Magnetic Neutron Scattering
Martin Rotter, University of Oxford
Martin Rotter
Magnetism in Complex Systems 2009
1
Contents
• Introduction: Neutrons and Magnetism
• Elastic Magnetic Scattering
• Inelastic Magnetic Scattering
Martin Rotter
Magnetism in Complex Systems 2009
2
Neutrons and Magnetism
Macro-Magnetism:
Solution of Maxwell´s
Equations – Engineering
of (electro)magnetic
MFM image
devices
Micromagnetism:
Domain Dynamics,
Hysteresis
Micromagnetic
10-1m
10-3m
Hall
Probe
VSM
SQUID
10-5m
MOKE
10-7m
MFM
simulation.
NMR
FMR
SR
-11
10 m NS
10-9m
Atomic Magnetism:
Instrinsic Magnetic
Properties
Martin Rotter
Magnetism in Complex Systems 2009
3
Single Crystal Diffraction
E2 – HMI, Berlin
neutrons:
S=1/2 μNeutron= –1.9 μN
τ = 885 s (β decay)
k=2π/ λ E=h2/2Mnλ2=81.1meV/λ2[Å2]
k
Q
O
   
Q  k  k '  G hkl
Martin Rotter
Magnetism in Complex Systems 2009
4
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
Magnetism in Complex Systems 2009
5
The Nobel Prize in
Physics 1994
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).
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
Magnetism in Complex Systems 2009
7
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
Magnetism in Complex Systems 2009
8
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
Magnetism in Complex Systems 2009
9
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
Magnetism in Complex Systems 2009
10
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
Magnetism in Complex Systems 2009
11
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Powder Diffraction
N
Rpnuc = 4.95%
Rpmag= 6.21%
Experimental data D4, ILL
Calculation done by McPhase
Goodness of fit
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Rp

 100
hkl
I calc (hkl)  I exp (hkl)

hkl
Martin Rotter
I exp (hkl)
Magnetism in Complex Systems 2009
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The Scattering Cross Section
Scattering Cross Sections

Number of scattered neutrons per sec  time1



area

1
1
Incident neutron flux
 time area

Total
 tot
Differential
d Number of scattered neutrons per sec into angle element d

d
Incident neutron flux . d
Double Differential
d
Number of ... and with energies between E' and E' dE'

ddE '
Incident neutron flux . dE' d
Scattering Law
d
k'
 S (Q,  )
ddE ' k
Units:
S .... Scattering function
1 barn=10-28 m2 (ca. Nuclear radius2)
Martin Rotter
Magnetism in Complex Systems 2009
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neutron mass
wavevector
Spin state of
the neutron
Psn
Polarisation
|i>, |f> Initial-,finalstate of the
targets
Ei, Ef Energies of –‘‘Pi
thermal
population
of state |i>
Hint
Interaction
-operator
M
k
|sn>
S. W. Lovesey „Theory of Neutron Scattering from
Condensed Matter“,Oxford University Press, 1984
d 2
k'  M 
 Martin Rotter 2 
ddE ' k  2 
2

2
P
P
|

s
;
i
|
H
(
Q
)
|
s
'
;
f

|
(  E14
2009
 sn i n Magnetism
int in Complex
n Systems 
i  Ef )
if , sn
(follows from Fermi`s golden rule)
Interaction of Neutrons with Matter
 

 3
iQrn
H (Q)   e
H (rn )d rn
H int  H nuc  H mag


 ~
2 2 j
j j 
H nuc (rn )  
(b  bN I  sn ) (rn  R j )
M
j
2
 ~


iQR j 2
j
j j 
H nuc (Q)   e
(b  bN I  sn )
M
j


2
2






 
1 
e
1 
e 

H mag (rn )  
 Pe  A n  A e  
 P e  A e   2 B se  B n
c
c 
 2m 
e 2m 
 ~

 ˆ 

i
Q
Rj
1
ˆ
H mag (Q)  8B  2 gF ( )j e
μN g n sn  Q  J j  Q

j
Martin Rotter



 
H int (Q)  ˆ (Q)  2αˆ (Q)  sn
Magnetism in Complex Systems 2009
15
Magnetic Diffraction
d 2
k '  e 2 

 N 
2 
ddE '
k  mc 
S nuc
S mag
el
el
coh
2
( 


1
  ( )
N
1
  ( )
N

1
2


k'

