Lecture 19

Today’s objectives - Magnetic Properties I Equations describing magnetic field strength, induction (several versions), relative magnetic permeability, magnetic susceptibility, magnetization of a solid, and saturation magnetization. Origins of magnetic moments. Magnetic types of materials, their relative magnetic permeabilities, and why they behave as they do (diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic). Temperature dependence of magnetization and why it occurs. Reading: All of Chapter 20 should be read.

Magnetic dipoles Magnetic forces develop when a charged particle moves. Magnetic dipoles exist within certain magnetic materials. Just like bar magnets with North and South poles. The magnetic dipoles point from South to North by convention. (Compare to Electric fields, which go from + to -) The force of a magnetic field exerts a torque on the dipole that tends to align it. Compass needle

Earth’s Magnetic Field A magnetic field is generated every time an electrically charged object moves. Most of the planets in the Solar System are known to generate magnetic fields. The Earth's magnetic field is generated in its fluid, outer core. This is because the heat of the inner core drives the fluid in the outer core up and around in a process called convection. Because this outer core is made of metal, which can be electrically charged, the convection causes a magnetic field to be generated.

The Stone from Magnesia - Magnetite Magnetite (or lodestone) : opaque, black, ceramic crystal. Magnetite (FeO · Fe 2 O 3 ) is an oxide of iron which, unlike Fe 2 O 3 , is strongly magnetic. Spinel Structure Atom x y z Fe(tet) .125 .125 .125 Fe(oct) .5 .5 .5 O .2549 .2549 .2549

Magnetic Vectors A magnetic field, either induced or permanent, generates a magnetic force. The direction of the force is drawn (blue lines). Density of field lines indicates the field strength. H=External magnetic field in units of H for Henry’s Also called magnetic field strength. a vector B=magnetic induction (Magnitude of internal magnetic field strength within a material exposed to an H field) in units of T for Tesla Also called magnetic flux density. also a vector

Comparison: magnets and dielectrics A good dielectric has charges which can polarize in an external field (opposite to it). electrons vs protons in nucleus cations and anions polar molecules interfaces µ=permeability (depends on the material, similar to a dielectric constant where ε was related to electronic polarizability). A strong permeability means the material is made of something which can align strongly to an external magnetic field. This leads to a strong magnetic induction (or flux density). ε (or k)

Magnetic Permeability Since the permeability influences the magnetic induction (flux density), It impacts how good of a magnet you can make. In a vacuum, the permeability is a universal constant, μ o = 1.257*10 -6 H/m.

Other magnetic terms The relative permeability ( µ r ) is sometimes used to describe the magnetic properties of a material (like ε for dielectrics) . The magnetization (M) represents the magnetic moments within a material in the presence of a magnetic field of strength H (akin to polarization, P, for a dielectric). The magnitude of M is proportional to the applied field according to the magnetic susceptibility (  m ) . There are thus four main ways to represent B, the magnetic induction (also called the flux density). Note that units get very confusing. Just stick with one system (SI).

Magnetic Orbital Moments Magnetic moments arise due to two mechanisms: Orbital motion of an electron around the nucleus. Essentially a small current loop, generating a very small magnetic field. A magnetic moment is established along the axis of rotation. m l is the magnetic quantum number for the electron. The magnetic quantum number indicates the type of orbital (shape and usually orientation). 5 3 1 Total orbitals 10 -2,-1,0,1,2 d 6 -1,0,1 p 2 0 s Total electrons m Orbital

Magnetic Spin Moments 2 nd source of a Magnetic Moment Direction that an electron spins. Only two directions are possible. The moment resulting from these spinning electrons are along the spin axis, either UP or DOWN. The combination of orbital and spin moments for every electron throughout a crystal define its magnetic properties.

How do we Classify Magnetic Properties of Materials? The Faraday Experiment

The Faraday Experiment The coil induces an inhomogeneous magnetic field. The sample is suspended from one of the arms of a sensitive balance into the magnetic field H . Certain materials are weakly expelled from this field (along x direction). These are DIAMAGNETS . PARAMAGNETS are weakly attracted to this field (along –x direction). FERROMAGNETIC , ANTI-FERROMAGNETIC and FERRIMAGNETIC materials are strongly attracted to this field .

