Magnetic Neutron Scattering

Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scatteringSummary

Collin Broholm*Johns Hopkins University and NIST Center for Neutron Research

*Supported by the NSF through DMR-9453362 and DMR-0074571

CRNL 6/20/00

Magnetic properties of the neutron

m

meBn

The neutron has a dipole moment

n is 960 times smaller than the electron moment

960913.1

1836

en

e

m

m

A dipole in a magnetic field has potential energy rBr V

Correspondingly the field exerts a torque and a force

B BF

driving the neutron parallel to high field regions

CRNL 6/20/00

The transition matrix element

The dipole moment of unfilled shells yield inhomog. B-field

2

ˆ

4 R

g B RSB 0

The magnetic neutron senses the field

220

ˆ

4 Rm

mgV B

em

RSrBr

The transition matrix element in Fermi’s golden rule llm iF

grV

mrSkk

exp

22 02

Magnetic scattering is as strong as nuclear scatteringcm 1054.0

412

20

0

em

er

It is sensitive to atomic dipole moment perp. to

lll SSS

CRNL 6/20/00

The magnetic scattering cross section

rrr dexp isF

Spin density spread out scattering decreases at high

The magnetic neutron scattering cross section

t

eedteFg

rk

k

EEVpm

k

k

ll

ll

itiW

m

ll

SS

kkEdd

d

RR

2

0

2

2

22

20

22

2

For unspecified incident & final neutron spin states

Edd

d

Edd

d 2

21

2

CRNL 6/20/00

Un-polarized magnetic scattering

llll

iti

W

teedt

eFg

rk

k

ll

SSrr 0

ˆˆ2Edd

d 22

20

2

llll

iti

W

teedt

eFg

rk

k

ll

SSrr 0

ˆˆ2Edd

d 22

20

2

Spin correlation functionSpin correlation function

Squared form factorSquared form factor Polarization factorPolarization factor

Fourier transformFourier transform

DW factorDW factor

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Magnetic neutron diffraction

Time independent spin correlations elastic scattering

llll

iW lleeFg

r

SSrrˆˆ

2d

d 22

20

Periodic magnetic structures Magnetic Bragg peaks

mm

mmv

Nr

22

32

0 ˆ2

d

dFF

The magnetic vector structure factor is

d

dd

2d

d S2

d iW eeF

gF

Magnetic primitive unit cell greater than chemical P.U.C.

Magnetic Brillouin zone smaller than chemical B.Z.

CRNL 6/20/00

Simple cubic antiferromagnet

ab

*a*b

Real sp

ace

Reci

pro

cal S

pace

ma

mb

*ma

*mb

zS ˆB

smg

zS ˆB

smg

lkheFm W

B

s

sinsinsin82

ˆ 2

z

F

m

mzW

B

s

vFe

mrN

3222

2

0

21

2d

d ‘

No magnetic diffraction for S

SS

Not so simple Heli-magnet : MnO2

llll iS RQyRQxRwS sinˆcosˆexp

Insert into diffraction cross section to obtain

QwQ-w

d

d

vF

geSrN z

W3

22

220

21

2

111w and 7200Q characterize structure

*a

*c

a bc

CRNL 6/20/00

Understanding Inelastic Magnetic Scattering:

Isolate the “interesting part” of the cross section

,ˆˆ2

22

20

2 s

WeF

grN

k

k

Edd

d

The “scattering law” is defined as

tSSeedt llll

-iN

ti

0, ll rr1

S

for a wide class of systems It satisfies useful sum-rules

,exp,

SS

1,ddd

1

SS

Sq

q

''2 cos1

1

3

1),( llll

llllJN

d rrSS

S

Detailed balanceDetailed balance

Total moment

Total moment

First moment sum-ruleFirst moment sum-rule

CRNL 6/20/00

Scattering from a quantum spin liquid

Dimerized spin-1/2 system: copper nitrate

JTkB

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A spin-1/2 pair has a singlet - triplet gap:

Weak inter-dimer coupling cannot close gap

Bond alternation is relevant operator for quantum critical uniform spin chaininfinitesimal bond alternation yields gap

Why a gap in spectrum of dimerized spin system

J0totS

1totS

JJ

J

CRNL 6/20/00

Spin waves in a ferromagnet

nnS

12

,S

Magnon creationMagnon creation Magnon destructionMagnon destruction

JJ 02S

1exp

1

TkE

En

B

Dispersion relation

Magnon occupation prob.

Gadolinium

CRNL 6/20/00

Spin waves in an antiferromagnet

nn

eJS iz

1

1

2

1d

d

,S

2202 JJ S

Dispersion relation

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and the magnetic susceptibility

Compare to the generalized susceptibility

0,2

llll

-itiB StSeedtN

g

ll rrq

They are related by the fluctuation dissipation theorem

eg B 1

1Im,q 2

qS

tSSeedt llll

-iN

ti

0, 1 ll rr

S

,S

We convert inelastic scattering data to• Compare with bulk susceptibility data• Isolate non-trivial temperature dependence• Compare with theories

q to

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Polarized magnetic neutron scattering

tll SS

0

Specify the incident and final neutron spin state

tSS zl

zl 0

tSS zl

zl 0

tSS ll

0

tSS ll

0

Non spin flip:SHS

Spin flip: SHS

CRNL 6/20/00

Polarized neutron scatteringH// H perp

Type of scattering SF NSF SF NSFNuclear coherent 0 1 0 1Nuclear isotope incoherent 0 1 0 1Nuclear spin incoherent 2/3 1/3 2/3 1/3Magnetic Sxx+Syy 0 Sxx Syy

Nuclear isotope incoherent scatteringNuclear isotope incoherent scatteringParamagnetic scattering MnF2Paramagnetic scattering MnF2

H// H//H H

SF

NSF

SF

NSF

CRNL 6/20/00

SummaryThe neutron has a small dipole moment that

causes it to scatter from inhomogeneous internal fields produced by electrons

The magnetic scattering cross section is similar in magnitude to the nuclear cross section

Elastic magnetic scattering probes static magnetic structure

Inelastic magnetic scattering probes spin dynamics through

Polarized neutrons can distinguish magnetic and nuclear scattering and specific spin components

,

S