MSE-630 Magnetism MSE 630 Fall, 2008

H = Ni/l (amp-turns/m) MSE-630 Magnetic field strength = H N = # turns i = current, amps l = conductor length Magnetic field lines of force around a current loop and a bar magnet B = Magnetic Induction or Magnetic flux density (Wb/m2) MSE-630

m is the “permeability” In a vacuum, Bo = moH where mo is the permeability in a vacuum =4p x 10-7 (1.257 x 10-6) H/m or T-m/A Units: T: Tesla 1T = 1 V-s/m2 B = mH m is the “permeability” Analogous to dielectrics, the “relative permeability” is mr = m/mo MSE-630

The Magnetization is another field quantity defined by the equation: B = moH + moM And M = cmH B = mH = moH + moM Or M = (m-mo)/mo * H Thus cm = (m-mo)/mo cm is called the magnetic susceptibility, and is a measure of how easily a material is magnetized Thus B = mo(1+cm)H MSE-630

The Origin of Magnetic Moments Magnetic moments come both from the electrons orbiting the nucleus and its spin The most fundamental magnetic moment is the Bohr magnetron, mB mB = 9.27 x 10-24 A-m2 For each electron, the spin magnetic moment is ±mB Furthermore, the orbital magnetic moment contribution is equal to ml*mB, where ml is the magnetic quantum number Only unpaired electrons contribute to the total magnetic moment in an atom MSE-630

Diamagnetism and Paramagnetism Diamagnetism is extremely small, nonpermanent, opposes external field, and only persists while the external field is applied The atomic dipole configuration for a diamagnetic material with and without a magnetic field. In the absence of an external field, no dipoles exist; in the presence of a field, dipoles are induced that are aligned opposite to the field direction. (b) Atomic dipole configuration with and without an external magnetic field for a paramagnetic material MSE-630

MSE-630

Ferromagnetism Ferromagnetism is displayed by large and permanent magnetizations. These occur in transition metals (BCC iron, nickel, and cobalt) and some rare earth elements Susceptibility is as high as 106 – thus, H<<M, and B = moM Schematic illustration of the mutual alignment of atomic dipoles for a ferromagnetic material, which will exist even in the absence of an external magnetic field In ferromagnets, magnetic moments remain aligned when external fields are removed, resulting in permanent magnetization MSE-630

Example: Calculate the saturation magnetization and saturation flux density for nickel, which has a density of 8.9 g/cm3: Ms = 0.60mBN N = rNA/ANi = (8.90 g/cm3)*(6.023 x 1023 atoms/mol)/58.71 g/mol = 9.13 x 1028 atoms/m3 Ms = 0.60 x (9.27 x 10-24) x (9.13 x 1028) = 5.1 x 105 A/m Bs = moMs = 4p x 10-7 H/m * 5.1 x 105 A/m = 0.64 Tesla The maximum possible magnetization, or magnetic saturation, Ms of a ferromagnetic material represents the magnetization that results when all the magnetic dipoles in a solid piece are mutually aligned to the external field. There is a corresponding saturation flux density, Bs. The saturation magnetization is equal to the product of the net magnetic moment for each atom and the number of atoms present. For iron, cobalt and nickel, the net magnetic moments per atom are 2.22, 1.2 and 0.60 Bohr magnetrons, respectively MSE-630

Antiferromagnetism MSE-630 Antiparallel alignment of spin magnetic moments in antiferromagnetic manganese oxide results in complete cancellation of magnetic moments and no net magnetism MSE-630

Ferrimagnetism Some ceramics can show permanent magnetism called Ferrimagnetism. Ferrimagnetism is similar to ferromagnetism, though the source of the net magnetic moments is different MSE-630

MSE-630

MSE-630 Example: Saturation Magnetization determination for Fe3O4 Calculate the saturation magnetization for Fe3O4 given that each cubic unit cell contains 8 Fe2+ and 16 Fe3+ ions, and that the unit cell edge length is 0.839 nm The saturation magnetization is equal to the product of the number, N’, of Bohr magnetrons per cubic meter of Fe3O4 and the magnetic moment per Bohr magnetron, mB: Ms = N’mB N’ is the number of Bohr magnetrons per unit cell nB divided by the unit cell volume Vc, or: N’ = nB/Vc Net magnetization results from the Fe2+ ions only. Each cell has 8 Fe2+ , and each Fe2+ has 4 Bohr magnetrons, thus nB = 32, and Vc = a3, thus Ms = (32 Bohr magnetrons/unit cell * 9.27 x 10-24 A/-m2/Bohr magnetron)/(0.839 x 10-9 m)3/unit cell Ms = 5.0 x 105 A/m MSE-630

Temperature and Magnetization Elevated temperatures cause magnetic dipoles to become unaligned. Magnetism is completely destroyed at the Curie Temperature, Tc MSE-630

Domains and Hysteresis There is a gradual change in magnetic dipole orientation across a domain wall, as shown below: Dipoles are aligned in each domain, but vary from one domain to the other The B-versus-H behavior for a ferromagnetic or ferrimagnetic material that was initially unmagnetized. MSE-630

Hysteresis H increases until Bs and Ms are reached. Upon removal, some magnetism, call remanence, remains at Br. H field must be reversed to –Hc to eliminate residual magnetism. This is called the coercivity. The area in the hysteresis loop represents work or energy expended in going from (+) to (-) H and back. The product of B*H is measured in kJ/m3 or gauss-oersted (MGOe) 1 MGOe = 7.96 kJ/m3 MSE-630

Magnetization and crystal alignment Magnetization curves for single crystals of iron and nickel. Magnetization varies with crystallographic directions Magnetization curves for single crystals of cobalt. MSE-630

Hard vs. Soft magnets MSE-630 Hard magnetic materials retain magnetism after field is removed; soft magnetic materials do not. Soft magnetic materials require small H to reach Bs, and Hc is small. This means less energy is wasted in hysteresis loop. Soft magnetic materials are desirable in transformer cores and other applications where residual magnetization is undesirable Defects, such as nonmagnetic phases and voids restrict easy movement of domain walls and are to be avoided in soft magnetic materials. Sometimes Si or Ni is added to Fe to minimize eddy currents that rob energy in hysteresis loops MSE-630

MSE-630

Magnetic storage Small domains in materials can be magnetically aligned in one of two ways, corresponding to a 0 or a 1 in digital storage. This same technique of magnetic alignment being “written” and “read” is used in recording tapes, VCR and other media Hysteresis loops for particulate magnetic storage media. Saturation flux density is typically 0.4-0.6 Tesla, and the hysteresis loop should be relatively large and square, to ensure that storage will be permanent and magnetization reversal will occur over a narrow range of applied field strengths. For coercivity is typically ~2 x 105 A/m. MSE-630

Scanning electron micrograph showing the microstructure of a magnetic storage disk. Needle-shaped particles of g-Fe2O3 are oriented and embedded within an epoxy phenolic resin. 8000X Each particle is a single domain that may be magnetized only with its magnetic moment lying along this axis. Only two states are allowed, corresponding to digital storage of 1’s and 0’s MSE-630

CoPtCr or CoCrTa alloy are applied to a substrate as a “thin film” for magnetic storage. This provides higher storage capacity than g-Fe2O3 at lower cost. The film is usually 10 – 50-nm thick, and is applied over a layer of pure chromium or a Cr alloy. The thin film is polycrystalline, with a grain size o ~10 – 30-nm. Each grain is a single magnetic domain. Storage density in particulate media is ~1.5 x 105 bits/mm2, while storage density for thin films exceeds 108 bits/mm2. MSE-630