Elements of magnetism and magnetic measurements Ruslan Prozorov Oct 2009, Feb 2014, Oct 2018 Physics 590B
magnetic moment of free currents Magnetic moment of a closed loop carrying current I: Magnetic field on the axis of a loop of radius R at a distance z is: Total magnetic moment: M= M i (superposition principle) I Mi = d = IS 2c r l n C H z = 2M i 2 2 ( R + z ) 3/ 2 Remember, M is a vector! October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 2
atomic and electronic moments M = J = g J ion J = L + S spin orbital total angular momentum B - gyromagnetic ratio g Landé factor e B = 9.27410 10 2mc 21 erg G ( + 1) + ( + 1) ( + 1) 2J( J + 1) J J S S L L g = 1+ Bohr magneton free electron: g = 2.0023 2.00 Magnetic moment: M e (J=S=1/2) B October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 3
magnetic moment in numbers it measures a total magnetic moment in cgs (emu) 1 emu is: M of a 1 m 2 loop carrying a 1 ma current M of a loop of radius 1.78 cm carrying a 1 A current Typical permanent magnet (1 mm 3 ) ~ 1 emu M of a neutron star ~ 10 30 emu The Earth s magnetic moment ~ 8x10 25 emu An electron spin: B ~10-20 emu Proton and neutron: N ~10-23 emu One Abrikosov vortex (0.1 mm long) ~ 10-10 emu Change in M due to d-wave gap < 10-10 emu/k Hard superconductors ~ 0.1 emu October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 4
the basics emu B Gauss = H Oersted + 4 M 3, cm B = H magnetization B= H H = H + 4 H M = H total magnetic moment of a sample of volume V m erg M = V 3 G cm = 1+ 4 a cow? magnetic permeability magnetic susceptibility October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 5
in SI B B A m B H m = + A m = H m Tesla, erg m = M / V G = H H = H + H = H = 1+ magnetic permeability magnetic susceptibility October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 6
magnetic susceptibility Magnetic moment 2 J 3erg 1 Amp m = 1 = 10 T G Magnetic field 3 A 4 10 A A = Oe Oe Oe = 3 m 10 4 m m 1 0.013, 1 80 Magnetic induction 4 T = G 1 10 NOTE: QD MPMS measures M [emu], not 4M October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 7
= m/ H magnetic susceptibility - dimensionless! SI = 4 cgs erg m = M / V G some other quantities are used: cc m = g = cc g mol m m 1 g = M = g mol mol 1 B H x non-magnetic: = 0, = 1 paramagnetic: 0, 1 diamagnetic: 0, 1 = 4 + 1 1 perfect diamagnetism: =, = 0 4 October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 8
magnetism in a periodic table October http://www.1tvg.com/magsep_u_pu.htm 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 9
superconductivity in a periodic table superconductor.org October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 10
more general form of magnetic susceptibility October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 11
October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 12
Global measurement: TOTAL magnetic moment Local measurement: magnetic induction DC measurement: frequency independent part M 0 AC measurement: frequency - resolved M() October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 13
what is measured magnetic moment? more specifically, - does it depend on the sample shape? How to evaluate intrinsic magnetic susceptibility,, from the measurements of the total magnetic moment, m? October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 14
definitions (all SI) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 15
October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 16
effective demagnetization factors October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 17
Total magnetic moment October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 18
simplest example current ring the integral depends on the integration domain!! 1.0 0.8 cylindrical 0.6 spherical 0.4 0.2 current 0.0 0.0 0.3 0.6 0.9 r (m) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 19
B z (G) B z (G) magnetic induction inside magnetic samples 1.4 r =0.6 spheroid c/a=0.2 r =1.4 1.2 1.0 1.0 0.9 z x 0.8 1.2 cylinder c/a=0.2 1.0 1.0 0.8 0.9 z x 0.6-3 -2-1 0 1 2 3 x (mm) -3-2 -1 0 1 2 3 z (mm) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 20
finite shape makes linear material non-linear N( r )-N(0) N 0.6 0.4 0.2 0.00 cylinder any ellipsoid c/a=0.2 c/a=1 c/a=5-0.05-0.10 0 10 20 30 40 50 r 0.0 0 10 20 30 40 50 r October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 21
October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 22
DC = Direct Current AC = Alternating Current https://en.wikipedia.org/wiki/direct_current#/media/file:types_of_current.svg DC Magnetization magnetic moment does not change (much) during the time window of the measurement October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 23
