photon correlation spectroscopy with...

PHOTON CORRELATION

SPECTROSCOPY WITH

COHERENT X-RAYS

Razib Obaid

PHYS 570 : Introduction to Synchroton Radiation

Overview

Introduction : What is XPCS?

Content

Light scattering

Scattering with coherent X-rays

Disorder under coherent illumination

X-ray Photon Correlation Spectroscopy

Static and Dynamic Properties of Colloidal Suspensions

Slow Dynamics in Magnetic Systems

Conclusion

Introduction (as well as summary)

X-ray Photon Correlation Spectroscopy (XPCS) is a method to characterize the

equilibrium dynamics of condensed matter by determining the intensity

autocorrelation function, g2(Q,t), of the scattered X-ray intensity (x-ray speckle

pattern) versus delay time, t , and wavevector, Q.

Quantities to keep in mind while viewing this talk:

𝑓 𝑸, 𝑡 , normalized scattering function

𝐹(𝑄, 𝑡), structure factor given by:

𝐹 𝑸, 𝑡 =1

𝑁(𝑓(𝑄))2

𝑛

𝑚

𝑓𝑛 𝑸 𝑓𝑚 𝑸 exp{𝑖𝑸[𝑟𝑛 0 − 𝑟𝑚 𝑡 ]}

𝐿𝐿, longitudinal coherence length, and 𝐿𝑇 , transverse coherence length.

B, brilliance and undulator flux.

Interesting for any condensed matter system.

Dynamic Light scattering (Photon

Correlation Spectroscopy)

Rayleigh scattering if size, 2r <<




Diffusion velocity, D is given by:

𝐷 =𝑘𝑇

6𝜋𝑎





𝐷 =𝑘𝑇

6𝜋𝑎

In Brownian motion, the constantly changing

distance between particles changes the

phase overlap between the scattered light

and incoming light.





𝐷 =𝑘𝑇

6𝜋𝑎




and incoming light.

Coherent light scattered by the disordered

system will create interference pattern.





𝐷 =𝑘𝑇

6𝜋𝑎




and incoming light.



This projected on a screen will depict a

random ‘speckle’ pattern.





𝐷 =𝑘𝑇

6𝜋𝑎




and incoming light.



This projected on a screen will depict a

random ‘speckle’ pattern.

‘Speckle’ pattern are related to the exact

spatial arrangement of the disorder.

If the spatial arrangement of the disorder changes as a function of time the

“speckle” pattern will also change.

A measurement of the temporal intensity fluctuations of a single or equivalent

speckle is thus a measure of the underlying dynamics.

The temporal intensity fluctuations can be characterized by: Correlation

Spectroscopy Techniques.

Coherent visible light from a laser source ( ~ 500 nm or 5000 Å ): Photon

Correlation Spectroscopy (PCS) or Dynamic Light Scattering (DLS)

Coherent light from a synchrotron source ( ~ 1Å): X-Ray Photon Correlation

Spectroscopy (XPCS)

Pop-Quiz

What is the order of magnitude of Brilliance in a

3rd generation synchrotron ?


In a third generation synchrotron, the brilliance, B, is of the order of

1020ph/s/𝑚𝑟𝑎𝑑2/𝑚𝑚2/ 0.1% bandwidth.




Using undulators, a fraction of the flux is used as the source for the coherent beam.





The fraction of the ID flux transversely coherent is:

𝐹𝑐 =2

2

𝐵






𝐹𝑐 =2

2

𝐵

From the textbook (Ch1), we know that:

𝐿𝑇 =2

1

∆where ∆ = 𝑠/𝑅 , is the angular source size

𝐿𝐿 =2

∆

Typically for a third generation sources, 𝐿𝑇 = 10𝜇𝑚 (horizontally) and 100𝜇𝑚(vertically).






𝐹𝑐 =2

2

𝐵

From the textbook (Ch1), we know that:

𝐿𝑇 =2

1

∆where ∆ = 𝑠/𝑅

𝐿𝐿 =2

∆

Typically for a third generation sources, 𝐿𝑇 = 10𝜇𝑚 (horizontally) and 100𝜇𝑚(vertically).

