Small Angle Scattering on Metals with Neutrons and X-rays SYNEW/vanDijk.pdf · Close to an...
Transcript of Small Angle Scattering on Metals with Neutrons and X-rays SYNEW/vanDijk.pdf · Close to an...
3-6-2015
Delft University of Technology
SyNeW School 2 June 2015: Small-Angle Scattering
Small Angle Scattering on Metals with Neutrons and X-rays Niels van Dijk Fundamental Aspects of Materials and Energy, TU Delft
2 SyNeW School 2 June 2015: Small-Angle Scattering
Outline
• Introduction: Nanoscale structures in metals • Experimental Methods: Summary SAS When to use SAS Comparison SANS & SAXS • Examples: SANS on precipitates SAXS on precipitates Small-angle diffraction with SANS
3 SyNeW School 2 June 2015: Small-Angle Scattering
Improved functionality by nanostructuring materials
Precipitation hardening Al-Cu (4 at.%)
Murray, Int. Metals Rev. 30 (1985) 5.
Strength during aging precipitation
4 SyNeW School 2 June 2015: Small-Angle Scattering
Nucleation: formation of new phase particles - nm size clusters - occurs on short time scales - positioned within bulk materials - strongly dependent on interface/defect properties Growth: increase in size of nucleated grain - controlled by diffusion of alloying elements and/or heat - interaction between neighboring growing particles - dependent on microstructure of the parent phase
Structure evolution during phase transformations in structural materials
Nucleus hard-sphere colloid
Need for time-dependent in-situ measurements
5 SyNeW School 2 June 2015: Small-Angle Scattering
Nanoscale probes
Small-Angle X-ray Scattering
Transmission Electron Microscopy Atom probe Tomography
Small-Angle Neutron Scattering
Hutchinson et al., Acta Materialia 74 (2014) 96.
6 SyNeW School 2 June 2015: Small-Angle Scattering
Small-Angle X-ray Scattering + Non destructive + In situ / time resolved + Contrast near absorption edge + High flux - Sample thickness heavy elements - No spatial information - Indirect chemical information
Transmission Electron Microscopy + Up to atomic spatial resolution + Can provide chemical information - Destructive - Probes limited volume
Atom probe Tomography + Near atomic spatial resolution + Precise chemical information - Destructive - Probes limited volume
Small-Angle Neutron Scattering + Non destructive + In situ / time resolved + Magnetic information + Can probe a large volume - Flux limited - No spatial information - Indirect chemical information
Combining techniques will provide complementary information
Nanoscale probes
7 SyNeW School 2 June 2015: Small-Angle Scattering
Small Angle Scattering: Elastic Scattering
= + ∆
+
= −
=
Momentum conservation:
out in
out i
o
n
ut inQ k
p p pQ
k
k k= + ∆
Energy conservati : on
E E Eout in2 2
2
2 2
=
=
= → =
= → =
= =
Neutrons:
X-rays:
E
k
m mc c
πλ
in
out inout in
out in out i
i
n
out n
p pp p
p p
k k
p p
k = wave vector Q = wave vector transfer λ = wavelength 2θ = scattering angle
inp
outp
2θ
∆p
n, X
n, X
4 sin( )=Q π θλ
Wave vector transfer :
ink
outkθ Qθ
8 SyNeW School 2 June 2015: Small-Angle Scattering
Interference pattern: Fourier transform of object
2R
2( )f Q
sphere block
zQ
yQ
L
cube
L
yQ
zQ
2( )f Q
2L
yQ
zQ
2( )f Q
L
( ) ( ) 2I f∝Q Q
( ) ( ) 3if e dρ ⋅= ∫ Q rQ r r
Note: patterns are in log scale
9 SyNeW School 2 June 2015: Small-Angle Scattering
Scattered Intensity:
( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )
/
2
22
0
2
22
0
,
in
( )
, s
1.
particle matr
N
i
N
x
D Rd Q dRd
P Q R
P
D
V R
R
Q d
RV
R fπ
ρ
ρ ρ
α
ρ
α
∞Σ = Ω∆
∆
=
= −
∫
∫
Contrast :
Orientation-averaged square of formfactor :
Particle volume :Number distribution of size particles :
Assumptions : dilute limit (low volume fraction of particles within the matrix)
2. weak scattering (<10% of the X-rays or neutrons are scattered)
matrix
particle
( )ρ r
10 SyNeW School 2 June 2015: Small-Angle Scattering
Contrast X-rays:
( ) ( )22 particle matrixρ ρ ρ∆ = −
Sensitive to variations in: - Chemical composition - Density
matrix
particle
( )ρ r
11 SyNeW School 2 June 2015: Small-Angle Scattering
Contrast Neutrons:
Nuclear contrast: ( ) ( )22
0 with i i
particle matrix ci
N bρ ρ ρ ρ∆ = − = ∑N0 = number density bc = scattering length ρ = scattering length density
matrix
particle
( )ρ r
Magnetic contrast: ( ) ( ) ( )2 22 2 2 2
0 0 0sin with i i
particle matrix particle matrixi
p M M p M M M Nρ α µ⊥ ⊥∆ = − = − = ∑N0 = number density μ = magnetic moment M= magnetisation
M
Qα
Sensitive to variations in: - Chemical composition - Density
Sensitive to variations in: - Size magnetic moment - Density - Orientation magnetic moment
⊥M
12 SyNeW School 2 June 2015: Small-Angle Scattering
Considerations SAXS on metals
Anomalous SAXS: Close to an absorption edge X-ray scattering depends on the energy: This gives additional contrast and can provide additional chemical information.
