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Magnetic Neutron Scattering
Neutron spin meets electron spin Magnetic neutron diffraction Inelastic magnetic neutron scattering Polarized neutron scattering Summary+Impurities in spin-1 chains
Collin Broholm*Johns Hopkins University and NIST Center for Neutron Research
*Supported by the NSF through DMR-9453362 and DMR-0074571
CRNL 6/20/00
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 V
Correspondingly the field exerts a torque and a force
B BF
driving the neutron parallel to high field regions
CRNL 6/20/00
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
220
ˆ
4 Rm
mgV B
em
RSrBr
The transition matrix element in Fermi’s golden rule llm iF
grV
mrSkk
exp
22 02
Magnetic scattering is as strong as nuclear scatteringcm 1054.0
412
20
0
em
er
It is sensitive to atomic dipole moment perp. to
lll SSS
CRNL 6/20/00
The magnetic scattering cross section
rrr dexp isF
Spin density spread out scattering decreases at high
The magnetic neutron scattering cross section
t
eedteFg
rk
k
EEVpm
k
k
ll
ll
itiW
m
ll
SS
kkEdd
d
RR
2
0
2
2
22
20
22
2
For unspecified incident & final neutron spin states
Edd
d
Edd
d 2
21
2
CRNL 6/20/00
Un-polarized magnetic scattering
llll
iti
W
teedt
eFg
rk
k
ll
SSrr 0
ˆˆ2Edd
d 22
20
2
llll
iti
W
teedt
eFg
rk
k
ll
SSrr 0
ˆˆ2Edd
d 22
20
2
Spin correlation functionSpin correlation function
Squared form factorSquared form factor Polarization factorPolarization factor
Fourier transformFourier transform
DW factorDW factor
CRNL 6/20/00
Magnetic neutron diffraction
Time independent spin correlations elastic scattering
llll
iW lleeFg
r
SSrrˆˆ
2d
d 22
20
Periodic magnetic structures Magnetic Bragg peaks
mm
mmv
Nr
22
32
0 ˆ2
d
dFF
The magnetic vector structure factor is
d
dd
2d
d S2
d
iW eeFg
F
Magnetic primitive unit cell greater than chemical P.U.C.
Magnetic Brillouin zone smaller than chemical B.Z.
CRNL 6/20/00
zS ˆB
smg
Simple cubic antiferromagnet
ab
*a*b
Real sp
ace
Reci
pro
cal S
pace
ma
mb
*ma
*mb
lkheFm W
B
s
sinsinsin82
ˆ 2
z
F
m
mzW
B
s
vFe
mrN
3222
2
0
21
2d
d ‘
No magnetic diffraction for S
SS
Not so simple Heli-magnet : MnO2
llll iS RQyRQxRwS sinˆcosˆexp
Insert into diffraction cross section to obtain
QwQ-w
d
d
vF
geSrN z
W3
22
220
21
2
111w and 7200Q characterize structure
*a
*c
a bc
CRNL 6/20/00
Understanding Inelastic Magnetic Scattering:
Isolate the “interesting part” of the cross section
,ˆˆ2
22
20
2 s
WeF
grN
k
k
Edd
d
The “scattering law” is defined as
tSSeedt llll
-iN
ti
0, ll rr1
S
for a wide class of systems It satisfies useful sum-rules
,exp,
SS
1,ddd
1
SS
Sq
q
''2 cos1
1
3
1),( llll
llllJN
d rrSS
S
Detailed balanceDetailed balance
Total moment
Total moment
First moment sum-ruleFirst moment sum-rule
CRNL 6/20/00
Scattering from a quantum spin liquid
Dimerized spin-1/2 system: copper nitrate
JTkB
CRNL 6/20/00
A spin-1/2 pair has a singlet - triplet gap:
Weak inter-dimer coupling cannot close gap
Bond alternation is relevant operator for quantum critical uniform spin chaininfinitesimal bond alternation yields gap
Why a gap in spectrum of dimerized spin system
J0totS
1totS
JJ
J
CRNL 6/20/00
Sum rules and the single mode approximation
When a coherent mode dominates the spectrum:
)(
cos11
3
1
)(
),()(
''2
q
JN
q
qdq
llllll
ll
rrSS
S
S
Sum-rules link S(q) and (q)
)()(),( qqq
SS
E (
meV
)
0.4
0.5
0 2 4 0 2 4 6q ()
CRNL 6/20/00
Spin waves in a ferromagnet
nnS
12
,S
Magnon creationMagnon creation Magnon destructionMagnon destruction
JJ 02S
1exp
1
TkE
En
B
Dispersion relation
Magnon occupation prob.
