic moment results from the following contributions: - The spin or intrinsic moments of the electrons. - The orbital motion of the electrons. - The spin magnetic moment of the nucleus. DKR-JIITN-PH611-MAT-SCI-2011

  3. Examples of paramagnetic materials: - Metals. - Atoms, and molecules possessing an odd number of electrons, viz., free Na atoms, gaseous nitric oxide (NO) etc. - Free atoms or ions with a partly filled inner shell: Transition elements, rare earth and actinide elements. Mn2+, Gd3+, U4+ etc. - A few compounds with an even number of electrons including molecular oxygen. DKR-JIITN-PH611-MAT-SCI-2011

  4. CLASSICAL (LANGEVIN’S) THEORY OF PARAMAGNETISM Let us consider a medium containing N magnetic dipoles per unit volume each with moment . In presence of magnetic field, potential energy of magnetic dipole Where,  is angle between magnetic moment and the field. B =0, M=0 B ≠0, M≠0 It shows that dipoles tend to line up with the field. The effect of temperature, however, is to randomize the directions of dipoles. The effect of these two competing processes is that some magnetization is produced.

  5. Suppose field B is applied along z-axis, then  is angle made by dipole with z-axis. The probability of finding the dipole along the  direction is f() is the Boltzmann factor which indicates that dipole is more likely to lie along the field than in any other direction. The average value of z is given as Where, integration is carried out over the solid angle, whose element is d. The integration thus takes into account all the possible orientations of the dipoles. DKR-JIITN-PH611-MAT-SCI-2011

  6. Substituting z =  cos and d = 2 sin d Let cos = x, then sin d = - dx and Limits -1 to +1 DKR-JIITN-PH611-MAT-SCI-2011

  7. Langevin function, L(a) In most practical situations a<<1, therefore, The magnetization is given as Variation of L(a) with a. ( N = Number of dipoles per unit volume) DKR-JIITN-PH611-MAT-SCI-2011

  8. This equation is known as CURIE LAW. The susceptibility is referred as Langevin paramagnetic susceptibility. Further, contrary to the diamagnetism, paramagnetic susceptibility is inversely proportional to T Above equation is written in a simplified form as: Curie constant DKR-JIITN-PH611-MAT-SCI-2011

  9. Self study: 1. Volume susceptibility () 2. Mass susceptibility (m) 3. Molecular susceptibility (M) Reference: Solid State Physics by S. O. Pillai DKR-JIITN-PH611-MAT-SCI-2011

  10. QUANTUM THEORY OF PARAMAGNETISM (With outderivation) Recall the equation of magnetic moment of an atom, i. e. Where g is the Lande’ splitting factor given as, Let N be the number of atoms or ions/ m3 of a paramagnetic material. The magnetic moment of each atom is given as, In presence of magnetic field, according to space quantization. Where MJ = –J, -(J-1),…,0,…(J-1), J i.e. MJ will have (2J+1) values.

  11. The magnetic moment of an atom along the magnetic field corresponding to a given value of MJ is thus, If dipole is kept in a magnetic field B then potential energy of the dipole would be Therefore, Boltzmann factor would be, Represents fraction of dipoles with energy MjgBB. The magnetic moment of such atoms would be Thus, average magnetic moment of atoms of the paramagnetic material would be DKR-JIITN-PH611-MAT-SCI-2011

  12. Average magnetic moment Therefore, magnetization would be Case 1: Let, DKR-JIITN-PH611-MAT-SCI-2011

  13. Since Mj = -J, -(J-1),….,0,….,(J-1), J, therefore, Simplifying this equation, we get (consult Solid State Physics by S.O. Pillai, see next two slides), DKR-JIITN-PH611-MAT-SCI-2011

  14. Let a = xJ, above equation may be written as, Here, BJ(a) = Brillouin function. DKR-JIITN-PH611-MAT-SCI-2011

  15. The maximum value of magnetization would be Thus, For J = 1/2 For J , B(a) = L(a) DKR-JIITN-PH611-MAT-SCI-2011

  16. Case 2: But Thus above equation becomes, DKR-JIITN-PH611-MAT-SCI-2011

  17. Thus where, where, This is curie law. Further, Thus Peff is effective number of Bohr Magnetons. C is Curie Constant. Obtained equation is similar to the relation obtained by classical treatment. DKR-JIITN-PH611-MAT-SCI-2011

  18. Calculation of peff: 1. Write electronic configuration. Partially filled sub-shell Say for 6C 2. Find orbital quantum number (l) for partially filled sub-shell. In the given case: 3. Obtain magnetic quantum number. In the given example: In the given case: 4. Accommodate electrons in d sub shell according to Pauli’s exclusion principle. For the given case: DKR-JIITN-PH611-MAT-SCI-2011

  19. 5. Apply following three Hund’s rules to obtain ground state: • Choose maximum value of S consistent with Pauli’s exclusion principle. In the given example: (ii) Choose maximum value of L consistent with the Pauli’s exclusion principle and rule 1. In the given example:

  20. (iii) If the shell is less than half full, J = L – S and if it is more than half full the J = L + S. 5. Obtain J. Since, shell is less than half filled therefore, 6. Obtain g. In the given example: Now calculate peff using For 6C, peff = 0, hence it does not show paramagnetism. DKR-JIITN-PH611-MAT-SCI-2011

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