B. Farago Polarized Neutrons
BNC Neutron school
Outline
1
Outline
•Basics •Magnetic scattering
•Spin manipulation •Instruments
This presentation might differ from the one which really will be presented
B. Farago Polarized Neutrons
BNC Neutron school 3
Incoherent scattering
Reminder: The scattering cross section:
dsdW
= Âi, j
bib j exp
⇣i~q
⇣~Ri�~Rj
⌘⌘
Suppose that at position Ri we can have different scattering length with a certain probability distribution
hbib ji = hbiihb ji+di j
⇣⌦b2
i↵�hbii2
⌘
dsdW
= Âi, jhbiihb jiexp
⇣i~q
⇣~Ri�~Rj
⌘⌘+Â
i
⇣⌦b2
i↵�hbii2
⌘
I
coherent
(~q)+ I
incoherent
(~q)
B. Farago Polarized Neutrons
BNC Neutron school
Isotop and spin incoherence
4
bi can have a distribution because:
- different isotopes exist
- the nucleus has a spin I => with the neutron 1/2 spin it forms two possible states
the nuclear spin is usually randomly oriented EXCEPT very
low T or very high B
I+1/2 => 2(I+1/2)+1 states with scattering length b+ I-1/2 => 2(I-1/2)+1 states with scattering length b-
Spin Incoherence
Isotope Incoherence
hbii = b+I +1
2I +1+b�
I2I +1
Coherent
Isotope Incoherence
⇣⌦b2
i↵�hbii2
⌘= (b+�b�)2 I(I +1)
(2I +1)2
Incoherent
B. Farago Polarized Neutrons
BNC Neutron school
Polarized beam scattering
5
σx and σy changes the spin state of the neutron => spin flip scattering σz does not => non spin flip scattering
If the nuclear spin I is randomly oriented in space each one has 1/3 probability thus: Spin Incoherent scattering 2/3 spin flip 1/3 non spin flip P => -1/3P
Let’s define the polarization of the beam as: ~P = 2h~si = h~siIn terms of Pauli matrices
sx
=✓
0110
◆sy =
✓0�ii 0
◆sz =
✓1 00�1
◆
Scattering length on a nucleus with spin I:
b = A+12
BsI
A = b+I +1
2I +1+b�
I2I +1
B =2(b+�b�)
2I +1
B. Farago Polarized Neutrons
BNC Neutron school
Magnetic Interactions 1
7
The neutron has a 1/2 spin => magnetic moment µn= γn µN
µN nuclear Bohr magneton and γn = -1.913
With a magnetic field the interaction potential Lovesey 1986:
V (~R) =�~µn
~B = g
n
µ
N
h2µ
B
curl
⇣~s⇥~
R
|R3|
⌘� e
2m
e
c
⇣~p
e
~s⇥~R
|R3| +~s⇥~R
|R3| ~pe
⌘i
~s = eletron spin operator,~pe
= eletron momentum operator
~s = neutron spin operator
B. Farago Polarized Neutrons
BNC Neutron school
Magnetic Interactions 2
8
The matrix element (scattering probability) becomes:
or in terms of the magnetization and changing to integral to account for the spatial extent of the electrons
Of the sample spins (magnetization) ONLY THE COMPONENT PERPENDICULAR to q contributes !!This
is fundamentally different from the nuclear spin!
