Magnetic Neutron Scattering
Martin Rotter, University of Oxford
Martin Rotter
NESY Winter School 2009
1
Contents
• Introduction: Neutrons and Magnetism
• Elastic Magnetic Scattering
• Inelastic Magnetic Scattering
Martin Rotter
NESY Winter School 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
NESY Winter School 2009
3
Bragg’s Law in Reciprocal Space (Ewald Sphere)
2/l
O
k
c*
2q
q
a*
k‘
τ=Q
Q  2 sin k
Single Crystal Diffraction
E2 – HMI, Berlin
k
Q
O
Martin Rotter
NESY Winter School 2009
5
The Scattering Cross Section
Scattering Cross Sections

Number of scatteredneutronsper sec  tim e1



area

1
1
Incidentneutronflux
 tim e 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
NESY Winter School 2009
6
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 
k'  M 
 Martin Rotter 2 
ddE ' k  2 
2
2

2
P
P
|

s
;
i
|
H
(
Q
)
|
s
'
;
f

|
 sn i n NESY
int Winter nSchool 2009  (  Ei 7 E f )
if , s n
(follows from Fermis golden rule)
Interaction of Neutrons with Matter
 

 3
 iQrn
H (Q)   e
H (rn )d rn
Hint  Hnuc  Hmag


 ~
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  An  Ae  
 P e  Ae   2B se  Bn
c
c 
 2m 
e 2m 
 ~

 ˆ 

i
Q
Rj
1
ˆ
H mag (Q)  8B  2 gF ( )j e
μ N g n sn  Q  J j  Q

j
Martin Rotter



 
Hint (Q)  ˆ (Q)  2αˆ (Q)  sn
NESY Winter School 2009
8
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  m c  
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
it 1
S nuc ( Q,  ) 
dte
2 
N
 (b
j*
j * j' 1
N N 4
b b b
j'
 jj ' I j ( I j  1))  e
jj '
Snuc  Snuc  Snuc
inel
Smag  Smag  Smag
el
inel
S nuc
S mag
el
el
e
T


~
R j (t )  R j  u j (t )
Splitting of S into elastic and inelastic part
el
 ~
 ~
 i Q  R j ( t ) iQ  R j ' ( 0 )
 
 
1
iQR j iQR j ' W j W j '
j* j'
j * j' 1
  ( )  (b b  bN bN 4  jj ' I j ( I j  1))e
e
N jj '
1
  ( )  12 gF(Q)j  J j T
N jj'

1
2
gF(Q)j '  J j ' T e
 
 
iQR j iQR j ' W j W j
e
L/2

A short
f ( x)   f n e inx 2 / L ...with... f n   f ( x' )e i 2nx '/ L dx'
Excursion
n 0
L / 2
to Fourier
...
L/2
1  inx 2 / 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)
e
kk '
Martin Rotter
 
 
iκ G k iκ G k '
 
(2 )3
 NG
 (κ  τ)

v0 rez .latt.τ
NESY Winter School 2009
10
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



R j  Gk  Bd
 1


S nuc   ( )   Q , τ 
 τ
 N B
2
1
2
  ( )
bd  bd 

NB d
el
1
  ( )
NB
b
one element(NB=1):
d 2 1
N
4
NB
b
d , d '1
bd 'e
 

iQ( B d  B d ' ) Wd Wd '
e



Independent of Q:
„Isotope-incoherent-Scattering“
I d ( I d  1)
d
 nuc el inc
d
*
„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

Magnetic Diffraction
S nuc
S mag
el
el
coh


*
1

i
Q

R
i
Q
R j '
j
j'
j
  ( )  b b e
e
N jj '
1
  ( )  12 gF(Q)j  J j T
N jj '
d 
k '  e 2 

 N 
2 
ddE'
k  mc 
2
2
( 


12 gF(Q)j '  J j ' T e

iQR j
e

iQR j '


k'

ˆ
ˆ
 Q Q  )S mag (Q,  )  N S nuc (Q,  )
k
Difference to nuclear scattering:
Formfactor
12 gF( )j
Polarisationfactor
... no magnetic signal at high angles
ˆ Q
ˆ ) ... only moment components
(  Q
 