𝐿𝐿 is also the measure of temporal coherence of the beam and thusly depends on

the monochoromaticity and for perfect crystal ~ 5𝜇𝑚 .


(cont…)

Requirement of coherent illumination implies that there is a maximum path length

difference (PLD). For XPCS, the two important requirements are:

PLD ≤ 𝐿𝐿

Incident beam size, 𝑑 ≤ 𝐿𝑇

𝑃𝐿𝐷 = 2𝑊 sin2+ 𝑑 sin 2


(cont…)

Requirement of coherent illumination implies that there is a maximum path length

difference (PLD). For XPCS, the two important requirements are:

PLD ≤ 𝐿𝐿

Incident beam size, 𝑑 ≤ 𝐿𝑇

𝑃𝐿𝐷 = 2𝑊 sin2+ 𝑑 sin 2

This sets a limit for the maximum momentum transfer, Q = 4𝜋

sin𝑚𝑎𝑥

Some details of the setup in ESRF

Slits-Mirror = 44.2 m, Mirror-Piezo mirror = 0.8 m, Piezo-Collimating aperture =

0.5 m, Collimator-sample = 0.1 m, sample-detector = 2m.

The integrated coherent flux through a 12m pinhole is ~ 109 photons/s for =1Å.

The sample distance, 𝑅𝑐 < 𝑑2/ .(?)

Angular size of the individual speckle, 𝐷𝑠 =𝑑

2

+ ∆21/2


The far field instantaneous intensity,

𝐼 𝑄, 𝑡 = 𝑛 exp[𝑖𝑸 ∙ 𝑹𝑛 (𝑡)] 𝑓𝑛(𝑸)2,

with 𝑓𝑛 𝑸 = −𝑟0 𝑅𝑛 𝑒𝑖𝑸. 𝑹𝑛. (Chap 4

of textbook)

Any measurement will be a time

average, 𝐼(𝑸, 𝑡) 𝑇.

If the sample has static random disorder,

𝐼(𝑸, 𝑡) 𝑇 will show ‘speckle’.


The far field instantaneous intensity,

𝐼 𝑄, 𝑡 = 𝑛 exp[𝑖𝑸 ∙ 𝑹𝑛 (𝑡)] 𝑓𝑛(𝑸)2,

with 𝑓𝑛 𝑸 = −𝑟0 𝑅𝑛 𝑒𝑖𝑸. 𝑹𝑛. (Chap 4

of textbook)

Any measurement will be a time

average, 𝐼(𝑸, 𝑡) 𝑇.

If the sample has static random disorder,

𝐼(𝑸, 𝑡) 𝑇 will show ‘speckle’

Else, 𝐼(𝑸, 𝑡) 𝑇 will just show the envelop

(ensemble average).

X-ray photon correlation Spectroscopy

Time dependent normalized intensity autocorrelation function,

𝑔2 𝑸, 𝑡 =𝐼 𝑸, 0 𝐼 𝑸, 𝑡

𝐼(𝑸) 2



𝑔2 𝑸, 𝑡 =𝐼 𝑸, 0 𝐼 𝑸, 𝑡

𝐼(𝑸) 2

= 1 + 𝛽(𝑸)𝐸 𝑸,0 𝐸(𝑸,𝑡) 2

𝐼(𝑸) 2

= 1 + 𝛽 𝑸 [𝑓(𝑸, 𝑡]2



𝑔2 𝑸, 𝑡 =𝐼 𝑸, 0 𝐼 𝑸, 𝑡

𝐼(𝑸) 2

= 1 + 𝛽(𝑸)𝐸 𝑸,0 𝐸(𝑸,𝑡) 2

𝐼(𝑸) 2

= 1 + 𝛽 𝑸 [𝑓(𝑸, 𝑡]2

where normalized intermediate scattering function 𝑓 𝑸, 𝑡 = 𝐹(𝑸, 𝑡)/𝐹(𝑸, 0)



