Lecture #14 Program: 1. Introduction Magnetization and it ...

1 1 Lecture #14 Program: 1. Introduction 2. Magnetization and it s relation to free energy 3. Measuring the Magnetization the force exerted by a spatially varying magnetic field. 4. Atomic susceptibility 5. Key equation of atomic Magnetization 6. Larmor Diamagnetism, susceptibility of insulators filled shells 7. Calculating the total angular momenta of atoms in ground states: Hund s rules 8. paramagnetism : susceptibility of insulators with partially filled shells 9. Curie s Law: T dependence of paramagnets 10. Susceptibly of metals: Pauli paramagnetism Handouts: 1. Payne et al Iterative minimization techniques for ab-initio total-energy calculations.

2 2. Ground states of ions with partially filled f and d shells as constructed from Hund s rules (page 652 out of Solid State Physics Ashcroft and Mermin). 2 Introduction : Today we will discuss the magnetic properties of materials, Fe is probably the most well known magnetic material. Other materials exhibit appreciable magnetic properties such as Ni, Co and some alloys such as Ni-Fe and Ni-Co. This type of magnetism is called Ferromagnetism we will learn about it in the next Lecture . It turns out that all substances exhibit magnetic properties albeit these might be exceedingly weak. Materials that exhibit this weak magnetic response can be classified into two: those that are attracted by the magnetic field (like Aluminum are called paramagnetic) and those that are repelled (these are called diamagnetic one example is Bismuth) Magnetism is an entirely QM phenomenon strictly classical system at equilibrium will not display any magnetic moment even in the presence of a magnetic field.

3 The magnetic moment of a free atom has three principle sources: 1. intrinsic spin angular momentum (nucleus and electronic 1/1000 ratio) 2. orbital angular momentum 3. field induced orbital angular momentum The first two effects lead to paramagnetic behavior whereas the third leads to diamagnetic contributions. We will focus on atomic contributions (non-interacting particles) neglecting electron-electron interactions these should allow us to explain the origin of diamagnetism and paramagnetism . Then we will analyze the effects of interactions and in particular the phenomenon of magnetic ordering which leads to ferromagnetism and antiferromagnetism.

4 The Magnetization and it s relation to the total energy 3 The definition of the Magnetization density for a QM system of volume V at T=0 in a uniform magnetic field is defined as: ()()01 EHMHVH = rrrr where ()0 EHris the ground state energy of the system in the presence of the field H which is the field acting on the individual magnetic moments. At temperatures other than 0 the system at thermal equilibrium will be in an excited state (or a superposition of excited states) the average Magnetization density is given by: (),nBnBEkTnnEkTnMeMHTe = r ()()1nnEHMHVH = rrrr Which can be seen as a weighted average of the Magnetization corresponding to the individual excited states.

5 In terms of a thermodynamic quantities one can define the partition function and the corresponding Helmholtz free energy F: nBBEFkTkTnee = ()1,nBnBEkTnnEkTnMeFMHTVHe == r Measuring the Magnetization The Magnetization can be measured by recording the force exerted on a specimen by an inhomogenous field that is varying slowly over the length of the sample, the corresponding change in free energy would be: 4 ()()()()FHHFHxdxFHxdxVMdxHxxfMHx + == = Definition of magnetic susceptibility 202100 MFVHH diamagneticparamagnetic == < > rr Calculation of atomic susceptibility: We would like to now explore the atomic origin of the magnetic susceptibility 1.

6 As we saw in the previous Lecture , in the presence of a uniform magnetic field the Hamiltonian of an atom is modified in the following way: ()()()21 2eHPqARVRm= + in particular the effects of a magnetic field Hr are incorporated via a vector potential A related to the magnetic field by: HA= rrr The simplest and commonly referred to case is that of a uniform magnetic field which can be shown to have a vector potential equal to 12 ArH= rrr In this case Hris a constant and therefore all operators commute with it which allow us to simplify the form of the Hamiltonian: ()()()222221 2 28iieiBiiiiieeparamagneticHPqARVRmPqHVLH Rmm = +=+ + rrh14243 2.

7 The interaction of the field with the individual electron spin is given by: 5 0 2 BzmagneticmomentizziizigHSHSss = == h14243h ; Combining both contributions leads to the identification of those terms in the Hamiltonian which depend on the magnetic field: ()()2220 8magneticBiiqHLgSHHRm = + + rrrh = h Since the relative magnitude of the magnetic energy levels are very small compared to excitation energies as we discussed in the previous Lecture we can compute the changes to the atomic levels using a perturbation approach: atomicmagneticnnperturbationHHH=+14243 where, atomicnnHnEn= The corrections to the energy eigenvalues are given by, 2 atomicmagneticnnnmagneticmagneticmagneti cnatomicatomicnnnnEEEnHnEnHnEE =+ =+ The key equation of magnetism By substituting the form of the magnetic Hamiltonian into the perturbation expression for the magnetic energy we obtain.

8 ()()()22200202222 8magneticBnBiiatomicatomicnninnqOHmamEnL gSnHnLgSHnqHnxynEEm = + + +++ rhrrh14444244443 6 This equation is used as a basis for theories that explain the magnetic susceptibility of atoms, ions and molecules regardless of whether they have filled or partially filled outer shells. This equation is also used to describe solids which can be described as a collection of individual ions such as ionic or molecular solids in such cases the susceptibility is calculated ion by ion. The first term is the dominant term and is on the order of unity which means that if you apply a filed of 410 gauss the correction is about 410eV.

9 The last term has is typically smaller than the first by approximately 5 orders of magnitude 7 Larmor Diamagnetism, susceptibility of insulators with all shells filled In an atom such as He, Ne, Ca which has all the electronic shells filled the ground state will be of spin and orbital angular momentum of 0 (follows from the spherical symmetry), 0000 JLS=== therefore only the last term in the equation for the energy shift will be non-zero. ()22220008magneticiiiqEHxym=+ The susceptibility of a solid composed of N such ions is given by: 22202006magneticiiENqNrVHmV = = Comments: 1. This is known as the Larmor diamagnetic susceptibility (or the Langevin susceptibility) where diamagnetic refers to 0 <which means that the induced moment is opposite to the field.

10 2. Alkali-halide salts such as NaCl exhibit Larmor diamagnetism where the susceptibility of the solid can be taken to be the sum of the independent susceptibility for the negative and positive ions. 3. Diamagnetic susceptibilities are on the order of 510 which means that the Magnetization is very small compared to the field. 4. The trend of diamagnetic susceptibility in the periodic table is: Li+ He Na+ Ne K+ Ar Cs+ Kr -28 Atoms with partially filled shells: Hund s Rules The Hund rules apply to the occupation of levels by electrons in a given shell at ground state, 1. The total electron spin, S is maximized which means that zsiiSm= is maximized consistent with pauli s exclusion principle.