magnetic neutron scattering

Martin Rotter NESY Winter School 2009 1

Magnetic Neutron Scattering

Martin Rotter, University of Oxford


Contents

• Introduction: Neutrons and Magnetism

• Elastic Magnetic Scattering

• Inelastic Magnetic Scattering


Neutrons and MagnetismMacro-Magnetism: Solution of Maxwell´sEquations – Engineering of (electro)magnetic devices

Atomic Magnetism: Instrinsic Magnetic Properties

Micromagnetism: Domain Dynamics,Hysteresis

MFM image

Micromagnetic simulation.

10-1m

10-3m

10-5m

10-7m

10-9m

10-11m

HallProbeVSMSQUID

MOKE

MFM

NMRFMRSRNS

Oa*

c*

Bragg’s Law in Reciprocal Space (Ewald Sphere)

In

com

ing

Neutro

n

τ=Q

Scatte

red

Neutro

n

k

k‘

kQ sin2


Single Crystal DiffractionE2 – HMI, Berlin

Q

O

k


The Scattering Cross Section

Scattering Cross Sections

Total

Differential

Double Differential

Scattering Law S .... Scattering function

Units: 1 barn=10-28 m2 (ca. Nuclear radius2)

areaareatime

timetot 11

1

fluxneutron Incident

secper neutrons scattered ofNumber

d .flux neutron Incident

delement angle into secper neutrons scattered ofNumber

d

d

ddE' .flux neutron Incident

dE'E' and E'between energies with and ... ofNumber

'dEd

d

),('

'

QSk

k

dEd

d


M neutron massk wavevector|sn> Spin state of

the neutronPsn Polarisation|i>,|f> Initial-,final-

state of the targets

Ei,Ef Energies of –‘‘-Pi thermal

populationof state |i>

Hint Interaction -operatorS. W. Lovesey „Theory of Neutron Scattering from

Condensed Matter“,Oxford University Press, 1984

n

nsif

finnis EEfsHisPPM

k

k

dEd

d

,

2int

2

2

2

)(|;'|)(|;|2

'

'

Q

(follows from Fermis golden rule)


Interaction of Neutrons with Matter

magnuc HHH int

j

jnnjj

Nj

nnuc bbM

H )~

()(2

)(2

RrsIr

nαH sQQQ )(ˆ2)(ˆ)(int

jn

jjN

jinuc bb

MeH j )(

2)(

2~

sIQ RQ

neBeee

enenmag cmcmH BsAAAr

2

e

2

1e

2

1)(

22

PP

jjnnN

jijBmag gμegFH QJQsQ RQ ˆˆ)(8)(

~

21

nni rdHeH n 3)()( rQ rQ

Unpolarised Neutrons - Van Hove Scattering function S(Q,ω)

)|ˆ||ˆ|||ˆ|(|)(2

'

'2

2

2

iffifiPEE

M

k

k

dEd

d

ififi αα

• for the following we assume that there is no nuclear order - <I>=0:

'

)0(~

)(~

'41'*'*

)0(~

')(

~

''2

121

2

2

22

'

'

))1((1

2

1),(

)0()()()(1

2

1),(

),('

),()ˆˆ('

'

jjT

itijjjj

jN

jN

jjtinuc

Ti

jti

jjj

jjti

mag

nucmag

jj

jj

eeIIbbbbN

dteS

eJetJQgFQgFN

dteS

Sk

kNS

mc

e

k

kN

dEd

d

RQRQ

RQRQ

Q

Q

QQQQ

'

'41'*'*

''))1((1

)(jj

WWiijjjj

jN

jN

jjelnuc

jjjj eeIIbbbbN

S RQRQ

)()(~

ttjjj

uRR

''

''21

'21 )()(

1)( jjjj WWii

TjjTjjj

j

elmag eeJQgFJQgF

NS RQRQ

Splitting of S into elastic and inelastic part

inelmag

elmagmag

inelnuc

elnucnuc

SSS

SSS


)(2

.../2

)()(

1)'(

')'(1

)(

...

')'(......)(

0

0

/2)'(

2/

2/

/'2

0

/2

2/

2/

/'2

0

/2

qa

eLxqa

c

xcx

eL

xx

dxexfeL

xf

dxexffwithefxf

n

iqna

n

Lxxin

L

L

Lnxi

n

Linx

L

L

Lnxin

n

Linxn

A shortExcursionto Fourierand DeltaFunctions ....

it follows by extending the range of x to more than –L/2 ...L/2 andgoing to 3 dimensions (v0 the unit cell volume)

τ

GκGκ τκ

..0

3

'

