today’s outline - september 18, 2013csrri.iit.edu/~segre/phys570/13f/lecture_09.pdf · today’s...
TRANSCRIPT
![Page 1: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/1.jpg)
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
![Page 2: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/2.jpg)
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
![Page 3: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/3.jpg)
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
![Page 4: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/4.jpg)
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
![Page 5: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/5.jpg)
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
![Page 6: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/6.jpg)
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
![Page 7: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/7.jpg)
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
![Page 8: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/8.jpg)
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
![Page 9: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/9.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 10: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/10.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 11: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/11.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 12: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/12.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 13: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/13.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 14: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/14.jpg)
Surface scattering cross-section
(dσ
dΩ
)=
(roρ
Qz
)2 ∫S
∫S ′e iQz (h(x ,y)−h(x ′,y ′))e iQx (x−x ′)e iQy (y−y ′)dxdydx ′dy ′
=
(roρ
Qz
)2 ∫S ′dx ′dy ′
∫S
⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1
∫ ⟨e iQz (h(0,0)−h(x ,y))
⟩e iQx xe iQy ydxdy
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈[h(0,0)−h(x ,y)]2〉/2e iQx xe iQy ydxdy
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 2 / 11
![Page 15: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/15.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 16: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/16.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 17: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/17.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 18: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/18.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 19: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/19.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 20: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/20.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 21: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/21.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
![Page 22: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/22.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1δ(Qx )δ(Qy )
Isc =
(IoAo
)(dσ
dΩ
)∆Qx ∆Qy
k2 sin θ2
R(Qz ) =Isc
Io=
(Q2
c /8
Qz
)2(1
Qz/2
)2
=
(Qc
2Qz
)4
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 3 / 11
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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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
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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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 25: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/25.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 26: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/26.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 27: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/27.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 28: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/28.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 29: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/29.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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
(dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
![Page 30: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/30.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−Q2
z 〈h2〉2/2e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
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(
dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 4 / 11
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Surface Roughness Effect
(dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
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
)
10-16
10-12
10-8
10-4
100
RS
lab
σ=0 Å
∆=68 Å
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 5 / 11
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Surface Roughness Effect
(dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
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
)
10-16
10-12
10-8
10-4
100
RS
lab
σ=0 Å
∆=68 Å
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 5 / 11
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Surface Roughness Effect
(dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
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
)
10-16
10-12
10-8
10-4
100
RS
lab
σ=0 Å
σ=3 Å
∆=68 Å
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 5 / 11
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Surface Roughness Effect
(dσ
dΩ
)=
(dσ
dΩ
)Fresnel
e−Q2z σ
2
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
)
10-16
10-12
10-8
10-4
100
RS
lab
σ=0 Å
σ=3 Å
∆=68 Å
σ=3 Å
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 5 / 11
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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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 36: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/36.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 37: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/37.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 38: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/38.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 39: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/39.jpg)
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
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 40: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/40.jpg)
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(
dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1
∫e−AQ2
z |x |2h/2 cos(Qxx)dx
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 6 / 11
![Page 41: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/41.jpg)
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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 7 / 11
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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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 7 / 11
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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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
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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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 7 / 11
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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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 7 / 11
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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
dΩ
)=
(Aor
2oρ
2
2 sin θ1
)A
(Q2x + (A/2)2Q4
z )
Lorentzian with half-width AQ2z /2
h = 1(dσ
dΩ
)=
(2√πAor
2oρ
2
2 sin θ1
)1
Q4z
e− 1
2
(Q2
xAQ2
z
)
Gaussian with variance AQ2z
-0.01 -0.005 0 0.005 0.01
Qx
0
0.02
0.04
0.06
0.08
0.1
Qz
1000
10000
100000
1e+06
1e+07
1e+08
1e+09
-0.04 -0.02 0 0.02 0.04
Qx
0
0.002
0.004
0.006
0.008
0.01
Qz
1e-300
1e-250
1e-200
1e-150
1e-100
1e-50
1
1e+50
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 7 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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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)
2h
(dσ
dΩ
)=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫
eQ2z C(x ,y)e iQx xe iQy ydxdy
=
(roρ
Qz
)2 Ao
sin θ1e−Q2
z σ2∫ [
eQ2z C(x ,y) − 1 + 1
]e iQx xe iQy ydxdy
=
(dσ
dΩ
)Fresnel
e−Q2z σ
2+
(roρ
Qz
)2 Ao
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 8 / 11
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The MRCAT Mirror
50 cm
x-rays
Rh Pt
glass
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 9 / 11
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The MRCAT Mirror
50 cm
x-rays
Rh Pt
glass
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 9 / 11
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The MRCAT Mirror
50 cm
x-rays
Rh Pt
glass
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 9 / 11
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The MRCAT Mirror
50 cm
x-rays
Rh Pt
glass
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 9 / 11
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The MRCAT Mirror
50 cm
x-rays
Rh Pt
glass
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
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 9 / 11
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 10 / 11
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 10 / 11
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 10 / 11
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 10 / 11
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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)
0.0
0.2
0.4
0.6
0.8
1.0
Reflectivity
12 keV
glass
Rh
Pt
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 10 / 11
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Mirror Performance II
0.1 0.2 0.3 0.4 0.5
α (degrees)
0.0
0.2
0.4
0.6
0.8
1.0
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.
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Mirror Performance II
0.1 0.2 0.3 0.4 0.5
α (degrees)
0.0
0.2
0.4
0.6
0.8
1.0
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.
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 11 / 11
![Page 67: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/67.jpg)
Mirror Performance II
0.1 0.2 0.3 0.4 0.5
α (degrees)
0.0
0.2
0.4
0.6
0.8
1.0
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.
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 11 / 11
![Page 68: Today’s Outline - September 18, 2013csrri.iit.edu/~segre/phys570/13F/lecture_09.pdf · Today’s Outline - September 18, 2013 Models of Surfaces Re ectivity from the MRCAT mirror](https://reader034.vdocument.in/reader034/viewer/2022052004/60170e962597816425022722/html5/thumbnails/68.jpg)
Mirror Performance II
0.1 0.2 0.3 0.4 0.5
α (degrees)
0.0
0.2
0.4
0.6
0.8
1.0
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.
C. Segre (IIT) PHYS 570 - Fall 2013 September 18, 2013 11 / 11