ˆ
ˆ
 Q Q  )S mag (Q,  )  N S nuc (Q,  )
k
b
j*

j '  iQ  R j
b e
e

iQ  R j '
jj '
gF (Q)j  J j T
12 gF (Q)j '  J j '
T e

 iQ  R j
e

iQ  R j '
jj '
Difference to nuclear scattering:
Formfactor
 12 gF (Q) j
Polarisationfactor
... no magnetic signal at high angles
ˆ Q
ˆ ) ... only moment components
(  Q
 
normal to Q contribute
Martin Rotter
Magnetism in Complex Systems 2009
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Formfactor
Q=
2 g
 j2 (Q) 
Dipole Approximation (small Q): F (Q)  j0 (Q)  
g
Martin Rotter
Magnetism in Complex Systems 2009
17
A caveat on the Dipole
Approximation
S mag
el
1
  ( )
N
  Qˆ  
j
T
ˆ  e
Q
j '
T

 iQ  R j
e

iQ  R j '
jj '
1
ˆ
Q j  
M j (Q)
2 B
Dipole Approximation (small Q):
ˆ  ~ 1 gF (Q)  J 
Q
j
T
j
T
2
j
2 g
F (Q)  j0 (Q)  
 j2 (Q) 
g
E. Balcar derived accurate formulas for the
 Q̂ j T
S. W. Lovesey „Theory of Neutron Scattering from
Condensed Matter“,Oxford University Press, 1984
Page 241-242
Martin Rotter
Magnetism in Complex Systems 2009
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NdBa2Cu3O6.97
superconductor TC=96K
orth YBa2Cu3O7-x structure
Space group Pmmm
Nd3+ (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, 79 (2009) R140405
Martin Rotter
Calculation done by McPhase
Magnetism in Complex Systems 2009
19
Measuring Spin Density Distributions
• polarized neutron beam
• sample in magnetic field to induce ferromagnetic moment
-> magnetic intensity on top of nuclear reflections
-> nuclear-magnetic interference term:





 2
 d 
PnB

  FN (Q)  FM (Q)
d




 2
 d 

  FN (Q)  FM (Q)
Pn B
d



Nuclear
Magnetic
Structure Factor
d   d 
 

 d    d  
“Flipping Ratio”: R  
Forsyth, Atomic Energy Review 17(1979) 345
FM (Q) 1  R

FN (Q) 1  R
Fourier Transform
FM (Q) 
 M (r)
• 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 !
Martin Rotter
Magnetism in Complex Systems 2009
20
Inelastic Magnetic Scattering
• Dreiachsenspektometer – PANDA
• Dynamik magnetischer Systeme:
1. Magnonen
2. Kristallfelder
3. Multipolare Anregungen
Martin Rotter
Magnetism in Complex Systems 2009
21
Three Axes Spectrometer (TAS)
k‘
k
q
Q
Ghkl
2
2

k  k '
 

2M
2M
   

Q  k  k '  G hkl  q
• constant-E scans
• constant-Q scans
•
Martin Rotter
Magnetism in Complex Systems 2009
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PANDA – TAS for Polarized Neutrons at the
FRM-II, Munich
beam-channel
monochromatorshielding with platform
Cabin with
computer work-places
and electronics
secondary spectrometer
with surrounding
radioprotection,
15 Tesla / 30mK Cryomagnet
Martin Rotter
Magnetism in Complex Systems 2009
23
Martin Rotter
Magnetism in Complex Systems 2009
24
Movement of Atoms [Sound, Phonons]
Brockhouse 1950 ...
The Nobel Prize in
Physics 1994
E
π/a
Phonon Spectroscopy: 1) neutrons
2) high resolution X-rays
Martin Rotter
Magnetism in Complex Systems 2009
Q
25
Movement of Spins - Magnons
153
1
H    J (ij )Si  S j
2 ij
MF - Zeeman Ansatz
(for S=1/2)
Martin Rotter
T=1.3 K
Magnetism in Complex Systems 2009
26
Movement of Spins - Magnons
153
1
H    J (ij )Si  S j
2 ij
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
Martin Rotter
Magnetism in Complex Systems 2009
27
Movement of Spins - Magnons
1
H    J (ij )Si  S j
2 ij
153
a
T=1.3 K
Bohn et. al.
PRB 22 (1980) 5447
Martin Rotter
Magnetism in Complex Systems 2009
28
Movement of Charges - the Crystal Field Concept
+
+
+
+
+
charge density
of unfilled shell
+
+
+
+
+
Hamiltonian H cf 
E
m m
B
 l Ol (J i )
lm,i
Martin Rotter
Q
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”
Magnetism in Complex Systems 2009
29
1950
Movements of Atoms [Sound, Phonons]
1970
Movement of Spins [Magnons]
?
Movement of Orbitals [Orbitons]
aa
ττorbiton
orbiton
Description: quadrupolar
(+higher order) interactions
Martin Rotter
H Q    C (ij )  Olm (J i )  Olm (J j )
ij ,lm
Magnetism in Complex Systems 2009
30
PrNi2Si2
bct ThCr2Si2 structure
Space group I4/mmm
Pr3+ (4f2) J=4
-CF singlet groundstate
-Induced moment system
-Ampl mod mag. structure
TN=20 K
q=(0 0 0.87), M=2.35 μB/Pr
10meV
Blancoet. al. PRB 45 (1992) 2529
Martin Rotter
Magnetism in Complex Systems 2009
31
PrNi2Si2 excitations
Neutron Scattering Experiment
Blanco et al. PRB 56 (1997) 11666
Blanco et al. Physica B 234 (1997) 756
Calculations done by McPhase
Martin Rotter
Magnetism in Complex Systems 2009
32
Calculate Magnetic Excitations and the Neutron
Scattering Cross Section
1
m m
H   Bl Ol ( J i )   g Ji  B J i H   J i J (ij )J j
2 ij
lm,i
i
2

