magnetism an itroduction, Lecture notes of Magnetic Resonance Imaging (MRI)

Download magnetism an itroduction and more Lecture notes Magnetic Resonance Imaging (MRI) in PDF only on Docsity! UEEP2024 Solid State Physics Topic 5 Magnetism Magnetic properties of solids • The magnetic moment of a free atom has three principal sources : 1. The spin with which electrons are endowed 2. The electron orbital angular momentum about the nucleus 3. The change in the orbital moment induced by an applied magnetic field Magnetic properties of solids Magnetization is defined as magnetic moment per unit volume. For certain magnetic materials, it is found empirically that the magnetization M is proportional to magnetic field strength H HM  Magnetic susceptibility  > 0 -- paramagnetism  < 0 -- diamagnetiism Atomic Magnets • In classical picture of an atom, the electrons describe a circular orbit around the nucleus. • Each orbit can therefore be thought of as a loop of electric current. • From electromagnetic theory, a loop of current produces a magnetic field, so electrons in an atom also generate a magnetic field. • Quantum theory predicts that electrons in an atom produce a magnetic field. • The quantum number n, l, ml and ms label the electrons in an atom. Atomic Magnets • The orbital magnetic number ml takes values between –l and +l. • Spin magnetic number ms takes values of 1/2. • A single electron the component of the spin magnetic moment, µs, in the direction parallel to the magnetic field is given by • The quantity is known as the Bohr magneton. ee s s m e m em 2     e B m e 2   Total angular momentum J • The total angular momemtum J is obtained by combining L and S as follows:  If the subshell is less than half filled then J = L-S; If the subshell is more than half filled then J = L + S; If the subshell is exactly half filled then L = 0 and so J = S. • By applying these rules, the values of S, L and J can be determined. Landé splitting factor g • The maximum component of the magnetic moment of the atom in the direction parallel to the magnetic field is where g is the Landé splitting factor • Most atom exhibit a magnetic moment. Jgm Bj  )1(2 )1()1()1(3    JJ LLSSJJ g Example Determine the values of S, L and J for Cr3+ which has three electrons in the 3d subshell. (All lower energy subshells are filled) Which materials have magnetic moment? • In covalent material the outer subshell is only partially filled, so these materials have finite magnetic moment. • However, each covalent bond is formed by a pair of electrons with opposite spin and with a net orbital angular momentum of zero. Covalent solids have a net magnetic moment zero. • Actually, this is not quite true. The present of a magnetic field also affect the orbital motion of the electrons in an atom in such a way that the atom generate a magnetic field which opposes the external field. This is referred to as diamagnetism. Which materials have magnetic moment? • Filled electron subshells in an atom do not affect the magnetic momentum of the atom because they have a net angular momentum of zero. (S=L=0, so J = 0). • This means that inert atoms have a magnetic moment of zero because they have only filled electron subshells. • Ionic materials have a magnetic moment of zero because the electrons are transferred from one atom type to another so the resulting ions have only filled subshells. Which materials have magnetic moment? • In covalent materials the outer subshell is only partially filled, and so these materials have a finite magnetic moment. • However, each covalent bond is formed by a pair of electrons with opposite spin and with net orbital angular momentum of zero, covalent solids have a net magnetic moment of zero. • Although most atoms have a non-zero magnetic moment, it appears that in majority of solids the effects cancel and the resultant magnetization is zero. Which materials have magnetic moment? • Free electron paramagnetism is a weak effect. • Paramagnetism is caused by the magnetic dipole moments becoming aligned with the magnetic field, whereas diamagnetism results from an induced magnetic field which opposes the applied field. • Paramagnetism gives rise to a positive susceptibility, whilst diamagnetism produces a negative susceptibility. Which materials have magnetic moment? • A group of metallic elements known as the transition metal, and the so called rare earth and actinide elements have more promising characteristics. • These materials have an electronic