hk0|VM |ki = �r0sQ? with r0 =gne2
mec2
ˆQ? = Âi
exp(i~q~ri)✓
q⇥ (~s⇥ q)� ih |~q| (q⇥~pi)
◆with q =
~q|~q|
~Q =� 1
2µB
Zd~r exp
⇣i~k~r
⌘~M (~r) and ~Q? = ~Q� q
⌘
B. Farago Polarized Neutrons
BNC Neutron school
Interference term
9
The scattered intensity is the Fourier transform of the self correlation function of the scattering length density
Alternatively if the Fourier transform of the scattering length density is F(q)
S(q) = F(q)F*(q)
If F(q) = FN(q)+FM(q) in the most generic case there will be four terms: • nuclear
• magnetic
• nuclear magnetic interference
• chiral
B. Farago Polarized Neutrons
BNC Neutron school
D20
10
3000
2000
1000
0
coun
ts /
5 m
in
140120100806040202Theta/deg
369.5Å^3
369.0
368.5
368.0
367.5
367.0
366.5
366.0
365.5
unit
cel
l vol
ume
250K200150100500sample temperature
5
4
3
2
1
0
Bohr magnetons
Unpolarized neutrons - comparable intensity to
nuclear - identified by a priory
knowledge - temperature dependence
B. Farago Polarized Neutrons
BNC Neutron school
3D Polanal X
11
neutron kinneutron kout Q scattering vector
neutron beam polarization direction
50% non spin flip scattering
50% spin flip scattering
B. Farago Polarized Neutrons
BNC Neutron school
3D Polanal Y
12
neutron kinneutron kout Q scattering vector
neutron beam polarization direction
spin flip scattering
spin flip scattering
B. Farago Polarized Neutrons
BNC Neutron school
3D Polanal Z
13
neutron kinneutron kout Q scattering vector
neutron beam polarization direction
50% spin flip scattering
50% non spin flip scattering
B. Farago Polarized Neutrons
BNC Neutron school
3D pol anal simple
15
UPX = N +12
M +13
I
DOWNX =12
M +23
I
UPZ = N +12
M +13
I
DOWNZ =12
M +23
I
UPY = N +13
I
DOWNY = M +23
I
N the nuclear scattering M the magnetic scattering I the incoherent scattering
B. Farago Polarized Neutrons
BNC Neutron school
Inelastc corrections
16
neutron kin
neutron beam polarization direction
50% non spin flip scattering
50% spin flip scattering
Q scattering vectorneutron kout
When the scattering is non negligibly quasielastic
with distribution
not quite !
B. Farago Polarized Neutrons
BNC Neutron school
3D pol anal complicated
17
UPX = N1+ f g
2+
12
M� g f2
M sin2 e+3�g f
6I
DOWNX = N1�g
2+
12
M +g2
M sin2 e+3+g
6I
UPZ = N1+ f g
2+
12
M +3�g f
6I
DOWNZ = N1�g
2+
12
M +3+g
6I
UPY = N1+ f g
2+
1�g f2
M +g f2
M sin2 e+3�g f
6I
DOWNY = N1�g
2+
1+g2
M� g2
M sin2 e+3+g
6I
g is the polarizing efficiency f the flipper efficiency N the nuclear scattering M the magnetic scattering I the incoherent scattering ε is the angle of Q to Qelastic
B. Farago Polarized Neutrons
BNC Neutron school
Larmor precession
19
|ci= a |+i+b |�i where a and b can be complex and p
a2 +b2 = 1
and conveniently |ci= e�ij/
2
cos
q2
|+i+ eij/
2
sin
q2
|�i
Time evolution of the neutron spin in magnetic field
∂s∂t
=1h[s,Hs] =�
g2
hs,
⇣s~H
⌘i= ... =�g
⇣~H⇥ s
⌘
The Polarization of the beam P=<σ> behaves as a “classical” Larmor precession
γL =2957 Hz/Gauss DfDx
⇥deg
�cm
⇤= 2.65l
⇥A⇤
H [Gauss]
Quantum mechanical description of the neutron spin state:
this leads to h ˆsx
i = sinqcosj, h ˆsy
i = sinqsinj, h ˆsx
i = cosq
B. Farago Polarized Neutrons
BNC Neutron school
Adiabaticity
20
What happens if the direction of the magnetic field changes in space?
