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Martin Rotter Magnetism in Complex Systems 2009 1
Magnetic Neutron Scattering
Martin Rotter, University of Oxford
Martin Rotter Magnetism in Complex Systems 2009 2
Contents
• Introduction: Neutrons and Magnetism
• Elastic Magnetic Scattering
• Inelastic Magnetic Scattering
Martin Rotter Magnetism in Complex Systems 2009 3
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
Martin Rotter Magnetism in Complex Systems 2009 4
Single Crystal DiffractionE2 – HMI, Berlin
Q
O
k
neutrons: S=1/2 μNeutron= –1.9 μN τ = 885 s (β decay) k=2π/ λ E=h2/2Mnλ2=81.1meV/λ2[Å2]
hklGkkQ
'
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
Martin Rotter Magnetism in Complex Systems 2009 7
TN= 42 K M [010]TR= 10 K q = (2/3 1 0) Magnetic Structure from
Neutron Powder Diffraction
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Experimental data D4, ILLCalculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 8
TN= 42 K M [010]TR= 10 K q = (2/3 1 0)
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Magnetic Structure from Neutron Powder Diffraction
Experimental data D4, ILLCalculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 9
TN= 42 K M [010]TR= 10 K q = (2/3 1 0)
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Magnetic Structure from Neutron Powder Diffraction
Experimental data D4, ILLCalculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 10
TN= 42 K M [010]TR= 10 K q = (2/3 1 0) Magnetic Structure from
Neutron Powder Diffraction
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Experimental data D4, ILLCalculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 11
TN= 42 K M [010]TR= 10 K q = (2/3 1 0) Magnetic Structure from
Neutron Powder Diffraction
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Experimental data D4, ILLCalculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 12
TN= 42 K M [010]TR= 10 K q = (2/3 1 0) Magnetic Structure from
Neutron Powder Diffraction
GdCu2
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Experimental data D4, ILLCalculation done by McPhase
Goodness of fit
Rpnuc
= 4.95%
Rpmag= 6.21%
hkl exp
hkl expcalc
phklI
hklIhklIR
)(
)()(100
Martin Rotter Magnetism in Complex Systems 2009 13
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
Martin Rotter Magnetism in Complex Systems 2009 14
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 Fermi`s golden rule)
Martin Rotter Magnetism in Complex Systems 2009 15
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
Martin Rotter Magnetism in Complex Systems 2009 16
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 Q contribute
12 ( )
jgF Q
)ˆˆ( QQ
Martin Rotter Magnetism in Complex Systems 2009 17
Formfactor
)(2
)()( 20 Qjg
gQjQF
Q=
Dipole Approximation (small Q):
Martin Rotter Magnetism in Complex Systems 2009 18
A caveat on the Dipole Approximation
'
''
ˆˆ1)( jj ii
jjTjTj
elmag ee
NS RQRQ
)(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
Martin Rotter Magnetism in Complex Systems 2009 19
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
Calculation done by McPhase
... using the dipole approximation may lead to a wrong magnetic structure !
M. Rotter, A. Boothroyd, PRB, 79 (2009) R140405
Martin Rotter Magnetism in Complex Systems 2009 20
• nuclear structure factor has to be known with high accuracy• only for centrosymmetric structure (no phase problem)• spin density measurements are made in external magnetic field, • comparison to results of ab initio model calculations desirable !
2
2
)()(
)()(
MN
MN
FFd
d
FFd
d
Measuring Spin Density Distributions
PnB
Pn BNuclear Magnetic
Structure Factor
“Flipping Ratio”:d d
Rd d
( ) 1
( ) 1M
N
F R
F R
Q
Q
Fourier Transform( ) ( )MF M Q r
Forsyth, Atomic Energy Review 17(1979) 345
• polarized neutron beam• sample in magnetic field to induce ferromagnetic moment -> magnetic intensity on top of nuclear reflections -> nuclear-magnetic interference term:
Martin Rotter Magnetism in Complex Systems 2009 21
• Dreiachsenspektometer – PANDA• Dynamik magnetischer Systeme:
1. Magnonen2. Kristallfelder3. Multipolare Anregungen
Inelastic Magnetic Scattering
Martin Rotter Magnetism in Complex Systems 2009 22
k
k‘
Q
Ghkl
q
qGkkQ
hkl
M
k
M
k
'
2
'
2
22
Three Axes Spectrometer (TAS)
• constant-E scans • constant-Q scans•
Martin Rotter Magnetism in Complex Systems 2009 23
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
Martin Rotter Magnetism in Complex Systems 2009 25
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
Martin Rotter Magnetism in Complex Systems 2009 26
Movement of Spins - Magnons
ij
jiijJH SS)(2
1
153
T=1.3 KMF - Zeeman Ansatz (for S=1/2)
Martin Rotter Magnetism in Complex Systems 2009 27
Movement of Spins - Magnons
ij
jiijJH SS)(2
1
Bohn et. al. PRB 22 (1980) 5447
T=1.3 K
153
Martin Rotter Magnetism in Complex Systems 2009 28
Movement of Spins - Magnons
ij
jiijJH SS)(2
1
Bohn et. al. PRB 22 (1980) 5447
T=1.3 Ka
153
Martin Rotter Magnetism in Complex Systems 2009 29
Movement of Charges - the Crystal Field Concept
+
+
+
+
+
+
+
+
+
+
Hamiltonian ilm
iml
mlcf OBH
,
)(JE
Q
charge densityof unfilled shell
Neutrons change the magnetic moment in an inelastic scattering process: this is correlated to a change in the charge density by the LS coupling …”crystal field excitation”
Martin Rotter Magnetism in Complex Systems 2009 30
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]
Martin Rotter Magnetism in Complex Systems 2009 31
PrNi2Si2
bct ThCr2Si2 structure
Space group I4/mmmPr3+ (4f2) J=4-CF singlet groundstate-Induced moment system-Ampl mod mag. structureTN=20 Kq=(0 0 0.87), M=2.35 μB/Pr
10meV
Blancoet. al. PRB 45 (1992) 2529
Martin Rotter Magnetism in Complex Systems 2009 32
PrNi2Si2 excitations
Blanco et al. PRB 56 (1997) 11666Blanco et al. Physica B 234 (1997) 756
Neutron Scattering Experiment
Calculations done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 33
Calculate Magnetic Excitations and the Neutron Scattering Cross Section
),()ˆˆ('
'
2
2
22
QQQ
magS
mc
e
k
kN
dEd
d
'
')(
'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
''/1
12 kTe
S
Martin Rotter Magnetism in Complex Systems 2009 34
Summary• Magnetic Diffraction• Magnetic Structures• Caveat on using the Dipole Approx.
