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Magnetic Neutron Scattering
Collin Broholm
*Supported by U.S. DoE, Basic Energy Sciences, Materials Sciences & Engineering DE-FG02-08ER46544
Neutron spin meets electron spin
Structure : Magnetic diffraction
Excitations: Inelastic magnetic scattering
Tomorrow : Entangled Magnetism
Overview
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
VCorrespondingly the field exerts a torque and a force
B
BF
driving the neutron parallel to high field regions
Magnetic Neutron Optics The combined nuclear and magnetic potential for neutron
V =
2p 2
mrb ± r0M ×s( )
The index of refraction is
n = 1-
V
E= 1-
l 2
prb ± r0M ×s( )
Critical angle for total external reflection:
qc = l
rb ± r0M ×s
p
M
Polarized beam!
Q
k ¢k
Structure of vortex matter: CeCoIn5
Bianchi et al., Science (2008)
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
2
20ˆ
4 Rm
mgV B
em
RSrBr
The transition matrix element in Fermi’s golden rule
m
2p 2¢k ¢s ¢l V
mksl = -r
0
g
2F q( ) ¢s ¢l s ×S
^lsl exp iq ×r
l( )Magnetic scattering length ~ nuclear scattering lengths
r0 = gm0
4p
e2
me
= 0.54 ´10-12 cm
It is sensitive to atomic dipole moment perp. to q
S
^ l= S
l- S
l×q̂( )q̂
The magnetic scattering cross section
exp dF s i q r q r r
Spin density spread out scattering decreases at high q
The magnetic neutron scattering cross section
d2s
dWd ¢Es® ¢s
=¢k
k
m
2p 2
æ
èçö
ø÷
2
pl
l ¢l
å ¢k ¢s ¢l Vm
ksl2
d El
- E¢l- w( )
=¢k
kr
0
2 g
2F q( )
2
e-2W k( ) 1
2pdt e- iw t
ò eiq× R
l-R
¢l( )
l ¢l
å
´ s s ×S^ l
0( ) ¢s ¢s s ×S^ ¢l
t( ) s
For unspecified incident & final neutron spin states
Edd
d
Edd
d 2
21
2
Un-polarized magnetic scattering
Spin correlation function
Squared form factor Polarization factor
Fourier transform
DW factor
d2s
dWd ¢E=
¢k
kr
0
2 g
2F q( )
2
e-2W q( )
dab
- q̂aq̂
b( )ab
å
´1
2pdt e- iwt e
iq× rl-r
¢l( )S
l
a 0( )S¢l
b t( )l ¢l
åò
Online lectures accessible at:
http://www.sns.gov/education/qcmp/
Magnetic neutron diffraction Time independent spin correlations elastic scattering
ds
dW= r
0
2 g
2F q( )
2
e-2W q( ) d
ab- q̂
aq̂
b( )ab
å eiq× r
l-r
¢l( )S
l
a S¢l
b
l ¢l
å
Periodic magnetic structures Magnetic Bragg peaks
The magnetic vector structure factor is
Magnetic primitive unit cell greater than chemical P.U.C. Magnetic Brillouin zone smaller than chemical B.Z.
Diffraction from Ferromagnet
Nuclear & magnetic scattering interferes. Effective scatt. length:
b +s r0S^
The structure factor for reciprocal lattice vector, t
Fs
2= bd +s r0S^d( )eit ×d
då2
= bdeit ×d
då2
+ s r0S^deit ×d
då2
+ 2s bdr0S^deit × d- ¢d( )
d ¢dåNuclear Magnetic Nuclear-magnetic
The last term vanishes for un-polarized neutrons ( average). A welcome consequence is that the diffracted beam is polarized!
s
bd » r0S^d Þ F±
2»
4 bd
2 for s = 1
0 for s = -1
ì
íï
îï
Simple cubic antiferromagnet
zS ˆB
smg
ab
*a
*b
ma
mb
*
ma
*
mb
ds
dW= N r
0
ms
2mB
æ
èç
ö
ø÷
2
e-2W q( )
F q( )2
1- q̂z
2( )2p( )
3
vd q -t
m( )t
m
å ‘
No magnetic diffraction for q S
qS
S
Not so simple Heli-magnet : MnO2
ˆ ˆexp cos sinl l m l m l
S i S w R x Q R y Q R
Insert into diffraction cross section to obtain
ds
dW= N r
0S( )
2
e-2W q( ) g
2F q( )
2
1+ q̂z
2( )2p( )
3
v
´ d q + w - Qm
-t( ) +d q + w + Qm
-t( ){ }t
å
111w and 27
00m
Q characterize structure
*a
*c
a b c
Diffuse Magnetic Scattering in FeTe0.6Se0.4 K
(r
.l.u
.)
L (
r.l.
u.)
Thampy et al PRL(2012)
Hypothesis: Ising spins?
