Paramagnetism and Diamagnetism Paramagnets (How do paramagnets differ fundamentally from ferromagnets?) The study of paramagnetism allows us to investigate the atomic magnetic moments of atoms almost in isolation, since unlike ferromagnetism, paramagnetism is not a cooperative phenomenon. Materials exhibiting paramagetism are usually atoms and molecules with odd number of electrons so that there is an unpaired electron spin, giving rise to a net magnetic moment. These include atoms and ions with partially filled inner shells, such as transition elements. Some elements with even number of electrons are paramagnetic. Paramagnetic materials (1) Pt, Al, O, Sn (2) salt of transition metals (Mn 2+, Cr 3+ Fe 2+, Cu 2+ ) chlorides sulphates carbonates in which the paramagnetic moment reside on the Cr 3+, Mn 2+, Fe 2+, and Cu 2+ respectively. (3) hydrated salt KCr(SO 4 ) 2 12H 2 O These salts obey thecurie law C: constant T:Kelvin Because the magnetic moments are located on the metal ions, while the presence of the water molecules in the hydrated salts ensures that the interactions between these electrons on neighboring metal ions are weak. C T 1
Salt and oxide of rare earth (La 系 ) elements are strongly paramagnetic. In these solids the magnetic properties are determined by highly localized 4f electrons. These are closely bond to the nucleus, and are effectively shielded by the outer electrons from the magnetic field at the ionic site caused by the other atoms in the crystal lattice, that is the crystal field. All ferromagnetic metals such as Fe, Co, Ni become paramagnetic above their T c. Paramagnetic metals which do not exhibit a ferromagnetic state. include 1. all the alkali metals (Na series) 2. alkaline earth metals (Ca series) with the exception of Be 3d 4d 5d transition metals are all paramagnetic with the exception of The elements O, Al, Sn are also paramagnetic Cu Zn Ag Cd Hg Temperature dependence of paramagnetic susceptibility (1) In many paramagnet the χ 1/T this dependence is known as the Curie law χ = C/T (2) In other paramagnet the χ is independent of T Two theories have evolved to deal with these two types of paramagnetism (1) The localized moment model χ = C/T (2) The conduction band electron model χ is independent of T (due to Pauli) 2
Field dependence of paramagnetic susceptibility ( what effect does a magnetic field have on the susceptibility of a paramagnet?) In paramagnet subjected to magnetic fields at vey high fields the M is proportional to the field. χ is constant and lies in the range 10-3 to10-5 In most cases, the spins are not coupled or are only weakly coupled. The reason for this is the thermal energy is sufficiently great to cause random alignment of the moments in H=0. When a field is applied the atomic moments begin to align, but the fraction deflected into the field direction remains small for all practical field strengths. 3
At moderate to strong fields, the χ is still constant and saturation only occurs at very high field strengths. The dependence can be expressed classically using the Langevin function: M nm = coth(μ 0mH k B T ) ( k BT μ 0 mh ) n: no. of atoms per unit volume, m: the magnetic moment per atom, k B : Boltzmann s constant, T: the temperature in kelvin. The approximate expression for χ, χ = μ 0nm 2 3k B T works well at high T where is the Curie constant, C. 4
Diamagnets (How do diamagnets differ fundamentally from paramagnets and ferromagnets?) Elements without permanent atomic electronic magnetic moments are unable to exhibit paramagnetism or ferromagnetism. These atoms have filled electron shells and therefore no net magnetic moment. When subjected to a magnetic filled their induced magnetization opposes the applied field so they have negative susceptibility. Diamagnetic χ is substantially independent of T 0 Ze 6 2 n( r 2 ) m e The monatomic rare gases He, Ne, Ar, etc, which have closed-shell electronic structures, are all diamagnetic. Most polyatomic gases, such as H 2, N 2, etc., because the process of molecule formation usually leads to filled electrons shells and no net magnetic moment per molecule. 5
ʘ exchange Forces For the hydrogen molecule For a particular pair of atoms, situated at a certain distance apart, there are certain electrostatic attractive forces and repulsire forces which can be calculated by Coulomb s law. But there is still another force, entirely non-classical, which depends on the relative orientation of the spins of the two electrons -> exchange force. If the spins are antiparallel, the sum of all the forces is attractive and a stable molecule is formed; the total energy of the atoms is then less for a particular distance of separation than it is for smaller or larger distance. If the spins are parallel, the two atoms repel one another. The exchange force is a consequence of the Pauli exclusion principle, applied to the two atoms as a whole. The consideration introduces an additional term, the exchange energy, into the expression for the total energy of the two atoms. 6
