NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
NANO106 is UCSD Department of NanoEngineering's core course on crystallography of materials taught by Prof Shyue Ping Ong. For more information, visit the course wiki at http://nano106.wikispaces.com.
X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.
X-raydiffraction has a very significant role in crystal determination.. specially in the field of Pharmaceutical analysis.
It contains the requirement for M.pharm 1st year according to RGUHS syllabus.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
2. Chapter Topics
1. Wave Diffraction by Crystals
2. Bragg Law
3. Reciprocal Lattice
5. Brillouin Zones
3. • We know that a crystal is a periodic structure with unit
cells that are repeated regularly.
Crystal Structure Information
can be obtained by understanding the diffraction
patterns of waves (of appropriate wavelengths)
interacting with the solid. Analysis of such diffraction
patterns is a main topic of this chapter.
Diffraction of Waves by Crystalline Solids
It is common to study crystal structures with
X-rays, Neutrons & Electrons.
Of course, the general principles are the same for each type of wave.
4. • The results of crystal diffraction depend on the
crystal structure and on the wavelength.
• At optical wavelengths such as 5,000 Ǻ, the
superposition of waves scattered elastically by
the individual atoms of a crystal
results in ordinary optical refraction.
• When the wavelength of the radiation is
comparable to or smaller than the lattice
constant, diffracted beams occur in
directions quite different from the direction
of the incident radiation.
5. Wavelength vs Energy
Quantum Mechanical Result
The energy & momentum of a
particle with De Broglie
Wavelength λ are
E = (hc/λ) & p = (h/ λ)
(h = Planck’s constant)
Diffraction from crystal planes
requires λ to be of the same
order of magnitude as the
distance d between planes:
d a few Ångstroms
so λ must also be in that range.
This gives
Photons: E keV
Neutrons: E 0.01 eV
Electrons: E 100 eV
7. • Crystal Structure can be found by studying the Diffraction
Pattern of a beam of radiation incident on the crystal.
• Beam diffraction takes place only in certain specific
directions, much as light is diffracted by a grating.
• By measuring the directions of the diffraction and the
corresponding intensities, information about the Crystal
Structure responsible for the diffraction.
X-Ray Diffraction
• W.H. & W.L. Bragg (father & son!) were the first
to develop a simple explanation of the X-Ray
diffracted beams from a crystal.
• The Bragg derivation is simple but it is convincing
since only it reproduces the result that agrees with
observations.
8. X-Ray Diffraction & The Bragg Equation
• English physicists Sir W.H. Bragg
& his son W.L. Bragg developed
a theory in 1913 to explain why the
cleavage faces of crystals
appear to reflect X-rays
ONLY at certain angles of
incidence θ.
This is an example of
X-Ray Diffraction
Sir William H. Bragg
(1862-1942)
Sir William L. Bragg
(1890-1971)
In 1915, the father & son were awarded the Nobel prize in physics
“For their services in the analysis of
crystal structure by means of X-Rays".
(The younger Bragg was fighting in WWI when he received the Nobel Prize!)
9. Crystal Structure Determination
A Crystal behaves as a 3-D
Diffraction Grating for X-Rays
• In a diffraction experiment, the spacing of lines on
the grating can be found from the separation of the
diffraction maxima. Also, information about the
structure of the lines on the grating can be obtained
by measuring the relative intensities of different orders
• Similarly, measurement of the separation of the X-
Ray diffraction maxima from a crystal enables the
determination of the unit cell size. Also, from the
intensities of diffracted beams, information can be
obtained about the arrangement of atoms within the cell.
10. • For X-Rays, the wavelength λ
is typically a few Ångstroms,
which is comparable to the
interatomic spacing (distances
between atoms or ions) in crystals.
• So, for crystal structure
determination, the X-Rays
have to be of energy:
eV
x
m
x
hc
hc
h
E ray
x
3
10
10
3
.
12
10
1
X-Ray Crystallography
11. Bragg Law
• Consider crystals as made up of parallel planes of
atoms. Incident waves are reflected specularly from
parallel planes of atoms in the crystal, with each
plane reflecting only a very small fraction of the
radiation, like a lightly silvered mirror.
• In mirrorlike reflection, the angle of incidence is
equal to the angle of reflection.
ө
ө
12. • The diffracted beams are found to have maximum intensity when the
Reflections from Planes of Atoms
Interfere Constructively.
Assume elastic scattering, in which the X-Ray energy isn’t changed on reflection.
• So, when X-Rays strike a crystal, we want the condition for
constructive interference between reflected rays from different planes.
That is, we want the condition for the reflected X-rays to be in-
phase with one another so that they that add together constructively.
Incident Angle θ
Reflected angle θ
X-ray Wavelength λ
Total Diffracted
Angle 2θ
2
Diffraction Condition
13. • The two X-Ray beams travel different distances. The difference in the
distances traveled is related to the distance between the adjacent layers.
• See Figure. Connecting the two beams with perpendicular lines shows
the difference in the distance traveled between top & bottom beams.
• In the figure, the length DE is the same as EF, so the total distance
traveled by the bottom wave is expressed by:
• Constructive interference of the radiation from successive planes
occurs when the path difference is an integral number of wavelengths.
