Luigi Paolasini paolasini@esrf · - Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series in Condensed Matter Physics. L. Paolasini - LECTURES ON MAGNETISM-

Luigi Paolasini [email protected]

L. Paolasini - LECTURES ON MAGNETISM- LECT.2

-  Systems of electrons -  Spin-orbit interaction and LS coupling -  Fine structure - Hund’s rules -  Magnetic susceptibilities

T

χ

Pauli paramagnetism

Curie paramagnetism

Langevin diamagnetism


LECTURE 3: “LOCAL PERTURBATIONS”

-  Magnetic moments -  Crystal field and orbital quenching in transition metals. - Jahn-Teller effect. - Nuclear spins and magnetic resonance techniques. - Electron spin resonance, Mössbauer and muon-spin rotation.

Reference books:

-  Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series in Condensed Matter Physics.


Rare-Earths ions behave in the crystals as in an isolated atoms. The Hund-Rules can predict the values of the effective magnetic moments µeff

peff=µeff /µB=

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

pJpexp

p eff

N (4f electrons)

Rare Earths

Sm3+

Gd3+

Eu3+

*Paramagnetic salts, with kBT>>ECF

pJ

* pJ


The LS-coupling is weak for Sm3+ and Eu3+ (of the order of 300K~25 meV)

The ground state and the next high order term are close in energy and levels mix as a function of magnetic field and temperature

Example Sm3+: 4f5 => S=5/2 L=5 pexp= 1.74

Ground state: J=L-S=5/2 => 6H5/2 p5/2 = 0.85 Next higher term: J=L-S+1=7/2 => 6H7/2 p7/2 = 3.32

0

2

4

6

8

10

12

0 2 4 6 8 10 12 14

pJpexp

p eff

N (4f electrons)

Rare Earths

Sm3+

Gd3+

Eu3+


In rare earths the 4f orbitals are much less extended (more localized) and lie beneath the 5s and 5p shells.

The crystal field terms are less important with respect to the spin-orbit interaction, and Hund third rule is obeyed.

For the transition metals the crystal field plays a major role, and the spin-orbit interaction appear to be negligible.


In transition metals the 3d-electrons participate to the chemical bonding, in contrast to the rare earts where the 4f-electrons are screened by 5d-electrons.

The experimantal values of the magnetic moments pexp of transition metals are closer to the calculed with J=S pS than the calculated pJ:

0

2

4

6

8

10

0 2 4 6 8 10

pJpSpexp

p eff

N (3d electrons)

Transition metals

Ti3+

V4+

Cu2+

Fe3+

Mn2+

Mn3+

Cr2+

pJ pS

pS pJ


Electric field derived from neighboring atoms in a crystal

Crystal field theory: purely electrostatic metal–ligand

interactions between negative point charge distribution around the positive cation

Ligand field theory: Take into account the the role of

orbitals on the central ion and their overlap with the ligand orbitals

Bethe(1929) and VanVleck (1932)

Crystal field play a major role in transition metal oxides, rising or lowering the energy levels of 3d shells due to strong electrostatic repulsions produced by p orbital of neighbouring anions


If an orbital can be transformed into another degenerate orbital (same energy) by a rotation, it is considered to contribute to the electronic angular momentum.

For example, an electron moving from one orbital dx2-y2 to another orbital dxy corresponds to orbital motion about the z axis.

If however there is an “energy cost” associated with this transformation, we say the orbital component (L) of the angular momentum has been QUENCHED and will not contribute to the total magnetic moment. The time average is <L>=0.

The “energy cost” includes a large crystal field splitting or an electron electron repulsion (Pauli exclusion principle). Notice that Hund’s rules do not apply for a non-spherical environment!


The interaction Hamiltonian is the sum of different terms

Hi = H0 + Hso + Hcf + Hz

Coulomb |L,S>

Spin-orbit λ L.S

Crystal field ∫ ρ0(r) φcf(r) d3r

Zeeman gLµB B0

.L+ gSµB B0.S

Notice that the spin-orbit and crystal field interactions are generally much larger than Zeeman interactions!

As a result the magnetic moment is always a small perturbation on the electronic problem, and the magnetic properties are solved by perturbation methods.


The inhomogeneous electric crystal field from the neighbors ligands cause a strong internal Stark effect on 3d-shells, lifting the (2L+1) degeneracy of the 3d-electron levels. mL is split in an upper and a lower band by the crystal field associated to the eg and t2g orbitals of different symmetry


The overlap of eg levels with the ligands (σ-bonding) is stroger than for the t2g levels (π-bonding) -  t2g orbitals have a low overlap with neighbors => less electrostatic repulsion decreases the level energy -  eg orbitals have a strong overlap with neighbors => strong electrostatic repulsion increases the level energy


- The situation is reversed if we consider a tetrahedral C.F. symmetry -  We can prove that in tetrahedral complexes the C.F. splitting is smaller with respect the octrahedral case, and

Δtetr. = 4/9 Δoct.


Competition between the crystal field and the spin-orbit coupling: ΔEcf << ΔEso The Hund’s rules are applicable ΔEcf >> ΔEso only lowest levels are occupied !!

High spin state Electrons occupy t2g and eg levels according to Hund’s rules.

Low spin state Electrons accommodate first in the lower t2g level and pair up to 6 electrons, then eventually occupy the eg levels .

