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2016 AIMag School, Milano 18-22 April 2016 [email protected]
Joint Research Centre The European Commission’s in-house science service
Neutrons: a quantum probe for magnetism
Roberto Caciuffo
EC JRC, Institute for Transuranium Elements,
Karlsruhe (Germany)
2
• Electrically neutral
• Microscopically magnetic
• Sensitive to light atoms
Neutrons – a tailor-made probe
H Li C O S Mn Zr Cs
X-rays
neutrons
mass mN = 1.67510-27 kg
charge qN = 0; spin = 1/2
magnetic dipole moment n = -1.913N
k = 2 = mNv/
E(meV) = 2.072 k2 (Å-1) = 81.797/2 (Å)
(Å) = 3956/v(ms-1)
3
Neutrons interact with nuclei via the short-range strong nuclear force, and with electrons via a dipole-dipole interaction between magnetic moments.
The interaction of neutrons with matter is weak: • Theoretically easy to model • Large penetration depth
Neutrons penetrate matter much more deeply than X-rays or charged particles,
as the size of neutron scattering centres
is typically 10-5 times smaller than the distance between those centres.
Neutrons – a gentle probe
4
100
104
108
1012
1016
1020
1024
1028
1032
1036
Pe
ak B
rilli
ance
(pa
rtic
les s
-1 m
rad
-2 m
m-2
0.1
% B
W)
Neutrons – a scarce and expensive probe
5
Weak interactions combined with low fluxes make neutron scattering a signal-limited technique. Its use is justified by the uniqueness of the information it provides.
a probe that lets you see different things...
making obvious the unexpected...
Neutrons – a unique probe
Courtesy P. G. Radaelli, Oxford University
Neutrons with energy comparable with that of elementary excitations in condensed matter have a wavelength matching interatomic distances.
E(meV) T(K) (Å)
Cold 0.1-10 1-120 0.4-3
Thermal 5-100 60-103 0.1-0.4
Hot 102-103 103-104 0.04-0.1
• Appropriate length and energy scale
New chemicals
Novel materials
Fundamental understanding of Nature
This makes energy-, momentum-, and spin-resolved neutron scattering a powerful tool for probing structure and dynamics of materials at different level of complexity, energy and length scales.
Neutrons – probing structure and dynamics of condensed matter
7
Hard
Soft multidisciplinary condensed matter science
1960
1970
1990
1980
2000
1960
1970
1990
1980
No One Experiment
Courtesy Andrew Taylor, STFC
8
Neutron sources: nuclear fission and spallation
Fission
Spallation
9
Pulsed versus Steady State Sources
Spallation 30 MeV/n
Fission 190 MeV/n
200 kW ISIS
58 MW ILL
Inte
ns
ity
time
Pulsed Sources
•More neutrons at high energy
•Neutrons produced in burst
•Pulsed operation
•Sharp pulses give high resolution
•Resolution function asymmetric
•Must use time of flight methods
•Horizons still to be explored
•Seen as environment friendly
Nuclear Reactors
•More neutrons at low energy
•Easier to shield
•Continuous operation
•Resolution can be adapted
•Resolution function Gaussian
•More flexibility
•Further development limited
•Seen as environmen unfriendly
10
Scattering experiments
Momentum transfer
Energy transfer
Definition of cross-section:
[s] = barn (1 barn = 10-28 m2)
s is the effective area presented by the target particle
to the passing neutron, which is scattered if it hits this area. s is then related to the scattering probability.
2
S
D
∆𝐼 = 𝑁Φ 𝐸𝑖 ∆𝐸𝑖𝑑2𝜎
𝑑Ω𝑑𝜔 ∆ΩΔ𝐸𝑓
𝑸 = (𝒌𝑖 − 𝒌𝑓) 𝒌𝑓 ± ∆𝒌𝑓
𝐸𝑓 ± ∆𝐸𝑓
𝑷𝑓
𝒌𝑖 ± ∆𝒌𝑖
𝐸𝑖 ± ∆𝐸𝑖
𝑷𝑓
∆𝐼 = neutrons recorded per unit time
𝑁 = number of scattering centres
= incident flux (n/time/area)
DW= accepted solid angle
DEf = accepted energy window
𝜔 = 𝐸𝑖 − 𝐸𝑓 = 2
2𝑚 (𝑘𝑖
2 − 𝑘𝑓2)
𝑁Φ 𝐸𝑖 ∆𝐸𝑖
11
Elastic scattering Inelastic scattering