0 ( ) '( ) ''( )f f f E if E= + +Q
Sample transmission: The transmission X-rays with E<30 keV leads to restrictions in the allowed sample thickness (especially for heavy elements).
Additional scattering for crystalline materials (metals): X-rays with 10-30 keV have a wavelength of λ = 0.4-1.2 Å. This allows for a possible diffraction signal.
13 SyNeW School 2 June 2015: Small-Angle Scattering
Considerations SANS on metals
Contrast: As the coherent scattering length strongly varies from element to element the contrast strongly depends on the composition.
Sample transmission: The large penetrating power of neutrons generally allow a large sample thickness (and sample volume to be probed).
No additional diffraction signal: In SANS neutrons have a wavelength of λ = 6-10 Å. This generally does not allow Bragg scattering.
Magnetic SANS: For magnetic materials the particles have both nuclear and magnetic contrast that probe the sample particle size distribution. The ratio between them may provide chemical information.
14 SyNeW School 2 June 2015: Small-Angle Scattering
Data analysis
Data reduction: Transform the raw intensity data on the 2D detector I(x,y) into instrument independent 1D SAS data of (dΣ/dΩ)(Q) versus Q.
Model fitting: (SASfit, Grasp, GNOM, …) Construct a model of the scattering objects and obtain the relevant model parameters by fitting. Additional information from other methods (TEM, Atom probe) is generally very useful to obtain reliable results.
15 SyNeW School 2 June 2015: Small-Angle Scattering
Example: SANS on Cu precipitation in deformed Fe-Cu alloys
16 SyNeW School 2 June 2015: Small-Angle Scattering
scattering length density
Magnetic scattering from only!
I(Q⊥)
I(Q||)
Sample
2 2( )p mρ ρ ρ∆ = −
2. Magnetic contrast:
1. Nuclear contrast:
Scattering from inhomogeneities
22 2 2 20( ) ( sin ) ( )I Q p M f Qρ α∝ ∆ + ∆
⊥ΔM Q
2 2( )p mM M M∆ = −
particle matrix
particle matrix
ρ
magnetisation M
Neutron Intensity:
17 SyNeW School 2 June 2015: Small-Angle Scattering Fe-Cu, AQ, 8% deformed, aged for 96h at 550oC
2m, B=0 6m, B=0 18m, B=0
2m, B=1.1 T 6m, B=1.1 T 18m, B=1.1 T
SANS 2D Patterns with & without Magnetic Field
B B B
18 SyNeW School 2 June 2015: Small-Angle Scattering
0.1 1
1E-3
0.01
0.1
1
10
100
after aging before aging
dΣ/d
Ω (c
m-1sr
-1)
Q (nm-1)
Fe-Cu0% pre-strain
0.1 1
0.01
0.1
1
10
100
1000
after aging before aging
dΣ/d
Ω (c
m-1sr
-1)
Q (nm-1)
Fe-Cu0%pre-strain
0.1 1
1E-3
0.01
0.1
1
10
100
before aging after aging
dΣ/d
Ω (c
m-1sr
-1)
Q (nm-1)
Fe-Cu24% pre-strain
0.1 1
0.01
0.1
1
10
100
1000
after aging before aging
dΣ/d
Ω (
cm-1sr
-1)
Q (nm-1)
Fe-Cu24% pre-strain
Isotropic scattering
Isotropic scattering
Anisotropic scattering
0% 24%
24% 0% Anisotropic scattering
Isotropic (nuclear) & anisotropic (magnetic) SANS intensity before/after aging at 550 oC for 12 h with 0% and 24% deformation
Fe-Cu
19 SyNeW School 2 June 2015: Small-Angle Scattering
TEM after 12 h of aging at 550 oC 0% deformation
8% deformation
He et al., Phys. Rev. B 82 (2010) 174111.