structure which is very different to that of the simple metals. • These elements give rise to Curie paramagnetism. • Under certain circumstances they can form permanent magnets. Diamagnetism • The value of B is smaller in the region of the diamagnetic material than it would be if the material were absent • The origin of diamagnetism is Lenz’s law When the flux through an electrical circuit is changed, an induced current is set up in such a direction as to oppose the flux change. Diamagnetism (Classical theory)  d d   22 qIqI  Current formed by the loop = charge x revolution/second For an electron in an atom m Be I m Bqq I  d d d  d 422 2  The magnetic moment  = (curent) x (area of the loop) m Be m Be 44 22 2 2 d   d   Diamagnetism (Classical theory) 2 2 1 4  d  m BZeZ i i   For an atom with Z electrons If r = radius of the 3-D electron shell 2222 zyxr  For a spherically symmetrical distribution of charge 22222 2 3  rzyx  is the radius of the electron loops 222 yx  Diamagnetism (Classical theory) 2 2 6 r m BZe atom d   The magnetic moment of an atom with Z electrons Diamagnetic susceptibility per unit volume B N atomo d    Classical Langevin equation for diamagnetism 2 2 6 r m NZeo  Number of atoms per unit volume Paramagnetism • Only unfilled subshells can have unpaired electrons, so that we expect paramagnetism only in material containing atoms whose electronic subshells are partly filled. • The basic requirement for paramagnetism in solids is that the individual magnetic dipole moments have some degree of isolation If wave functions of atoms overlap significantly, they will tend to pair up the magnetic dipole moments Paramagnetism • Unfilled inner subshells • Isolation of the individual moments results from the shielding of these inner subshells by the filled outer subshells Transition elements Rare earth elements Paramagnetic solids Paramagnetism Transition elements : an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell (defined by IUPAC ) Sc 21 Ti 22 V 23 Cr 24 Mn 25 Fe 26 Co 27 Ni 28 Cu 29 Zn 30 Y 39 Zr 40 Nb 41 Mo 42 Tc 43 Ru 44 Rh 45 Pd 46 Ag 47 Cd 48 Lu 71 Hf 72 Ta 73 W 74 Re 75 Os 76 Ir 77 Pt 78 Au 79 Hg 80 Lr 103 Rf 104 Db 105 Sg 106 Bh 107 Hs 108 Mt 109 Ds 110 Rg 111 Uub 112 Rare earth elements : a collection of seventeen chemical elements in the periodic table, namely scandium, yttrium, and the fifteen lanthanoids (defined by IUPAC ) Paramagnetism • Magnetic field – line up the magnetic dipole moments • Thermal motion – make the directions of the magnetic dipoles random The susceptibility to decrease with increasing temperature T C Curie law Positive constant Paramagnetism JgJ B   The magnetic moment of an atom in free space is Total angular momentum = sum of the orbital and spin angular momentum , gyromagnetic ratio – the ratio of the magnetic moment to the angular momentum B, Bohr magneton – is define as eh/4m g – g factor or spectroscopic splitting factor Paramagnetism        12 111 1    JJ LLSSJJ g The g factor comes about during the calculation of the first- order perturbation in the energy of an atom when a weak uniform magnetic field is applied to the system J : the total electronic angular momentum L : the orbital angular momentum S : the spin angular momentum The energy levels of the system in an applied magnetic field B are BE    Paramagnetism       F F E B E B dEBEDN dEBEDN     )( 2 1 )( 2 1 The total magnetic moment B E N NNM F2 3 )( 2    Pauli paramagnetism Ordered magnetic materials • Interaction between the 3d electrons in these materials have two effect on the magnetic moments of the ions 1. Interaction affects the orbital angular momentum in such a way that the average orbital angular momentum on neighbouring ions cancels. Magnetic moment due to orbital angular momentum become zero. 2. The spins of the 3d electrons interact in such a way that there is a correlation between the spins of 3d electrons on neighbouring ions. This is called the exchange interaction. Ordered magnetic materials • If the interaction between the electrons is sufficiently strong, it is energetically favourable for the electrons in the two ions to have the same spin. • Each ion affects the dipole moment of each of its neighbouring ions, then all of the atomic dipoles in the crystal will be aligned in a common direction. • Such a material is called a ferromagnet. Three simplest type of ordering of atomic magnetic moments . (a) Ferromagnetic. (adjacent magnetic moment are aligned) (b) Antiferromagnetic (adjacent magnetic