ki kf
-kf
Q
Scattering triangle
Structure of
materials
Dynamics of
materials
𝐸𝑖 = 𝐸𝑓 𝑘𝑖 = 𝑘𝑓 = 2𝜋/𝜆
𝑄 = 2𝑘𝑖 sin𝜃 𝑄2 = 𝑘𝑖2 + 𝑘𝑓
2 − 2𝑘𝑖𝑘𝑓 𝑐𝑜𝑠𝜙
𝐸𝑖 ≠ 𝐸𝑓 𝑘𝑖 ≠ 𝑘𝑓
ki
kf
Q
12
𝑸 = 𝒌𝑖 − 𝒌𝑓
𝑄𝑥 = 𝑘𝑖 − 𝑘𝑓𝑠𝑖𝑛𝜗 𝑐𝑜𝑠𝜑
𝑄𝑦 = − 𝑘𝑖𝑠𝑖𝑛𝜗 𝑠𝑖𝑛𝜑
𝑄𝑧 = − 𝑘𝑓 𝑐𝑜𝑠𝜑
x
y
z
ki
kf
Q
Ei = 300 meV Ei = 600 meV
=2 deg
=110
D=4
Kinematical constraints in measuring scattering processes
Each detector traces a parabolic trajectory trough (Q,w) space
13
Selecting neutrons energy
Crystal monochromators
2dsin = n
∆𝜆
𝜆
2
= 𝛼𝑡𝑜𝑡 𝑐𝑜𝑡𝜃 2 +
∆𝑑
𝑑
2
Velocity selectors and choppers
G=2/d
G/2
14
Counting neutrons
Detectors tank of LET@ISIS
3He linear PSDs of MERLIN @ISIS (3 m long, 69632 pixels)
nuclear reactions used to convert neutrons into charged particles: • gas proportional counters • ionization chambers • scintillation detectors • semiconductor detectors
Scintillator detector Module of GEM @ISIS
prototype of a Gas-Electron Multiplier detector using 10B4C-coated alumina lamellae for LOKI @ESS (G. Gorini)
15
Manipulating the neutron spin
• The neutron wavefunction is a two-component spinor
• The neutron-nucleus interaction VN is spin-dependent
• The magnetic interaction between neutrons and atomic magnetic moment is spin dependent
Polarization can be achieved by spin-dependent absorption from a nuclear spin polarized target (3He spin filter), by scattering from a ferromagnetic crystal or by reflection by a birefringent medium
Ψ = Ψ + Ψ = 𝑐𝑒𝑖𝑘 ∙𝑟
10 + 𝑐𝑒
𝑖𝑘 ∙𝑟 01
𝑘 , = 𝑛, 𝑘 0
𝑛,2 = 1 −
2𝑚
( k0)2 𝑉𝑁 ± 𝜇𝐵 = 1 −
4𝜋
k02 𝜚𝑁 ± 𝜚𝐵
𝑠 =
2𝜎 =
2
0 11 0
, 0 −𝑖𝑖 0
, 1 00 −1
𝜇 = −1.913e
2𝑚 𝜎 = 𝛾𝐿𝑠
𝛾𝐿 = −1.832 × 108 𝑟𝑎𝑑 𝑠−1𝑇−1
16
Manipulating the neutron spin
Beam polarization vector P: statistical average of an ensemble of spins 𝑃 = < 𝜎 > = 𝑇𝑟(𝜚 𝜎 )
𝜚 = 1
2
1 00 1
+ 𝜎 ∙ 𝑃 = 1
2
1 + 𝑃𝑧 𝑃𝑥 − 𝑖𝑃𝑦𝑃𝑥 + 𝑖𝑃𝑦 1 − 𝑃𝑧
Component of P along an arbitrary direction n:
𝑃𝑛 = 𝑇𝑟[𝜚 (𝑛𝑥𝜎𝑥 + 𝑛𝑦𝜎𝑦 + 𝑛𝑧𝜎𝑧)]
Experimentally one measures the component of P along an applied magnetic field:
𝑃 = 𝐼 − 𝐼𝐼 + 𝐼
17
Manipulating the neutron spin
Heussler Cu2MnAl monochromators. Spin-dependent Bragg diffraction (95% polarization)
Polarized 3He filters. Spin-dependent absorption (~99-100% polarization)
Fe/Si supermirrors. Spin-dependent reflection(96-99% polarization)
© ILL
18
Manipulating the neutron spin
In a magnetic field B, the neutron feels a potential
𝜔𝐿 𝑟𝑎𝑑 𝑠−1 = 18325 𝐵(𝐺)
𝑉𝑚 = −𝜇 ∙ 𝐵 = −𝛾𝐿𝟐
𝜎𝑥𝐵𝑥 + 𝜎𝑦𝐵𝑦 + 𝜎𝑧𝐵𝑧 = −𝛾𝐿𝟐
𝐵𝑧 𝐵𝑥 − 𝑖𝐵𝑦
𝐵𝑥 + 𝑖𝐵𝑦 −𝐵𝑧
The polarization undergoes a Larmor precession with an angular frequency
© ILL
19
If the direction of B rotates with a frequency wB << wL , P is
transported adiabatically (𝜇 ∙ 𝐵 conserved)
Manipulating the neutron spin
Otherwise 𝜇 ∙ 𝐵 is not conserved
For a /2 rotation over 5 cm = 1 Å B = 204 G = 10 Å B = 20 G
20
Courtesy of E. Lelièvre-Berna, ILL
Spherical polarimetry with CRYOPAD
21
Instruments without energy analysis (diffractometers)
WISH@ISIS
D9@ILL
D3@ILL
Atomic and magnetic structures
Crystals
Gasses
Liquids
Amorphous solids
VIVALDI@ILL
GEM @ ISIS
D2B @ ILL
22
D22@ILL
SANS2D@ISIS
Small Angle Neutron Scattering
Measuring structures on the scale of 1 to 100 nm
Flux lattices in superconductors
Magnetic correlations in
nanoparticles
23
Neutron reflectometry
• Magnetic thin films and nanostructures (m-nm)
• Spintronic
• Magnetization at the interface
• Spin injection
• Kinetics (s-ms)
• Excitations
CRISP@
ISIS
FIGARO@ILL
Characterization of interfacial phenomena on the microscopic length-
scale
Fe
Si
Vn
Vn + Vm
Vn - Vm
24
Three-axis spectrometers
Collective motion of
atoms and magnetic
moments in single
crystalline samples.