d 
k '  e 

ˆ
ˆ

 N 
(   Q Q  )S mag (Q,  )
2  
ddE '
k  mc  

inel

S mag (Q,  )  21N  { 12 gF (Q)}d { 12 gF (Q)}d ' e  iκ ( B
2
2
d
B d ' )
b

 dd
' ' ' ( z) 

dd '

1 
 dd ' ( z )   d'd ( z*)
2i

S  2
e
Wd Wd '
1
1 e
  / kT

S dd ' (Q,  )

 ''


 1
 (Q,  )   0 ( ) 1   0 ( ) J (Q) Linear Response Theory, MF-RPA
 0 ( )  

 i | J    J   H ,T | j  j | J    J   H ,T | i 
ij
 j   i  
(ni  n j )
.... High Speed (DMD) algorithm: M. Rotter Comp. Mat. Sci. 38 (2006) 400
Martin Rotter
Magnetism in Complex Systems 2009
33
Summary
• Magnetic Diffraction
• Magnetic Structures
• Caveat on using the Dipole Approx.
•
•
•
•
Martin Rotter
Magnetic Spectroscopy
Magnons (Spin Waves)
Crystal Field Excitations
Orbitons
Magnetism in Complex Systems 2009
34
Epilogue
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
Martin Rotter, University of Oxford
Martin Rotter
Magnetism in Complex Systems 2009
36
McPhase - the World of Magnetism
McPhase is a program package for the calculation of magnetic properties
! NOW AVAILABLE with INTERMEDIATE COUPLING module !
Magnetization
Magnetic Phasediagrams
Magnetic Structures
Martin Rotter
Elastic/Inelastic/Diffuse
Neutron Scattering
Cross Section
Magnetism in Complex Systems 2009
37
Crystal Field/Magnetic/Orbital Excitations
McPhase runs on
Linux & Windows
it is freeware
www.mcphase.de
Magnetostriction
and much more....
Martin Rotter
Magnetism in Complex Systems 2009
38
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
Thanks to ……
M. Doerr, M. Loewenhaupt, TU-Dresden
R. Schedler, HMI-Berlin
P. Fabi né Hoffmann, FZ Jülich
S. Rotter, Wien
M. Banks, MPI Stuttgart
Duc Manh Le, University of London
J. Brown, B. Fak, ILL, Grenoble
A. Boothroyd, Oxford
P. Rogl, University of Vienna
E. Gratz, E. Balcar, G.Badurek
TU Vienna
J. Blanco,Universidad Oviedo
Martin Rotter
University of Oxford
……. and thanks to you !
Magnetism in Complex Systems 2009
39
Bragg’s Law in Reciprocal Space (Ewald Sphere)
2/l
O
k
c*
2q
q
a*
k‘
τ=Q
Q  2 sin  k
Unpolarised Neutrons - Van Hove Scattering
function S(Q,ω)
d 2
k'  M 
ˆ | f |2   i | αˆ  | f    f | αˆ | i )
 

(



E

E
)
P
(|

i
|



i
f
i
ddE ' k  2 2 
if
• for the following we assume that there is no nuclear order - <I>=0:
2
2 2



d 
k ' e 
k'

ˆ
ˆ

 N 
(   Q Q  )S mag (Q,  )  N S nuc (Q,  )
2  
ddE '
k  mc  
k

 ~
 ~

1

i
Q

R
(
t
)
i
Q
R j ' ( 0 )

it 1
j
1
1




S mag (Q,  ) 
dte
gF
(
Q
)
gF
(
Q
)

J
(
t
)
e
J
(
0
)
e
T

j
j '
2
j 2
j'

2 
N jj '

 ~
 ~

1
iQR j ( t ) iQR j ' ( 0 )
it 1
j* j'
j * j' 1
S nuc ( Q,  ) 
dte
(b b  bN bN 4  jj ' I j ( I j  1))  e
e
T


2 
N jj '


~
R
(
t
)

