PHYSICS 208 – 2nd TERM

Outline Notes (continued) Section 5. Magnetic Properties (This part of the notes will be relatively brief, because the material is based rather closely on parts of Callister chapter 20). The properties to be considered are “weak” magnetic materials (paramagnets, diamagnets); “strong” magnetic materials (ferromagnets, antiferromagnets, ferrimagnets); applications of strong magnets (e.g., in magnetic recording, computer memory storage,

fast switching, etc.); relation to superconductivity.

5.1 Basic definitions and concepts (see also Callister, sections 20.1 and 20.2) In electricity we have electric charges (electric monopoles) q, and an electric dipole can be formed from +q and −q close together. In magnetism we have magnetic dipoles, but no magnetic monopoles under normal conditions. A magnetic dipole is equivalent to a small current loop (it produces an equivalent distribution of magnetic field lines) − see Fig. 20.1 in Callister.

(from Callister) We can define two different magnetic field vectors H and B as follows:-

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H is usually defined first in terms of an electric current flowing along a wire. For example, if we take a long solenoid of length , N turns, and current I, the field inside the solenoid is (see Fig. 20.3)

(from Callister)

NIH =

This is just a special case of Ampere’s law, which says that The line integral (or sum) of H round a closed path = The total current enclosed

Therefore H = NI, provided we assume no field outside the solenoid and ignore end effects. The units of H are Amp/m. It is called the magnetic field strength (or intensity). It is the same for whatever medium is filling the solenoid. We define another field quantity B (called the magnetic induction) to represent the resultant internal field within a material that is subjected to a H field. In general we write B = µ H where µ is a property of the material called the permeability. The units of B are Tesla (T), basically kg/s-C, and so the units of µ can be deduced. In the case of a vacuum we write B0 = µ0 H where µ0 = permeability of vacuum = 4π × 10-7 kg-m/C2 It is convenient to define a few other quantities from the above definitions:- The difference between B0 (in vacuum) and B (when a material is present) is used to define the magnetization M in the material:

B = B0 + µ0 M = µ0 (H + M) M is the magnetic moment per unit volume in the material. It has the same dimensions (and units) as H. Some other relations are to write: M = χm H where χm is called the magnetic susceptibility

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and B = µ H = µ0 µr H where µr is called the relative permeabilityBoth χm and µr are dimensionless numbers. Substituting into B = µ0 (H + M) gives µ0 µr H = µ0 (H + χm H) , which simplifies to µr = (1 + χm) What is the origin of the magnetic moments in materials? In other words, what makes B different from the vacuum B0? Basically, it comes from more current loops (but this time on an atomic scale or smaller):-

1. Electrons may have an orbital motion around the nucleus (orbital angular momentum → like a small current loop → magnetic moment)

2. Electrons also have an intrinsic spin (spin angular momentum) → also gives a magnetic moment (see Callister, Fig. 20.4)

Add these together to get the total contribution from each electron, and then add again for all the electrons in the atom. Usually this is nonzero (except in a few cases where there is cancellation).

(from Callister) The basic unit on the atomic scale is taken as that due to the electron spin: it is called the Bohr magneton and denoted by µB. µB = 9.27 × 10-24 A-m2 (from QM, it is e /2m, where m = electron mass). In “weak” magnetic materials, χm > 1 typically. 5.2 Weak magnetic materials (see also Callister, section 20.3) There are two cases:

χm is small and negative (µr is slightly less than 1) → diamagnetism χm is small and positive (µr is slightly greater than 1) → paramagnetism

Diamagnetism This is observed in materials when The magnetic moment of each atom = 0

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This occurs if the contributions from all the different electrons add up (vectorially) to zero, e.g., because of complete shell (as in He, Ne, etc) or just because of cancellation (as in Si, Hg, etc). When a field H ≠ 0 is applied, the electron orbits distort slightly because electrons experience a Lorentz force. This changes the current loops and induces a small magnetic moment (in the opposite direction to H) − see Fig 20.5.

(from Callister)

∴ If H > 0, it implies induced M < 0, and since B = µ0 (H + M) it implies B < µ0 H. Hence χm < 0 (typically around −10−5)and µr < 1 (but only slightly). (See Fig. 20.6 and Table 20.2).

(from Callister)

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(from Callister) In principle, the diamagnetic effect occurs in all materials, but it is so small that it is only observed when other magnetic effects are zero (i.e., when the atomic magnetic moments are zero). Paramagnetism This occurs when The magnetic dipole moment of each atom is nonzero, and There are negligible magnetic interactions between neighbouring atoms.

When H = 0 the dipole moments are pointing randomly in all directions and cancel out, so M = 0. When H ≠ 0 the dipole moments preferentially line up in the direction of H (in order to reduce their potential energy, which is given by −M⋅H).