IN20@ILL
𝑄2 = 𝑘𝑖2 + 𝑘𝑓
2 − 2𝑘𝑖𝑘𝑓 𝑐𝑜𝑠𝜙 kf ki
Q
G
q
kf
Constant-Q scan at fixed Ei Constant-E scan
𝑘𝑖 = 𝜋
𝑑𝑀𝑠𝑖𝑛𝜃𝑀
𝑘𝑓 = 𝜋
𝑑𝐴𝑠𝑖𝑛𝜃𝐴
𝜔 ≥2𝑄2
2𝑚 𝑠𝑖𝑛𝜙
25
Time-of-Flight Chopper Spectrometers
MERLIN@ISIS
LET@ISIS
IN5@ILL
26
Time-of-flight measurements
Direct geometry Indirect geometry
Time
source
monochromator
sample
detector
Time
source
E analyzer
sample
detector
source
chopper
detector L0
L1 L2
sample
Fixed initial energy All final energies -∞< w<Ei
source detector
L 0
L 1
L 2
2
analyzer
sample
All incident energies Fixed final energy -Ef < w<∞
ti t
𝐸𝑓 = 1
2 𝑚
𝐿22
(𝑡 − 𝑡𝑖)2
𝑘𝑓 = 𝜋
𝑑𝐴𝑠𝑖𝑛𝜃𝐴
𝑘𝑖 = 𝜋 𝑚 𝐿0
𝜋𝑡 − 𝑚 (𝐿1+𝐿2) 𝑑𝐴𝑠𝑖𝑛𝜃𝐴
t=0
E(meV) = 5.227106 t-2 (m2/s2)
3 m
MERLIN – Large detector Coverage
• steradians of solid angle
• 3m long position sensitive detectors
• position resolution along tube ~ 20mm
• Total of 69000 ‘pixel’ elements
• 2000 time channels/pixel (14x107 bins)
• detectors in evacuated 30 m3 tank
• virtually gap free coverage
±30o
+135o -45o 3o
28
(meV)
Inte
nsity (
arb
. units)
0 2 4 6 8 100
100
200
300
400
500
•Visualisation software
Combine ~200 datasets full map of S(Q,) Bespoke visualisation
40GB 109 pixels software (“HORACE”)
29
Spin-Echo Spectrometers
slow relaxation in magnetic materials elastic paramagnetic scattering phonon linewidths
IN11@ILL
Neutron energy encoded into
a Larmor precession angle
Energy range: 1.3x10-5 ... 0.01 meV
Interaction of a neutron with a nucleus at position Rj:
Neutron-nucleus scattering
The Fermi length b has different values for neutron and nucleus spins parallel or antiparallel
b(238U) = 0.8417 x 10-12 cm b(H) = -0.374 x 10-12 cm
Coherent scattering due to the “average” nucleus; intensity as the square of the sum of individual amplitudes: interference Incoherent scattering due to deviation from the average “nucleus”; intensity as the sum of individual intensities: no interference
𝑉𝑁 = 2𝜋2
𝑚 𝑏𝑗 𝛿(𝑟 − 𝑅 𝑗 )
𝑏𝑗 = 𝐼𝑗 + 1 𝑏𝑗
↑ + 𝐼𝐽𝑏𝑗↓
2𝐼𝑗 + 1+
𝑏𝑗↑ − 𝑏𝑗
↓
2𝐼𝑗 + 1 𝜎 ∙ 𝐼 𝑗
31
Neutron interaction with a magnetic field
𝑉𝑚 = −𝜇 ∙ 𝐵 = −𝛾𝐿𝟐
𝜎𝑥𝐵𝑥 + 𝜎𝑦𝐵𝑦 + 𝜎𝑧𝐵𝑧 = −𝛾𝐿𝟐
𝐵𝑧 𝐵𝑥 − 𝑖𝐵𝑦
𝐵𝑥 + 𝑖𝐵𝑦 −𝐵𝑧
The magnetic field that scatters the neutrons is due to currents and magnetic dipole moments of electrons.