hysteresis M(H) loops - ferromagnet October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 24
influence of domain structure October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 25
4m (G) type-i superconductor 400 300 200 100 0-100 -200-300 -400 a "perfect" Pb sphere T = 4.5 K -600-400 -200 0 200 400 600 T (K) H c (1-N)=327 Oe N=1/3 H c (1-N) H c =490 Oe October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 26
M (emu) hysteresis is a generic feature 1.2 0.8 Pb single crystal T = 4.5 K H p = H c (1-N)=220 Oe N=0.55 0.4 0.0 H c =500 Oe -0.4-0.8 full M(H) loop partial M(H) loops field cooling -1.2-600 -400-200 0 200 400 600 H (Oe) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 27
M (emu) typical type-ii superconductor 0.15 0.10 0.05 0.00-0.05-0.10-0.15-20000 -10000 0 10000 20000 H (Oe) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 28
B/H m(r) in a superconductor after ZFC+H(pulse) 1.0 1.0 0.5 0.5 0.0 0.0-1.0-0.5 0.0 0.5 1.0 x/d October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 29
local measurements of the magnetic induction undergrad experiment October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 30
magnetometer Popular definition (Wikipedia): A magnetometer is a scientific instrument used to measure the strength and/or direction of the magnetic field in the vicinity of the instrument. Magnetism varies from place to place and differences in Earth's magnetic field (the magnetosphere) can be caused by the differing nature of rocks and the interaction between charged particles from the Sun and the magnetosphere of a planet. Magnetometers are often a frequent component instrument on spacecraft that explore planets. Rotating coil magnetometer Hall effect magnetometer Proton precession magnetometer Gradiometer Fluxgate magnetometer Lab definition: A device to measure magnetic moment or magnetic induction of finite volume samples at fixed temperature and magnetic field. Magnetic moment is a vector, but only one component is measured at a time. October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 31
types of GLOBAL magnetic measurements Use Faraday law of induction, extraction from the coil MPMS SQUID Vibrating sample magnetometer (VSM) AC susceptibility pulse Use the force or torque Faraday balance (force) torque magnetometer Use sample as a nonlinear element of the cirquit cavity perturbation (MW) torque: tunnel diode resonator (RF) d V = dt τ = μb B Energy: Force: W = μb = Bcos( ) F = grad ( W ) = grad ( μb) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 32
local probes Surface probes Kerr effect magneto-optics Faraday effect magneto-optics miniature Hall-probes Bitter decoration magnetic force microscope miniature SQUID Bulk local probes muon spin rotation polarized neutron reflectometry nuclear magnetic resonance neutron scattering Menascale probes mechanical resonant imaging nanosquids fluorescent confocal microscope optical magnetometry of NV-centers in diamonds (now here at AmesLab!) October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 33
PASSIVE! pick-up coils (no current) AC/DC measurements H H M(t) V ac = t M=const lock-in October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 34
use the force Faraday balance magnetometer Faraday pole caps have the property that in vertical direction z on the symmetry axis of the magnet, where the field, let us say, is in x- direction, the product B x *db x /dz is constant over a considerable range in z. U = MB if F = B = MB ( ) ( B ( z),0,0) x d d ( ) ( 2 db ) x Fz = M xbx = Bx = 2B x dz dz dz October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 35
examples October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 36
extraction coil magnetometer October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 37
QD extraction - coil magnetometer October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 38
torque magnetometer advantages? - fast - small samples τ = MB October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 39
Vibrating-sample magnetometer October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 40
VSM QD versalab October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 41
Lakeshore cryotronics VSM October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 42
QD SQUID-VSM magnetometer October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 43
October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 44
QD MPMS October 2018 Prof. Ruslan Prozorov: Magnetic Measurements Part I - Basics. 45