Upload
ngothien
View
227
Download
2
Embed Size (px)
Citation preview
Neutron Spin and Magnetic Moment
• Neutron is a fermion with spin ½ & the usual spin operators
• The Pauli spin operators are given by
• Because of its spin, the neutron also has a magnetic moment
• Where γn = -1.913 and μN is the nuclear magneton
• μn is 3 orders of magnitude smaller than the Bohr magneton and antiparallel to the neutron spin
1001
2 ,
00
2 ,
0110
2 ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡ −=⎥
⎦
⎤⎢⎣
⎡=
hhhzyx s
ii
ss
1-17 J.T 1066.92 −−== xsNnnr
h
rμγμ
srh
r 2=σ
The Neutron Spin Wavefunction & Polarization
• The neutron spin wavefunction is a superposition of “up” and “down”states
• The polarization components of an individual neutron are the expectation values of the appropriate Pauli matrix i.e.
• The polarization P of a neutron beam is the mean value of p in the beam
• Note that:
↓+↑= baχ
22
)*Im(2
)*Re(2
bap
bap
bap
zz
yy
xx
−==
==
==
χσχ
χσχ
χσχ
↓↑
↓↑
+−
=nnnnPz
Behavior of the Neutron’s Spin in a Magnetic Field
• The time evolution of any two-state quantum system can be represented by the movement of a classical vector
where γL is the gyromagnetic ratio of the neutron
• The solutions are
BPdtPd
L
rrr
∧−= γ
1-1-8L T.rad.s 1.832x10 where
)0()()0()sin()0()sin()()0()sin()0()cos()(
−==
=
+=
−=
B
PtPPtPttPPtPttP
L
zz
xLxLy
yLxLx
γω
ωω
ωω
Guiding the Neutron Polarization
• If the direction of a magnetic field varies sufficiently slowly in space, the component of neutron polarization parallel to the applied field is preserved. This is adiabatic polarization rotation.
Drawing from Bob Cywinski
How does a Neutron Spin Behave when the Magnetic Field Changes Direction?
• Distinguish two cases: adiabatic and sudden• Adiabatic – tan(δ) << 1 – large B or small ω – spin and field
remain co-linear – this limit used to “guide” a neutron spin• Sudden – tan(δ) >> 1 – large ω – spin precesses around new
field direction – this limit is used to design spin-turn devices
When H rotates withfrequency ω, H0->H1->H2…, and the spin trajectory is described by a cone rolling on the plane in which H moves
Adiabatic rotation of neutron spin
0
50
100
150
200
250
0 5 10 15 20 25 30 35 40
Hx
Hz
Hx a
nd H
z [Oe]
distance [cm]
φ
0 cm 30 cm
0
20
40
60
80
100
0 5 10 15 20 25 30 35 40
[°]
distance [cm]
φ
x
z
0
2 105
4 105
6 105
8 105
1 106
1.2 106
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
|ωL|/4
|dφ/dt|
0 5 10 15 20 25 30 35
frequ
ency
[rad
/s]
time [ms]
distance [cm]
Note, the relation between distance and time is valid for a neutron with λ = 4 Å.
L
L
dtd
B
ϖφ
γϖ
<<
×−== s-rad/G10833.1 4r
Condition to maintain polarization of the neutron beam.
M. R. Fitzsimmons
Non-Adiabatic Transitions
• If the guide field direction is suddenly changed (i.e. the adiabaticityparameter tan δö¶, the neutron polarization vector will precess about the new field direction.
• If the field is reversed, the neutron polarization is flipped with respect to the field
Graphic courtesy of Bob Cywinski
Spin Flippers based on Larmor Precession
Sini
B
Send
Sini
BSend
B BBπ/2
BπB B
Direct current spin flippers combine adiabatic and non-adiabatic spin rotation.