𝐵 𝑒 = 2 𝜇𝐵 (𝑐𝑢𝑟𝑙𝑠 × 𝑟
𝑟3+
1
𝑝 × 𝑟
𝑟3)
The total field B(r,t) in a sample is the sum of the fields generated by all the electrons and depends on the wavefunction of the system.
e
n
R
r
ri
s
p
32
Scattering cross section
Born approximation and Fermi golden rule
𝑑𝜎
𝑑𝛺𝑑𝜔 𝑘𝑖𝜎𝑖→𝑘𝑓𝜎𝑓
= 𝑘𝑓
𝑘𝑖 𝑚
2𝜋ℏ
2
𝑝𝑓𝑝𝑖 Ψ𝑓 𝑉 Ψ𝑖 2
𝑖 ,𝑓
𝛿(ℏ𝜔 + 휀𝑓 − 휀𝑖)
Ψ ∝ exp(𝑖 𝑘 ∙ 𝑟 ) σ Υ𝑡𝑎𝑟𝑔𝑒𝑡
𝑘𝑓 𝑉 𝑘𝑓 = 𝑉( 𝑟 ) exp −𝑖 𝑘 𝑓 − 𝑘 𝑖 ∙ 𝑟 𝑑𝑟 = 𝑉( 𝑟 ) exp −𝑖 𝑄 ∙ 𝑟 𝑑𝑟
Born approximation and Fermi golden rule:
The matrix elements contains the physics of the neutron-sample system with wavefunction
The scattering amplitude is the Fourier transform of the interaction potential in the reciprocal space
33
For the magnetic interaction the scattering amplitude aM(Q) is:
Q
s
sQ
QsQ
Q
p
pQ
Magnetic scattering amplitude
−𝜇0𝑀 ⊥ 𝑄 = 𝐵 𝑟 exp 𝑖 𝑄 ∙ 𝑟 𝑑𝑟
Ψ𝑓 −𝜇 ∙ 𝐵 Ψ𝑖 = 𝜇0 𝑠𝑓 𝜇 𝑠𝑖 Υ𝑓 𝑀 ⊥ 𝑄 Υ𝑖
The total field B(r, t) in a sample of condensed matter is the sum of the fields generated by all the electrons. Defining
𝑎𝑀 𝑄 = 𝑘𝑓 𝑉𝑚 (𝑟 ) 𝑘𝑓 = 4𝜋
𝑄2exp 𝑖 𝑄 ∙ 𝑟 𝑄 × 𝑠 × 𝑄 +
4𝜋𝑖
𝑄2exp 𝑖 𝑄 ∙ 𝑟 𝑝 × 𝑄
𝑑𝜎
𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓𝜎𝑓
= 𝑘𝑓
𝑘𝑖 𝛾𝑟0
2𝜇𝐵
2
4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥ 𝑄 λ𝑖 ⋅ 𝑠𝑓 𝜎 ↑ 2
𝑖 ,𝑓
𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)
34
magnetic scattering amplitude
The magnetic scattering of neutrons depends only on the component
of the magnetisation perpendicular to the scattering vector Q.
Q
M(q)
M
MQ
The magnetic neutron scattering cross section measures correlations in
magnetization, that is how the magnetization on a given site influence the
magnetization of the surrounding.
𝑑𝜎
𝑑𝛺𝑑𝜔 ∝ 𝑀 ⊥
∗ 𝑄 𝑀 ⊥ 𝑄
35
𝑑𝜎
𝑑𝛺𝑑𝜔 ∝ 𝑀 ⊥
∗ 𝑄 𝑀 ⊥ 𝑄
Flux Line Lattice in superconductors
SANS
ki
kf
Q = Ki-Kf almost to M J. S. White et al., PRL 102, 097001 (2009)
B
Diffraction patterns from the FLL in YBa2Cu3O7, as a function of field applied perpendicular to the CuO2 planes. SC in the chains is suppressed by increasing field and the d-wave nodes move closer to 45° angles.
𝜇 = −𝜇𝐵 𝐿 + 2𝑆 = −𝑔𝐽𝜇𝐵 𝐽
𝑎𝑀 𝑄 = −𝑝𝑔𝐽𝑓 𝑄 𝐽 ⊥ ∙ 𝜎
Russell-Sanders coupling, ground multiplet J:
f(Q) is the magnetic form factor. It arises from the spatial distribution
of unpaired electrons around a magnetic atom. If r is the normalized
density of spin around the equilibrium position:
𝑓 𝑄 = 𝜌↑ 𝑟 − 𝜌↓ 𝑟 exp 𝑖 𝑄 ∙ 𝑟 𝑑𝑟 𝑎𝑡𝑜𝑚 𝑣𝑜𝑙𝑢𝑚𝑒
magnetic form factor
r
Pu3+ Pu3+
Fe3+
𝑓 𝑄 = 𝑗0(𝑄) +𝐿
𝐿 + 𝑠 𝑗2(𝑄)
Pu has 5 5f electrons:
S=–5/2; L=5; J=5/2
Strong cancellation of spin and
orbital components.
Almost equal contributions
from <jo> and <j2>
37
Elastic scattering from a crystal.
Constructive interference results in Bragg peaks
only if 𝑄 is equal to a reciprocal lattice vector 𝜏
𝑑𝜎
𝑑𝛺=
𝑑𝜎
𝑑𝛺𝑑𝜔 𝑑𝜔 = 𝑁
2𝜋 3
𝑉0 𝐹𝑁 𝑄
2
𝜏
𝛿(𝑄 − 𝜏 )
𝜏 = ℎ𝑎 ∗ + 𝑘𝑏 ∗ + ℓ𝑐 ∗
FN are the coefficients of the Fourier transform of the particle density.
A Fourier synthesis is possible if FN are known in amplitude and phase. However, measurements give
𝐹𝑁 𝑄2 not amplitude and phase.