∴ M > 0 (with magnitude depending on the strength of H). Hence χm > 0 (typically 10−5 to 10−2)and µr > 1 (slightly). See Callister, Figs. 20.5b and 20.6. 5.3 Strong magnetic materials (see also Callister, sections 20.4 and 20.5) These occur when

• The magnetic dipole moment of each atom is nonzero, and • There are strong magnetic interactions between neighbouring atoms (by contrast with

paramagnetism). The interactions are QM in origin (and are called exchange interactions)

• They depend on the overlap of wave functions on neighbouring atoms • They give a potential energy contribution that makes neighbouring magnetic moments

align either parallel (like ↑ ↑) or antiparallel (like↑ ↓) There are three important types of strong magnetic behaviour:-

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Ferromagnetism: this is when all atoms have the same magnitude of magnetic moment and the interactions favour parallel alignment (even when H = 0) → Large M ≠ 0 Antiferromagnetism: same as above but the interactions favour antiparallel alignment (even when H = 0) → M = 0 (by cancellation) Ferrimagnetism: this occurs in compounds where there are two (or more) types of atoms having different magnetic moments, and the interactions favour antiparallel alignment of some of these (even when H = 0) → M ≠ 0 (only partial cancellation), but not as large as in ferromagnetism.

Ferromagnetism M is large, so that typically M >> H, i.e. χm >> 1 (can be up to about 106), and so B = µ0 (H + M) ≈ µ0 M This can occur even when H = 0 (unlike both diamagnetism and paramagnetism). (See Fig. 20.7 in Callister).

(from Callister) The max value of M in any material occurs when all the atomic dipoles in the solid line up parallel − this is called the saturation magnetization Ms The corresponding value of B is denoted by Bs (≈ µ0 Ms). Example: Calculate Bs and Ms for Ni given that the magnetic moment of each atom is 0.6µB

and the density of Ni is 8.9 g/cm3. Answer: Ms = max magnetic moment/unit volume = 0.6µBN

where N = no. of atoms/unit volume (m3)

But N = ρNi NA /ANi , where ρNi = density, NA = Avogadro’s number, and ANi = atomic weight

∴ N = 6 3 23(8.9 10 g/m )(6 10 atoms/mol)

58.7 g/mol× × = 9.1 × 1028 atoms/m3

so Ms = 0.6 × 9.27 × 10-24 × 9.1 × 1028 A/m = 5.1 × 105 A/m Then Bs = µ0Ms = 4π × 10-7 × 5.1 × 105 Tesla (T) = 0.64 T Other common ferromagnets are Fe and Co (and metallic alloys).

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Antiferromagnetism These materials have a high degree of magnetic order (antiparallel arrangement of nearest neighbours) but M = 0 due to cancellation. Occurs in many insulating compounds and ceramics. An example is MnO where the Mn2+ ions have canceling magnetic moments and the O2− ions have zero magnetic moment (see Callister, Fig. 20.8).

(from Callister) Ferrimagnetism Again there are antiparallel alignment of some magnetic moments, but cancellation is only partial and so M is nonzero. Some of the most common ferrimagnets are: Cubic ferrites M Fe2O4 , where M is a metal e.g. Fe3O4 (when M = Fe), which is magnetite Iron garnets M3Fe5O12 , where M is a rare earth ion such as Sa, Eu, Gd, Y, etc e.g. Y3Fe5O12 , which is yttrium iron garnet (or YIG). See Callister, Fig. 20.9 to illustrate the behaviour in Fe3O4.

(from Callister)

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5.4 Temperature dependence and domains (see also Callister, sections 20.6 − 20.10) Temperature dependence

In ferromagnets and ferrimagnets, M decreases as T is increased, because thermal effects reduce the magnetic order caused by the exchange interactions. The M value starts at Ms at low T (with T ≈ 0) and steadily decreases and becomes zero (if H = 0) at a transition temperature TC (called the Curie temperature). For T > TC the material becomes paramagnetic. [See Callister, Fig. 20.10].

(From Callister) In antiferromagnets, M is zero because there is the good antiparallel ordering at low T. This ordering is reduced by thermal fluctuations as T is increased. Again it disappears at a transition temperature TN (called the Neel temperature). For T > TN the material becomes paramagnetic. Domains

These are small regions of uniform magnetic alignment (in ferromagnets and antiferromagnets). Their size depends very much on the type of material (i.e., whether it is in a field H of given strength) and on the temperature − it might often be of ~ µm or mm − domains are separated by small boundary regions (domain walls) where there is a

transition from one direction of M to another (see Callister, Figs. 20.11 and 20.12).

