MAGNETISM IN SOLIDS

Types of Magnetic Material The magnetic behavior of materials can be classified into the following five major groups:  1. Diamagnetism   2. Paramagnetism   3. Ferromagnetism 4. Ferrimagnetism 5. Antiferromagnetism Magnetic properties of Materials

© 2011 Cengage Learning Engineering. All Rights Reserved. Chapter 20: Magnetic Materials Ferromagnetism Alignment of the magnetic moments of atoms in the same direction so that a net magnetization remains after the magnetic field is removed Ferrimagnetism Magnetic behavior obtained when ions in a material have their magnetic moments aligned in an antiparallel arrangement such that the moments do not completely cancel out and a net magnetization remains Diamagnetism The effect caused by the magnetic moment due to the orbiting electrons, which produces a slight opposition to the imposed magnetic field Paramagnetism The net magnetic moment caused by the alignment of the electron spins when a magnetic field is applied © 2011 Cengage Learning Engineering. All Rights Reserved. 20 - 3

© 2011 Cengage Learning Engineering. All Rights Reserved. Chapter 20: Magnetic Materials Antiferromagnetism Arrangement of magnetic moments such that the magnetic moments of atoms or ions cancel out causing zero net magnetization Superparamagnetism In the nanoscale regime, materials that are ferromagnetic or ferrimagnetic but behave in a paramagnetic manner (because of their nano-sized grains or particles) Permanent magnets A hard magnetic material © 2011 Cengage Learning Engineering. All Rights Reserved. 20 - 4

Magnetization Magnetization refers to the process of converting a non-magnetic material into a Magnetic material. The intensity of Magnetization is directly related to the applied field H.

RELATION BETWEEN RELATIVE PERMEABILITY AND MAGNETIC SUSCEPTIBLITY

Magnetic properties of Materials Diamagnetism Diamagnetism is a fundamental property of all matter, although it is usually very weak. It is due to the non-cooperative behavior of orbiting electrons when exposed to an applied magnetic field. Diamagnetic substances are composed of atoms which have no net magnetic moments (ie., all the orbital shells are filled and there are no unpaired electrons). However, when exposed to a field, a negative magnetization is produced and thus the susceptibility is negative. If we plot M vs H, we see: Magnetic properties of Materials

LANGEVIN THEORY OF DIAMAGNETISM Langevin gave a satisfactory explanation of diamagnetism on the basis of electron theory the basic principle of which ia Lenz’s law in electromagnetic induction which states that when a magnetic flux linked with electric current due to revolving electrons is changed, an induced current is set up in such a direction as to oppose the change in flux. It is manifested by the very small and negative value of magnetic susceptibility. If o be the frequency of electron in the absence of applied field and r is the radius of the loop then: Fo = mo2 r = Ze2/4or2 Lorentz force acting on the electron moving with velocity v is given by FL = -Bev = -Ber  = o – eB/2m The –ve sigm indicates that these electrons whise orbital magnetic moments are parallel to the magnetic field are slowed downand those with moments antiparallel are speeded by an amount eB/2m : LARMOR THEOREM.

The additional current produced due to change in frequency of the electron is given by I = -e2B/4m and the change in magnetic moment is given by Ma = -e2r2B/4m <r2> = <x2> +<y2> and <r02> = <x2> +<y2> +<z2> <r2> = 2/3 <r02> Magnetization, M = -e2Z0HN <r02> /6m Susceptibility,  = M/H = -e2Z0N <r02> /6m Since  is independent of temperature so the diamagnetic behavior of the material does not change into temperature.

Magnetic properties of Materials Paramagnetism This class of materials, some of the atoms or ions in the material have a net magnetic moment due to unpaired electrons in partially filled orbitals. One of the most important atoms with unpaired electrons is iron. However, the individual magnetic moments do not interact magnetically, and like diamagnetism, the magnetization is zero when the field is removed. In the presence of a field, there is now a partial alignment of the atomic magnetic moments in the direction of the field, resulting in a net positive magnetization and positive susceptibility. Magnetic properties of Materials

