B – H Curve : Hysteresis in ferro-/ferrimagnets 1. Diamagnetism – no permanent dipole moment 2. Paramagnetism – permanent non-interacting magnetic dipoles exist 3. Magnetically ordered materials – interaction between moments causes magnetic order leading to ferro-, antiferro- and ferrimagnetism (and non-zero spontaneous magnetization in ferro- and ferrimagnets).
Types of magnetism 1. Diamagnetism Absence of Permanent magnetic moments Fully filled orbitals On application of magnetic field (H), magnetic moment is induced in such a way to oppose the effect of H Occurs through distortion of orbitals It is proportional to the total no. of electrons . χ dia ~ 10 -6 and is independent of Temperature(T)
2. Paramagnetism : Unpaired electrons existing in any orbital Permanent magnetic moments exist on atoms or ions Applied H orients the moments already present, against thermal agitation to randomize them Magnetization M = m/V ; i.e. M is magnetic moment per unit volume Magnetic Susceptibility: ~ 10 -3 = M/H
Ferromagnetism : M is large M = Ng B J B J (y) ; y = g B H m /k B T Molecular field H m = H + M, is the Weiss molecular field constant B J (y) is the Brillouin function At high T, = C/(T- ) …Curie –Weiss Law At low T, large H : M saturates Induction B = 0 ( H+M) ; Permeability = B/H Antiferromagnetism : M = 0, = C/(T+ ) y
Magnetic order from T dependence of Susceptibility
Domain Theory- Competing Energies 1. Magnetostatic energy H d = -N d M ; E d = ( 0 /2)N d M 2 N d is demagnetizing factor 2. Magnetocrystalline energy 3. Exchange Energy J = S i .S J Minimization of total energy leads to formation of domains separated by domain walls. When H is applied, M develops due to movement of Domain walls Irreversible wall motion causes hysteresis and losses
Anisotropy energy : is due to preference of certain crystallographic directions for the alignment of atomic moments M vs. H for Fe ( Cubic) and Co (Hexagonal) Anisotropy constants Cubic : K Uniaxial : K 1 , K 2 E a = K 1 Sin 2 + K 2 Sin 4 : / M, EA
B – H hysteresis Curve using an AC Inductance method <ul><li>For a secondary coil of N S turns : = N s A B, A- area of coil </li></ul><ul><li>Emf = - / t = - N S ( B A ) / t = - N S A B / t </li></ul><ul><li>Emf α B / t ; </li></ul><ul><li>H = n p I, n p = N p /l p , No. of turn in primary/unit length </li></ul>This works on the principle of “ Faraday's Law of Induction.” as in Transformers Induced Emf = - / t – from one of the Maxwell’s equations With B , the Magnetic Induction = µ 0 (H+M) , = BA,
Integrator C R2 R1 P s S s P S Variac Step down Transformer Circuit : V x ( H ) V Y ( B) Sample V x to X – plates of CRO V y to Y- plates of CRO One Primary and one secondary
Loop Tracer for Toroidal Sample Mutual Inductance set up One primary &Two secondaries – directly gives M – H loops Single primary and single secondary Emf 1 = - N S A µ 0 (H+M) / t Emf 2 = - N S A µ 0 (H) / t
Hysteresis loops Domain theory : Irreversible motion of domain walls causes hysteresis , large coercivity Presence of defects in domains, domain boundaries causes pinning of the walls & large hysteresis Barkhausen effect - Wall diplacement occurs by jerks
Barkhausen Effect Cilcks can be heard on a loud speaker Domain walls need to cross potential wells in the magnetization process Flexible Domain Wall - Contributes to Reversible M
Steps during Magnetization Process H = 0, Random directions of domains Low fields,Reversible Easy dirns. Close to H expand Medium fields, Domains in other equivalent Easy dirns. High fields Domain rotation to H dirn .
Formulae : 1. Magnetic field H = n p I in (A/m) ; I = V R /R Here n p = N p /l p - i.e.no. of turns in primary per unit length. N p is the no. of turns in the Primary a) Solenoid : l p = length of primary b) Toroid : l p = 2 r , where r = (r 1 +r 2 ) / 2 r 2 r 1 1 kA/m = 4 Oe,1 Tesla / 0 = 1 A/m 1 T = 10 4 Oe ; 0 = 4 * 10 -7 H / m Toroid
<ul><li>Magnetic Induction :B = H = 0 ( H + M ) </li></ul><ul><li>Here M is the Magnetization = Magnetic moment / volume </li></ul><ul><li> is the permeability of the sample </li></ul><ul><li> </li></ul><ul><li>Applied current I, as well as M, H and B are time dependent </li></ul><ul><li>in this method. </li></ul>H M T 3 < T 2 < T 1 H B
where = 2 f where f is the frequency of ac signal used. Here f = 50 Hz ( the line frequency) Secondary Output Let B = B 0 Sin t
C R 2 R 1 1. To determine the calibration constant of Integrator 2. Measure the value of R in the primary circuit to determine current I, I = V R /R
Calculate remanance ( B r ), coercivity ( H c ), saturation magnetization ( M s ), field required for saturation ( H s ), Saturation Induction ( B s ) = 4 M s AC Hysteresis loss (W) from the measured B-H curves. The ((1/4 )*area under the B-H curve) gives the information about hysteresis loss in CGS units ( erg / cc / cycle )
Corrections 1. Demagnetisation The rod shaped ( l / d > 10) and Toroidal shaped samples are chosen to minimize the demagnetizing factors (N) so that H int = H / (1-N) H is a valid approximation. 2. Filling Factor : Here, the sample is assumed to fill the volume of the secondary coil completely. If not, we introduce a Filling Factor ( ) correction which defines the extent of sample filling in the secondary coil.