Classification via Magnetic Susceptibility  m  r  m

Diamagnetism Nonmagnetic (only occurs in the presence of an external magnetic field, H) Even in an external magnetic field, very weak form of magnetism Non-permanent Occurs opposite to external field. Relative permeability < 1 ( ≈0.99999) Found in all materials, just usually too weak to matter. So weak that only noticed if no other form of magnetism exists for the atom and/or crystal. Most common for atoms with completely filled orbitals (no unmatched electrons that could have spin moments). Inert gases Some ionic structures (H 2 O, Al 2 O 3 ) Noble metals (Au, Cu, Ag, Hg, Zn)

Paramagnetism If the orbitals are not completely filled or spins not balanced, an overall small magnetic moment may exist. Without an external magnetic field, the moments are randomly oriented. No net macroscopic magnetization. “ NonMagnetic” In an external field, the moments align with the field, thus enhancing it (only a very small amount, though). There is no interaction between adjacent dipoles. Permeability ( μ r ) > 1 (barely, ≈ 1.00001 to 1.01. Examples include Al, Cr, Cr 2 Cl 3 , MnSO 4 )

Ferromagnetism Unlike paramagnetism with incompletely balanced orbital or spin moments which are randomly aligned, for some materials unbalanced spin can lead to significant permanent magnetic moments . Fe (BCC alpha), Co, Ni, Gd. The permanent moments are further enhanced by coupling interactions between magnetic moments of adjacent atoms so that they tend to align even without an external field. Maximum possible magnetization for these materials is the saturation magnetization (M s , usually quoted per volume). There is a corresponding saturating flux density (B s ). 0.6 1.72 2.22  B Ni Co Fe M per atom

Anti-Ferromagnetism Magnetic moment coupling (for each individual atom) does not always align constructively as for ferromagnetism. For some materials, the alignment of the spin moments of adjacent atoms is in opposite directions. MnO O 2- has no net moment. Mn 2+ have a spin based net magnetic moment. Overall, there is no net magnetic moment even though at the atomic level there is a local moment.

Anti-Ferromagnetism Ordered arrangement of spins of the Mn 2+ ions in MnO determined by neutron diffraction. The 0 2- ions are not shown.

Ferrimagnetism Ferrimagnets are similar to ferromagnets . There is a net magnetic moment . They are also similar to antiferromagnets . The net magnetic moment is not as large as if all of the magnetic atoms coupled constructively. Essentially, ferrimagnetism entails some of the magnetically active atoms coupling constructively, and an unequal number coupling destructively. Examples include Fe 3 O 4 (Fe.Fe 2 O 4 ), NiFe 2 O 4 , ZnFe 2 O 4 . Unlike ferromagnets, they are not electrically conductive. Used in high frequency applications such as microwave devices, circulators, phase shifters.

Ferrimagnetism All Fe 2+ have a spin magnetic moment. Half of Fe 3+ have a spin moment in on direction, the other half in the other (decreasing the overall moment to just that contributed by the Fe 2+ ions). Common for inverse spinel materials and garnets . Usually, 2+ ions of Ni, Mn, Co, and Cu are the active ones. Simpler picture showing a net magnetic moment.

Comparisons To be quantitative, there are 4 options (magnetic permeability, relative permeability, or susceptibility): μ r =1 μ r >1.001 μ r> >>1 μ r =.99999 Large, with H Small, with H Small, opposite H Mag Induction (B) Positive, 10 -3 to 10 -5 >1 (this time  ≈  o + ) paramagnetic >>1 - >>1 ferromagnetic Negative, -10 -5 <1 (barely, so  ≈  o - ) diamagnetic Susceptibility (  m ) Relative Permeability (  r ) Type  measures the material response relative to a vacuum.

Temperature dependence Saturation magnetization M S is the maximum magnetization in a material assuming perfect magnetic dipole alignment. This happens only at T=OK . Increasing T increases thermal vibrations and decreases M S due to diminished (exchange) coupling between dipoles. This is VERY important for ferro- , ferri- , and anti-ferromagnets . Thermal vibrations also cause the dipoles to spend more time pointing in the ‘wrong’ direction, reducing M s . Above a critical temperature called the Curie (or Ne è l) point ( T C or T n ), ferro- and ferrimagnetic materials no longer possess a spontaneous magnetization. They become PARAELECTRIC .

The Curie (or Ne è l) Temperature T n (Fe 3 O 4 ) T C (Fe)

Temperature dependence T C or T n Above a critical temperature called the Curie point ( TC ), ferro- and ferrimagnetic materials no longer possess a spontaneous magnetization. They become PARAMAGNETIC . So do anti-ferromagnetic materials. ferromagnetic anti-ferromagnetic ferrimagnetic T=0K paramagnetic

MAGNETIC MOMENTS FOR 3 TYPES Adapted from Fig. 20.5(a), Callister 6e . Adapted from Fig. 20.5(b), Callister 6e . Adapted from Fig. 20.7, Callister 6e .

Wht about Ferri- and Anti-FerroMagnets ? What about ferrimagnetic? Similar to Ferromagnets What about antiferromagnetic? Similar to Paramagnets

SUMMARY Reading for next class Magnetic properties II Chapter sections: 20.7-11 Equations describing magnetic field strength, induction (several versions), relative magnetic permeability, magnetic susceptibility, magnetization of a solid, and saturation magnetization. Origins of magnetic moments. Magnetic types of materials, their relative magnetic permeabilities, and why they behave as they do (diamagnetic, paramagnetic, ferromagnetic, antiferromagnetic, ferrimagnetic). Temperature dependence of magnetization and why it occurs.

Lecture 19

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Lecture 19