High Sensitivity Spectroscopy at Reactor Neutron Sources
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Transcript High Sensitivity Spectroscopy at Reactor Neutron Sources
Magnetic Neutron Scattering
Collin Broholm*
Johns Hopkins University and NIST Center for Neutron Research
Neutron spin meets electron spin
Magnetic neutron diffraction
Inelastic magnetic neutron
scattering
Polarized neutron scattering
Summary
*Supported by the NSF through DMR-9453362 and DMR-0074571
Magnetic properties of the
neutron
The neutron has a dipole moment
me
n B
m
n is 960 times smaller than the electron moment
e
n
m
m e
1836
960
1 . 913
A dipole in a magnetic field has potential energy
V r B r
Correspondingly the field exerts a torque and a force
F B
B
driving the neutron parallel to high field regions
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The transition matrix element
The dipole moment of unfilled shells yield inhomog. B-field
0 g B S Rˆ
B
2
4
R
The magnetic neutron senses the field
S Rˆ
V m r B r
g
R2
4
m
0
me
2
B
The transition matrix element in Fermi’s golden rule
m
g
k V m k r0 F S l exp i rl
2
2
2
Magnetic scattering is 2as strong as nuclear scattering
r0
0 e
4 m e
0 . 54 10
12
cm
It is sensitive to atomic dipole moment perp. to
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S l S l S l
The magnetic scattering cross section
scattering decreases at high
Spin density spread out
F s r exp i r d r
The magnetic neutron scattering cross section
2
k m
p k V m k
2
k 2
d
2
d d E
k
k
r0
2
g
2
F
2
e
2 W
dt e
i t
2
E E
e
i R l R l
ll
S l 0 S l t
For unspecified incident & final neutron spin states
d
2
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d dE
d
2
1
2
d dE
Un-polarized magnetic scattering
Squared form factor
d
2
d dE
k
k
r0
DW factor
2
dt e
g
2
F
i t
Fourier transform
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e
ll
2
e
Polarization factor
2 W
ˆ ˆ
i rl rl
S l 0 S l t
Spin correlation function
Magnetic neutron diffraction
Time independent spin correlations
d
d
r0
g
2
2
2
F
e
2 W
ˆ ˆ e
d
d
r0 N m
2
2
vm
F
m
i rl rl
ll
Periodic magnetic structures
3
elastic scattering
Sl
S l
Magnetic Bragg peaks
2
ˆ F
2
m
Magnetic primitive unit cell greater than chemical P.U.C.
Magnetic Brillouin zone smaller than chemical B.Z.
The magnetic vector structure factor is
F
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d
gd
2
F d e
2 W d
Sd e
i d
Simple cubic antiferromagnet
g S
F
m s zˆ
2 B
F e
ms
N r0
d
2 B
d
ms
B
zˆ
bm
2 W
2
8 sin h sin k sin l
2 W
2
2
e
F
1
z
No magnetic diffraction for
S
S
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2 3
v
b
a
am
‘
m
m
S
b*
b
a*
*
m
a
*
Not so simple Heli-magnet :
MnO2
c*
c
b
a
S l S exp i w R l xˆ cos Q R l yˆ sin Q R l
a*
w 111 and Q 00 72 characterize structure
Insert into diffraction cross section to obtain
d
d
N r0 S e
2
2 W
g
2
F
2
1
2
z
2
v
3
w - Q w Q
Understanding Inelastic Magnetic
Scattering:
Isolate the “interesting part” of the cross section
d
2
d dE
k
k
N r0
2
g
2
F
2
e
2 W
ˆ ˆ s
,
The “scattering law” is defined as
S
,
dt e
i t 1
N
e
i rl - r l
ll
S l 0 S l t
for a wide class of systems It satisfies useful sum-rules
Detailed balance
Total moment
S , exp S ,
1
d q d S , S S 1
dq
d S ( , )
2
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1 1
3 N
J l l S l S l ' 1 cos rl rl '
ll
First moment sum-rule
Scattering from a quantum spin liquid
Dimerized spin-1/2 system: copper nitrate
k B T J
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Why a gap in spectrum of dimerized spin system
A spin-1/2 pair has a singlet - triplet gap:
J
S tot 1
S tot 0
Weak inter-dimer coupling cannot close
gap
J
J
J
Bond alternation is relevant operator for
quantum critical uniform spin chain
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Spin waves in a ferromagnet
S
,
S
n 1 n
2
Magnon creation
Magnon destruction
Gadolinium
Dispersion relation
2 S J 0 J
Magnon occupation prob.
1
nE
exp E
1
k
T
B
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Spin waves in an antiferromagnet
S
,
S J 1
1
z
d e
i d
n 1 n
2
Dispersion relation
2 S
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J 0 J
2
2
and the magnetic
susceptibility
S ,
S
,
dt e
i t 1
N
e
i rl - rl
ll
S l 0 S l t
Compare to the generalized susceptibility
2
g B
i r -r
i t
S l t , S l 0
q
e
dt e
l
N
l
ll
They are related by the fluctuation dissipation theorem
S
q ,
Im q
1
2
1
e
g B
We convert inelastic scattering data to q
• Compare with bulk susceptibility data
• Isolate non-trivial temperature dependence
• Compare with theories
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to
Polarized magnetic neutron scattering
Specify the incident and final neutron spin state
S l 0 S l t
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S
z
l
0 S z l t
S
z
l
0 S t
S
l
0 S t
z
l
l
S l 0 S l t
Non spin flip:
S
H
S
Spin flip:
S H
S
Polarized neutron scattering
H per p
H //
T y pe of scattering
N uclear coherent
N uclear isotope incoherent
N uclear spin incoherent
M agnetic
SF
0
0
2/3
yy
xx
S +S
SF
0
0
2/3
xx
S
N SF
1
1
1/3
0
N SF
1
1
1/3
yy
S
Nuclear isotope incoherent scatteringParamagnetic scattering MnF2
H//
H
H//
H
SF
SF
NSF
NSF
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Summary
The 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 S ,
Polarized neutrons can distinguish magnetic
and
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6/20/00 nuclear scattering and specific spin