today’s outline - february 27, 2020segre/phys570/20s/lecture_14.pdf · c. segre (iit) phys 570 -...
TRANSCRIPT
![Page 1: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/1.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 2: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/2.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 3: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/3.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 4: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/4.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 5: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/5.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 6: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/6.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 7: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/7.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 8: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/8.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 9: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/9.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 10: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/10.jpg)
Today’s Outline - February 27, 2020
• Lattices & space groups
• Ewald sphere
• XRayView demonstration
• Crystal Truncation Rods
• Diffuse Scattering
• Modulated structures
• Lattice vibrations
Reading assignment: Chapter 5.2
Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 10, 2020
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 1 / 22
![Page 11: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/11.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 12: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/12.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 13: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/13.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 14: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/14.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 15: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/15.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 16: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/16.jpg)
BCC structure factor
In the body centered cubic structure, there are 2 atoms in theconventional, cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2 +~a3)
the unit cell structure factor is thus
F bcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k+l))
= f (~G )×
{2 h + k + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 2 / 22
![Page 17: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/17.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 18: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/18.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 19: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/19.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 20: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/20.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 21: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/21.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 22: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/22.jpg)
FCC structure factor
In the face centered cubic structure, there are 4 atoms in the conventional,cubic unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3), ~r4 =
1
2(~a1 +~a3)
the unit cell structure factor is thus
F fcchkl = f (~G )
∑j
e i~G ·~rj
= f (~G )(
1 + e iπ(h+k) + e iπ(k+l) + e iπ(h+l))
= f (~G )×
{4 h + k , k + l , h + l = 2n
0 otherwise
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 3 / 22
![Page 23: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/23.jpg)
Diamond structure
This is a face centered cubic structure with two atoms in the basis whichleads to 8 atoms in the conventional unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3) ~r4 =
1
2(~a1 +~a3)
~r5 =1
4(~a1 +~a2 +~a3), ~r6 =
1
4(3~a1 + 3~a2 +~a3)
~r7 =1
4(~a1 + 3~a2 + 3~a3), ~r8 =
1
4(3~a1 +~a2 + 3~a3)
F diamondhkl = f (~G )
(1 + e iπ(h+k) + e iπ(k+l)
+ e iπ(h+l) + e iπ(h+k+l)/2 + e iπ(3h+3k+l)/2
+ e iπ(h+3k+3l)/2 + e iπ(3h+k+3l)/2)
This is non-zero when h,k,l all even and h +k + l = 4n or h,k ,l all odd
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 4 / 22
![Page 24: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/24.jpg)
Diamond structure
This is a face centered cubic structure with two atoms in the basis whichleads to 8 atoms in the conventional unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3) ~r4 =
1
2(~a1 +~a3)
~r5 =1
4(~a1 +~a2 +~a3), ~r6 =
1
4(3~a1 + 3~a2 +~a3)
~r7 =1
4(~a1 + 3~a2 + 3~a3), ~r8 =
1
4(3~a1 +~a2 + 3~a3)
F diamondhkl = f (~G )
(1 + e iπ(h+k) + e iπ(k+l)
+ e iπ(h+l) + e iπ(h+k+l)/2 + e iπ(3h+3k+l)/2
+ e iπ(h+3k+3l)/2 + e iπ(3h+k+3l)/2)
This is non-zero when h,k,l all even and h +k + l = 4n or h,k ,l all odd
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 4 / 22
![Page 25: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/25.jpg)
Diamond structure
This is a face centered cubic structure with two atoms in the basis whichleads to 8 atoms in the conventional unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3) ~r4 =
1
2(~a1 +~a3)
~r5 =1
4(~a1 +~a2 +~a3), ~r6 =
1
4(3~a1 + 3~a2 +~a3)
~r7 =1
4(~a1 + 3~a2 + 3~a3), ~r8 =
1
4(3~a1 +~a2 + 3~a3)
F diamondhkl = f (~G )
(1 + e iπ(h+k) + e iπ(k+l)
+ e iπ(h+l) + e iπ(h+k+l)/2 + e iπ(3h+3k+l)/2
+ e iπ(h+3k+3l)/2 + e iπ(3h+k+3l)/2)
This is non-zero when h,k,l all even and h +k + l = 4n or h,k ,l all odd
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 4 / 22
![Page 26: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/26.jpg)
Diamond structure
