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
CRNL 6/20/00
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


CRNL 6/20/00

 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
CRNL 6/20/00

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
CRNL 6/20/00
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
CRNL 6/20/00
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
nE  


exp  E
 1
k
T
B


CRNL 6/20/00
Spin waves in an antiferromagnet
S


 ,   

S J 1
1
z
d e


i  d




         n      1          n    
2
 
Dispersion relation

    2 S
CRNL 6/20/00

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
CRNL 6/20/00


to
Polarized magnetic neutron scattering
Specify the incident and final neutron spin state


   S  l  0        S  l  t   
 
 
 
 
CRNL 6/20/00
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
CRNL 6/20/00
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
CRNL
6/20/00 nuclear scattering and specific spin