Larmor Precession allows the Neutron Spin to be Manipulated using π or π/2 Spin-Turn Coils: Both are Needed for NSE
• The total precession angle of the spin, φ, depends on the time the neutron spends in the B field
v/BdtL γωφ ==
Neutron velocity, v
d
B
][].[].[.135.65
1 turnsofNumber AngstromscmdGaussB λ=
Using Soft Magnetic Films to Rotate Neutron Spins
• A thin film of soft magnetic material is placed in a polarized neutron beam at an angle θ to the beam
• As θ is changed the polarization of the neutron beam changes
220 240 260 280 300 320 340 360-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Data: Data54_polM odel: p i_on_2_fitW eighting: y No w eighting Chi^2/DoF = 0.00092R^2 = 0 .9965 K 2.01933 ±0.00239x0 271.17802 ±0
Pol
ariz
atio
n
R o ta tion S tage A ng le (degrees)
po l10 m icron fo il tuning curve
-1
-0.5
0
0.5
1-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
B
H,σ Bθ neutron
H
-1
-0.5
0
0.5
1-1 -0.5 0 0.5 1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Thin Magnetic Films used as π/2 and π Rotators
M
Hg
χneutron
MHg
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Hg
M
χneutron
MHg
M
χγφsinvMd
=
dv
Rekvedlt
“Wavelength Independent” Flippers can be Made with Permanently Magnetized Films (or Coils)
neutron
B BB1
B2
neutron
B BB1
B2
B
B2
B1
B
B2
B1
1 2 3 4 5 6
-0.2
-0.1
0.1
0.2
C x neutron wavelengthPola
rizat
ion
1 2 3 4 5 6
-0.2
-0.1
0.1
0.2
C x neutron wavelengthPola
rizat
ion
1 2 3 4 5 6
-0.04
-0.02
0.02
0.04
C x neutron wavelength
Neu
tron
Pola
rizat
ion
1 2 3 4 5 6
-0.04
-0.02
0.02
0.04
C x neutron wavelength
Neu
tron
Pola
rizat
ion
2 films
3 films
White Beam Spin Flipper –A rf Gradient-Field Spin Flipper
Hs
H1
Heff
Hs
H1
Hs
H1
HRF=H1cos(ωt) Hs(z) = H0+∇H•z − ω/γ
Viewgraph from M. R. Fitzsimmons
Realization – The ASTERIX rf Flipper
40
60
80
100
120
140
160
-10 0 10 20 30 40
Sta
tic fi
eld,
Hs(z
) [O
e]
Distance along neutron path [cm]
RF field region15 cm
∇H = 3 Oe cm-1
50 feet70 W @ 296kHz
Viewgraph from M. R. Fitzsimmons
Performance of the Asterix rf Flipper
10-7
10-6
10-5
10-4
10-3
10-2
10-1
4 6 8 10 12 14
Polarized beam spectra in 2001
(++)(--)(-+)(+-)
Inte
nsity
[a.u
.]
Wavelength [Å]
proton flash
• (++) = (--) ⇒flippers 99+% efficient for λ > 4Å.
• (++)/(SF) ⇒polarization ~94% for λ > 4Å.
Viewgraph from M. R. Fitzsimmons
Production of Polarized Beams
• Polarizing filters– Usually 3He these days, although pumped protons and rare earths have been
tried– Good for polarizing large, divergent neutron beams– Depend on good polarization of filter material
• Polarizing monochromators– Mainly Heusler alloy (Cu2MnAl) these days– Good when beam monochromatization is also required (e.g. TAS)
• Supermirrors– Very efficient, broad-band polarizers– “cavities” or “benders” are excellent for preparing polarized beams– Disadvantage is that wide angular beam divergence requires devices with
non-uniform transmission
Polarizing Filters
• The polarization and transmission are given by:
2/)( and 2/)( where)cosh( );tanh(
0 −+−+
−
+=−=
=−=
σσσσσσ
σσ σ
p
pNt
p NteTNtP o
Production of polarized 3He at ILL
Making a Polarizing Neutron Monochromator• In a Bravais crystal, the magnetization may be written as a Fourier series:
• This may be combined with the matrix element for nuclear scattering to give an effective scattering length given by:
• Consider a situation in which the magnetization at each lattice site is along z, the quantization direction of the neutron spin, and ┴ to Q
• From the results on the previous slide, the effective scattering lengths are:
and there is no spin flip scattering
• If we can find a FM crystal for which the nuclear and magnetic scattering lengths are equal we can use it to monochromate and polarize neutrons
Factor Form Magnetic theis ).exp()()( where).exp()()( ∑ ∫== ⊥⊥j
jj rdrQirmQFMRQiQFQM
rrrrrrrrrrr
j
Bjj MQF
rbb ⊥+=
rrr.)(
2ˆ 0 σ
μγ
∑j
RQij
jebrr
.