Polarized neutrons:
isotropic static spin correlations
Diffuse Magnetic Scattering in FeTe0.6Se0.4 K
(r
.l.u
.)
L (
r.l.
u.)
Thampy et al PRL(2012)
Interstitial Iron in Fe1+yTe1-xSex
Extract Local Spin Structure
Model by coupling itinerant spin to 5-band model through
Kang & Tesanovic
Nakatsuji et al. Science (2005)
Unusually weak interlayer coupling
Dominant 3rd neighbor interactions
Exact triangular lattice
Weak easy plane anisotropy
J3 : AFM
Elastic neutron scattering from NiGa2S4 powder
2 * 22
2
0 22 2
/ˆ 1 2 cos2
d g Ar F Q N
d
q q
τ
m Q m Q c
Q τ q
2 Surprises:
12 120oQ Q
1 25(3) Å
Nakatsuji et al. (2005) ds
dW powder
=dWQ
4pòds
dW xtal
Multi crystal assembly of high quality NiGa2S4
19 NiGa2S4 crystals Grown by Yusuke Nambu co-aligned by Chris Stock
Y. Nambu et al., PRB (2009).
NiGa2S4: Elastic Magnetic Scattering
1.5 meV
Short range magnetic ordering
C. Stock et al. PRL (2010)
Summary: Diffraction
• The neutron has a small dipole moment causing scattering from electrons
• Magnetic scattering is similar in magnitude to nuclear scattering
• Elastic magnetic scattering probes static magnetic structure
• Spin direction gleaned from polarization factor (see spin perpendicular to Q)
• Periodic structure determined through Bragg intensities
• Generally probe magnetic structure on 1-104 Å scale:
– flux line lattice
– short range order
− powder, single crystals, thin films
Features of the cross section
The dynamic correlation function
Fluctuation Dissipation theorem
Sum-rules
Examples
Summary
Inelastic Magnetic Neutron Scattering
Understanding Inelastic Magnetic Scattering:
Isolate the “interesting part” of the cross section
The dynamic correlation function (“scattering law”) :
In thermodynamic equilibrium (detailed balance):
Useful exact sum-rules:
Total moment sum-rule (general result from Fourier analysis)
Correlations in space & time rearrange intensity in space leaving the total scattering scattering unaffected
Q-w
Definition of the equal time correlation function:
The wave vector dependence of the energy-integrated intensity probe a snap-shot of the fluctuating spin configuration
Fluctuation Dissipation Theorem
FQ t( ) =
iSQ t( ),S-Qéë ùû
Define the following t-dependent “response” function:
c Qw( ) = - gmB( )2
lime®0+
dte-i w-ie( )t
0
¥
ò FQ t( )
The generalized susceptibility can be expressed as:
The “scattering law” can be expressed as:
From which, the “fluctuation-dissipation” theorem:
First Moment Sum-rule Fourier transform of expression for scattering law:
Equating these leads to the first moment sum-rule:
Time derivative of left side evaluated for t=0:
Time derivative of right side evaluated at t=0:
2
1
,2
1,
0totS
1totS
, ,
,
Weakly Interacting spin-1/2 pairs in Cu-nitrate
Sum rules and the single mode approximation When a coherent mode dominates the spectrum:
Sum-rules link and :
0.4
0.5
0 2 4 0 2 4 6
Exp. data Single mode model
e(Q)
Q p( )
Spin waves in a ferromagnet
ACNS 6/29/10
Magnon creation Magnon destruction
n w( ) =
1
eb w -1
Dispersion relation
Occupation number
Gadolinium
Spin waves in an antiferromagnet
ACNS 6/29/10
S^ Q,w( ) =S
2
J 1- 1z eiQ×d
då( )e Q( )
´ d e Q( ) - w( ) n w( ) +1( ) +d e Q( ) + w( )n w( ){ }
Dispersion relation
J = 4.38 meV ¢J = 0.15 meV D = 0.107 meV
Probing matter through scattering
fp
ip
Single Particle Process
Multi Particle Process
fi
fi
EE
21
21
ppQQ
1Q
2Q fi
fi
EE
Q
ppQ
Q
Q
ћ
Intensity
Intensity
Q
ћ
Disintegration of a spin flip in 1D
Spinon Spinon
From spinon band-structure
to bounded continuum
2
2, QSGQS
Q (p )q (p )
e/J
w
/J
Quantum critical spin-1/2 chain
Hammar et al. (1999)
Cu(C4H4N2)(NO3)2
Copper Pyrazine dinitrate
Summary
• Inelastic Magnetic scattering probes dynamic spin
correlation function:
• Exact results are critical to interpret data
• The full dynamic correlation function as opposed
to just the dispersion relation tells the story
– Intensity gives equal time correlation function
– Energy width in resonant excitations probes life time
– Broad continuum reveals multiparticle excitations
• Know your sum-rules and happy scattering!
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