• Magnetic Spectroscopy• Magnons (Spin Waves)• Crystal Field Excitations• Orbitons
How much does an average European citizen spend on Neutron Scattering per year ?
• NESY- Fachausschuss “Forschung mit Neutronen und Synchrotron-strahlung” der Oesterr. Physikalischen Gesellschaft, http://www.ati.ac.at/~nesy/welcome.html
• CENI – Central European Neutron Initiative (Austria, Czech Rep., Hungary) – membership at ILL (Institute Laue Langevin) www.ill.eu
• Funding is strongly needed to build the ESS, the European Spallation Source
Epilogue
Martin Rotter Magnetism in Complex Systems 2009 37
McPhase - the World of MagnetismMcPhase is a program package for the calculation of magnetic properties
! NOW AVAILABLE with INTERMEDIATE COUPLING module ! Magnetization Magnetic Phasediagrams
Magnetic Structures Elastic/Inelastic/Diffuse Neutron Scattering
Cross Section
Martin Rotter Magnetism in Complex Systems 2009 38
and much more....
Magnetostriction
Crystal Field/Magnetic/Orbital Excitations McPhase runs on Linux & Windows it is freeware
www.mcphase.de
Martin Rotter Magnetism in Complex Systems 2009 39
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
M. Banks, MPI StuttgartDuc Manh Le, University of London
J. Brown, B. Fak, ILL, GrenobleA. Boothroyd, Oxford
P. Rogl, University of ViennaE. Gratz, E. Balcar, G.Badurek
TU ViennaJ. Blanco,Universidad Oviedo
University of Oxford
Thanks to ……
……. and thanks to you !
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
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
Martin Rotter Magnetism in Complex Systems 2009 42
)(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)
'
3
' . .0
(2 )( )k ki i
Gkk rez latt
e Nv
Q G Q G
τ
Q τ
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 |F|2
one element(NB=1): )1()(44 41222
dd
dN
incel
nucinc
elnuc IIbbbN
d
d
i 2||4 bc
Martin Rotter Magnetism in Complex Systems 2009 44
k
k‘
Q Ghkl
q
qGkkQ
hkl
M
k
M
k
'
2
'
2
22
Three Axes Spectrometer (TAS)
Martin Rotter Magnetism in Complex Systems 2009 45
Arrangement of Magnetic Moments in Matter
Paramagnet
Ferromagnet
Antiferromagnet
And many more ....Ferrimagnet, Helimagnet, Spinglass ...collinear, commensurate etc.
Martin Rotter Magnetism in Complex Systems 2009 46
0 2 4 6 80
2
4
F2
AF2 AF3
FM
F1
AF1
0H (
T)
T (K)
NdCu2 Magnetic Phasediagram(Field along b-direction)
Martin Rotter Magnetism in Complex Systems 2009 47
Complex Structures
AF1
Q=
μ0Hb=0
μ0Hb=1T
μ0Hb=2.6T
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter Magnetism in Complex Systems 2009 48
Complex Structures
F1
Q=
μ0Hb=0
μ0Hb=1T
μ0Hb=2.6T
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter Magnetism in Complex Systems 2009 49
Complex Structures
F2
Q=
μ0Hb=0
μ0Hb=1T
μ0Hb=2.6T
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter Magnetism in Complex Systems 2009 50
NdCu2 Magnetic PhasediagramH||b
F3
AF1
F1
ab
c
F1
Lines=ExperimentColors=Theory
Calculation done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 51
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
Martin Rotter Magnetism in Complex Systems 2009 52
T=10 KT=40 KT=100 K
NdCu2 - 4f Charge Density
)()(|)(|)(ˆ
,...,06,4,2,0
24
nmT
nmn
mnnnmf ZOecrR Jr
NdCu2F3
AF1
F1
F3: measured dispersion was fitted to get exchange constants J(ij)
Calculations done by McPhase
Martin Rotter Magnetism in Complex Systems 2009 54
M. Rotter & A. Boothroyd 2008
did some calculations
E. Balcar
Martin Rotter Magnetism in Complex Systems 2009 55
M. Rotter, A. Boothroyd, PRB, submitted
CePd2Si2
Calculation done by McPhase
Comparison toexperiment
(|F
M|2
-|F
Md
ip|2
)/ |F
Md
ip|2
(%)
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
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