The moving neutron will see a B field which changes its direction in time
Two limiting cases
if ωB << ωL it will follow adiabatically
if ωB >> ωL it will start to precess around the new field direction
neutron trajectory
magnetic field directionadiabatic example:
B. Farago Polarized Neutrons
BNC Neutron school
Mezei flipper
21
Non adiabatic example : Mezei flipper
B. Farago Polarized Neutrons
BNC Neutron school
RF flipper
23
RF flipper
Brf oscillating ωRF radiofrequency field
P0 beam polarizationB0 static field
Makes up a π flipper • if B0γL = ωRF
• and BRF is just enough for a π turn during the flight time
B. Farago Polarized Neutrons
BNC Neutron school
Haussler
24
Haussler Xtal Cu2MnAl (111):
• FM(q) = FN(q) for one of the Bragg reflections • easy to saturate • grow single crystal • controlled mosaicity • low λ/2 contamination (or filter)
B. Farago Polarized Neutrons
BNC Neutron school
Multilayer
25
} d
λ
θ®
® ΔQ
0
0.2
0.4
0.6
0.8
1
1.2
0 0.02 0.04 0.06 0.08 0.1
Ref
lect
ivity
Q (Å-1 )
Δλλ
= 2d2 (f
1- f
2)
π
4N2d4(f1- f
2)2
π2n4R =
θ ~ λ2d
B. Farago Polarized Neutrons
BNC Neutron school
He3
27
He3 absorbs only neutrons with spin antiparallel to the nuclear spin
B. Farago Polarized Neutrons
BNC Neutron school
NSE basics
312
Selector
Polarizer
pi/2 flipper
Precession(1)
pi flipper
Precession(2)
pi/2 flipper
Analyser
Detector
Beam polarisation
Magnetic field direction
Sample
ωL = γL ·B
�
B1dℓ =π / 2
π
∫ B2dℓπ
π / 2
∫
Echo condition: The measured quantity is: S(q,t)/S(q,0) where
�
t ∝λ3 Bdℓ∫
= 0ϕtot = γB1l1v1 �
γB2l2v2For elastic scattering:
ϕtot = hγBlmv3ω+o
✓⇣ω
1/2mv2⌘2◆For omega energy
exchange:
S(q,ω)The probability of omega energy exchange:
hcosϕi =Rcos(hγBl
mv3ω)S(q,ω)dω
RS(q,ω)dω = S(q, t)The final polarization:
B. Farago Polarized Neutrons
BNC Neutron school
resolution correction
32
1.0
0.8
0.6
0.4
0.2
0.0
Echo
2520151050time [nsec]
sample (resolution corrected)
1.0
0.8
0.6
0.4
0.2
0.0
Echo
2520151050time [nsec]
sample resolution
As mesures the fourier transforme
S(q,t) the instrumental resolution is a
simple division instead of deconvolution
B. Farago Polarized Neutrons
BNC Neutron school
When NOT to use NSE
33
Say you have a weak but well defined excitation
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
Intensity
-100 -50 0 50 100omega
S (q,w) = 0.95 · d(w)+0.025 · (d(w�w0)+d(w+w0))
1.0
0.8
0.6
0.4
0.2
0.0
echo
100806040200time
S (q, t) = 0.95+0.05 · cos(w0
)
Bad signal to noise, Δλ/λ => no good for Spin Echo
B. Farago Polarized Neutrons
BNC Neutron school
Reptation
34
1.0
0.8
0.6
0.4
0.2
0.0
S(q,t)
0.120.100.080.060.040.020.00
q2sqrt(t)
q values
0.49 nm-1
0.78 nm-1
0.97 nm-1
1.16 nm-1
1.36 nm-1
1.0
0.8
0.6
0.4
0.2
0.0
S(q,t)
3530252015105
time [nsec]
0.49 nm-1
0.78 nm-1
0.97 nm-1
1.16 nm-1
1.36 nm-1
10% marked polymer chain(H) in deuterated matrix of the same polymer melt
at short time => Rouse dynamics 1/tau ~ q4 at longer times starts to feel the “tube” formed by the other chains (deGennes)
1.0
0.8
0.6
0.4
0.2
0.0
S(q,t)
160140120100806040200time [nsec]
1.0
0.8
0.6
0.4
0.2
0.0
S(q,t)
0.250.200.150.100.05
q^2sqrt(time)
