Post on 20-May-2020
NEUTRON GRATING INTERFEROMETRY PART II
QUANTITATIVE DFI
Alexander Backs alexander.backs@frm2.tum.de
Dark Field Images
10.07.2017 - Combining Neutron Scattering and Imaging to Investigate Domain Structures in Superconductors- alexander.backs@frm2.tum.de
nGI reveals scattering
โข from small structures (~ ฮผm)
โข under very small angles
What does that mean exactly ?
Quantitative DFI: Size Matters
โข varying particle diameter โข constant volume fraction
โข constant absorbtion โข varying DFI value
Polysterene Colloids in Solution
Particle Size (ฮผm)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Reimann et al., J.Appl.Cryst. (2016)
Origin of the DFI
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Model Case: One Single Scattering Angle
without sample:
๐๐๐ = cos ๐ฅ
Origin of the DFI
without sample:
๐๐๐ = cos ๐ฅ
with sample:
๐๐ =1
2cos ๐ฅ + ๐ +
1
2cos(๐ฅ โ ๐)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Model Case: One Single Scattering Angle
Origin of the DFI
๐ = ฮพ๐บ๐ผ๐๐ฅ ฮพ๐บ๐ผ = ฮป๐ฟ๐ ,๐๐๐
๐2
๐๐ = cos ๐ฅ cos(ฮพ๐บ๐ผ๐๐ฅ)
๐ท๐น๐ผ =๐๐ ๐๐๐
= cos ฮพ๐บ๐ผ๐๐ฅ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Model Case: One Single Scattering Angle
without sample:
๐๐๐ = cos ๐ฅ
with sample:
๐๐ =1
2cos ๐ฅ + ๐ +
1
2cos(๐ฅ โ ๐)
Origin of the DFI
Real Case: Distribution of Scattering Angles
๐ท๐น๐ผ = ๐ ๐๐ฅ cos ฮพ๐บ๐ผ๐๐ฅ ๐๐๐ฅ
๐ท๐น๐ผ = cos ฮพ๐บ๐ผ๐๐ฅ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Origin of the DFI
Real Case: Distribution of Scattering Angles
๐ท๐น๐ผ = ๐ ๐๐ฅ cos ฮพ๐บ๐ผ๐๐ฅ ๐๐๐ฅ
๐ท๐น๐ผ = cos ฮพ๐บ๐ผ๐๐ฅ
What defines ๐ ๐ช๐ฑ ?
๐ ๐๐ฅ โprobability of scattering a neutron with a certain momentum transfer ๐๐ฅ โ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
S qx : A Bit of Scattering Theory
๐ฯ(๐)
๐ฮฉ=
neutrons scattered into ๐ฮฉ
incomingneutronsperunitarea
๐ฮฉ
Reciprocal Space: the differential cattering cross section (looking at momenum)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
S qx : A Bit of Scattering Theory
๐ฯ(๐)
๐ฮฉ=
neutrons scattered into ๐ฮฉ
incomingneutronsperunitarea
๐ฮฉ
Reciprocal Space: the differential cattering cross section (looking at momenum)
Real space: the pair correlation function (looking at space coordinates)
ฮณ(๐) = ฮฯ ๐ ฮฯ ๐ + ๐ ๐๐
๐
ฮณ(๐)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Distribution of scsattering strength
S qx : A Bit of Scattering Theory
๐ฯ(๐)
๐ฮฉ=
neutrons scattered into ๐ฮฉ
incomingneutronsperunitarea
๐ฮฉ
Reciprocal Space: the differential cattering cross section (looking at momenum)
Real space: the pair correlation function (looking at space coordinates)
ฮณ(๐) = ฮฯ ๐ ฮฯ ๐ + ๐ ๐๐
๐
ฮณ(๐)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Distribution of scsattering strength
fourier transformation
๐ฯ
๐ฮฉ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
S qx : Reducing the Dimensions
Step 1: no scattering along the beam direction ๐๐ง = 0
๐ฯ(๐)
๐ฮฉโ๐ฯ(๐๐ฅ, ๐๐ฆ , 0)
๐ฮฉ
๐บ(๐ฅ, ๐ฆ) = ๐พ(๐ฅ, ๐ฆ, ๐ง)๐๐ง
๐ฯ(๐)
๐ฮฉโ๐ฯ(๐๐ฅ, ๐๐ฆ , 0)
๐ฮฉ
๐บ(๐ฅ, ๐ฆ) = ๐พ(๐ฅ, ๐ฆ, ๐ง)๐๐ง