20 SyNeW School 2 June 2015: Small-Angle Scattering
Phase fraction of Cu precipitates
Invariant:
0 2 4 6 8 10 120.0
0.1
0.2
0.3
0.4
0.5
Prec
ipita
tes
Volu
me
Frac
tion
(%)
Aging Time (h)
0% pre-strain 8% pre-strain 24% pre-strain
Fe-CuT = 550 oC
21 SyNeW School 2 June 2015: Small-Angle Scattering
Profile fitting of the SANS curve
( )( )
( ) ( )
2 28
2 28
2 2
2.2 10
15.5 10
-4
-4
Theoretical estimate contrast:
Nuclear: m
Magnetic: m
NUC
MAG
MAG NUC
ρ
ρ
ρ ρ
∆ = ×
∆ = ×
∆ ∆
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )( )
2 2
0
3
2
2
22
3
,
( ) 4 / 3
ln lnexp
22
sin cos, , 3
Particle volume:
Log-normal number distribution of particles:
Square of formfactor:
N
mpN
d Q D R V R P Q R dRd
V R R
R RND R
R
QR QR QRP Q R F Q R
QR
ρ
π
σσ π
∞Σ = ∆ Ω
=
− = −
−= =
∫
22 SyNeW School 2 June 2015: Small-Angle Scattering
Time-resolved SANS measurements Fe-Cu
23 SyNeW School 2 June 2015: Small-Angle Scattering
Time-resolved SANS measurements Fe-Cu Structure evolution Cu precipitates during growth: bcc → 9R → 3R → fcc bcc : R < 5 nm 9R : 5 nm < R < 16 nm 3R & fcc : R > 16 nm
24 SyNeW School 2 June 2015: Small-Angle Scattering
Time-resolved SANS measurements Fe-Cu
4lim[ ( )] pQI Q K Q B−
→∞= +
2p V2 ( )K Sπ ρ= ∆
Porod constant Kp:
0 2 4 6 8 10 120
1x10-3
2x10-3
3x10-3
4x10-3
K p
Aging Time (h)
24% pre-strain 8% pre-strain 0% pre-strain
Fe - CuT = 550 oC
Network of Cu along dislocations/interfaces
with specific surface: Sv = interface area volume
Pre-strain leads to a strong Cu precipitation at dislocations/interfaces
25 SyNeW School 2 June 2015: Small-Angle Scattering
Time-resolved ASAXS measurements undeformed Fe-Cu (8x enhanced contrast)
E = 7.106 keV L = 20-30 μm
Perez et al., Philos. Mag. 85 (2005) 2197.
26 SyNeW School 2 June 2015: Small-Angle Scattering
Example: SANS on Au precipitation in deformed Fe-Au alloys
27 SyNeW School 2 June 2015: Small-Angle Scattering
Nuclear SANS
Magnetic SANS
Initial state After 12 h at 550 oC
Au precipitation in deformed Fe-Au (1 at.%)
Zhang et al., Acta Mater. 61 (2013) 7009.
grain boundaries Q -4
precipitates Q -2
incoherent incoherent
28 SyNeW School 2 June 2015: Small-Angle Scattering
TEM of Au precipitates in Fe-Au after aging
Zhang et al., Acta Mater. 61 (2013) 7009.
Fe-Au 0%, Pre-strain
Fe-Au 24%, Pre-strain
Undeformed: only along grain boundaries Deformed: 1. along grain boundaries 2. disk-like, closely connected to the dislocations
Zhang et al., Acta Mater. 61 (2013), p7009.
29 SyNeW School 2 June 2015: Small-Angle Scattering
Disk-shaped precipitates
L 2R
constant ( )Σ
Ω d Qd
Q
Q -2
Q -4
2π/L π/R
incoherent
30 SyNeW School 2 June 2015: Small-Angle Scattering
Phase fraction of Au precipitates
Invariant:
L 2R
31 SyNeW School 2 June 2015: Small-Angle Scattering
Specific surface (S/V) circular caps Au precipitates
( ) ( )22 22 0 2
/ 2 / :
4 /−Σ = + = ∆ Ω
For disk-shaped precipitates with
where V vc
R Q Ld Q C Q C C f Sd
π π
π ρ
( )2222 2 / 2= = → = ∆
Specific surface for monodisperse disks:
vc p V VS R N f L C f Lπ π ρ
L 2R
L 2R
32 SyNeW School 2 June 2015: Small-Angle Scattering
Profile fitting of the SANS curve
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
2 2
0
2 3
2
2
,
( ) 2 / 2 /
ln lnexp
22
Particle volume: with aspect ratio
Log-normal number distribution of particles:
Orientation-averaged square o
N
mpN
d Q D R V R P Q R dRd
V R R L R R L
R RND R
R
ρ
π π ε ε
σσ π
∞Σ = ∆ Ω
= = =
− = −
∫
( ) ( ) ( )( )( )
( )( )( ) ( )
2/2 /2
2 1
0 0
2 sin sin sin / 2, sin sin
sin sin / 2
f formfactor:
J QR QLP Q R F d d
QR QL
π π α αα α α α
α α
= =
∫ ∫
L 2R
33 SyNeW School 2 June 2015: Small-Angle Scattering
Profile fitting of the SANS curve
SANS TEM
12 h at 550 oC 24% deformation
12 h at 550 oC 24% deformation
2 / 8R Lε = ≈
34 SyNeW School 2 June 2015: Small-Angle Scattering
Au segregation at (sub)grain boundaries
( ) ( )242 0 4
/ 2 / :
2−Σ = + = ∆ Ω
For Porod scattering: and
where
Specific surface = interfacial area per unit volume
v
v
Q R Q Ld Q C Q C C Sd
S
π π
π ρ
35 SyNeW School 2 June 2015: Small-Angle Scattering
Example: SAXS on (Fe,Cr)7C3 carbides and dislocation structures in low-Cr steel
36 SyNeW School 2 June 2015: Small-Angle Scattering
SAXS pattern:
Heat treatment:
Gözde Dere et al., J. Appl. Cryst. 46 (2013) 181.
