Magnetic Neutron Scattering

Collin Broholm

*Supported by U.S. DoE, Basic Energy Sciences, Materials Sciences & Engineering DE-FG02-08ER46544

Neutron spin meets electron spin

Structure : Magnetic diffraction

Excitations: Inelastic magnetic scattering

Tomorrow : Entangled Magnetism

Overview

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

VCorrespondingly the field exerts a torque and a force

B

BF

driving the neutron parallel to high field regions

Magnetic Neutron Optics The combined nuclear and magnetic potential for neutron

V =

2p 2

mrb ± r0M ×s( )

The index of refraction is

n = 1-

V

E= 1-

l 2

prb ± r0M ×s( )

Critical angle for total external reflection:

qc = l

rb ± r0M ×s

p

M

Polarized beam!

Q

k ¢k

Structure of vortex matter: CeCoIn5

Bianchi et al., Science (2008)

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

2

20ˆ

4 Rm

mgV B

em

RSrBr

The transition matrix element in Fermi’s golden rule

m

2p 2¢k ¢s ¢l V

mksl = -r

0

g

2F q( ) ¢s ¢l s ×S

^lsl exp iq ×r

l( )Magnetic scattering length ~ nuclear scattering lengths

r0 = gm0

4p

e2

me

= 0.54 ´10-12 cm

It is sensitive to atomic dipole moment perp. to q

S

^ l= S

l- S

l×q̂( )q̂

The magnetic scattering cross section

exp dF s i q r q r r

Spin density spread out scattering decreases at high q

The magnetic neutron scattering cross section

d2s

dWd ¢Es® ¢s

=¢k

k

m

2p 2

æ

èçö

ø÷

2

pl

l ¢l

å ¢k ¢s ¢l Vm

ksl2

d El

- E¢l- w( )

=¢k

kr

0

2 g

2F q( )

2

e-2W k( ) 1

2pdt e- iw t

ò eiq× R

l-R

¢l( )

l ¢l

å

´ s s ×S^ l

0( ) ¢s ¢s s ×S^ ¢l

t( ) s

For unspecified incident & final neutron spin states

Edd

d

Edd

d 2

21

2

Un-polarized magnetic scattering

Spin correlation function

Squared form factor Polarization factor

Fourier transform

DW factor

d2s

dWd ¢E=

¢k

kr

0

2 g

2F q( )

2

e-2W q( )

dab

- q̂aq̂

b( )ab

å

´1

2pdt e- iwt e

iq× rl-r

¢l( )S

l

a 0( )S¢l

b t( )l ¢l

åò

Online lectures accessible at:

http://www.sns.gov/education/qcmp/

Magnetic neutron diffraction Time independent spin correlations elastic scattering

ds

dW= r

0

2 g

2F q( )

2

e-2W q( ) d

ab- q̂

aq̂

b( )ab

å eiq× r

l-r

¢l( )S

l

a S¢l

b

l ¢l

å

Periodic magnetic structures Magnetic Bragg peaks

The magnetic vector structure factor is

Magnetic primitive unit cell greater than chemical P.U.C. Magnetic Brillouin zone smaller than chemical B.Z.

Diffraction from Ferromagnet

Nuclear & magnetic scattering interferes. Effective scatt. length:

b +s r0S^

The structure factor for reciprocal lattice vector, t

Fs

2= bd +s r0S^d( )eit ×d

då2

= bdeit ×d

då2

+ s r0S^deit ×d

då2

+ 2s bdr0S^deit × d- ¢d( )

d ¢dåNuclear Magnetic Nuclear-magnetic

The last term vanishes for un-polarized neutrons ( average). A welcome consequence is that the diffracted beam is polarized!

s

bd » r0S^d Þ F±

2»

4 bd

2 for s = 1

0 for s = -1

ì

íï

îï

Simple cubic antiferromagnet

zS ˆB

smg

ab

*a

*b

ma

mb

*

ma

*

mb

ds

dW= N r

0

ms

2mB

æ

èç

ö

ø÷

2

e-2W q( )

F q( )2

1- q̂z

2( )2p( )

3

vd q -t

m( )t

m

å ‘

No magnetic diffraction for q S

qS

S

Not so simple Heli-magnet : MnO2

ˆ ˆexp cos sinl l m l m l

S i S w R x Q R y Q R

Insert into diffraction cross section to obtain

ds

dW= N r

0S( )

2

e-2W q( ) g

2F q( )

2

1+ q̂z

2( )2p( )

3

v

´ d q + w - Qm

-t( ) +d q + w + Qm

-t( ){ }t

å

111w and 27

00m

Q characterize structure

*a

*c

a b c

Diffuse Magnetic Scattering in FeTe0.6Se0.4 K

(r

.l.u

.)

L (

r.l.

u.)

Thampy et al PRL(2012)

Hypothesis: Ising spins?