LANGEVIN’S CLASSICAL THEORY Magnetic moments (spins*) in paramagnetic material aligned in a internal (Weiss) field: Hw HW = wM w=Weiss or molecular field coefficient Average total magnetization is: H (applied) M = atomic magnetic dipole moment *Orbital angular momentum gives negligible contribution to magnetization in solids (quenching)

Langevin function Consider graphical solution: Tc is Curie temperature M/Ms 1 At Tc, spontaneous magnetization disappears and material become paramagnetic 1 T/Tc (see Chikazumi, Chp. 6)

QUANTUM THEORY OF PARAMAGNETISM According to quantum theory, the magnetic moments are quantized and in free space is given by ,  = -g B J B is called Bohr magneton g = 1+(J(J+1)+S(S+1)-L(L+1))/2J(J+1) Using Maxwell Boltzmann statistics, magnetization is given as M= N  mJ g B emJ g B/kβ T/ e mJ g B/kβ T CASE I At ordinary temperatures BmJ g B /kβT << 1 Using the exponential series and  mJ2 = (1/3)(J+1)(2J+1) M = Ng2B20 H J(J+1)/3kβT  = M/H para = Ng2B20 Peff 2 Peff = gJ(J+1) CASE II At low temperature and strong magnetic field BmJ g B /kβT is not less than unity

Let x= B g B /kβT M =  BmJ g B emJx As mJ x is a geometric progression with (2J+1) terms M= NgB d/dx (ln( eJx (1-e –(2J+1)x/(1-e-x))) = NgJB BJ(a) BJ(a) = (2J+1)/2J coth (2J+1)a/2J - 1/2J coth (a/2J), Brillouin function a= JgB/kβT In the limit J tends to  Brillouin function approaches the Langevin function i.e., infinite number of possible orientations are allowed.

Magnetic properties of Materials Ferromagnetism When you think of magnetic materials, you probably think of iron, nickel or magnetite. Unlike paramagnetic materials, the atomic moments in these materials exhibit very strong interactions. These interactions are produced by electronic exchange forces and result in a parallel or antiparallel alignment of atomic moments. Exchange forces are very large, equivalent to a field on the order of 1000 Tesla, or approximately a 100 million times the strength of the earth's field. Magnetic properties of Materials

WEISS THEORY OF FERROMAGNETISM The Weiss theory is centered about the following two hypothesis : A ferromagnetic substance contains a number of small regions called domains which are spontaneously magnetized. The value of spontaneous magnetization of the specimen is determined by the vector sum of the magnetic mpments of the individual domains. The spontaneous magnetization within each domain is due to the existence of a molecular filed which produces a parallel alignment of the atomic dipoles. The field is assumed to be proportional to the magnetization of each domain, H=wM. The magnetization of a ferromagnetic material containing N atoms per unit volume placed in the magnetic field Heff is given as M = NgJ B BJ(a) with a = JgBeff/kβT = Jg(B+m)/kβT

Magnetic properties of Materials Hysteresis In addition to the Curie temperature and saturation magnetization, ferromagnetism can retain a memory of an applied field once it is removed. This behavior is called hysteresis and a plot of the variation of magnetization with magnetic field is called a hysteresis loop. Magnetic properties of Materials

P Q -Hs R Hs S Saturation Magnetization Ms Residual o Hc Coercivity Ferro Magnetic Material Hs -Hs o Hc Ms Mr -Ms

If we start with no Magnetized specimen (M= 0) with the increasing values of magnetizing field H. The Magnetization of the specimen increases from zero to higher values and attains its maximum value at a point P, at this point the Magnetization referred as Saturation Magnetization. When we increase Magnetic field H there is no further increment in Magnetic moment. When we decrease Magnetic field H to Zero, the Magnetization M attains point Q. At this point Magnetization referred as residual Magnetization Mr or RETENTIVITY.

Further if we increase the Magnetic field from zero to negative values, the Magnetization of material becomes zero at a point R, at that point the Magnetic field Hc is referred as COERCIVITY of the specimen. If we increase Magnetic field H in reverse direction Magnetization of material reaches its peak value at a points S. On reversing the polarities of Magnetic field and increasing its strength the Magnetization slowly decreases first to residual value then to zero and finally increases to saturation state and touches the original saturation curve. The area of loop indicates the amount of energy wasted in one cycle of operation.