physics 590

ruslan prozorov

magnetic measurementsNov 9, 2009 - …

Magnetic moment of a closed loop carrying current I:2i C

I d ISc

= × =∫M r l n

Total magnetic moment: i=∑M M

Magnetic field on the axis of a loop of radius R at a distance z is:( )3/ 22 2

2 iz

MHR z

=+

(superposition principle)

magnetic moment of free currents

ion Bgγ µ= = −= +

M J JJ L S

total angular momentumorbital

spin

γ - gyromagnetic ratiog – Landé factor

( ) ( ) ( )( )

1 1 11

2 1J J S S L L

gJ J

+ + + − += +

+

free electron:

2.0023 2.00g = ≈

219.27410 102Be ergmc G

µ −= ≈ × Bohr magneton

Magnetic moment:

e BM µ≈

(J=S=1/2)

atomic moments

the basics

[ ] [ ] emuGauss Oersted 3cm

4 ,B H mπ = + B Hµ=

4

1 4

B HH H H

m Hµ

µ πχχ

µ πχ

= ⇒ = + ⇒=

⇒ = +

this is ONLY true for homogeneous, uniform para- or dia- magnetic systems

erg /G

m M V =

magnetic susceptibility

/M Hχ = - dimensionless!some other quantities are used:

1m

cc gcc g

χχρ

− = = ⋅

1mol m m

gM molg mol

χ χ −

= = ⋅

4SI cgsχ πχ=

H

B

x

2 3J erg1 Amp m 1 10T G ⋅ = =

Magnetic moment

Extensions – global (total) magnetic moment

( ) ( )4B r H m rπ= +B Hµ=

M HV

χ=

( )( ) 34V

M B r H d rπ = −∫

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 cases1. B is not spatially uniform2. there is magnetic hysteresis

( )4V

M B H dVπ = −∫

/M Hχ = - has no meaning

/dM dHχ = - differential magnetic susceptibility can be used

M

B(x)

M

rectangular slab

M(H) loops - ferromagnet

AC/DC magnetization loop

influence of domain structure

type-I superconductor

-600 -400 -200 0 200 400 600-400

-300

-200

-100

0

100

200

300

400

4πm

(G)

T (K)

a "perfect"Pb sphereT = 4.5 K

Hc=490 Oe

Hc(1-N)=327 Oe

Hc(1-N)

N=1/3

hysteresis is a generic feature

-600 -400 -200 0 200 400 600

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

full M(H) loop partial M(H) loops field cooling

M (e

mu)

H (Oe)

Pb single crystalT = 4.5 K

Hc=500 Oe

Hp = Hc(1-N)=220 OeN=0.55

typical type-II superconductor

-20000 -10000 0 10000 20000

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15M

(em

u)

H (Oe)

types of M(H) loops (Co-122)

-6 -4 -2 0 2 4 6

-0.02

0.00

0.02

0.04M

(em

u)

H (Oe)

fishtail

-6 -4 -2 0 2 4 6

-0.02

0.00

0.02

0.04

M (e

mu)

H (Oe)

distribution of the magnetic induction

-1.0 -0.5 0.0 0.5 1.00.0

0.5

1.0

0.0

0.5

1.0B/

H

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 m2 loop carrying a 1 mA current• M of a loop of radius 1.78 cm carrying a 1 A current• Typical permanent magnet (1 mm3) ~ 1 emu

• M of a neutron star ~ 1030 emu• The Earth’s magnetic moment ~ 8x1025 emu• An electron spin: µB~10-20 emu• Proton and neutron: µN~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 atime.

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)

B

µ

torque:

= ×τ μ B

in inhomogeneous magnetic field

( )cosW Bµ θ= − = −μB

for example, for ( ) ( ),0,0 , , ,0x x yB x µ µ= = Bμ

Energy:

( ) ( )F grad W grad= − = μBForce:

x xBµ=μBx

xdBFdx

µ=

F F changes signhowever torque aligns along the field

and

0xµ > 0xµ <

B B

θ

magnetic moment in a magnetic field

AC/DC measurements

H

lock-in

acV t∂Φ

= −∂

H

PASSIVE! pick-upcoils (no current)

M(t) M=const

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 Bx*dBx/dz is constant over a considerable range in z.

( )U = −= ∇

MBF MB

( ) ( )2 2 xz x x x x dBd dF M B B Bdz dz dzχ χ = = =

( )( ),0,0xB z=Bif

examples

extraction coil magnetometer

QD extraction - coil magnetometer

torque magnetometer

= ×τ M B

advantages?- fast- small samples

Vibrating-sample magnetometer

VSM – QD versalab

Lakeshore cryotronics VSM

QD SQUID-VSM magnetometer

QD MPMS

Josephson effect

Superconducting Quantum Interference Device (SQUID)

DC SQUID AC (RF) SQUID

flux-voltage convertors

DC SQUID

rf-squid – one junction

rf-SQUID

rf-SQUID

rf (10-20 MHz) current source

rf-amp

resonant tank circuit

flux transformer

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