This is a face centered cubic structure with two atoms in the basis whichleads to 8 atoms in the conventional unit cell. These are located at
~r1 = 0, ~r2 =1
2(~a1 +~a2), ~r3 =
1
2(~a2 +~a3) ~r4 =
1
2(~a1 +~a3)
~r5 =1
4(~a1 +~a2 +~a3), ~r6 =
1
4(3~a1 + 3~a2 +~a3)
~r7 =1
4(~a1 + 3~a2 + 3~a3), ~r8 =
1
4(3~a1 +~a2 + 3~a3)
F diamondhkl = f (~G )
(1 + e iπ(h+k) + e iπ(k+l)
+ e iπ(h+l) + e iπ(h+k+l)/2 + e iπ(3h+3k+l)/2
+ e iπ(h+3k+3l)/2 + e iπ(3h+k+3l)/2)
This is non-zero when h,k,l all even and h +k + l = 4n or h,k ,l all odd
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 4 / 22
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Heteroatomic structures
← bcc
sc →
← diamondfcc →
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 5 / 22
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Heteroatomic structures
← bccsc →
← diamondfcc →
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 5 / 22
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Heteroatomic structures
← bccsc →
← diamond
fcc →
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 5 / 22
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Heteroatomic structures
← bccsc →
← diamondfcc →
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 5 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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The Ewald sphere
The Ewald sphere is a construct which per-mits the enumeration of reflections which ful-fill the Laue diffraction condition.
The sphere radius is set by the length of the ~kand ~k ′ vectors which characterize the incidentand scattered (where the detector is placed)x-rays and ∆~k being the bandwidth of theincident x-rays
As the detector moves, ~k ′ rotates but theEwald sphere remains constant.
k
Q
∆k
k
The xrayview program can be used to gain a more intuitive understandingof the Ewald sphere.
http://www.phillipslab.org/software
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 6 / 22
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Ewald sphere & the reciprocal lattice
a*1
a*2
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays. The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald sphere & the reciprocal lattice
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays. The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald sphere & the reciprocal lattice
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays. The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald sphere & the reciprocal lattice
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays.
The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald sphere & the reciprocal lattice
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays. The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald sphere & the reciprocal lattice
The reciprocal lattice isdefined by the unit vec-tors ~a∗1 and ~a∗2.
The key parameter is therelative orientation of theincident wave vector ~k
As the crystal is rotatedwith respect to the inci-dent beam, the reciprocallattice also rotates
When the Ewald sphere intersects a reciprocal lattice point there will be adiffraction peak in the direction of the scattered x-rays. The diffractionvector, ~Q, is thus a reciprocal lattice vector
~Ghlk = h~a∗1 + k~a∗2C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 7 / 22
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Ewald construction
It is often more con-venient to visualize theEwald sphere by keep-ing the reciprocal latticefixed and “rotating” theincident beam to visual-ize the scattering geome-try.
In directions of ~k ′ (detec-tor position) where thereis no reciprocal latticepoint, there can be nodiffraction peak.
If the crystal is rotated slightly with respect to the incident beam, ~k , theremay be no Bragg reflections possible at all.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 8 / 22
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Ewald construction
It is often more con-venient to visualize theEwald sphere by keep-ing the reciprocal latticefixed and “rotating” theincident beam to visual-ize the scattering geome-try.
In directions of ~k ′ (detec-tor position) where thereis no reciprocal latticepoint, there can be nodiffraction peak.
If the crystal is rotated slightly with respect to the incident beam, ~k , theremay be no Bragg reflections possible at all.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 8 / 22
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Ewald construction
It is often more con-venient to visualize theEwald sphere by keep-ing the reciprocal latticefixed and “rotating” theincident beam to visual-ize the scattering geome-try.
In directions of ~k ′ (detec-tor position) where thereis no reciprocal latticepoint, there can be nodiffraction peak.
If the crystal is rotated slightly with respect to the incident beam, ~k , theremay be no Bragg reflections possible at all.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 8 / 22
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Polychromatic radiation
If ∆~k is large enough,there may be more thanone reflection lying onthe Ewald sphere.
With an area detector,there may then be multi-ple reflections appearingfor a particular orienta-tion (very common withprotein crystals wherethe unit cell is verylarge).
In protein crystallography, the area detector is in a fixed location withrespect to the incident beam and the crystal is rotated on a spindle so thatas Laue conditions are met, spots are produced on the detector at thediffraction angle
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 9 / 22
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Polychromatic radiation
If ∆~k is large enough,there may be more thanone reflection lying onthe Ewald sphere.