𝐹𝑁 𝑄 = 𝑏𝑑𝑑
𝑒−𝐵𝑄2 𝑒2𝜋𝑖 (ℎ𝑥𝑑+𝑘𝑦𝑑+ℓ𝑧𝑑 )
ki kf
Q
38
The phase problem
The dog b
and its F-transform
A duck
and its F-transform
-F[b] m-F[b]
-F[duck] m-F[duck]
anti F-transform
anti F-transform
-F[b] m-F[b]
-F[duck] m-F[duck]
39
The phase problem
-F[b]
m-F[b] -F[duck]
m-F[duck]
anti F-transform
anti F-transform
As phases are not measured, structural determinations require parametric modelling
40
Magnetic crystallography
Magnetic structure factor (vector)
𝐹 𝑀 𝑄 = 𝑝 𝑚 𝑑𝑑
𝑓 𝑄 𝑒−𝐵𝑄2 𝑒2𝜋𝑖 (ℎ𝑥𝑑+𝑘𝑦𝑑+ℓ𝑧𝑑 )
𝑑𝜎𝑀𝑑𝛺
= 𝑁 2𝜋 3
𝑉0 𝐹𝑀⊥ 𝑄
2
𝜏
𝛿(𝑄 − 𝜏 𝑀)
𝑚 𝑣𝑑 = 𝜇 𝑣𝑑 = 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑡𝑜𝑚 𝑑 𝑖𝑛 𝑐𝑒𝑙𝑙 𝑣
The magnetic moment distribution can be Fourier expanded
𝑚 𝑣𝑑 = 𝑚 𝑘𝑑 exp −𝑖 𝑘 ∙ 𝑅 𝑣
𝑘
k is the propagation vector of the magnetic structure. Only k
vectors confined within the first Brillouin zone of the Bravais
lattice of the crystallographic cell enter into the summation.
41
The propagation vector describes the propagation direction and
Wavelength of the ordering waves: it defines the relation between
the moments in neighbouring cell of the crystal.
The set of symmetry equivalent k vectors is defined as {k} and is
called the vector star. Each individual k in the set is called an arm of
the star.
The propagation vector
𝑚 𝑣𝑑 = 𝑚 𝑘𝑑 exp −𝑖 𝑘 ∙ 𝑅 𝑣
𝑘
𝑑𝜎𝑀𝑑𝛺
= 𝑁 2𝜋 3
𝑉0 𝐹𝑀⊥ 𝑄
2
𝜏
𝛿(𝑄 − 𝑘 − 𝜏 𝑀)
𝑘
Magnetic Bragg peaks occur at
𝑄 = 𝑘 + 𝜏 𝑀
Within the Brillouin zone defined by there is a number of magnetic peaks equal to the number of distinct wave vectors k in the sum
42
Fluorite-type structure Long-range order (Jones et al. 1952, Osborne & Westrum 1953)
0
4
8
12
300
340
380
0 40 80 120 160
Cp (
cal K
-1 m
ol-1
)
Temperature (K)
Magnetic Structure of UO2
Propagation vector k = (0, 0, 1) {k} = (1, 0, 0) (0, 1, 0) (0, 0, 1)
43
Type I, 3-k transverse structure 0 = 1.74 B
n m exp(i kj•Rn) kj
j = 1
j = 3
k1 = (1, 0, 0) etc.
Magnetic Structure of UO2
(0 0 0) m1 = (1 1 1) (1/2 1/2 0) m2 = (1 -1 -1) (1/2 0 1/2) m3 = (-1 -1 1) (0 1/2 1/2 ) m4 = (-1 1 -1)
(0 0 0) m1 = (1 1 1) (1/2 1/2 0) m2 = (-1 1 -1) (1/2 0 1/2) m3 = (1 -1 -1) (0 1/2 1/2 ) m4 = (-1 -1 1)
𝐷𝑜𝑚𝑎𝑖𝑛 𝐵: 𝑚 100 = 0, 0, 1 𝑒𝑡𝑐. 𝐷𝑜𝑚𝑎𝑖𝑛 𝐴: 𝑚 100 = 0, 1, 0 𝑒𝑡𝑐.
44
A cold-neutron high-resolution powder neutron diffractometer:
WISH@ISIS
d-spacing range: 0.7-50 Å
LC Chapon et al, Neutron News 22, 22 (2011)
a) the magnetic structure can have a large number of degrees of freedom (three components of the magnetic moment on several inequivalent atoms);
b) the d-range available to observe Bragg peaks is limited due to fall-off of f(Q) c) magnetic and nuclear Bragg peaks are often nearly overlapping d) powder averaging of the magnetic structure factor for quasi-degenerate reflections
may prevent the determination of the direction of the magnetic moments:
high-resolution data are required to solve the structure.
