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

[email protected]

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


Improved functionality by nanostructuring materials

Precipitation hardening Al-Cu (4 at.%)

Murray, Int. Metals Rev. 30 (1985) 5.

Strength during aging precipitation


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


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.


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


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


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


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


Contrast X-rays:

( ) ( )22 particle matrixρ ρ ρ∆ = −

Sensitive to variations in: - Chemical composition - Density

matrix

particle

( )ρ r


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


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.


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.


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.


Example: SANS on Cu precipitation in deformed Fe-Cu alloys


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


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


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


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


TEM after 12 h of aging at 550 oC 0% deformation

8% deformation

He et al., Phys. Rev. B 82 (2010) 174111.


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


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

=

− = −

−= =

∫


Time-resolved SANS measurements Fe-Cu


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


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


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.


Example: SANS on Au precipitation in deformed Fe-Au alloys


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


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.


Disk-shaped precipitates

L 2R

constant ( )Σ

Ω d Qd

Q

Q -2

Q -4

2π/L π/R

incoherent


Phase fraction of Au precipitates

Invariant:

L 2R


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



( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

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



SANS TEM

12 h at 550 oC 24% deformation

12 h at 550 oC 24% deformation

2 / 8R Lε = ≈


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


Example: SAXS on (Fe,Cr)7C3 carbides and dislocation structures in low-Cr steel


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



Isotropic SAXS profile

Lognormal size distribution:

TEM


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



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.



Streaks in SAXS profiles Correlation to simultaneous X-ray Diffraction


Example: SANS on the magnetic flux-line lattice of superconducting UPt3


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


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


Huxley et al., Nature 406 (2000) 160.

Unconventional superconductivity in UPt3

(B||c)

superconducting phase B

superconducting phase C

superconducting phase A


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+


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.


SANS instrument under development in Delft

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