• Note that line CE = d = distance between the 2 layers
• So:
• Giving:
Bragg Law
2 sin
DE EF d
sin
DE d
sin
EF d
2 sin
n d
This is called the Bragg Law
14. Bragg Law (Bragg Equation)
d = Spacing of the Planes
n = Order of Diffraction.
• Because sin θ ≤ 1,
Bragg reflection can only occur for
wavelengths satisfying:
• This is why visible light can’t be used. No diffraction
occurs when this condition is not satisfied.
• The diffracted beams (reflections) from any set of
lattice planes can only occur at particular angles
predicted by Bragg’s Law.
n
d
sin
2
d
n 2
15. • Now, a similar, but slightly different treatment:
• See Figure: Consider X-Rays incident at angle θ
on one of the lattice planes. Look at the
Scattering of these X-Rays from
Adjacent Lattice Points
• There will be Constructive Interference of the
waves scattered from the two successive lattice points A
& B in the plane if the distances AC and DB are equal.
A B
C
D
2
16. So, look at the conditions for
Constructive Interference
of Waves Scattered from the same plane.
• If the scattered wave makes the same angle with the
plane as the incident wave (see figure on the previous slide):
The diffracted wave will look as if it was reflected from the plane.
• It is common to consider
Scattering from Lattice Points Rather than Atoms
because it is the basis of atoms associated with each
lattice point that is the true repeat unit of the crystal.
• The lattice point is an analogue of the line on an optical diffraction
grating. The basis represents the structure of the line.
17. Diffraction Maxima
• Coherent scattering from a single plane is not sufficient
to obtain a diffraction maximum. It is also necessary
That Successive Planes also
Scatter in Phase.
• This will be the case if the path difference for
scattering off of two adjacent planes is an
integral number of wavelengths. That is, if
n
d
sin
2
18. Additional Notes on Bragg Reflections
• Although the reflection from each plane is
specular,
Only for certain values of will the
reflections from all planes add up in
phase to give a strong reflected beam.
• Each plane reflects only 10-3 to 10-5 of the
incident radiation, i.e. it is not a perfect reflector.
• So, 103 to 105 planes contribute to the formation
of the Bragg-reflected beam in a perfect crystal.
• The composition of the basis determines the relative
intensity of the various orders of diffraction.
19. • Now, consider X-Ray Scattering from
crystals & analyze the
Amplitude of the Scattered Waves.
• The electronic number density n(r) in the
crystal is a periodic function in space:
n(r) = n(r +T)
with period T equal to a
Direct Lattice Translation Vector:
T = n1a1 + n2a2 + n3a3
Scattered Wave Amplitude
Reciprocal Lattice Vectors
20. • The electronic number density n(r) in the crystal is
periodic in space: n(r) = n(r +T), with T equal to a
Direct Lattice Translation Vector:
T = n1a1 + n2a2 + n3a3
• So, n(r) can be expressed as a (spatial) Fourier series
expansion. So, for a one-dimensional model crystal, n(x)
can be represented as
where the p’s are integers and the Fourier coefficient of
the number density can be written as:
a
px
i
p
p
p
p
p e
n
a
px
S
a
px
C
n
x
n /
2
0
0 )
/
2
sin(
)
/
2
cos(
)
(
a
a
px
i
a
p e
x
dxn
n
0
/
2
1 )
(
21. • In 3 Dimensions, the Fourier coefficient of the
number density has the form:
(1)
a
i
V
i
e
dVn
n
e
n
n
c
0
1 )
(
)
( r
G
G
r
G
G
G r
r
1
)
(
)
( )
(
T
G
T
G
r
G
G
G
T
r
G
G
G r
T
r i
i
i
i
e
when
n
e
e
n
e
n
n
The vectors G are called
Reciprocal Lattice Vectors
• As we said, the electronic density n(r) is required
to be invariant (periodic) under lattice translations:
n(r) = n(r +T) (2)
• That is, it must satisfy:
(3)
22. • Only The Set of Reciprocal Lattice Vectors G that
satisfy both (1) & (3) (previous slide) lead to an electronic
number density n(r) that is invariant under lattice translations.
• It’s not too hard to show that the set of G’s that meet this
requirement are of the form
3
3
2
2
1
1 b
b
b
G v
v
v
ij
j
i
k
j
i
k
j
i z
y
x
i
2
,
,
,
2 a
b
a
a
a
a
a
b
• The aj’s are the primitive lattice vectors for the crystal
structure. It also can be shown that
The Set of Reciprocal Lattice Vectors
G is a Bravais Lattice!
where υ1, υ2 & υ3 are integers & the bi’s are vectors
which are defined as:
23. The Diffraction Condition (Bragg’s
Law) in the Reciprocal Lattice
• An X-Ray diffraction pattern of the lattice
Can be interpreted as a map of the
reciprocal lattice of the crystal.
This statement is consistent with the following theorem:
The Set of Reciprocal Lattice
Vectors G determines
the possible X-ray reflections.
24. • An X-Ray diffraction pattern of the lattice
Can be interpreted as a map of the
reciprocal lattice of the crystal.
In other words
The Set of Reciprocal Lattice Vectors G
determines the possible X-ray reflections.
Wavevector Representation
of X-ray Scattering: k k´
d w
k
k´
r
25. 0
2 2
G
G
k
k
k'
k
G
This result is called
The Laue Condition.
It’s not too difficult to show that
it is 100% equivalent to
The Bragg Law!
•Now, look at this condition for elastic
scattering (specular reflection):