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Today’s Outline - March 01, 2018 C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

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Page 1: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 2: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 3: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 4: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 5: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 6: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 7: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 8: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 9: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 10: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Today’s Outline - March 01, 2018

• Final presentation

• Ewald sphere (continued)

• Modulated Structures

• Crystal Truncation Rods

• Diffuse Scattering

• Debye-Waller factor

No class on Tuesday, March 06, 2018 or Thursday, March 08, 2018

Homework Assignment #04:Chapter 4: 2, 4, 6, 7, 10due Tuesday, March 20, 2018

Homework Assignment #05:Chapter 5: 1, 3, 7, 9, 10due Thursday, March 29, 2018

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 1 / 16

Page 11: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Final presentation

1. Choose paper for presentation

2. Clear it with me!

3. Do some background research on the technique

4. Prepare a 15 minute presentation

5. Be ready for questions!

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 2 / 16

Page 12: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Final presentation

1. Choose paper for presentation

2. Clear it with me!

3. Do some background research on the technique

4. Prepare a 15 minute presentation

5. Be ready for questions!

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 2 / 16

Page 13: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Final presentation

1. Choose paper for presentation

2. Clear it with me!

3. Do some background research on the technique

4. Prepare a 15 minute presentation

5. Be ready for questions!

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 2 / 16

Page 14: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Final presentation

1. Choose paper for presentation

2. Clear it with me!

3. Do some background research on the technique

4. Prepare a 15 minute presentation

5. Be ready for questions!

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 2 / 16

Page 15: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

Final presentation

1. Choose paper for presentation

2. Clear it with me!

3. Do some background research on the technique

4. Prepare a 15 minute presentation

5. Be ready for questions!

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 2 / 16

Page 16: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 17: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 18: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 19: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 20: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 21: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 3 / 16

Page 22: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 4 / 16

Page 23: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 4 / 16

Page 24: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 4 / 16

Page 25: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 5 / 16

Page 26: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 5 / 16

Page 27: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 5 / 16

Page 28: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 6 / 16

Page 29: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 6 / 16

Page 30: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 6 / 16

Page 31: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 6 / 16

Page 32: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 6 / 16

Page 33: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 7 / 16

Page 34: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 7 / 16

Page 35: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 7 / 16

Page 36: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 8 / 16

Page 37: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 9 / 16

Page 38: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 10 / 16

Page 39: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

C. Segre (IIT) PHYS 570 - Spring 2018 March 01, 2018 11 / 16

Page 40: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 12 / 16

Page 41: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 12 / 16

Page 42: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 12 / 16

Page 43: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 12 / 16

Page 44: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 45: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 46: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 47: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 48: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 49: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 50: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 51: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 13 / 16

Page 52: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 53: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 54: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 55: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 56: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 57: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 58: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 14 / 16

Page 59: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 60: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 61: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 62: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 63: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 64: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 15 / 16

Page 65: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 66: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 67: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 68: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 69: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 70: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16

Page 71: Today’s Outline - March 01, 2018phys.iit.edu/~segre/phys570/18S/lecture_16.pdf · Today’s Outline - March 01, 2018 Final presentation Ewald sphere (continued) Modulated Structures

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 2018 March 01, 2018 16 / 16