𝑔2 𝑸, 𝑡 =𝐼 𝑸, 0 𝐼 𝑸, 𝑡

𝐼(𝑸) 2

= 1 + 𝛽(𝑸)𝐸 𝑸,0 𝐸(𝑸,𝑡) 2

𝐼(𝑸) 2

= 1 + 𝛽 𝑸 [𝑓(𝑸, 𝑡]2


Revisiting the example of DLS, 𝑓 𝑸, 𝑡 = exp(−𝐷 𝑄2𝑡)



𝑔2 𝑸, 𝑡 =𝐼 𝑸, 0 𝐼 𝑸, 𝑡

𝐼(𝑸) 2

= 1 + 𝛽(𝑸)𝐸 𝑸,0 𝐸(𝑸,𝑡) 2

𝐼(𝑸) 2

= 1 + 𝛽 𝑸 [𝑓(𝑸, 𝑡]2


Revisiting the example of DLS, 𝑓 𝑸, 𝑡 = exp(−𝐷 𝑄2𝑡)

But, in the presence of particle interaction, 𝑓 𝑸, 𝑡 = exp(−𝐷(𝑄) 𝑄2𝑡)

A plot of ln f(Q,t) vs t.

A useful quantity is the slope of this graph

at t → 0 is . It is define by:

𝑄 = −𝐷 𝑄 𝑄2

Structure and Dynamics of Colloidal

Suspensions

Structure is given by:

𝑆 𝑄 = 1 + 4 𝜋 𝜌 𝑔 𝑟 − 1sin 𝑄𝑟

𝑄𝑟𝑟2𝑑𝑟

with g(r) being the radial distribution of

the potential between two spheres

and 𝜌 being the number density.

Dynamics: Interaction between colloids,

colloid-solvent, hydrodynamics etc.

For 𝜌 ≪ 1, 𝑆 𝑄 = 1, and D(Q) = 𝐷0

Fig: Colloidal Silica at 1% vol. in

a glycerol/water mixture

Structure and Dynamics of Colloidal

Suspensions

Structure is given by:

𝑆 𝑄 = 1 + 4 𝜋 𝜌 𝑔 𝑟 − 1sin 𝑄𝑟

𝑄𝑟𝑟2𝑑𝑟

with g(r) being the radial distribution of

the potential between two spheres

and 𝜌 being the number density.

Dynamics: Interaction between colloids,

colloid-solvent, hydrodynamics etc.

For 𝜌 ≪ 1, 𝑆 𝑄 = 1, and D(Q) = 𝐷0

For higher concentration, D(Q) = 𝐷0 H(Q) /

S(Q)

Fig: Colloidal PMMA at 37%

vol. in cis-decalin mixture

Slow dynamics in Magnetic systems

Magnetic speckles were also

observed in resonant small angle

scattering with soft X-rays.

Soft-xrays were used because of

higher coherence length, thus

higher scattering cross-sections

associated with magnetic

contributions.

Fig: Magnetic speckles observed in resonant

SAXS with soft x-rays from magnetic domain

in 350 Angstrom film of GdFe2 .

Conclusion

Applications:

Condensed matter dynamics

Dynamics of complex fluids

Ultra-slow and non-equilibrium dynamics

2D systems

Conclusion

Applications:

Condensed matter dynamics

Dynamics of complex fluids

Ultra-slow and non-equilibrium dynamics

2D systems

XPCS – Cons:

Need (partially) coherent beam, thus lesser photons than DLS. Low SNRs.

Need fast detectors. Area detectors are too slow, ergo, the title ‘Slow’ dynamics.

X-ray scattering cross-sections are far smaller than laser.

XPCS – Pros:

X-ray wavelengths are much shorter, thus allows larger momentum transfers.

No multiple scattering effects since refractive index is very close to 1.

References

Grubel,G., Zontone, Z., “Correlation spectroscopy with Coherent X-rays”, J. Alloys. Comp. 362

(2004) 3-11.

Sutton, M., Mochrie S., et.el. “Observation of Speckle by Diffraction of X-rays”, Nature 352

(1991) 608.

Grubel,G., Als-Nielsen, J., J. Phys. IV C9(4) (1994) 27.

Als-Nielsen, J., McMorror, D., “Elements of Modern X-ray Physics”, Wiley (2001).

Berne, B., Pecora, R., “Dynamic Light Scattering with Applications”, Wiley (1976).

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