Long range magnetic order of the quasicrystal approximant in the Tb-Au-Si system (G. Gebresenbut et al., JPCM, 26, 322202 (2014))
45
Nature 442, 797-801 (2006)
Complex magnetic structures
46
Polarised neutron diffraction
Unpolarised neutrons
𝐼 𝑄 = 𝐹𝑁(𝑄 ) 2
+ 𝐹 𝑀⊥ 𝑄 2
𝐼 𝑄 = 𝐹𝑁(𝑄 ) 2
+ 2𝑃 ∙ 𝐹 𝑀⊥ 𝑄 𝐹𝑁 𝑄 + 𝐹 𝑀⊥ 𝑄 2
Polarised neutrons
𝐼+ = 𝐹𝑁 𝑄 + 𝐹𝑀(𝑄 ) 2
𝐼− = 𝐹𝑁 𝑄 − 𝐹𝑀(𝑄 ) 2
𝑃 = 𝑛↑ − 𝑛↓ 𝑧
𝑅 = 𝐼+
𝐼−=
1 + 𝛾
1 − 𝛾
2
𝛾 = 𝐹𝑀𝐹𝑁
𝐹 𝑀 = 𝐹𝑀 𝑧
𝑃 = ±1, 𝑄 ⊥ 𝑃
47
F D
S
B I+
I-
Polarised neutron diffraction
For small ,
Ex.: = 0.1
With unpolarised neutrons
𝑅 ≃ 1 + 4𝛾
𝑅 = 𝐼+
𝐼−=
1 + 𝛾
1 − 𝛾
2
𝐼+ = 𝐹𝑁 𝑄 2
1 + 𝛾 2 = 1.21 𝐹𝑁 𝑄 2
𝐼− = 𝐹𝑁 𝑄 2
1 − 𝛾 2 = 0.81 𝐹𝑁 𝑄 2
𝐼 = 𝐹𝑁 𝑄 2
1 + 𝛾2 = 1.01 𝐹𝑁 𝑄 2
48
Magnetic form factor of NpCoGa5
dipole approximation
f(Q) = [<j0> + C2 <j2>]
C2 = (2-g)/g = L/(L+ S)
C2 = 2.11 Np3+ I.C.
= 0.091(1) B (B = 9.6 T) (Å-1)
magnetic f
orm
facto
r (
B/f
.u.
𝑗𝑛(𝑄) = 𝑈2(𝑟 )∞
0
𝑗𝑛 𝑄𝑟 𝑑𝑟
49
Structure of the Cr8Cd molecular ring. Cr atoms are represented in green, Cd in purple, O in red, F in yellow and C in black. H ions are omitted for simplicity.
Finite size effects in chains of antiferromagnetically coupled spins
Top: Spin density map for Cr8Cd (scale in μB Å
−2) obtained by the refinement of PND experimental data for applied fields 9 tesla at T=1.8 K (projection along the crystal b axis). Bottom: Classical spin ground state configuration for an even-open chain (a) and an even-closed chain (b) of AF-coupled spins under an external magnetic field.
T. Guidi et al., Nat. Commun. 6:7061 doi: 10.1038/ncomms8061 (2015).
50
Magnetic inelastic neutron scattering
• Single ion excitations
• crystal field measurements
• quasi 0-D clusters of spins
• 1-D spin chains, 2-D square lattices, 3-D systems
• frustrated magnetic systems……
INS offers the ability to measure directly the interactions of magnetic moments with other magnetic moments and with the local environment. A variety of problems can be investigated in a variety of systems:
Magnetic inelastic neutron scattering (INS) probes spin dynamics
The partial differential cross section at non-zero temperature is
proportional to the space-time Fourier transform of a spin-spin
correlation function
51
Magnetic inelastic neutron scattering cross-section
The cross-section for scattering of n0 in the |> initial spin state into the final spin state <sf| at T = 0 is (r0 = 0.5410-12 cm):
𝑑𝜎
𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓↑
= 𝑘𝑓
𝑘𝑖 𝛾𝑟0
2𝜇𝐵
2
4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥𝑧 λ𝑖 2
𝑖 ,𝑓
𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)
𝑑𝜎
𝑑𝛺𝑑𝜔 𝑘𝑖↑→𝑘𝑓↓
= 𝑘𝑓
𝑘𝑖 𝛾𝑟0
2𝜇𝐵
2
4𝜋 2 𝑝𝑓𝑝𝑖 λ𝑓 𝑀 ⊥𝑥 λ𝑖 + 𝑖 λ𝑓 𝑀 ⊥𝑦 λ𝑖 2
𝑖 ,𝑓
𝛿(휀𝑓 − 휀𝑖 − ℏ𝜔)
Non Spin Flip scattering probes the components of M along the quantization axis of the n0 spin, while Spin Flip scattering probes the components of M perpendicular to it.
