magnetic neutron scattering neutron spin meets electron spin magnetic neutron diffraction inelastic...

33
Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering Collin Broholm* Hopkins University and NIST Center for Neutron Rese *Supported by the NSF through DMR-9453362 and DMR-0074571

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Page 1: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

Magnetic Neutron Scattering

Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering Summary+Impurities in spin-1 chains

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

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

Page 2: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 3: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 4: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 5: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 6: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

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 eeFg

F

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

Magnetic Brillouin zone smaller than chemical B.Z.

Page 7: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

zS ˆB

smg

Simple cubic antiferromagnet

ab

*a*b

Real sp

ace

Reci

pro

cal S

pace

ma

mb

*ma

*mb

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

Page 8: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 9: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 10: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Scattering from a quantum spin liquid

Dimerized spin-1/2 system: copper nitrate

JTkB

Page 11: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

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

Page 12: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Sum rules and the single mode approximation

When a coherent mode dominates the spectrum:

)(

cos11

3

1

)(

),()(

''2

q

JN

q

qdq

llllll

ll

rrSS

S

S

Sum-rules link S(q) and (q)

)()(),( qqq

SS

E (

meV

)

0.4

0.5

0 2 4 0 2 4 6q ()

Page 13: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 14: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Spin waves in an antiferromagnet

nn

eJS iz

1

1

2

1d

d

,S

2202 JJ S

Dispersion relation

Page 15: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Continuum magnetic inelastic scattering

Inelastic scattering is not confined to disp. relations when• There is thermal ensemble of excitations present• and do not uniquely specify excited state

- electron hole pair excitations in metals - spinon excitations in quantum magnets

spinon continuum in spin-1/2 AFM chain

Page 16: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

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

Page 17: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

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

Page 18: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 19: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

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

Page 20: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Low T excitations in spin-1 AFM chain

Y2BaNiO5 T=10 KMARI chain ki

•Haldane gap =8 meV

•Coherent mode

•S(q,)->0 for Q->2n

pure

Page 21: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

AKLT state for spin-1 chain

• This is exact ground state for spin projection Hamiltonian

• Magnets with 2S=nz have a nearest neighbor singlet covering with full lattice symmetry.

• Excited states are propagating bond triplets separated from the ground state by an energy gap .J

Haldane PRL 1983Affleck, Kennedy, Lieb, and Tasaki PRL 1987

i

iii

iiiii

toti SP 12

131

12 SSSSSSH

Page 22: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Impurities in Y2BaNiO5

Ca2+

Y3+

• Mg2+on Ni2+ sites finite length chains• Ca2+ on Y3+ sites mobile bond defectsMg

Ni

Kojima et al. (1995)

Page 23: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Zeeman resonance of chain-end spinsI(

H=

9 T

)-I(

H=

0 T

) (

cts.

per

min

.)

Hg B

h (

meV

)

H (Tesla)0 2 4 6 8

g=2.16

0 0.5 1 1.5 2

-5

0

10

15

20

Hg B

meV)(

Page 24: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Form factor of chain-end spins

Q-dependence revealsthat resonating objectis AFM.

The peak resemblesS(Q) for pure system.

Chain end spin carryAFM spin polarizationof length back into chain

Y2BaNi1-xMgxO5 x=4% Hg B

Page 25: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

New excitations in Ca-doped Y2BaNiO5

Pure 9.5% Ca

•Ca-doping creates states below the gap

•sub-gap states have doubly peaked structure factor

Y2-xCaxBaNiO5:

Page 26: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Why a double ridge below the gap in Y2-xCaxBaNiO5 ?

Charge ordering yields incommensurate spin order

Quasi-particle Quasi-hole pair excitations in Luttinger liquid

Anomalous form factor for independent spin degrees of freedom associated with each donated hole

xq

q is single impurity prop.Indep. of

x

Page 27: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Does q vary with calcium concentration?

q not strongly dependent on x

Double peak is single impurity effect

Page 28: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

(b)

Ca

Ba

Ni

Y

c

b

a

(e)

(f)

(c)

(d)

a

O

Bond Impurities in a spin-1 chain: Y2-

xCaxBaNiO5

(a)

Page 29: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

Form-factor for FM-coupled chain-end spins

2/)(Re2)( iqeqMqS

A symmetric AFM droplet

22/*2/ )()()(

ll

iql

iqlll eqMeqMPqS

Ensemble of independent randomly truncated AFM droplets

Page 30: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Calcium doping Y2BaNiO5

Experimental facts:Ca doping creates sub-gap excitations with

doubly peaked structure factor and bandwidth The structure factor is insensitive to

concentration and temperature for 0.04<x<0.14 (and T<100 K)

Analysis:Ca2+ creates FM impurity bonds which nucleate AFM droplets with doubly peaked structure factorAFM droplets interact through intervening chain

forming disordered random bond 1D magnet

Page 31: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Incommensurate modulations in high TC superconductors

YBa2Cu3O6.6 T=13 K E=25 meV

h (rlu)

k

(rl

u)

Hayden et al. 1998

Page 32: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

What sets energy scale for sub gap scattering ?

0

5

10

Possibilities:

•Residual spin interactions through Haldane state. A Random bond AFM.

•Hole motion induces additional interaction between static AFM droplets

•AFM droplets move with holes: scattering from a Luttinger liquid of holes.

How to distinguish:

• Neutron scattering in an applied field

• Transport measurements

• Theory

?

q~

h

(meV

)

Page 33: Magnetic Neutron Scattering Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering

CRNL 6/20/00

Conclusions on spin-1 chain

Dilute impurities in the Haldane spin chain create sub-gap composite spin degrees of freedom.

Edge states have an AFM wave function that extends into the bulk over distances of order the Haldane length.

Holes in Y2-xCaxBaNiO5 are surrounded by AFM spin polaron with central phase shift of

Neutron scattering can detect the structure of composite impurity spins in gapped quantum magnets.

The technique may be applicable to probe impurities in other gapped systems eg. high TC superconductors.

Microscopic details of gapped spin systems may help understand related systems where there is no direct info.