)()2(

'

lattrezG

kk

ii

vNe kk

Neutron – Diffraction

'412'*

)1(||11

)( ''

jj jjj

jN

WWiijjelnuc IIb

Neebb

NS jjjj RQRQ

Lattice G with basis B: dkjkdj BGR

)........(

„Isotope-incoherent-Scattering“

„Spin-incoherent-Scattering“

Independent of Q:

ddd

dN

B

ddd

B

N

dd

WWidd

B

elnuc

IIbN

bbN

eebbN

SB

dddd

)1(1

)(

1)(

1)(

41

2

22

1',

)('

*

,''

BBQ

ττQ

Latticefactor Structurefactor

one element(NB=1): )1()(44 41222

dd

dN

incel

nucinc

elnuc IIbbbN

d

d

i 2||4 bc


Magnetic Diffraction

'

'*'1

)(jj

iijjcoh

elnuc

jj eebbN

S RQRQ

),('

),()ˆˆ('

'

2

2

22

QQQQ

nucmag S

k

kNS

mc

e

k

kN

dEd

d

'

''21

'21 )()(

1)( jj ii

TjjTjjj

j

elmag eeJQgFJQgF

NS RQRQ

Difference to nuclear scattering: Formfactor ... no magnetic signal at high angles

Polarisationfactor ... only moment components

normal to κ contribute

j

gF )(21

)ˆˆ( QQ


Atomic Lattice

Magnetic Lattice

ferro

antiferro


Formfactor

)(2

)()( 20 Qjg

gQjQF

Q=

Dipole Approximation (small Q):

In 1949 Shull showed the magnetic structure of the MnO crystal, which led to the discovery of antiferromagnetism (where the magnetic moments of

some atoms point up and some point down).

The Nobel Prize in Physics 1994

http://www.ill.fr/dif/3D-crystals/images/mnom.gif


Arrangement of Magnetic Moments in Matter

Paramagnet

Ferromagnet

Antiferromagnet

And many more ....Ferrimagnet, Helimagnet, Spinglass ...collinear, commensurate etc.

19NESY Winter School 2009Martin Rotter

TN= 42 K M [010]TR= 10 K q = (2/3 1 0) Magnetic Structure from

Neutron Scattering

GdCu2

Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281

Experimental data D4, ILLCalculation done by McPhase


TN= 42 K M [010]TR= 10 K q = (2/3 1 0)

GdCu2


Magnetic Structure from Neutron Scattering




Neutron Scattering

GdCu2





Neutron Scattering

GdCu2



Goodness of fit

Rpnuc

= 4.95%

Rpmag= 6.21%

hkl exp

hkl expcalc

phklI

hklIhklIR

)(

)()(100


0 2 4 6 80

2

4

F2

AF2 AF3

FM

F1

AF1

0H

(T

)

T (K)

NdCu2 Magnetic Phasediagram(Field along b-direction)


Complex Structures

AF1

Q=

μ0Hb=0

μ0Hb=1T

μ0Hb=2.6T

Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499


Complex Structures

F1

Q=

μ0Hb=0

μ0Hb=1T

μ0Hb=2.6T



Complex Structures

F2

Q=

μ0Hb=0

μ0Hb=1T

μ0Hb=2.6T



NdCu2 Magnetic PhasediagramH||b

F3

AF1

F1

ab

c

F1

Lines=ExperimentColors=Theory

Calculation done by McPhase


A caveat on the Dipole Approximation

'

''

ˆˆ1)( jj ii

jjTjTj

elmag ee

NS RQRQ

QQ

)(2

)()(

)(~ˆ

)(2

1ˆ

20

21

Qjg

gQjQF

JQgF

QM

TjjTj

jB

j

Q

Q

Dipole Approximation (small Q):

E. Balcar derived accurate formulas for the

S. W. Lovesey „Theory of Neutron Scattering from Condensed Matter“,Oxford University Press, 1984Page 241-242

Tj Q̂


M. Rotter & A. Boothroyd 2008

did some calculations

E. Balcar


M. Rotter, A. Boothroyd, PRB, submitted

CePd2Si2


d

d

Comparison toexperiment

(σ-σ

dip)/

σd

ip (%

)

Goodness of fit:

Rpdip=15.6%

Rpbey=8.4 %

(Rpnuc=7.3%)

bct ThCr2Si2 structure

Space group I4/mmmCe3+ (4f1) J=5/2

TN=8.5 K

q=(½ ½ 0), M=0.66 μB/Ce


NdBa2Cu3O6.97

superconductor TC=96K

orth YBa2Cu3O7-x structure

Space group PmmmNd3+ (4f3) J=9/2

TN=0.6 K

q=(½ ½ ½), M=1.4 μB/Nd


... using the dipole approximation may lead to a wrong magnetic structure !