Examples: Verwey transition of Magnetite, Jahn-Teller effect, oxy-deoxy transition


Fe2+ is contained in the centre of a large heterocyclic organic ring called a porphyrin. When O2 binds to deoxymyoglobin, the iron is converted to low-spin Fe3+, which is smaller, allowing the iron to move into the plane of the four nitrogen atoms of the porphyrin to form an octahedral complex.


Jahn-Teller theorem: We cannot have an orbital degeneracy in the ground state of any quantum system.

-  Lifting of orbital degeneracy local symmetry breaking -  Deformation of crystal lower the symmetry and give rise to a more favorable energy state

Mn3+ 3d4


Energy cost to move an atom of mass M away from a thermal equilibrium position is characterized by Q and is associated to a a particular vibrational normal mode with frequency ω.

The total energy of the system is raised/ lowered by the local distortion of orbitals:

Elastic deformation (harmonic oscillator)

Rising or lowering the orbital energy level (supposed at first order linear)

Competition between elastic distorsion and orbital occupancy.

The system have two minima at Q0= ±A/(Mω2) with energy Emin=-A2/(2Mω2)

As a result if one of this orbital is occupied, the system save energy by spontaneously distorting


The correlated displacements give rise to a structural phase transition with orbital ordering at its origin, and which favour alternate orbital occupations

The orbital ordering influence the magnetic exchange interactions between magnetic ions, giving rise to complex magnetic structures.


Some nuclei possess a non-zero spin resulting from its angular momentum:

I = nuclear spin quantum number: represent the total angular momentum of the nucleus. Take the positive values (0, 1/2, 1, 3/2, 2 …) in unit of ħ

mI = components of nuclear angular momentum along the z-axis Take the integer values (-I, -I+1 … I-1, I) in unit of ħ

The unit of nuclear magnetism in the nuclear magneton: µN=eħ/2mp= 5.0508 x 10-27 A m2

The nuclear magnetic moment is defined as: µI=gI µN I

where gI is the nuclear g-factor and I is the largest value resolved along the z-axis


The interaction of nuclear magnetic moments with atomic electrons is very weak and is called hyperfine interaction. The energy splitting of nuclear levels is called hyperfine structure, in analogy with the fine structure due to the spin-orbit coupling.

The hyperfine interaction is due to the coupling of nuclear spin with the field Bel. generated by the atomic electrons, proportional to J

Hhf= A I.J

The total angular momentum of an atom is given by F=J+I and the quantum number F can take the values:

|J-I| , |J-I-1| … J+I-1, J+I

Because F2=J2+I2+2 J.I follow that the expected value of the hyperfine energy is:

<A J.I> = A/2 [F(F+1) - I(I+1) – J(J+1)


The hyperfine interaction split the electronic levels into different hyperfine structure levels labeled by F. Notice that the energy separation between adiacent levels is:

E(F) - E(F-1) = A F


Nuclei with a non-zero spin in a magnetic field absorb and re-emit electromagnetic radiation (1H-proton, 2H-deuteron, 13C …).

This energy is at a specific resonance frequency which depends on the strength of the magnetic field and the magnetic properties of the isotope of the atoms.

A static field B0 is applied perpendicularly to an oscillating field B1 (RF coils). The RF coils excite transition between adjacent pairs of levels.

For a proton in a field of B0=1T:

ΔE= ħω ~10-7 eV ~ 1mK ν=ω/2π ~42.5 MHz

At room temperature the nuclei are completely desordered!


Analogous to NMR, but involve the electron spins. Because the µe>>µI, the precession frequency is much larger than for NMR and in the microwaves range instead the RF.

The sample is inserted in the microwave cavity with a fixed frequency ħω and the magnetic field B0 is sweep. The microwave absorption is modified as a function of B0.

Notice that the tranition ΔmJ=±1 are allowed and that ħω=gµBB


γ-ray produced by radioactive nuclear decay excite a resonant transition in the sample being studied.

Source emitting the γ-rays must be of the same isotope as the atoms in the sample absorbing them.

Frequency tuning by moving the source and compensate the energy lost to recoil (Doppler effect).

v=1 mm s-1 => shift=νv/c ~ 12 MHz

-  Stability of the γ-frequency Related to the long lifetime of 57Fe (Δν~2 MHz) -  Low recoil energy recoil free emission due to the solid state and the energy transfer to the entire crystal.


Isomer (chemical) shift: slight change in the Coulomb interaction between nuclei and electronic charge distribution (s-orbitals).

Quadrupole splitting: Interaction between the nuclear energy levels and surrounding electric field gradient.

Magnetic splitting: Hyperfine splitting due to the interaction between the nucleus any surrounding magnetic field. Selection rules: ΔmI=0,±1


Muon is a spin-1/2 particle with mass mµ=250 me and charge µ±

Muons decay in positrons e+, which conserve signature of the original muon spin.

A magnetic field is applied to rotate the muon spin

The time evolution of muon polarization can be monitored by detecting the number of e+ forward NF(t) and backward NB(t) detectors.

The implanted muons in the in magnetically ordered materials precess in the internal magnetic field, and the oscillating frequency is directly proportional to the internal magnetic field. Strong sensitivity to weak magnetic moments (down to ~10-5 T)

Documents

Luigi Paolasini paolasini@esrf · - Stephen Blundell: “Magnetism in Condensed Matter”, Oxford Master series in Condensed Matter Physics. L. Paolasini - LECTURES ON MAGNETISM-