)(2
ˆ :vv For ; )(2
ˆ :uuFor 00 jz
Bjj
jz
Bjj MQFrbbMQFrbb ⊥⊥ −=→+=→
rr
μγ
μγ
• scattering potential is +ve if neutron spinis parallel to i.e. its magnetic moment is anti-parallel to
• thus, if polarized reflection occurs, the neutron spin must be parallel to
Schematic of a Polarizing Monochromator
⊥Mr
Q
u
vUnpolarizedIncidentNeutrons
PolarizedScatteredNeutrons
BSampleMagnetization
x y
zQ
u
⊥Mr⊥M
r
Vertically focussing Heusler alloy monochromator for polarized neutronsA. Freund, R. Pynn, W. G. Stirling and C. M. E. Zeyen; Physica 120B, 86 (1983)
Viewgraph from Bob Cywinski
Neutron Reflection from Magnetic Mirrors
• Since, for a saturated ferromagnet, we can write the magnetic scattering in terms of an effective scattering length, we can deduce a combined nuclear and magnetic SLD for such a material
• We can make a mirror that polarizes a neutron beam by choosing the nuclear and magnetic SLDs to be equal.
• Although the schematic of this process looks much like the polarizing monochromator, the reflection angles are much smaller.
– We also have more control over the mirror than the monochromator because only the average composition matters – we don’t need to rely on nature to hand us the answer
– The first polarizing mirror was demonstrated by in 1951 by Hughes and Burgy
)(1ˆ1 jmagnetic
j
jcoherent
jj bb
volumeb
volume±== ∑∑ρ
Polarizing supermirrors
10-5
10-4
10-3
10-2
10-1
100
0 0.02 0.04 0.06 0.08 0.1
R++ Fe/SiR-- Fe/SiR++ 3θ
c SM
R-- 3θc SM
0 1 2 3 4
Pol
ariz
ed n
eutro
n re
flect
ivity
Q [Å-1]⊥
m
0
0.2
0.4
0.6
0.8
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0 0.5 1 1.5 2 2.5 3 3.5
Pol
ariz
atio
n (tr
ansm
issi
on)
Q [Å-1]⊥
m
Qmin
= 0.01 Å-1
Qmax
= 0.065 Å-1
Si
100 nm 0.1 nm4
10)(
n
dnd =
F. Mezei and P.A. Dagleish, Comm. on Phys., 2, 41 (1977).
down
up
Partial Cross Sections
IIcohSImag
NSF
z
SImag
SF
z
IIcohSImag
NSF
y
SImag
SF
y
IIcohSImag
NSF
x
SImag
SF
x
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
dd
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
=⎟⎠⎞
⎜⎝⎛Ω
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
=⎟⎠⎞
⎜⎝⎛Ω
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+=⎟⎠⎞
⎜⎝⎛Ω
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+=⎟⎠⎞
⎜⎝⎛Ω
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+=⎟⎠⎞
⎜⎝⎛Ω
⎟⎠⎞
⎜⎝⎛Ω
+⎟⎠⎞
⎜⎝⎛Ω
+=⎟⎠⎞
⎜⎝⎛Ω
σσσσσ
σσσ
σσσσασ
σσασ
σσσσασ
σσασ
31
21
32
21
31)1(cos
21
32)1(sin
21
31)1(sin
21
32)1(cos
21
2
2
2
2
Blume, Phys. Rev. 130, 1670 (1963); Moon, Riste and Koehler Phys. Rev. 181, 920 (1969
Polarized Neutron Scattering – Various Cases
PG(004) – coherent scatteringIs always NSF
Isotopic incoherent scattering from Niis NSF. Flipper off is NSF.Flipper on is SF
Nuclear spin incoherent scatteringfrom Vanadium is 2/3 SF. Flipperoff is NSF cross section
Paramagnetic scattering from Mn2F. WithP//Q, all mag scattering is SF. With P^Q½ mag scattering is SF and ½ is NSF
Science with Polarized Neutrons
• Most measurements are made with 1-d polarization analysis – Magnetic field is applied to sample and neutron spin component is analyzed in
the field direction– Diffraction
• “Flipping ratio” measurements of form factors, electron spin density distributions usually with single crystals
• 3-directional polarization analysis of diffuse scattering• SANS
– Polarized Neutron Reflectometry (PNR)• Depth dependent vector magnetometry in thin films
– Inelastic scattering• Magnetic excitations and fluctuations
• Generalized polarization analysis is also possible– No magnetic field on sample; incident neutron spin in controlled direction;
analyze neutron spin along any direction• Vector distribution of magnetization in single crystals