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
S qx : Reducing the Dimensions
Step 1: no scattering along the beam direction
๐๐ฆ = โโ โ +โ
๐บ ๐ฅ, ๐ฆ โ ๐บ(๐ฅ, 0)
๐ฯ(๐๐ฅ)
๐ฮฉ๐ ๐๐๐ก
= ๐ฯ(๐๐ฅ , ๐๐ฆ , 0)
๐ฮฉ๐๐๐ฆ
Step 2: projection along the grating lines
๐๐ง = 0
๐๐๐ฆ
๐ฯ
๐ฮฉ
Quantitative DFI: The Formula
๐ท๐น๐ผ = ๐ ๐๐ฅ cos ฮพ๐บ๐ผ๐๐ฅ ๐๐๐ฅ
๐ ๐๐ฅ โ๐ฯ(๐๐ฅ)
๐ฮฉ๐ ๐๐๐ก
๐๐ง = 0 ๐๐ฆ = โโ โ +โ
Reciprocal Space:
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Reimann, Ph.D. Thesis (2016)
Quantitative DFI: The Formula
๐ท๐น๐ผ = ๐ ๐๐ฅ cos ฮพ๐บ๐ผ๐๐ฅ ๐๐๐ฅ
๐ ๐๐ฅ โ๐ฯ(๐๐ฅ)
๐ฮฉ๐ ๐๐๐ก
๐๐ง = 0 ๐๐ฆ = โโ โ +โ
Reciprocal Space:
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Reimann, Ph.D. Thesis (2016)
๐ท๐น๐ผ = ๐๐ฅ๐ ฮฃ๐ก๐บ(ฮพ๐บ๐ผ)
๐บ(0)โ 1
ฮฃ = ฮฯ 2ฮป2ฮฆ๐ฃ๐บ(0)
ฮณ(0)
ฮพ๐บ๐ผ
Total scattering crossection
Real Space:
sample thickness ๐ก
nGI correlation length ฮพ๐บ๐ผ
A Simple Example: Hard Spheres
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Pair Correlation Function
Polysterene Colloids in Solution
A Simple Example: Hard Spheres
ฮณ(๐ฅ, ๐ฆ, ๐ง)
calculate:
๐บ(๐ฅ, ๐ฆ)
๐บ(๐ฅ, 0)
๐ฯ(๐๐ฅ, ๐๐ฆ , ๐๐ง)
๐ฮฉ
๐ฯ(๐๐ฅ, ๐๐ฆ , 0)
๐ฮฉ
๐ฯ(๐๐ฅ)
๐ฮฉ๐ ๐๐๐ก
fourier transform:
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Real Space Reciprocal Space
Pair Correlation Function
Polysterene Colloids in Solution
Polysterene Colloids in Water
Real Space: Pair Correlation function
Reciprocal Space: Scattering cross section
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Reimann et al., J.Appl.Cryst. (2016)
Polysterene Colloids in Water
๐ท๐น๐ผ = ๐๐ฅ๐ ฮฃ๐ก๐บ(ฮพ๐บ๐ผ)
๐บ(0)โ 1
๐น dependence: โข G = ๐บ(๐ ) โข ฮฃ = ฮฃ(๐ )
ฮป dependence: โข ฮพ๐บ๐ผ = ฮพ๐บ๐ผ(ฮป)
โข ฮฃ = ฮฃ(ฮป)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Reimann et al., J.Appl.Cryst. (2016)
DFI depends only on one value of the pair correlation function
More information can be gained by:
Changing G
Different particle sizes
๐บ(ฮพ๐บ๐ผ)
Only possible in rare cases
Possibilities and Restrictions
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
DFI depends only on one value of the pair correlation function
More information can be gained by:
Changing ฮพ๐ฎ๐ฐ
Change the setup or
Sample position
Changing G
Different particle sizes
๐บ(ฮพ๐บ๐ผ)
Various geometric constraints
Tedious callibrations
Only possible in rare cases
Possibilities and Restrictions
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
DFI depends only on one value of the pair correlation function
More information can be gained by:
Changing ฮพ๐ฎ๐ฐ
Change the setup or
Sample position Vary the wavelength
Changing G
Different particle sizes
๐บ(ฮพ๐บ๐ผ)
Only a small range is accessible
Only possible in rare cases
Possibilities and Restrictions
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Various geometric constraints
Tedious callibrations
Inhomogeneous Scattering
๐๐๐ฆ
๐ฯ
๐ฮฉ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
homogenious inhomogenious
๐๐๐ฆ
๐ฯ
๐ฮฉ
Inhomogeneous Scattering
๐๐๐ฆ
๐ฯ
๐ฮฉ
๐๐๐ฆ
Rotate sample
