Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s...

Today’s Outline - September 18, 2013

• Models of Surfaces

• Reflectivity from the MRCAT mirror

APS Visits:10-BM: Friday, November 8, 201318-ID: TBD

Homework Assignment #02:Problems to be provideddue Monday, September 23, 2013

C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 1 / 11

Surface scattering cross-section

)2 ∫S

∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

⟩e iQx xe iQy ydxdy

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

where Ao/ sin θ1 is just the illuminated surface area and the term in theangled brackets is an ensemble average over all possible choices of theorigin within the illuminated area.Finally, it is assumed that the statistics of the height variation areGaussian and(

sin θ1

∫e−Q2

z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy

)2 ∫S

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫e−Q2

)2 ∫S

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫e−Q2

)2 ∫S

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

where Ao/ sin θ1 is just the illuminated surface area

and the term in theangled brackets is an ensemble average over all possible choices of theorigin within the illuminated area.Finally, it is assumed that the statistics of the height variation areGaussian and(

sin θ1

∫e−Q2

)2 ∫S

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

where Ao/ sin θ1 is just the illuminated surface area and the term in theangled brackets is an ensemble average over all possible choices of theorigin within the illuminated area.

Finally, it is assumed that the statistics of the height variation areGaussian and(

sin θ1

∫e−Q2

)2 ∫S

)2 ∫S ′dx ′dy ′

⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫ ⟨e iQz (h(0,0)−h(x ,y))

sin θ1

∫e−Q2

Limiting Case - Flat Surface

Define a function g(x , y) =⟨

[h(0, 0)− h(x , y)]2⟩

which can be modeled

in various ways.

For a perfectly flat surface, h(x , y) = 0 for all x and y .

by the definition of a deltafunction

2πδ(q) =

∫e iqxdx

the expression for the scat-tered intensity in terms ofthe momentum transfer wavevectors is

sin θ1

∫e iQx xe iQy ydxdy

sin θ1δ(Qx )δ(Qy )

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

[h(0, 0)− h(x , y)]2⟩

in various ways.

2πδ(q) =

∫e iqxdx

sin θ1

)∆Qx ∆Qy

k2 sin θ2

R(Qz ) =Isc

Uncorrelated Surfaces

For a totally uncorrelated surface, h(x , y) is independent from h(x ′, y ′) and

⟨[h(0, 0)− h(x , y)]2

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

This quantity is simply related to the rms roughness, σ by σ2 =⟨h2⟩

andthe cross-section is given by(

sin θ1

∫e−Q2

z 〈h2〉2/2e iQx xe iQy ydxdy

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

Which, apart from the term containing σ is simply the Fresnelcross-section for a flat surface(

)Fresnel

e−Q2z σ

For a totally uncorrelated surface, h(x , y) is independent from h(x ′, y ′) and⟨[h(0, 0)− h(x , y)]2

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

Which, apart from the term containing σ is simply the Fresnelcross-section for a flat surface

)Fresnel

e−Q2z σ

⟩= 〈h(0, 0)〉2 − 2 〈h(0, 0)〉〈h(x , y)〉+ 〈h(x , y)〉2

= 2⟨h2⟩

sin θ1

∫e−Q2

sin θ1e−Q2

z σ2∫

e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

Surface Roughness Effect

)Fresnel

e−Q2z σ

for a perfectly flat surface,we get the Fresnel reflectivityderived for a thin slab

for an uncorrelated roughsurface, the reflectivity isreduced by an exponentialfactor controlled by the rmssurface roughness σ 0 0.2 0.4 0.6 0.8 1

Q (Å-1

σ=0 Å

∆=68 Å

)Fresnel

e−Q2z σ

for an uncorrelated roughsurface, the reflectivity isreduced by an exponentialfactor controlled by the rmssurface roughness σ

0 0.2 0.4 0.6 0.8 1

Q (Å-1

σ=0 Å

∆=68 Å

)Fresnel

e−Q2z σ

Q (Å-1

σ=0 Å

σ=3 Å

∆=68 Å

)Fresnel

e−Q2z σ

Q (Å-1

σ=0 Å

σ=3 Å

∆=68 Å

σ=3 Å

Correlated Surfaces

Assume that height fluctuations are isotropically correlated in the x-yplane. Therefore, g(x , y) = g(r) = g(

√x2 + y2).