The exchange energy forms an important part of the total energy of many molecules and of the covalent bond in many solids. This interchange of electrons takes place at a very high frequency, about 10 18 times per second in H 2. If two atoms i and j have spin angular momentum S i h/2π and S j h/2π, respectively, the exchange energy between them is given by Eex 2JexSiSj JexSiSj cos J ex : exchange integral θ: the angle between the spins (cos θ=1) If J ex > 0, E ex is a minimum, When the spins are parallel E ex is a maximum, When the spins are anti-parallel (cos θ=-1) If J ex < 0, E ex is a minimum, When the spins are anti-parallel E ex is a maximum, When the spins are parallel Ferromagnetism is due to the alignment of spin moments on adjacent atoms, J ex > 0 7
Bethe-Slater curve Bethe-Slater curve, shows the postulated variation of the exchange integral with ratio r a /r 3d the radius of its 3d shell of electron. r r a When is large, J ex is small and positive. r r 3d a When is small, a further decrease in the interatomic distance brings the 3d electron so close together that their spins must becomes anti-parallel (J ex < 0) antiferromagnetism 3d When J ex > 0, it magnitude is proportional to T c, because spins which are held parallel to each other by strong exchange forces can be disordered only by large amounts of thermal energy. J ex (Co)> J ex (Fe)> J ex (Ni) Co, T c =1131 o C > Fe, T c =770 o C > Ni, T c =358 o C 8
Quantum numbers (1) The principle quantum number : n (integer) n=1,2,3 Related to the size and E (2) The angular quantum number : l (integer) For each n, l = 0 ~ n-1 Related to the shape l = 0 s orbital l = 1 p orbital l = 2 d orbital l = 3 f orbital n = 1 l = 0 1s n = 2 l = 0 2s n = 2 l = 1 2p (3) The magnetic quantum number: m l (integer) Related to the orientation in space m l = l -l (including 0) l = 1 m l = 1, 0, -1 p x, p y, p z l = 2 m l = 2, 1, 0, -1, -2 d z2, d x 2 -y2, d xy, d yz, d zx (4) The spin quantum number: m s m s = +1/2 or -1/2 Pauli principle: In a given atom, two electrons can not have the same n, l, m l, m s In the same orbital, n, l, m l must be the same m s must be different 9
Band theory The pauli principle: each energy level in an atom can contain a maximum of two electrons and they must have opposite spin. The 2P subshell is actually composed of 3 sub-subshells of almost the same energy, each capable of holding two electrons. 3d and 4s level have nearly the same energy and they shift their relative positive positions almost from atom to atom. The transition elements, those in which an incomplete 3d shell is being filled are the ones of most interest to us because they include 3 ferromagnetic metals. When atoms are brought close together to from a solid, the positions of the energy level are profoundly modified. 10
When two atoms approach so closely that their electron clouds begin to overlap. In the transition elements, the outermost electrons are the 3d and 4s; these electron clouds are the first to overlap as the atoms are brought together, and the corresponding level are the first to split. When the interatomic distance d d o, the 3d levels are spread into a band extending from B C, and 4s levels are spread into a much wider band from A D. (because the 4s electrons are farther from the nucleus) However, the inner core electrons (1s and 2s) are too far apart to have much effect on one another, and the corresponding energy levels show a negligible amount of splitting. 11
N(E) is not constant but a function of the energy E. density of state The product of the density N(E) and any given energy range gives the number of levels in that range; thus N(E)dE is the number of levels between E and E+dE. Since the 3d and 4s bands overlap in energy. the corresponding density curve as shown as following. 12
The density of 3d levels far greater than that of 4s levels, because there are five 3d levels per atom, with a capacity of a capacity of 2 electrons. Filled energy levels can t contribute a magnetic moment, because the two electrons in each level have opposite spin and thus cancel each other out. Suppose that 10 atoms are brought together to form a crystal. Then the single level in the free atom will split into 10 levels, and the lower 5 will each contain 2 electrons. If one electron reverses its spin, as in (b), then a spin imbalance of 2 is created, and the magnetic moment, μ H =2/10 μ B /atom. The force creating this spin imbalance in a ferromagnetic is just the exchange force. 13