Q values
0.50 nm-1
0.77 nm-1
0.96 nm-1
1.15 nm-1
1.45 nm-1
D. Richter, B. Ewen, B. Farago, et al., Physical Review Letters 62, 2140 (1989).
P. Schleger, B. Farago, C. Lartigue, et al., Physical Review Letters 81, 124 (1998).
B. Farago Polarized Neutrons
BNC Neutron school 35
Spin Ice
‘spin ice’ materials Ho2Ti2O2 , Dy2Ti2O7 and Ho2Sn2O7 the spin is equivalent to the H displacement vector in water ice
Letter to the Editor L11
Figure 1. NSE results for Ho2Ti2O7: (a) integrated over all Q and fitted to an exponential function;(b) as a function of Q, indicating negligible Q-dependence.
Grenoble, using a neutron wavelength of λ = 5.5 Å. The spectrometer was calibrated witha reference sample that does not show dynamics on the timescale of this experiment. At alltemperatures between 0.05 and 200 K the measured relaxation can be fitted with excellentprecision to a simple exponential function:
F(Q, t) = (1 − B) exp{−"(T )t}, (1)
where B = 0.09 ± 0.01. The fact that B is finite, that is F(Q, t) = 1 at ∼10−12 s, provesthe existence of relaxation processes at short timescales beyond the resolution of the NSEtechnique. We suggest that these very fast processes might be associated with small incoherentoscillations of the spins about their ⟨111⟩ easy axes. Such processes cannot relax the moreimportant magnetic fluctuations associated with spin reversals, and hence are unable to providecomplete relaxation of the system. We do not comment on them further in this letter.
The frequency "(T ) can be fitted to an Arrhenius expression "(T ) = 2"h exp(−Ea/kT )
with attempt frequency "h = 1.1 ± 0.2 × 1011 Hz (∼5 K) and activation energy Eh =293 ± 12 K. The relaxation was found to be Q-independent (figure 1(b)), an important clue toits origin (see below). For the rest of the letter we shall refer to this exponential relaxation asthe ‘high’ process, referring to its typical temperature range, with subscript ‘h’.
Single exponential thermally activated
Letter to the Editor L11
Figure 1. NSE results for Ho2Ti2O7: (a) integrated over all Q and fitted to an exponential function;(b) as a function of Q, indicating negligible Q-dependence.
Grenoble, using a neutron wavelength of λ = 5.5 Å. The spectrometer was calibrated witha reference sample that does not show dynamics on the timescale of this experiment. At alltemperatures between 0.05 and 200 K the measured relaxation can be fitted with excellentprecision to a simple exponential function:
F(Q, t) = (1 − B) exp{−"(T )t}, (1)
where B = 0.09 ± 0.01. The fact that B is finite, that is F(Q, t) = 1 at ∼10−12 s, provesthe existence of relaxation processes at short timescales beyond the resolution of the NSEtechnique. We suggest that these very fast processes might be associated with small incoherentoscillations of the spins about their ⟨111⟩ easy axes. Such processes cannot relax the moreimportant magnetic fluctuations associated with spin reversals, and hence are unable to providecomplete relaxation of the system. We do not comment on them further in this letter.
The frequency "(T ) can be fitted to an Arrhenius expression "(T ) = 2"h exp(−Ea/kT )
with attempt frequency "h = 1.1 ± 0.2 × 1011 Hz (∼5 K) and activation energy Eh =293 ± 12 K. The relaxation was found to be Q-independent (figure 1(b)), an important clue toits origin (see below). For the rest of the letter we shall refer to this exponential relaxation asthe ‘high’ process, referring to its typical temperature range, with subscript ‘h’.
Q independent relaxation
G. Ehlers , Cornelius ,A L, Orendac ,M, Kajnakova ,M, Fennell ,T, Bramwell ,S T, Gardner , Journal of Physics Condensed Matter 15, L9 (2003).
Paramagnetic echo => Only magnetic scattering gives echo !
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