With an area detector,there may then be multi-ple reflections appearingfor a particular orienta-tion (very common withprotein crystals wherethe unit cell is verylarge).
In protein crystallography, the area detector is in a fixed location withrespect to the incident beam and the crystal is rotated on a spindle so thatas Laue conditions are met, spots are produced on the detector at thediffraction angle
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 9 / 22
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Polychromatic radiation
If ∆~k is large enough,there may be more thanone reflection lying onthe Ewald sphere.
With an area detector,there may then be multi-ple reflections appearingfor a particular orienta-tion (very common withprotein crystals wherethe unit cell is verylarge).
In protein crystallography, the area detector is in a fixed location withrespect to the incident beam and the crystal is rotated on a spindle so thatas Laue conditions are met, spots are produced on the detector at thediffraction angle
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 9 / 22
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Multiple scattering
If more than one recipro-cal lattice point is on theEwald sphere, scatteringcan occur internal to thecrystal.
The xrays are first scat-tered along ~kint thenalong the reciprocal lat-tice vector which con-nects the two points onthe Ewald sphere, ~G andto the detector at ~k ′.
This is the cause of monochromator glitches which sometimes removeintensity but can also add intensity to the reflection the detector is set tomeasure.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 10 / 22
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Multiple scattering
If more than one recipro-cal lattice point is on theEwald sphere, scatteringcan occur internal to thecrystal.
The xrays are first scat-tered along ~kint
thenalong the reciprocal lat-tice vector which con-nects the two points onthe Ewald sphere, ~G andto the detector at ~k ′.
This is the cause of monochromator glitches which sometimes removeintensity but can also add intensity to the reflection the detector is set tomeasure.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 10 / 22
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Multiple scattering
If more than one recipro-cal lattice point is on theEwald sphere, scatteringcan occur internal to thecrystal.
The xrays are first scat-tered along ~kint thenalong the reciprocal lat-tice vector which con-nects the two points onthe Ewald sphere, ~G
andto the detector at ~k ′.
This is the cause of monochromator glitches which sometimes removeintensity but can also add intensity to the reflection the detector is set tomeasure.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 10 / 22
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Multiple scattering
If more than one recipro-cal lattice point is on theEwald sphere, scatteringcan occur internal to thecrystal.
The xrays are first scat-tered along ~kint thenalong the reciprocal lat-tice vector which con-nects the two points onthe Ewald sphere, ~G andto the detector at ~k ′.
This is the cause of monochromator glitches which sometimes removeintensity but can also add intensity to the reflection the detector is set tomeasure.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 10 / 22
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Multiple scattering
If more than one recipro-cal lattice point is on theEwald sphere, scatteringcan occur internal to thecrystal.
The xrays are first scat-tered along ~kint thenalong the reciprocal lat-tice vector which con-nects the two points onthe Ewald sphere, ~G andto the detector at ~k ′.
This is the cause of monochromator glitches which sometimes removeintensity but can also add intensity to the reflection the detector is set tomeasure.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 10 / 22
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Laue diffraction
The Laue diffractiontechnique uses a widerange of radiation from~kmin to ~kmax
These define two Ewaldspheres and a volumebetween them such thatany reciprocal latticepoint which lies inthe volume will meetthe Laue condition forreflection.
This technique is useful for taking data on crystals which are changing ormay degrade in the beam with a single shot of x-rays on a 2D detector.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 11 / 22
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Laue diffraction
The Laue diffractiontechnique uses a widerange of radiation from~kmin to ~kmax
These define two Ewaldspheres and a volumebetween them such thatany reciprocal latticepoint which lies inthe volume will meetthe Laue condition forreflection.
This technique is useful for taking data on crystals which are changing ormay degrade in the beam with a single shot of x-rays on a 2D detector.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 11 / 22
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Laue diffraction
The Laue diffractiontechnique uses a widerange of radiation from~kmin to ~kmax
These define two Ewaldspheres and a volumebetween them such thatany reciprocal latticepoint which lies inthe volume will meetthe Laue condition forreflection.