Using the Fourier representation of the function, the cross section can be written as the F-Transform of a spin-spin time correlation function
𝑆 ⊥ ∙ 𝑆 ⊥ 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )
𝑛 ,𝑛′
𝑒𝑖𝜔𝑡∞
0
𝑑𝑡 𝑝(𝐸𝜆) 𝜆 𝑠 ⊥𝑛(0) ∙ 𝑠 ⊥𝑛′(𝑡) 𝜆
𝜆
NSF:
SF:
52
Spin-spin correlation functions
𝑑𝜎
𝑑𝛺𝑑𝜔 ↑
= 𝐴 𝑄 𝑆 ⊥ ∙ 𝑧 𝑆 ⊥ ∙ 𝑧 𝑄 ,𝜔
𝑑𝜎
𝑑𝛺𝑑𝜔 ↓
= 𝐴 𝑄 (𝑆 ⊥)⊥ ∙ (𝑆 ⊥)⊥ 𝑄 ,𝜔 + 𝑖 𝑆 ⊥ × 𝑆 ⊥ ∙ 𝑧 𝑄 ,𝜔
For neutrons with initial spin
(𝑆 ⊥)⊥ = 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑝𝑖𝑛 ⊥ 𝑡𝑜 𝑄 𝑎𝑛𝑑 𝑡𝑜 𝑧
𝐴 𝑄 = 𝑘𝑓
𝑘𝑖 𝛾𝑟0
2𝜋ℏ
2
𝑓(𝑄 ) 2
𝑒−2𝑊(𝑄)
where,
NSF correlations of the S component to the initial direction of P
SF correlations of the S component to the initial direction of P
53
<Sn(0)Sn’
b(t)> is the thermal average of the time dependent spin operator and corresponds to the van Hove correlation function: the probability of finding a spin Sn’ at site n’ and at time t when the spin at position n is Sn at t=0
: quasielastic broadening : intrinsic linewidth
: lifetime
: correlation length
Spin-spin correlation function
Spin-Spin Corr. Function Magnetic neutron cross section
static moment oscillating moment
relaxing moment
Long range order
Short range order
54
Correlation function unpolarised neutrons
or
or, in terms of matrix elements:
Ex. #4. Derive the above result.
𝑑𝜎
𝑑𝛺𝑑𝜔 = 𝐴 𝑄 𝑆 ⊥ ∙ 𝑆 ⊥ 𝑄 ,𝜔
𝑑𝜎
𝑑𝛺𝑑𝜔 = 𝐴 𝑄 𝛿𝛼 ,𝛽 −
𝑄𝛼𝑄𝛽
𝑄2
𝛼 ,𝛽
𝑆𝛼 ,𝛽 𝑄 ,𝜔 , 𝛼,𝛽 = 𝑥,𝑦, 𝑧
𝑆𝛼 ,𝛽 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )
𝑛 ,𝑛′
𝑒𝑖𝜔𝑡∞
0
𝑑𝑡 𝑝(𝐸𝜆) 𝜆 𝑠 ⊥𝑛(0) ∙ 𝑠 ⊥𝑛′(𝑡) 𝜆
𝜆
𝑆𝛼 ,𝛽 𝑄 ,𝜔 = 𝑒𝑖𝑄 ∙(𝑟 𝑛 ′−𝑟 𝑛 )
𝑛 ,𝑛′
𝑝(𝐸𝜆) 𝜆 𝑠𝑛𝛼 𝜆′ ∙ 𝜆′ 𝑠𝑛
𝛽 𝜆
𝜆 ,𝜆′
55
Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering
Cr8 homometallic ring (Cr3+, s = 3/2, S = 0)
IN6 @ ILL
Data from a powder sample
56
Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering
Data from a single crystal sample
147° h; ±20˚ v
Z
Y
X’
Z’
Horace scan 4D S(Q,w) Step of 1° over 360°
= 5 Å T= 1.5K each 10’
Courtesy T. Guidi STFC-ISIS
57
Direct measurement of spin-spin correlation functions by four- dimensional inelastic neutron scattering
Courtesy T. Guidi STFC-ISIS
p=1
p=2
p=3
M. Baker et al Nature Physics (2012)
58
Squared form factor DW factor
Spin correlation function
Inelastic Magnetic Scattering
For ions with unquenched orbital moment and for q0
geometrical factor
𝑑𝜎
𝑑𝛺𝑑𝜔 =
𝑘𝑓
𝑘𝑖 𝛾𝑟0
2𝜋ℏ
2
𝑓(𝑄 ) 2
𝑒−2𝑊(𝑄) 𝛿𝛼 ,𝛽 −𝑄𝛼𝑄𝛽
𝑄2
𝛼 ,𝛽
𝑆𝛼 ,𝛽 𝑄 ,𝜔
𝑠𝑛𝛼 =
1
2𝑔 𝐽𝑛
𝛼 𝑔 = 1 +𝐽 𝐽 + 1 − 𝐿 𝐿 + 1 + 𝑆(𝑆 + 1)
2𝐽(𝐽 + 1)
𝐽𝑛𝛼 𝑏𝑒𝑖𝑛𝑔 𝑎𝑛 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟
59
For a wide class of systems Sb satisfies useful sum-rules
Detailed balance
Total moment
First moment sum-rule
Inelastic Magnetic Scattering
𝑆𝛼𝛽 𝑄 ,𝜔 = exp ℏ𝜔
𝑘𝐵𝑇 𝑆𝛼𝛽 −𝑄 ,−𝜔
ℏ
𝑑𝑄 𝑆𝛼𝛼
𝛼
𝑄 ,𝜔 𝑑𝑄 𝑑𝜔 = 𝑆(𝑆 + 1)
ℏ2 𝑆 𝑄 ,𝜔 𝜔 𝑑𝜔 = −1
3𝑁 𝐽𝑛𝑛 ′ 𝑠 𝑛 ∙ 𝑠 𝑛 ′
𝑛 ,𝑛 ′
1 − cos𝑄 ∙ (𝑟 𝑛′ − 𝑟 𝑛)
60
The scattering function Sb(q, w) is related to the generalized susceptibility b by the fluctuation-dissipation theorem:
b determines the response of the system to the magnetic field established by the neutron:
We convert inelastic scattering data to b to • Compare with bulk susceptibility data • Analyze the temperature dependence of the response • Compare with theories
Note that:
Generalised susceptibility
𝑆𝛼𝛽 𝑄 ,𝜔 = 𝑁ℏ
𝜋
1
1 − exp −ℏ𝜔𝑘𝐵𝑇
𝐼𝑚𝜒𝛼𝛽 𝑄 ,𝜔
𝑀𝛼 𝑄 ,𝜔 = 𝜒𝛼𝛽 𝑄 ,𝜔 𝐻𝛽 𝑄 ,𝜔
𝜒 𝑄 , 0 = 1
2𝜋𝑖 𝑑𝜔
𝐼𝑚𝜒𝛼𝛽 𝑄 ,𝜔
𝜔
61
Valence-fluctuating ground state of delta-plutonium
Experiments find no static magnetism in -Pu.