Rotate gratings
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
๐ฯ(๐๐ฅ)
๐ฮฉ ๐ ๐๐๐กdepends on the relative orientation of sample and gratings
homogenious inhomogenious
๐๐๐ฆ
๐ฯ
๐ฮฉ
Inhomogeneous Scattering
๐๐๐ฆ
๐ฯ
๐ฮฉ
๐๐๐ฆ
Rotate sample
Rotate gratings
๐ท๐น๐ผ(ฯ) = ๐๐ฅ๐ ฮฃ๐ก๐บ(ฮพ๐บ๐ผ cos ฯ ,โฮพ๐บ๐ผsin(ฯ))
๐บ(0)โ 1
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
๐ฯ(๐๐ฅ)
๐ฮฉ ๐ ๐๐๐กdepends on the relative orientation of sample and gratings
homogenious inhomogenious
๐๐๐ฆ
๐ฯ
๐ฮฉ
Anisotropic scattering in Brass
Extrusion Moulded Brass: anisotropic โข production โข crystallites โข scattering
DFI
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Neuwirth, Master Thesis (2017)
Anisotropic scattering in Brass
๐ท๐น๐ผ(ฯ) = ๐๐ฅ๐ โฮพ๐บ๐ผ2ฯ๐ฅ2 + ฯ๐ฆ
2 โ ฯ๐ฅ2 ๐ ๐๐2 ฯโ ฯ ฯ๐๐ = ฯ๐ฅ
2 โ ฯ๐ฆ2
Bi gaussian scattering cross section
Different distribution of crystallite size in x and y direction
๐๐๐ ๐๐
๐
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
T. Neuwirth, Master Thesis (2017)
Anisotropic scattering in Brass
๐๐๐ ๐
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
๐ท๐น๐ผ(ฯ) = ๐๐ฅ๐ โฮพ๐บ๐ผ2ฯ๐ฅ2 + ฯ๐ฆ
2 โ ฯ๐ฅ2 ๐ ๐๐2 ฯโ ฯ ฯ๐๐ = ฯ๐ฅ
2 โ ฯ๐ฆ2
Bi gaussian scattering cross section
Different distribution of crystallite size in x and y direction
T. Neuwirth, Master Thesis (2017)
Anisotropic scattering in Brass
๐ซ๐ญ๐ฐ(๐)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
๐ท๐น๐ผ(ฯ) = ๐๐ฅ๐ โฮพ๐บ๐ผ2ฯ๐ฅ2 + ฯ๐ฆ
2 โ ฯ๐ฅ2 ๐ ๐๐2 ฯโ ฯ ฯ๐๐ = ฯ๐ฅ
2 โ ฯ๐ฆ2
Bi gaussian scattering cross section
Different distribution of crystallite size in x and y direction
T. Neuwirth, Master Thesis (2017)
Magnetic Domains in Superconductors
Normal Conductor Superconductor
low field high field
๐ฉ ๐ฉ ๐ฉ
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Meissner state Schubnikov state
Magnetic Domains in Superconductors
Normal Conductor Superconductor
low field high field
๐ฉ ๐ฉ ๐ฉ
๐ฉโ
flux line lattice intermediate mixed state
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
E. H. Brandt & M. P. Das, J. Supercond. Nov. Magn. (2011) U. Essmann & H. Trรคuble, Phys. Lett. A (1967)
The Intermediate Mixed State in Niobium
B = 40mT
Field Cooling into the IMS
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
What changes? โข domain size โข domain density โข scattering contrast
The Intermediate Mixed State in Niobium
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
Supression of the IMS
Additional field enters the sample โข IMS in the center โข FLL at the edge โข pinning along crystal axes
Heterogeneous Phase transition T. Reimann, Ph.D Thesis (2016)
From USANS to nGI
๐บ(๐ฅ, 0)
๐ฯ(๐๐ฅ)
๐ฮฉ๐ ๐๐๐ก
๐ท๐น๐ผ = ๐๐ฅ๐ ฮฃ๐ก๐บ(ฮพ๐บ๐ผ)
๐บ(0)โ 1
USANS Measurement
Pair Correlation Function
DFI Signal
T. Reimann, Ph.D Thesis (2016)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de
From USANS to nGI
Magnetic Field Scan Wavelength Scan
Domain size changes (USANS result)
Probedฮพ๐บ๐ผ changes
T. Reimann, Ph.D Thesis (2016)
29.08.2017 - IAEA workshop AUNIRA - alexander.backs@frm2.tum.de