Crystals are composed of repeating unit cells that generate the entire crystal structure when translated through space. A crystal's symmetry is defined by symmetry elements like rotations, translations, and reflections that leave the crystal unchanged. There are 32 possible point groups and 14 Bravais lattices that combine to form 230 unique space groups describing all possible crystal symmetries. The asymmetric unit is the smallest portion of the unit cell that generates the full crystal structure through symmetry operations.
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.
Crystal Material, Non-Crystalline Material, Crystal Structure, Space Lattice, Unit Cell, Crystal Systems, and Bravais Lattices, Simple Cubic Lattice, Body-Centered Cubic Structure, Face centered cubic structure, No of Atoms per Unit Cell, Atomic Radius, Atomic Packing Factor, Coordination Number, Crystal Defects, Point Defects, Line Defects, Planar Defects, Volume Defects.
Solid state chemistry- laws of crystallography- Miller indices- X ray diffraction- Bragg equation- Spectrophotometer- Determination of interplanar distance- Types of crystal
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.
Crystal Material, Non-Crystalline Material, Crystal Structure, Space Lattice, Unit Cell, Crystal Systems, and Bravais Lattices, Simple Cubic Lattice, Body-Centered Cubic Structure, Face centered cubic structure, No of Atoms per Unit Cell, Atomic Radius, Atomic Packing Factor, Coordination Number, Crystal Defects, Point Defects, Line Defects, Planar Defects, Volume Defects.
Solid state chemistry- laws of crystallography- Miller indices- X ray diffraction- Bragg equation- Spectrophotometer- Determination of interplanar distance- Types of crystal
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.
This presentation will be helpful to beginners on chemical aspects of group theory. Also this ppt consists of videos on mirror plane symmetry and rotational axis of symmetry
Crystallography and X-ray diffraction (XRD) Likhith KLIKHITHK1
Atoms in materials are arranged into crystal structures and microstructures.
Periodic arrangement of atoms depends strongly on external factors such as temperature, pressure, and cooling rate during solidification. Solid elements and their compounds are classified into amorphous, polycrystalline, and single crystalline materials. The amorphous solid materials are isotropic in nature because their atomic arrangements are not regular and possess the same properties in all directions. In contrast, the crystalline materials are anisotropic because their atoms are arranged in regular and repeated pattern, and their properties vary with direction. The polycrystalline materials are combinations of several crystals of varying shapes and sizes. The properties of polycrystalline materials are strongly dependent on distribution of crystals sizes, shapes, and orientations within the individual crystal. Diffraction pattern or intensities of X-ray diffraction techniques are used for characterizing and probing arrangement of atoms in each unit cell, position of atoms, and atomic spacing angles because of comparative wavelength of X-ray to atomic size.The X-ray diffraction, which is a non-destructive technique, has wide range of material analysis including minerals, metals, polymers, ceramics, plastics, semiconductors, and solar cells. The technique also has wide industry application including aerospace, power generation, microelectronics, and several others. The X-ray crystallography remained a complex field of study despite wide industrial applications.
Space lattice, Unit cell, Bravais lattices (3-D), Miller indices, Lattice planes, Hexagonal closed packing (hcp) structure, Characteristics of an hcp cell, Imperfections in crystal: Point defects (Concentration of Frenkel and Schottky defects).
X – ray diffraction : Bragg’s law and Bragg’s spectrometer, Powder method, Rotating crystal method.
In this lecture , the focus is on geological aspects of minerals. We are going to discuss about the crystal symmetry of minerals,axis of rotation , axis of rotoinversion symmetry y, and mirror plane. The crystal classes may be sub-divided into one of 6 crystal systems namely Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric (cubic).
Crystallographic axis
The identification of specific symmetry operations enables one to orientate a crystal according g to an imaginary set of reference lines known as the crystallographic axis.With the exception of the hexagonal system, the axes are designated designated The ends of each a, b, and c. The ends of each axes are designated axes are designated + or -. This is important for the derivation of Miller Indices.
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.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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.
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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
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?
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
2. Crystals are made of infinite number of unit cells
Unit cell is the smallest unit of a crystal, which, if
repeated, could generate the whole crystal.
A crystal’s unit cell dimensions are defined by six numbers,
the lengths of the 3 axes, a, b, and c, and the three interaxial
angles, α, β and γ.
3. A crystal lattice is a 3-D stack of unit cells
Crystal lattice is an imaginative grid system in three dimensions in
which every point (or node) has an environment that is identical to that
of any other point or node.
4. Symmetry
A state in which parts on opposite sides of a plane,
line, or point display arrangements that are related to
one another via a symmetry operation such as
translation, rotation, reflection or inversion.
Application of the symmetry operators leaves the
entire crystal unchanged.
5. Symmetry Elements
Rotation
turns all the points in the asymmetric
unit around one axis, the center of
rotation. A rotation does not change
the handedness of figures. The center
of rotation is the only invariant point
(point that maps onto itself).
8. Symmetry Elements
Translation moves all the points in the
asymmetric unit the same
distance in the same direction.
This has no effect on the
handedness of figures in the
plane. There are no invariant
points (points that map onto
themselves) under a translation.
9. Symmetry Elements
Screw axes (rotation + translation)
rotation about the axis of
symmetry by 360°/n, followed
by a translation parallel to the
axis by r/n of the unit cell length
in that direction. (r < n)
12. Symmetry Elements
Inversion, or center of symmetry
every point on one side of
a center of symmetry has a
similar point at an equal
distance on the opposite
side of the center of
symmetry.