This technique is useful for taking data on crystals which are changing ormay degrade in the beam with a single shot of x-rays on a 2D detector.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 11 / 22
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Diffraction resources
XRayView
http://www.phillipslab.org/downloads
Bilbao Crystallography Server
http://www.cryst.ehu.es/
Hypertext Book of Space Groups
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
GSAS-II
https://subversion.xray.aps.anl.gov/trac/pyGSAS
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 12 / 22
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Diffraction resources
XRayView
http://www.phillipslab.org/downloads
Bilbao Crystallography Server
http://www.cryst.ehu.es/
Hypertext Book of Space Groups
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
GSAS-II
https://subversion.xray.aps.anl.gov/trac/pyGSAS
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 12 / 22
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Diffraction resources
XRayView
http://www.phillipslab.org/downloads
Bilbao Crystallography Server
http://www.cryst.ehu.es/
Hypertext Book of Space Groups
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
GSAS-II
https://subversion.xray.aps.anl.gov/trac/pyGSAS
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 12 / 22
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Diffraction resources
XRayView
http://www.phillipslab.org/downloads
Bilbao Crystallography Server
http://www.cryst.ehu.es/
Hypertext Book of Space Groups
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
GSAS-II
https://subversion.xray.aps.anl.gov/trac/pyGSAS
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 12 / 22
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XRayView demonstration
Exercise 1 - Ewald sphere
Exercise 4 - Wavelength
Exercise 8 - Laue diffraction
Exercise 9 - Serial crystallography
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 13 / 22
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C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 14 / 22
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C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 15 / 22
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C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 16 / 22
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C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 17 / 22
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Diffraction from a Truncated Surface
a1*
a3*
G
a1*
a3*
G
a3
a2
a1
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 18 / 22
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Diffraction from a Truncated Surface
a1*
a3*
G
a1*
a3*
G
a3
a2
a1
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 18 / 22
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Diffraction from a Truncated Surface
a1*
a3*
G
a1*
a3*
G
a3
a2
a1
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 18 / 22
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Diffraction from a Truncated Surface
a1*
a3*
G
a1*
a3*
G
a3
a2
a1
For an infinite sample, the diffractionspots are infinitesimally sharp.
With finite sample size, these spotsgrow in extent and become more dif-fuse.
If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.
The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 18 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3
=A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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CTR Scattering Factor
The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).
FCTR = A(~Q)∞∑j=0
e iQza3j
=A(~Q)
1− e iQza3=
A(~Q)
1− e i2πl
this sum has been discussed previ-ously and gives
or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3
since the intensity is the square of the scattering factor
ICTR =∣∣∣FCTR
∣∣∣2 =
∣∣∣A(~Q)∣∣∣2
(1− e i2πl) (1− e−i2πl)=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl)
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 19 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2
=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Dependence on Q
When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.
ICTR =
∣∣∣A(~Q)∣∣∣2
4 sin2 (Qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (πl + qza3/2)
=
∣∣∣A(~Q)∣∣∣2
4 sin2 (qza3/2)
≈
∣∣∣A(~Q)∣∣∣2
4(qza3/2)2=
∣∣∣A(~Q)∣∣∣2
q2za23
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 20 / 22
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Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 86: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/86.jpg)
Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3j
e−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 87: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/87.jpg)
Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 88: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/88.jpg)
Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 89: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/89.jpg)
Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
β=0.2
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 90: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/90.jpg)
Absorption Effect
Absorption effects can be included as well by adding a term for each layerpenetrated
FCTR = A(~Q)∞∑j=0
e iQza3je−βj
=A(~Q)
1− e iQza3e−β
This removes the infinity andincreases the scattering profileof the crystal truncation rod
10-1
100
101
102
0 2π 4π
β=0.2
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 21 / 22
![Page 91: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/91.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
![Page 92: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/92.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
![Page 93: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/93.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
![Page 94: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/94.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
![Page 95: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/95.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3
This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
![Page 96: Today’s Outline - February 27, 2020segre/phys570/20S/lecture_14.pdf · C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 20201/22. Today’s Outline - February 27, 2020 Lattices](https://reader034.vdocument.in/reader034/viewer/2022050504/5f961d44b205ea621347cdc8/html5/thumbnails/96.jpg)
Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3
This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
z0=+0.05
z0=-0.05
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22
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Density Effect
The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.
F total = FCTR + F top layer
=A(~Q)
1− e i2πl
+ A(~Q)e−i2π(1+z0)l
where z0 is the relative displace-ment of the top layer from thebulk lattice spacing a3This effect gets larger for largermomentum transfers
10-1
100
101
102
0 2π 4π
z0=+0.05
z0=+0.10
ICT
R
Qz
C. Segre (IIT) PHYS 570 - Spring 2020 Feburary 27, 2020 22 / 22