Polarized neutrons:

isotropic static spin correlations

Diffuse Magnetic Scattering in FeTe0.6Se0.4 K

(r

.l.u

.)

L (

r.l.

u.)

Thampy et al PRL(2012)

Interstitial Iron in Fe1+yTe1-xSex

Extract Local Spin Structure

Model by coupling itinerant spin to 5-band model through

Kang & Tesanovic

Nakatsuji et al. Science (2005)

Unusually weak interlayer coupling

Dominant 3rd neighbor interactions

Exact triangular lattice

Weak easy plane anisotropy

J3 : AFM

Elastic neutron scattering from NiGa2S4 powder

2 * 22

2

0 22 2

/ˆ 1 2 cos2

d g Ar F Q N

d

q q

τ

m Q m Q c

Q τ q

2 Surprises:

12 120oQ Q

1 25(3) Å

Nakatsuji et al. (2005) ds

dW powder

=dWQ

4pòds

dW xtal

Multi crystal assembly of high quality NiGa2S4

19 NiGa2S4 crystals Grown by Yusuke Nambu co-aligned by Chris Stock

Y. Nambu et al., PRB (2009).

NiGa2S4: Elastic Magnetic Scattering

1.5 meV

Short range magnetic ordering

C. Stock et al. PRL (2010)

Summary: Diffraction

• The neutron has a small dipole moment causing scattering from electrons

• Magnetic scattering is similar in magnitude to nuclear scattering

• Elastic magnetic scattering probes static magnetic structure

• Spin direction gleaned from polarization factor (see spin perpendicular to Q)

• Periodic structure determined through Bragg intensities

• Generally probe magnetic structure on 1-104 Å scale:

– flux line lattice

– short range order

− powder, single crystals, thin films

Features of the cross section

The dynamic correlation function

Fluctuation Dissipation theorem

Sum-rules

Examples

Summary

Inelastic Magnetic Neutron Scattering

Understanding Inelastic Magnetic Scattering:

Isolate the “interesting part” of the cross section

The dynamic correlation function (“scattering law”) :

In thermodynamic equilibrium (detailed balance):

Useful exact sum-rules:

Total moment sum-rule (general result from Fourier analysis)

Correlations in space & time rearrange intensity in space leaving the total scattering scattering unaffected

Q-w

Definition of the equal time correlation function:

The wave vector dependence of the energy-integrated intensity probe a snap-shot of the fluctuating spin configuration

Fluctuation Dissipation Theorem

FQ t( ) =

iSQ t( ),S-Qéë ùû

Define the following t-dependent “response” function:

c Qw( ) = - gmB( )2

lime®0+

dte-i w-ie( )t

0

¥

ò FQ t( )

The generalized susceptibility can be expressed as:

The “scattering law” can be expressed as:

From which, the “fluctuation-dissipation” theorem:

First Moment Sum-rule Fourier transform of expression for scattering law:

Equating these leads to the first moment sum-rule:

Time derivative of left side evaluated for t=0:

Time derivative of right side evaluated at t=0:

2

1

,2

1,

0totS

1totS

, ,

,

Weakly Interacting spin-1/2 pairs in Cu-nitrate

Sum rules and the single mode approximation When a coherent mode dominates the spectrum:

Sum-rules link and :

0.4

0.5

0 2 4 0 2 4 6

Exp. data Single mode model

e(Q)

Q p( )

Spin waves in a ferromagnet

ACNS 6/29/10

Magnon creation Magnon destruction

n w( ) =

1

eb w -1

Dispersion relation

Occupation number

Gadolinium

Spin waves in an antiferromagnet

ACNS 6/29/10

S^ Q,w( ) =S

2

J 1- 1z eiQ×d

då( )e Q( )

´ d e Q( ) - w( ) n w( ) +1( ) +d e Q( ) + w( )n w( ){ }

Dispersion relation

J = 4.38 meV ¢J = 0.15 meV D = 0.107 meV

Probing matter through scattering

fp

ip

Single Particle Process

Multi Particle Process

fi

fi

EE

21

21

QQ

ppQQ

1Q

2Q fi

fi

EE

Q

ppQ

Q

Q

ћ

Intensity

Intensity

Q

ћ

Disintegration of a spin flip in 1D

Spinon Spinon

From spinon band-structure

to bounded continuum

2

2, QSGQS

Q (p )q (p )

e/J

w

/J

Quantum critical spin-1/2 chain

Hammar et al. (1999)

Cu(C4H4N2)(NO3)2

Copper Pyrazine dinitrate

Summary

• Inelastic Magnetic scattering probes dynamic spin

correlation function:

• Exact results are critical to interpret data

• The full dynamic correlation function as opposed

to just the dispersion relation tells the story

– Intensity gives equal time correlation function

– Energy width in resonant excitations probes life time

– Broad continuum reveals multiparticle excitations

• Know your sum-rules and happy scattering!