13. Symmetry Elements
Mirror plane or Reflection
flips all points in the asymmetric unit
over a line, which is called the mirror,
and thereby changes the handedness of
any figures in the asymmetric unit.
The points along the mirror line
are all invariant points (points that map
onto themselves) under a reflection.
15. Symmetry Elements
Glide reflection (mirror plane + translation)
reflects the asymmetric unit
across a mirror and then
translates parallel to the mirror.
A glide plane changes the
handedness of figures in the
asymmetric unit. There are no
invariant points (points that map
onto themselves) under a glide
reflection.
16.
17.
18. Symmetries in crystallography
• Crystal systems
• Lattice systems
• Space group symmetry
• Point group symmetry
• Laue symmetry, Patterson symmetry
19. Crystal system
• Crystals are grouped into seven crystal
systems, according to characteristic
symmetry of their unit cell.
• The characteristic symmetry of a crystal is a
combination of one or more rotations and
inversions.
20. 7 Crystal Systems
orthorhombic hexagonal
monoclinic trigonal
cubic tetragonal triclinic
Crystal System External Minimum Symmetry Unit Cell Properties
Triclinic None a, b, c, al, be, ga,
Monoclinic One 2-fold axis, || to b (b unique) a, b, c, 90, be, 90
Orthorhombic Three perpendicular 2-folds a, b, c, 90, 90, 90
Tetragonal One 4-fold axis, parallel c a, a, c, 90, 90, 90
Trigonal One 3-fold axis a, a, c, 90, 90, 120
Hexagonal One 6-fold axis a, a, c, 90, 90, 120
Cubic Four 3-folds along space diagonal a, a, ,a, 90, 90, 90
21. Auguste Bravais
Lattices
(1811-1863)
• In 1848, Auguste Bravais demonstrated that
in a 3-dimensional system there are fourteen
possible lattices
• A Bravais lattice is an infinite array of
discrete points with identical environment
• seven crystal systems + four lattice centering
types = 14 Bravais lattices
• Lattices are characterized by translation
symmetry
22. Four lattice centering types
No. Type Description
1 Primitive Lattice points on corners
only. Symbol: P.
2 Face Centered Lattice points on corners as
well as centered on
faces. Symbols: A (bc
faces); B (ac faces); C
(ab faces).
3 All-Face Centered Lattice points on corners as
well as in the centers of
all faces. Symbol: F.
4 Body-Centered Lattice points on corners as
well as in the center of
the unit cell body.
Symbol: I.
23.
24. Tetragonal lattices are either primitive (P) or
body-centered (I)
C centered lattice
=
Primitive lattice
26. Point group symmetry
• Inorganic crystals usually have perfect shape
which reflects their internal symmetry
• Point groups are originally used to describe the
symmetry of crystal.
• Point group symmetry does not consider
translation.
• Included symmetry elements are rotation, mirror
plane, center of symmetry, rotary inversion.
29. N-fold axes with n=5 or n>6 does
not occur in crystals
Adjacent spaces must be completely filled (no gaps, no
overlaps).
30. Laue class, Patterson symmetry
• Laue class corresponds to symmetry of
reciprocal space (diffraction pattern)
• Patterson symmetry is Laue class plus
allowed Bravais centering (Patterson map)
31. Space groups
The combination of all available symmetry operations (32
point groups), together with translation symmetry,
within the all available lattices (14 Bravais lattices) lead
to 230 Space Groups that describe the only ways in which
identical objects can be arranged in an infinite lattice.
The International Tables list those by symbol and
number, together with symmetry operators, origins,
reflection conditions, and space group projection
diagrams.
33. Identification of the Space Group is called indexing the crystal.
The International Tables for X-ray Crystallography tell us a huge
amount of information about any given space group. For instance,
If we look up space group P2, we find it has a 2-fold rotation axis
and the following symmetry equivalent positions:
X , Y , Z
-X , Y , -Z
and an asymmetric unit defined by:
0≤x≤ 1
0≤y≤ 1
0 ≤ z ≤ 1/2
An interactive tutorial on Space Groups can be found on-line in Bernhard Rupp’s
Crystallography 101 Course: http://www-structure.llnl.gov/Xray/tutorial/spcgrps.htm
37. Space group P21
Point group 2 + Bravais lattice “primitive monoclinic”,
but consider screw axis
38. Coordinate triplets, equivalent positions
r = ax + by + cz,
Therefore, each point can be described by its fractional
coordinates, that is, by its coordinate triplet (x, y, z)
39. Space group determination
• Symmetry in diffraction pattern
• Systematic absences
• Space groups with mirror planes and
inversion centers do not apply to protein
crystals, leaving only 65 possible space
groups.
42. Asymmetric unit
Recall that the unit cell of a crystal is the smallest 3-D geometric
figure that can be stacked without rotation to form the lattice. The
asymmetric unit is the smallest part of a crystal structure from
which the complete structure can be built using space group
symmetry. The asymmetric unit may consist of only a part of a
molecule, or it can contain more than one molecule, if the molecules
not related by symmetry.
43. Matthew Coefficient
• Matthews found that for many protein crystals the
ratio of the unit cell volume and the molecular
weight is between 1.7 and 3.5Å3/Da with most
values around 2.15Å3/Da
• Vm is often used to determine the number of
molecules in each asymmetric unit.
• Non-crystallographic symmetry related molecules
within the asymmetric unit