Gadolinium
CRNL 6/20/00
Spin waves in an antiferromagnet
nn
eJS iz
1
1
2
1d
d
,S
2202 JJ S
Dispersion relation
CRNL 6/20/00
Continuum magnetic inelastic scattering
Inelastic scattering is not confined to disp. relations when• There is thermal ensemble of excitations present• and do not uniquely specify excited state
- electron hole pair excitations in metals - spinon excitations in quantum magnets
spinon continuum in spin-1/2 AFM chain
CRNL 6/20/00
and the magnetic susceptibility
Compare to the generalized susceptibility
0,2
llll
-itiB StSeedtN
g
ll rrq
They are related by the fluctuation dissipation theorem
eg B 1
1Im,q 2
qS
tSSeedt llll
-iN
ti
0, 1 ll rr
S
,S
We convert inelastic scattering data to• Compare with bulk susceptibility data• Isolate non-trivial temperature dependence• Compare with theories
q to
CRNL 6/20/00
Polarized magnetic neutron scattering
tll SS
0
Specify the incident and final neutron spin state
tSS zl
zl 0
tSS zl
zl 0
tSS ll
0
tSS ll
0
Non spin flip:SHS
Spin flip: SHS
CRNL 6/20/00
Polarized neutron scatteringH// H perp
Type of scattering SF NSF SF NSFNuclear coherent 0 1 0 1Nuclear isotope incoherent 0 1 0 1Nuclear spin incoherent 2/3 1/3 2/3 1/3Magnetic Sxx+Syy 0 Sxx Syy
Nuclear isotope incoherent scatteringNuclear isotope incoherent scatteringParamagnetic scattering MnF2Paramagnetic scattering MnF2
H// H//H H
SF
NSF
SF
NSF
CRNL 6/20/00
SummaryThe neutron has a small dipole moment that
causes it to scatter from inhomogeneous internal fields produced by electrons
The magnetic scattering cross section is similar in magnitude to the nuclear cross section
Elastic magnetic scattering probes static magnetic structure
Inelastic magnetic scattering probes spin dynamics through
Polarized neutrons can distinguish magnetic and nuclear scattering and specific spin components
,
S
CRNL 6/20/00
Low T excitations in spin-1 AFM chain
Y2BaNiO5 T=10 KMARI chain ki
•Haldane gap =8 meV
•Coherent mode
•S(q,)->0 for Q->2n
pure
CRNL 6/20/00
AKLT state for spin-1 chain
• This is exact ground state for spin projection Hamiltonian
• Magnets with 2S=nz have a nearest neighbor singlet covering with full lattice symmetry.
• Excited states are propagating bond triplets separated from the ground state by an energy gap .J
Haldane PRL 1983Affleck, Kennedy, Lieb, and Tasaki PRL 1987
i
iii
iiiii
toti SP 12
131
12 SSSSSSH
CRNL 6/20/00
Impurities in Y2BaNiO5
Ca2+
Y3+
• Mg2+on Ni2+ sites finite length chains• Ca2+ on Y3+ sites mobile bond defectsMg
Ni
Kojima et al. (1995)
CRNL 6/20/00
Zeeman resonance of chain-end spinsI(
H=
9 T
)-I(
H=
0 T
) (
cts.
per
min
.)
Hg B
h (
meV
)
H (Tesla)0 2 4 6 8
g=2.16
0 0.5 1 1.5 2
-5
0
10
15
20
Hg B
meV)(
CRNL 6/20/00
Form factor of chain-end spins
Q-dependence revealsthat resonating objectis AFM.
The peak resemblesS(Q) for pure system.
Chain end spin carryAFM spin polarizationof length back into chain
Y2BaNi1-xMgxO5 x=4% Hg B
CRNL 6/20/00
New excitations in Ca-doped Y2BaNiO5
Pure 9.5% Ca
•Ca-doping creates states below the gap
•sub-gap states have doubly peaked structure factor
Y2-xCaxBaNiO5:
CRNL 6/20/00
Why a double ridge below the gap in Y2-xCaxBaNiO5 ?
Charge ordering yields incommensurate spin order
Quasi-particle Quasi-hole pair excitations in Luttinger liquid
Anomalous form factor for independent spin degrees of freedom associated with each donated hole
xq
q is single impurity prop.Indep. of
x
CRNL 6/20/00
Does q vary with calcium concentration?
q not strongly dependent on x
Double peak is single impurity effect
CRNL 6/20/00
(b)
Ca
Ba
Ni
Y
c
b
a
(e)
(f)
(c)
(d)
a
O
Bond Impurities in a spin-1 chain: Y2-
xCaxBaNiO5
(a)
Form-factor for FM-coupled chain-end spins
2/)(Re2)( iqeqMqS
A symmetric AFM droplet
22/*2/ )()()(
ll
iql
iqlll eqMeqMPqS
Ensemble of independent randomly truncated AFM droplets
CRNL 6/20/00
Calcium doping Y2BaNiO5
Experimental facts:Ca doping creates sub-gap excitations with
doubly peaked structure factor and bandwidth The structure factor is insensitive to
concentration and temperature for 0.04<x<0.14 (and T<100 K)
Analysis:Ca2+ creates FM impurity bonds which nucleate AFM droplets with doubly peaked structure factorAFM droplets interact through intervening chain
forming disordered random bond 1D magnet
CRNL 6/20/00
Incommensurate modulations in high TC superconductors
YBa2Cu3O6.6 T=13 K E=25 meV
h (rlu)
k
(rl
u)
Hayden et al. 1998
CRNL 6/20/00
What sets energy scale for sub gap scattering ?
0
5
10
Possibilities:
•Residual spin interactions through Haldane state. A Random bond AFM.
•Hole motion induces additional interaction between static AFM droplets
•AFM droplets move with holes: scattering from a Luttinger liquid of holes.
How to distinguish:
• Neutron scattering in an applied field
• Transport measurements
• Theory
?
q~
h
(meV
)
CRNL 6/20/00
Conclusions on spin-1 chain
Dilute impurities in the Haldane spin chain create sub-gap composite spin degrees of freedom.
Edge states have an AFM wave function that extends into the bulk over distances of order the Haldane length.
Holes in Y2-xCaxBaNiO5 are surrounded by AFM spin polaron with central phase shift of
Neutron scattering can detect the structure of composite impurity spins in gapped quantum magnets.
The technique may be applicable to probe impurities in other gapped systems eg. high TC superconductors.
Microscopic details of gapped spin systems may help understand related systems where there is no direct info.
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