Isotropic → precipitate Streaks → dislocations
E = 17 keV λ = 0.729 Å
37 SyNeW School 2 June 2015: Small-Angle Scattering
Gözde Dere et al., J. Appl. Cryst. 46 (2013) 181.
Isotropic SAXS profile
Lognormal size distribution:
TEM
38 SyNeW School 2 June 2015: Small-Angle Scattering
Streaks in SAXS profiles Small-angle scattering from dislocations
Edge dislocation Strain field (= Density variation)
For dislocation walls (Long & Levine, Acta Cryst. A 61 (2005) 557):
( ) ( )2 , ,t t w wI A cL F Q L F Q L= +
Strongly anisotropic scattering along (Qw) and perpendicular (Qt) to the wall.
Dislocation wall
39 SyNeW School 2 June 2015: Small-Angle Scattering
Gözde Dere et al., J. Appl. Cryst. 46 (2013) 181.
Streaks in SAXS profiles Correlation to simultaneous X-ray Diffraction
The data provide direct information that links: the matrix phase structure, dislocations and nucleated precipitate.
40 SyNeW School 2 June 2015: Small-Angle Scattering
Gözde Dere et al., J. Appl. Cryst. 46 (2013) 181.
Streaks in SAXS profiles Correlation to simultaneous X-ray Diffraction
41 SyNeW School 2 June 2015: Small-Angle Scattering
Example: SANS on the magnetic flux-line lattice of superconducting UPt3
42 SyNeW School 2 June 2015: Small-Angle Scattering
pout 2θ
sample
Diffraction pattern (superconducting Nb)
Small Angle Neutron Scattering
pin
B
θ θ pin pout
Magnetic flux-line lattice
in superconductor
n
I(Q) ∝ | f (Q) |2
f (Q) = form factor field profile
43 SyNeW School 2 June 2015: Small-Angle Scattering
d10
d10
a
Magnetic Flux-Line Lattice
S
S = Φ0/B = (√3/2) a2 = (2/√3) d102
d10 = √(√3/2)(Φ0/B) = 423.4 Å/√B [T]
Flux quantum: Φ0 = h/2e = 2.07×10-15 Tm2
44 SyNeW School 2 June 2015: Small-Angle Scattering
Huxley et al., Nature 406 (2000) 160.
Unconventional superconductivity in UPt3
(B||c)
superconducting phase B
superconducting phase C
superconducting phase A
45 SyNeW School 2 June 2015: Small-Angle Scattering
Flux-Line Lattice UPt3 (B||c)
Normal State
Zero-Field Cooled
Field Cooled
TQ = 475 mK
Huxley et al., Nature 406 (2000) 160.
B
TQ
B
Tc+
B
Tc+
46 SyNeW School 2 June 2015: Small-Angle Scattering
Further reading SAXS/SANS on metals:
P. Fratzl, Small-angle scattering in materials science – a short review of applications in alloys, ceramics and composite materials, J. Appl. Cryst. 36 (2003) 397-404. G. Kostorz, Small-Angle Scattering Studies of Phase Separation and Defects in Inorganic Materials, J. Appl. Cryst. 24 (1991) 444-456. F. De Geuser, A. Deschamps, Precipitate characterisation in metallic systems by small-angle X-ray or neutron scattering, C. R. Physique 13 (2012) 246–256. A. Deschamps, On the validity of simple precipitate size measurements by small- angle scattering in metallic systems, J. Appl. Cryst. 44 (2011) 343–352. E. Eidenberger et al., Application of Photons and Neutrons for the Characterization and Development of Advanced Steels, Adv. Eng. Mater. 13 (2011) 664-673.
47 SyNeW School 2 June 2015: Small-Angle Scattering
SANS instrument under development in Delft