LDA + Exact Diagonalization of an impurity Anderson model suggests that Pu has an intermediate-valence state <n5f> = 5.21. Similar results from DMFT.
The hybridized ground state of the impurity is a nonmagnetic singlet (S = L = J = 0) The 5f shell magnetic moment fluctuates in time because of the intermediate valence, but is dynamically compensated by the moment of the conduction electron bath
62
Observation of magnetic fluctuations centred around 84 meV in agreement with theory.
Valence-fluctuating ground state of delta-plutonium
Experiment Theory
Crystal Field excitations in AnO2
E
Isolated magnetic ion with total angular momentum J: full rotational symmetry. the ground state is (2J+1)-fold degenerate.
Magnetic ion embedded in a solid:
local charge symmetry lifts partially or totally the degeneracy of the ground state multiplet.
Crystal Field excitations in UO2
5f2 3H4 Cubic symmetry: 2 parameters determine the CF
J = 4
3x
2x
3x
1x
PRB 40, 1865 (1989)
65
Crystal Field excitations in NpO2
5f3 4I9/2
J = 9/2
8
8
62x
4x
4x
E (meV)
5-ph
3x
270
54
0
CF-phonons bound state
0
S (
arb
. units)
Energy Transfer (meV)
2
1
20 40 60 80
Inelastic neutron scattering spectrum at 5K
66
Neptunium ions ground state in NpO2
The Np4+ ground state in the paramagnetic phase of NpO2 is a
quartet of 8 symmetry. In addition to 3 magnetic dipoles and
5 electric quadrupoles, the 8 quartet supports magnetic
octupoles (2, 4, 5), a triplet of magnetic triakontadipoles (5)
and two triplets of rank-7 multipoles degrees of freedom.
67
e-Q primary order parameter m-T primary order parameter
Splitting of the NpO2 ground state
68
Splitting of the NpO2 ground state
INS with polarization analysis on polycrystalline NpO2
Spin Flip (magnetic)
Non Spin Flip (vibrational)
Magnetic Scattering Theory
Splitting of the 15 meV peak due to magnetoelastic interactions?
69
0
5
10
15
20
25
30
0.00 0.25 0.50 0.75 1.00
(,,) (r.l.u.)
En
erg
y (
meV
)
<001> X
LA(D1)
TA(D5)
0
1
2
3
4
5
5 10 15 20
(6,1,0)
(6,0.8,0)
(6,0.6,0)
(6,0.4,0)
(6,0.2,0)
Photo
n In
ten
sity (
arb
. u
nits)
Energy transfer (meV)
0
2
4
6
8
0.4 0.6 0.8 1.0
Inte
nsity (
arb
. u
nits)
reduced wavevector q (rlu)
Magnetoeleastic ph-M25 interactions?
Anomalous behaviour of TA(D5) phonon intensity observed by IXS at energies close to the one predicted for the MM reversal excitation.
Magnetic excit. calculated
IXS measured TA phonon groups
Primary OP: 25 magnetic
multipoles
Magnetic field distribution around a Np ion
Triakontadipole order in NpO2
Lattice dynamics in UO2
Pure spin and quadrupole waves, together with mixed magneto quadrupolar and magneto-vibrational modes. The measured INS cross-section is reproduced by reasonable values of the 5 free parameters.
IN14@ILL
Q = G + q = (1, 1, -1) + (0, 0, )
Transverse constant-Q scans
Lattice dynamics in UO2
Magnon-phonon avoided crossing at q = (0, 0, 0.45)
Longitudinal scan reveals a transverse
50% phonon- 50% magnon mixed mode
Iph (Q·e)2 = 0
eTA
IN22 + CRYOPAD
Quadrupolar modes are visible in the
INS spectra through the associated spin
or vibrational fluctuations.
SA SO QO
SO-TA
TA
Lattice dynamics in UO2
75
INS accounts for the detailed atomic motions and magnetic excitations - individual or collective - within a many-body system.
Microscopic motions or excitations may occur in vastly different time and length scales, typically ps to ms and sub-nm to m: INS necessitates a wide coverage in the energy (E) and wavevector (Q) space with good resolutions.
Interpretation of INS data can be a challenge facing experimentalists. Researchers nowadays have to apply methods of theoretical modeling and simulations that require high degree of sophistication and substantial amount of computing resources.
Concluding remarks