(From Callister)

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(From Callister) Magnetization process and hysteresis

Suppose we start from an initially unmagnetized piece of ferromagnet (i.e., one where all the domains cancel out, so B =0 when H = 0). If H is gradually increased, the domains pointing roughly in the direction of H tend to grow in size. Those in the “wrong” direction shrink. So B increases as H increases, until there is saturation (B = Bs) when H is large enough to make all domains parallel. (See Callister, Fig. 20.13).

(From Callister) If H is now gradually reduced, B becomes smaller but does not follow the original magnetization curve (because the domains do not form again exactly as before) − this effect is known as hysteresis. By cycling H from large positive to large negative values, the B versus H plot traces out a closed loop − the hysteresis loop. See also Callister, Figs. 20.14and 20.15.

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(From Callister)

(From Callister) Note the definitions of remanence Br (the value of B when H is reduced to zero) and coercivity Hc (the magnitude of H required to make B = 0). Hard and soft magnetic materials

“Hard” − large area hysteresis loop, large Br and Hc. “Soft” − small area hysteresis loop, small Hc. Hard materials are suitable for applications as permanent magnets

(e.g., tungsten steel − see also Callister, Table 20.6).

(From Callister)

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Soft materials are suitable for applications where the magnetic state is constantly being changed, as in high-frequency magnetic devices, switches, transformer cores, etc.

(e.g. soft iron, Si-iron, Fe/Ni alloys, etc − see also Table 20.5).

(From Callister) It can be shown that the work needed to take a material through a complete hysteresis cycle is proportional to the area of the hysteresis loop. ∴ A small area loop is an advantage in soft materials, as is small Hc. Magnetic information storage The domain structure can be used for information storage (e.g. in computer memories or magnetic recording). Read section 20.10 of Callister for a brief account of magnetic storage in soft materials. The basis idea is for some magnetic materials (particularly when produced as a thin film) there are just two stable directions for the domain magnetizations (say ← and →). These can be used to represent “0” and “1” in the binary code. An external circuit (a “recording head” can be used to read or write the information). See Callister, Fig. 20.18)

(From Callister) In more recent developments it has become advantageous to use films magnetized in the perpendicular direction (rather than the parallel orientation shown above). Currently arrays of artificially produced magnetic nanometer-sized elements (such as nanowires) are being studied as a way to achieve higher-density magnetic memories.

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5.5 Superconductivity (see also Callister, section 20.11) Superconductivity (SC) is concerned with the low-T behaviour of the electrical conductivity σ or the resistivity ρ (= 1/σ). In 1911 it was discovered that for T < 4 K a sample of Hg has infinite σ (or zero ρ). Soon the SC effect was discovered in other materials, mainly metals, below some critical temperature TC. (See also Fig. 20.21).

(From Callister) In metals, TC ranges from a fraction of 1 K up to ~10 K. In metallic alloys, TC values up to ~25 K can be obtained. This how it remained for many years (and in fact it was believed that ~30 K was an upper limit on TC). Then in 1986 a new class of SC was discovered in some particular ceramic materials. Remarkably, TC values up to ~100 K or even 200 K were obtained. In the conventional SCs (metals, alloys) the ρ = 0 property is very well tested (e.g., by observing currents in SC loops remaining undiminished over large periods of time). The TC values can be reduced by either applying a large field H or making a large current density J pass through the material. For large enough H or J (exceeding some critical values) the SC property can be destroyed. (See Fig. 20.22).

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(From Callister) The occurrence of SC is closely related to some magnetic behaviour called the Meissner effect − the field lines of B are excluded from the interior of the SC material (see Fig. 20.23).

(From Callister) But B = µ H = µ0 µr H = µ0 (1 + χm) H , so B = 0 implies χm = −1, i.e., the SC is like a “perfect” diamagnet. The above property holds for H < HC. In some materials there is just one HC value

− these are called Type I SC. In other materials there are two values HC1 and HC2, such that for H < HC1 the Meissner effect is complete (B = 0), but for HC1 < H < HC2 there is a partial Meissner effect in which the B field is partially excluded from the SC (and so −1 < χm < 0)

− these are called Type II SC. Why does SC occur? A successful theory was developed in the 1950s by Bardeen, Cooper and Schrieffer (called the BCS theory). It depended on identifying an attractive component to the interaction forces between two electrons inside the SC (in addition to the Coulomb force of repulsion). This could come about from the distortion effect on the crystal lattice. In some circumstances the electrons could move as pairs through the material, giving rise to the SC effect.

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The new ceramic-based SC compounds, discovered in 1986, have a much higher TC (~ 100 to 200 K). It is believed that an electron-pairing effect still takes place, but it must occur in quite a different way from the original BCS theory. An important factor for applications involving the high-TC SC materials is that the TC values are above liquid-N2 temperatures (~ 77 K). See Table 20.7 for listings of SC materials.

(From Callister)

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Outline Notes (continued)Section 5. Magnetic Properties