R

u
(t )
Splitting of S into elastic and inelastic part
j
j
j
Snuc  Snuc  Snuc
el
inel
Smag  Smag  Smag
el
inel
S nuc
S mag
el
el
1
  ( )
N
1
  ( )
N
 (b b  b b
j*
 jj ' I j ( I j  1))e
j * j' 1
N N 4
 
 
 iQ  R j  iQ  R j '
e
W j W j '
jj '

1
2
jj '
j'
gF (Q)j  J j T

1
2
gF (Q)j '  J j ' T e
 
 
 i Q  R j  iQ  R j '
e
W j W j
L/2

A short
f ( x)   f n e inx2 / L ...with... f n   f ( x' )e i 2nx'/ L dx'
Excursion
n 0
L / 2
to Fourier
...
L/2
1  inx2 / L
and Delta
f ( x)    e
f ( x' )e i 2nx'/ L dx'
Functions ....
L n 0
L / 2
1  in( x  x ') 2 / L
 ( x  x' )   e
L n 0
 ( x)
 (cx) 
c 
2
qa  2x / L... e iqna 
 (q)
a
n 0
it follows by extending the range of x to more than –L/2 ...L/2 and
going to 3 dimensions (v0 the unit cell volume)
 iQG k  iQG k '
e

kk '
Martin Rotter
(2 )3
 NG
v0

 (Q  τ )
rez .latt . τ
Magnetism in Complex Systems 2009
42
Neutron – Diffraction
S nuc
el
 
 
1

*
1

i
Q

R

i
Q
R j ' W j W j '
j
j'
j 2 1
j
  ( )   b b e
e
  | bN | 4 I j ( I j  1)
N j
 N jj '

Lattice G with basis B: j  ( kd )........
Latticefactor
Structurefactor |F|2



R j  Gk  Bd
 1


S nuc   ( )   Q , τ 
 τ
 N B
2
1
2
  (  )
bd  bd 

NB d
el
1
  (  )
NB
b
d 2 1
N
4
NB
b
d , d '1
inc
e



„Isotope-incoherent-Scattering“
I d ( I d  1)
„Spin-incoherent-Scattering“

i
 c  4 | b |2
2
d nuc inc
 4
 N  4 b 2  b   (bNd ) 2 14 I d ( I d  1)
d
el
el
bd 'e
 

iQ( B d  B d ' ) Wd Wd '
Independent of Q:
d
one element(NB=1):  nuc
d
*

Three Axes
Spectrometer (TAS)
k
Q
Ghkl
k‘
q
2
2

k  k '
 

2M
2M
   

Q  k  k '  G hkl  q
Martin Rotter
Magnetism in Complex Systems 2009
44
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
45
NdCu2 Magnetic Phasediagram
(Field along b-direction)
4
FM
0H (T)
F2
2
F1
AF3
AF1 AF2
0
0
2
4
6
8
T (K)
Martin Rotter
Magnetism in Complex Systems 2009
46
Complex Structures
μ0Hb=2.6T
AF1
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
Magnetism in Complex Systems 2009
47
Complex Structures
μ0Hb=2.6T
F1
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
Magnetism in Complex Systems 2009
48
Complex Structures
μ0Hb=2.6T
F2
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
Magnetism in Complex Systems 2009
49
NdCu2 Magnetic Phasediagram
H||b
F1   
F3 
c
F1 
a
b
AF1 
Lines=Experiment
Colors=Theory
Calculation done by McPhase
Martin Rotter
Magnetism in Complex Systems 2009
50
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
51
NdCu2 - 4f Charge Density
 ˆ (r ) | R4 f (r ) |2
m
ec
q

O
 nm n n (J) T Z nm ()
n 0, 2, 4, 6
m 0 ,..., n
T=100
T=40
T=10 K
K
K
Martin Rotter
Magnetism in Complex Systems 2009
52
F3 
F3: measured
dispersion was
fitted to get
exchange
constants J(ij)
NdCu2
F1 
Calculations done by McPhase
AF1 
E. Balcar
M. Rotter & A. Boothroyd
2008
did some calculations
Martin Rotter
Magnetism in Complex Systems 2009
54
bct ThCr2Si2 structure
Space group I4/mmm
Ce3+ (4f1) J=5/2
TN=8.5 K
q=(½ ½ 0), M=0.66 μB/Ce
Goodness of fit:
(|FM|2-|FMdip|2)/ |FMdip|2
(%)
CePd2Si2
Comparison to
experiment
Rpdip=15.6%
Rpbey=8.4 %
(Rpnuc=7.3%)
Martin Rotter
Calculation done by McPhase
M. Rotter, A. Boothroyd, PRB, submitted
Magnetism in Complex Systems 2009
55