In the limit that the correlations are unbounded as r →∞, g(x , y) isgiven by

g(x , y) = Ar2h

where h is a fractal parameter which defines the shape of the surface.

jagged surface for h 1 smoother surface for h→ 1

If the resolution in the y direction is very broad (typical for a synchrotron),we can eliminate the y-integral and have(

sin θ1

∫e−AQ2

z |x |2h/2 cos(Qxx)dx

Correlated Surfaces

√x2 + y2).

g(x , y) = Ar2h

sin θ1

∫e−AQ2

Correlated Surfaces

√x2 + y2).

g(x , y) = Ar2h

jagged surface for h 1

smoother surface for h→ 1

sin θ1

∫e−AQ2

Correlated Surfaces

√x2 + y2).

g(x , y) = Ar2h

sin θ1

∫e−AQ2

Correlated Surfaces

√x2 + y2).

g(x , y) = Ar2h

If the resolution in the y direction is very broad (typical for a synchrotron),we can eliminate the y-integral and have

sin θ1

∫e−AQ2

Correlated Surfaces

√x2 + y2).

g(x , y) = Ar2h

sin θ1

∫e−AQ2

Unbounded Correlations - Limiting Cases

This integral can be evaluated in closed form for two special cases, bothhaving a broad diffuse scattering and no specular peak.

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

Lorentzian with half-width AQ2z /2

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

Gaussian with variance AQ2z

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

h = 1/2(dσ

2 sin θ1

(Q2x + (A/2)2Q4

h = 1(dσ

(2√πAor

2 sin θ1

e− 1

-0.01 -0.005 0 0.005 0.01

100000

-0.04 -0.02 0 0.02 0.04

1e-300

1e-250

1e-200

1e-150

1e-100

Bounded Correlations

If the correlations remain bounded as r →∞

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

whereC (x , y) = σ2e−(r/ξ)

sin θ1e−Q2

z σ2∫

eQ2z C(x ,y)e iQx xe iQy ydxdy

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

And the scattering exhibits both a specular peak, reduced by uncorrelatedroughness, and diffuse scattering from the correlated portion of the surface

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

g(x , y) = 2⟨h2⟩− 2 〈h(0, 0)h(x .y)〉 = 2σ2 − 2C (x , y)

sin θ1e−Q2

z σ2∫

sin θ1e−Q2

z σ2∫ [

eQ2z C(x ,y) − 1 + 1

]e iQx xe iQy ydxdy

)Fresnel

e−Q2z σ

sin θ1e−Q2

z σ2Fdiffuse(~Q)

The MRCAT Mirror

x-rays

Ultra low expansion glass polished to afew A roughness

One platinum stripe and one rhodiumstripe deposited along the length of themirror on top of a chromium buffer layer

A mounting system which permits angu-lar positioning to less than 1/100 of adegree as well as horizontal and verticalmotions

A bending mechanism to permit verticalfocusing of the beam to ∼ 60 µm

The MRCAT Mirror

x-rays

The MRCAT Mirror

x-rays

The MRCAT Mirror

x-rays

The MRCAT Mirror

x-rays

Mirror Performance I

When illuminated with 12 keVx-rays on the glass “stripe”, thereflectivity is measured as:

With the Rh stripe, the thinslab reflection is evident andthe critical angle is significantlyhigher.

The Pt stripe gives a higher crit-ical angle still but a lower reflec-tivity and it looks like an infiniteslab. Why?

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

The Pt stripe gives a higher crit-ical angle still but a lower reflec-tivity and it looks like an infiniteslab.

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

The Pt stripe gives a higher crit-ical angle still but a lower reflec-tivity and it looks like an infiniteslab.

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

The Pt stripe gives a higher crit-ical angle still but a lower reflec-tivity and it looks like an infiniteslab. Why? 0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

12 keV

Mirror Performance II

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

Pt stripe

12 keV

20 keV

30 keV

As we move up in energy thecritical angle for the Pt stripedrops.

The reflectivity at low anglesimproves as we are well awayfrom the Pt absorption edgesat 11,565 eV, 13,273 eV, and13,880 eV.

As energy rises, the Pt layer be-gins to show the reflectivity of athin slab.

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

Pt stripe

12 keV

20 keV

30 keV

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

Pt stripe

12 keV

20 keV

30 keV

0.1 0.2 0.3 0.4 0.5

α (degrees)

Reflectivity

Pt stripe

12 keV

20 keV

30 keV

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