physics 590 ruslan prozorov magnetic measurements Nov 9, 2009 -
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
atomic moments M = γ J = gµ J ion J = L+ S spin orbital total angular momentum B γ - gyromagnetic ratio g Landé factor e µ 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 µ e B (J=S=1/2)
the basics [ ] [ ] emu B Gauss = H Oersted + 4 π m 3, cm B B m = µ H = µ H µ H = H + 4πχ H = χh = 1+ 4 µ πχ erg m= M / V G this is ONLY true for homogeneous, uniform para- or dia- magnetic systems
magnetic susceptibility Magnetic moment 2 J 3 erg 1 Amp m = 1 = 10 T G χ = M / H - dimensionless! χ SI = 4πχ cgs some other quantities are used: χ χ m χ cc 1 = g ρ = cc g g = χ M = g mol mol m m mol 1 B H x
Extensions global (total) magnetic moment ( ) = + 4 ( ) B r H π m r B = µ H 4 ( ( ) ) 3 π = M Br H dr V M V = χh
most general form of magnetic susceptibility
demagnitization
spatial distribution of a magnetic induction
infinite cylinder (or slab) demagnetization = 0
ellipsoid (non-zero demag, H uniform)
non-zero demag, non-uniform H
confusing? in general, we cannot assume uniform field distribution validity of equations depend on geometry and particular system the process of measurement involves applying an external field, so you only measure properties at that field! there is ALWAYS a total magnetic moment, but its relation to the applied field may be very complicated Let s demonstrate: paramagnet: ½ + ½ = 1 superconductor ½ + ½ = ¾
critical state model
inhomogeneous B In many cases 1. B is not spatially uniform 2. there is magnetic hysteresis M 4π M = ( B H ) dv V B(x) χ = M χ = dm / H / dh - has no meaning - differential magnetic susceptibility can be used
rectangular slab
M(H) loops - ferromagnet
AC/DC magnetization loop
influence of domain structure
type-i superconductor 4πm (G) 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
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 M (emu) 0.4 0.0-0.4-0.8 full M(H) loop partial M(H) loops field cooling H c =500 Oe -1.2-600 -400-200 0 200 400 600 H (Oe)
typical type-ii superconductor 0.15 0.10 0.05 M (emu) 0.00-0.05-0.10-0.15-20000 -10000 0 10000 20000 H (Oe)
types of M(H) loops (Co-122) 0.04 0.02 M (emu) 0.00-0.02-6 -4-2 0 2 4 6 H (Oe)
fishtail 0.04 0.02 M (emu) 0.00-0.02-6 -4-2 0 2 4 6 H (Oe)
distribution of the magnetic induction 1.0 1.0 B/H 0.5 0.5 0.0 0.0-1.0-0.5 0.0 0.5 1.0 x/d
are the profiles real? undergrad experiment
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: µ B ~10-20 emu Proton and neutron: µ 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
magnetometer Popular definition: 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 of small samples at fixed temperature and magnetic field. Magnetic moment is a vector, but only one component is measured at a a time.
types of magnetic measurements global total magnetic moment local B(x) Magnetometers extraction Faraday balance SQUID Vibrating sample magnetometer AC susceptibility Surface probes magneto-optics Hall-probes decoration magnetic force microscope scanning probes (SQUIDs, Hall probes etc)
magnetic moment in a magnetic field torque: τ = μ B θ µ B Energy: Force: W = μb = µ Bcos( θ) F = grad ( W ) = grad ( μb) for example, for Bμ= Bx( x ),0,0, = ( µ x, µ y,0) μb = µ B x x and dbx F = µ x dx in inhomogeneous magnetic field µ > 0 µ < 0 x x F B F changes sign however torque aligns along the field B
PASSIVE! pick-up coils (no current) AC/DC measurements H H M(t) V ac Φ = t M=const lock-in
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 = MxBx = χ Bx = χ 2B x dz dz dz
examples
extraction coil magnetometer
QD extraction - coil magnetometer
torque magnetometer advantages? - fast - small samples τ= M B
Vibrating-sample magnetometer
VSM QD versalab
Lakeshore cryotronics VSM
QD SQUID-VSM magnetometer
QD MPMS
Josephson effect
Superconducting Quantum Interference Device (SQUID) flux-voltage convertors DC SQUID AC (RF) SQUID
DC SQUID
rf-squid one junction
rf-squid
rf-squid
rf (10-20 MHz) current source rf-amp flux transformer resonant tank circuit
gradiometers
Zimmerman rf SQUID
combinations
MPMS longitudinal and transverse coil sets
MPMS: longitudinal coil set
transverse coil set
transverse component
sensitivity of transverse coil set to longitudinal moment
compare to
gradiometer measurements
how does MPM measure magnetic moment?
regression
basic principles
measuring large moments
background subtraction
Practical MPMS - background
background is displaced from the signal by 1 mm
magnetic field variation
Remanent (remnant) field
field non-uniformity and superconductors
another example & literature will be available for download