moment are antiparallel) (c) Ferrimagnetic (adjacent magnetic moment are antiparalle and unequal magnitude) Long range magnetic ordering • Long range magnetic ordering is duo to exchange field from neighbors • Magnetic ordered states occurs only ay low temperatures (T < Tc). When T > Tc there will be no ordering and the material has to be paramagnetic • Three common types of magnetic ordering : Spontaneous magnetisation Ordered magnetic materials • Another possible way of ordering the dipoles is if the magnetic dipoles in adjacent planes are misaligned in such a way that the dipole form a helix. • Example, in magnesium dioxide the angle between the dipoles is about 129o. • These materials are called helimagnets. In a helimagnet the magnetic dipoles are rotated by a fixed angle Solution • The difference is therefore • Since each electron contributes a magnetic moment mm. • The magnetism is • Susceptibility .)( onF BmEgNN   omFm BmEgmNNM 2 )()(   2 )( mFo o o m mEg B M     Example the probability that an atomic dipole moment mj is given by where A is a contant as a function of temperature. Show that the temperature dependence of the magnetic susceptibility can be written as where C is a constant.(called Curie constant) ,kT Bm oj Ae T C m  Solution • If there are N magnetic ions per unit volume, then the total number of dipole in this state is and the magnetic moment of these atom is Magnetization M of the crystal kT Bm oj NAe kT Bm j oj NAem    j j kT Bm j oj NAemM Example Calculate the magnetization of magnetite, Fe3O4, assuming that the magnetization is due only to the Fe2+ ions (which have six 3d electrons) and that only the spin angular momentum of the electrons contributes to the magnetic moment of the ions. (The molar volume of Fe3O4 is 4.40×10 -5 m-3.) Solution • According to the Hund’s rule, the six 3d electrons in each Fe2+ ion are arranged so that the spins of five of the electrons are parallel to one another and the sixth electron is antiparallel. • This can be represented as ↑↑↑↑↑↓. • Since each electron has a spin magnetic moment of µB, the net magnetic dipole moment per ions is 4 µB, and so the dipole moment per mole is 4 µBNA where NA is avogadro’s number. Solution • The magnetization .mJT1007.5 molm1040.4 )mol1002.6)(T J1027.9)(4( memolar volu 4 315 135 123124         AB N M  Values of the saturation magnetization at 300 K shown as Ms (JT -1 m-3), µoMs(T) and Curie temperature c(K). Ms (×10 5 J T-1 m-3) µoMs (T) c (K) Iron 17.1 2.15 1043 Cobalt 14.0 1.76 1388 Nickel 4.85 0.61 627 Gadolinium 20.6 2.60 292 CrO2 5.18 0.65 386 Fe3O4 4.80 0.60 858 MnFe2O4 4.10 0.52 573 NiFe2O4 2.70 0.34 858 Temperature dependence of permanent magnets • In iron, cobalt and nickel the curie temperature is well above room temperature. • Some material, their Curie points are below room temperature, such materials are not normally used as permanent magnet. • The existence of the Curie temperature can be explained as follow. • As the temperature is increased, the thermal vibrations of the ions become so large that the alignment of the magnetic dipoles is destroyed. Temperature dependence of permanent magnets • This happens when the thermal energy kT is comparable with the interaction energy between the neighbouring dipoles. • Since thermal effects are random in nature, we might expect the transition to occur gradually over a temperature range of several degrees, but the Curie temperature is remarkably well defined. • To understand why this is so, let us consider what happens if we have a ferromagnetic material in the absence of any external magnetic field. Schematic diagram showing the process of recording information onto a magnetic medium Current in “1” “1” “1” “1”“0” “0” B Application of magnetic materials for information storage • The choice of material for magnetic recording medium depends on several factors. • The coercivity must be low enough so that the orientation of the particles can be altered by the write head, but high enough so that the orientation is not accidentally affected by other external magnetic fields. • The curie temperature should also be well above the temperature to which the material will normally be exposed. Application of magnetic materials for information storage • Magnetic hard disks are used in computer because they are considerably cheaper than semiconductor memory. • They are not volatile. • Magnetic system does not require a constant electric power supply in order to remain in the same memory state.