normal to κ contribute
Martin Rotter
NESY Winter School 2009
12
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
13
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
14
Atomic Lattice
Magnetic Lattice
ferro
antiferro
Martin Rotter
NESY Winter School 2009
15
Formfactor
Q=
2 g
 j2 (Q) 
Dipole Approximation (small Q): F (Q)  j0 (Q)  
g
Martin Rotter
NESY Winter School 2009
16
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).
Arrangement of Magnetic Moments in Matter
Paramagnet
Ferromagnet
Antiferromagnet
And many more ....
Ferrimagnet, Helimagnet, Spinglass ...collinear, commensurate etc.
Martin Rotter
NESY Winter School 2009
18
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
19
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
20
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
21
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
22
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
N
Experimental data D4, ILL
Calculation done by McPhase
Rotter et.al. J. Magn. Mag. Mat. 214 (2000) 281
Martin Rotter
NESY Winter School 2009
23
GdCu2 T = 42 K
M [010]
TR= 10 K q = (2/3 1 0)
Magnetic Structure from
Neutron Scattering
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
NESY Winter School 2009
I exp (hkl)
24
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
NESY Winter School 2009
25
Complex Structures
μ0Hb=2.6T
AF1
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
26
Complex Structures
μ0Hb=2.6T
F1
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
27
Complex Structures
μ0Hb=2.6T
F2
μ0Hb=1T
μ0Hb=0
Q=
Experimental data TAS6, Riso Loewenhaupt, Z. Phys. (1996) 499
Martin Rotter
NESY Winter School 2009
28
NdCu2 Magnetic Phasediagram
H||b
F1   
F3 
c
F1 
a
b
AF1 
Lines=Experiment
Colors=Theory
Calculation done by McPhase
Martin Rotter
NESY Winter School 2009
29
A caveat on the Dipole
Approximation
S mag
el


1

i
Q

R
i
Q
R j '
j
ˆ  Q
ˆ  e
  ( )   Q
e
j
T
j '
T
N 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
ˆ 
Q
j
T
S. W. Lovesey „Theory of Neutron Scattering from
Condensed Matter“,Oxford University Press, 1984
Page 241-242
Martin Rotter
NESY Winter School 2009
30
E. Balcar
M. Rotter & A. Boothroyd
2008
did some calculations
Martin Rotter
NESY Winter School 2009
31
CePd2Si2
(σ-σdip)/σdip (%)
d
bct ThCr2Si2 structure
Space group I4/mmm
d
3+
Ce (4f1) J=5/2
TN=8.5 K
q=(½ ½ 0), M=0.66 μB/Ce
Comparison to
experiment
Goodness of fit:
Rpdip=15.6%
Rpbey=8.4 %
(Rpnuc=7.3%)
Martin Rotter
Calculation done by McPhase
M. Rotter, A. Boothroyd, PRB, submitted
NESY Winter School 2009
32
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, submitted
Martin Rotter
Calculation done by McPhase
NESY Winter School 2009
33
Inelastic Magnetic Scattering
• Dreiachsenspektometer – PANDA
• Dynamik magnetischer Systeme:
1. Magnonen
2. Kristallfelder
3. Multipolare Anregungen
Martin Rotter
NESY Winter School 2009
34
Three Axes
Spectrometer (TAS)
k
Q
Ghkl
k‘
q
2
2

k  k '
 

2M
2M
   

Q  k  k '  G hkl  q
Martin Rotter
NESY Winter School 2009
35
PANDA – TAS for Polarized Neutrons
at the FRM-II, Munich
Martin Rotter
NESY Winter School 2009
36
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
NESY Winter School 2009
37
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
NESY Winter School 2009
Q
38
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
NESY Winter School 2009
39
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
NESY Winter School 2009
40
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
NESY Winter School 2009
41
Movement of Charges - the Crystal Field Concept
+
+
+
+
+
4f –charge density
+
+
+
E
+
+
Hamiltonian H cf 
m m
B
 l Ol (J i )
lm,i
Martin Rotter
Q
NESY Winter School 2009
42
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
NESY Winter School 2009
43
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
NESY Winter School 2009
44
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  m c  


inel
iκ( B d B d ' ) Wd Wd ' 

1
1
1
S mag (Q, )  2Nb { 2 gF(Q)}d { 2 gF(Q)}d ' e
e
Sdd ' (Q, )
2
2

 dd
' ' ' ( z) 

dd '

1 
 dd ' ( z )   d'd ( z*)
2i

S  2
1
1 e
  / kT
 ''


 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
NESY Winter School 2009
45
F3 
F3: measured
dispersion was
fitted to get
exchange
constants J(ij)
NdCu2
F1 
Calculations done by McPhase
AF1 
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
NESY Winter School 2009
47
Summary
• Magnetic Diffraction
• Magnetic Structures
• Caveat on using the Dipole Approx.
•
•
•
•
Martin Rotter
Magnetic Spectroscopy
Magnons (Spin Waves)
Crystal Field Excitations
Orbitons
NESY Winter School 2009
48
Martin Rotter, University of Oxford
Martin Rotter
NESY Winter School 2009
49
McPhase - the World of Rare Earth Magnetism
McPhase is a program package for the calculation of
magnetic properties of rare earth based systems.
Magnetization
Magnetic Phasediagrams
Magnetic Structures
Martin Rotter
Elastic/Inelastic/Diffuse
Neutron Scattering
Cross Section
NESY Winter School 2009
50
Crystal Field/Magnetic/Orbital Excitations
McPhase runs on
Linux & Windows
it is freeware
www.mcphase.de
Magnetostriction
and much more....
Martin Rotter
NESY Winter School 2009
51
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, Austria
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 TU Vienna
Martin Rotter
University of Oxford
……. and thanks to you !
NESY Winter School 2009
52