M. Rotter, A. Boothroyd, PRB, submitted


• Dreiachsenspektometer – PANDA• Dynamik magnetischer Systeme:

1. Magnonen2. Kristallfelder3. Multipolare Anregungen

Inelastic Magnetic Scattering


k

k‘

Q Ghkl

q

qGkkQ

hkl

M

k

M

k

'

2

'

2

22

Three Axes Spectrometer (TAS)


PANDA – TAS for Polarized Neutronsat the FRM-II, Munich


PANDA – TAS for Polarized Neutrons at the FRM-II, Munich

beam-channelmonochromator-shielding with platform

Cabin with computer work-placesand electronics

secondary spectrometerwith surrounding radioprotection,15 Tesla / 30mK Cryomagnet


Movement of Atoms [Sound, Phonons]

Brockhouse 1950 ...

E

Q

π/a

The Nobel Prize in Physics 1994

Phonon Spectroscopy: 1) neutrons 2) high resolution X-rays


Movement of Spins - Magnons

ij

jiijJH SS)(2

1

153

T=1.3 KMF - Zeeman Ansatz (for S=1/2)



ij

jiijJH SS)(2

1

Bohn et. al. PRB 22 (1980) 5447

T=1.3 K

153



ij

jiijJH SS)(2

1

Bohn et. al. PRB 22 (1980) 5447

T=1.3 Ka

153


Movement of Charges - the Crystal Field Concept

+

+

+

+

+

+

+

+

+

+

Hamiltonian ilm

iml

mlcf OBH

,

)(JE

Q

4f –charge density


NdCu2 – Crystal Field Excitations

orthorhombic, TN=6.5 K, Nd3+: J=9/2, Kramers-ion

Gratz et. al., J. Phys.: Cond. Mat. 3 (1991) 9297


T=10 KT=40 KT=100 K

NdCu2 - 4f Charge Density

)()(|)(|)(ˆ

,...,06,4,2,0

24

nmT

nmn

mnnnmf ZOecrR Jr


Calculate Magnetic Excitations and the Neutron Scattering Cross Section

),()ˆˆ('

'

2

2

22

QQQ

magS

mc

e

k

kN

dEd

d

''/1

12 kTe

S

'

')(

'21

21

21 ),()}({)}({),( ''

dddd

WWiddN

inelmag SeeQgFQgFS dddd

b

QQ BBκ

*)()(2

1)('' ''' zz

iz dddddd

)(

||||)(

)()(1)(),(

,,0

1

00

jiij ij

THTH nniJJjjJJi

J

QQ Linear Response Theory, MF-RPA

ij

jii

iBJiilm

iml

ml ijJgOBH JJHJJ )(

2

1)(

,

.... High Speed (DMD) algorithm: M. Rotter Comp. Mat. Sci. 38 (2006) 400

NdCu2F3

AF1

F1

F3: measured dispersion was fitted to get exchange constants J(ij)

Calculations done by McPhase


1950 Movements of Atoms [Sound, Phonons]

a

τorbiton

lmij

jmli

mlQ OOijCH

,

)()()( JJDescription: quadrupolar (+higher order) interactions

a

τorbiton

1970 Movement of Spins [Magnons]

? Movement of Orbitals [Orbitons]


Summary• Magnetic Diffraction• Magnetic Structures• Caveat on using the Dipole Approx.

• Magnetic Spectroscopy• Magnons (Spin Waves)• Crystal Field Excitations• Orbitons


Martin Rotter, University of Oxford


McPhase - the World of Rare Earth MagnetismMcPhase is a program package for the calculation of

magnetic properties of rare earth based systems. Magnetization Magnetic Phasediagrams

Magnetic Structures Elastic/Inelastic/Diffuse Neutron Scattering

Cross Section


and much more....

Magnetostriction

Crystal Field/Magnetic/Orbital Excitations McPhase runs on Linux & Windows it is freeware

www.mcphase.de


Important Publications referencing McPhase: • M. Rotter, S. Kramp, M. Loewenhaupt, E. Gratz, W. Schmidt, N. M. Pyka, B. Hennion, R.

v.d.Kamp Magnetic Excitations in the antiferromagnetic phase of NdCu2 Appl. Phys. A74

(2002) S751 • M. Rotter, M. Doerr, M. Loewenhaupt, P. Svoboda, Modeling Magnetostriction in RCu2

Compounds using McPhase J. of Applied Physics 91 (2002) 8885• M. Rotter Using McPhase to calculate Magnetic Phase Diagrams of Rare Earth

Compounds J. Magn. Magn. Mat. 272-276 (2004) 481

M. Doerr, M. Loewenhaupt, TU-DresdenR. Schedler, HMI-Berlin

P. Fabi né Hoffmann, FZ Jülich S. Rotter, Wien, AustriaM. Banks, MPI Stuttgart

Duc Manh Le, University of LondonJ. Brown, B. Fak, ILL, Grenoble

A. Boothroyd, OxfordP. Rogl, University of ViennaE. Gratz, E. Balcar TU Vienna

University of Oxford

Thanks to ……

……. and thanks to you !

magnetic neutron scattering

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