The document summarizes key concepts about crystalline and amorphous solids, including:
- Crystalline solids exhibit long-range order while amorphous solids only have short-range order.
- Ionic crystals like NaCl and CsCl form face-centered cubic or body-centered cubic structures to maximize interactions between oppositely charged ions.
- The cohesive energy of ionic crystals can be calculated by considering contributions from Coulomb attraction, electron overlap repulsion, ionization energies, and electron affinities.
Solids are characterized by their definite shape and also their considerable mechanical strength and rigidity. The particles that compose a solid material(with few exceptions), whether ionic, molecular, covalent or metallic, are held in place by strong attractive forces between them.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
Intermediate state of mesophases & halfway between isotropic liquid &solid crystal.
In solid crystal, basic unit display translational long range order, with center of molecule located on crystal lattice &display orientational order.
In isotropic liquid, basic unit do not preset positional or orientational long rang order.
this is about center of mass, center of mass for complicated shapes, center of mass of hemisphere, center of mass of many particles, center of mass of solids, center of mass of uniform cylinder, center of mass of uniform rod
Solids are characterized by their definite shape and also their considerable mechanical strength and rigidity. The particles that compose a solid material(with few exceptions), whether ionic, molecular, covalent or metallic, are held in place by strong attractive forces between them.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
Intermediate state of mesophases & halfway between isotropic liquid &solid crystal.
In solid crystal, basic unit display translational long range order, with center of molecule located on crystal lattice &display orientational order.
In isotropic liquid, basic unit do not preset positional or orientational long rang order.
this is about center of mass, center of mass for complicated shapes, center of mass of hemisphere, center of mass of many particles, center of mass of solids, center of mass of uniform cylinder, center of mass of uniform rod
“Beyond Experiments: General Equilibrium Simulation Methods for Impact Evaluation" presented by Xinshen Diao, IFPRI and Edward Taylor, University of California at the ReSAKSS-Asia Conference, Nov 14-16, 2011, in Kathmandu, Nepal.
ELECTRICAL PROPERTIES OF NI0.4MG0.6FE2O4 SYNTHESIZED BY CONVENTIONAL SOLID-ST...IAEME Publication
Ni0.4Mg0.6Fe2O4 samples are prepared by conventional double sintering approach and sintered at 1300oC/ 2 h. These ferrites are characterized using X-ray diffractometer. The diffraction study reveals that the present compound shows perfect single phase cubic spinel structure. In addition, the behavior of distinct electrical properties such as dielectric constant (ε'), dielectric loss (ε") and ac-conductivity (σac) as a function frequency as well as temperature is analyzed using the LCR controller
14/09/2017
1
Crystal Structure
1
Crystalline Solid
• Crystalline Solid is the solid form of a substance in
which the atoms or molecules are arranged in a
definite, repeating pattern in three dimension.
• Single crystals, ideally have a high degree of order, or
regular geometric periodicity, throughout the entire
volume of the material.
Crystalline Solids
2
Macroscopic form reflects underlying atomic structure
14/09/2017
2
Crystal Structure
3
Polycrystalline Solid
Polycrystalline
Pyrite form
(Grain)
Polycrystal is a material made up of an aggregate of many small single crystals
(also called crystallites or grains).
Polycrystalline material have a high degree of order over many atomic or molecular
dimensions.
These ordered regions, or single crytal regions, vary in size and orientation wrt one
another.
These regions are called as grains ( domain) and are separated from one another
by grain boundaries. The atomic order can vary from one domain to the next.
The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains
that are <10 nm in diameter are called nanocrystalline
Crystal Structure
4
Amorphous Solid
• Amorphous (Non-crystalline) Solid is composed of randomly
orientated atoms , ions, or molecules that do not form defined
patterns or lattice structures.
• Amorphous materials have order only within a few atomic or molecular
dimensions.
• Amorphous materials do not have any long-range order, but they have
varying degrees of short-range order.
• Examples to amorphous materials include amorphous silicon,
plastics, and glasses.
• Amorphous silicon can be used in solar cells and thin film transistors.
http://www.alaskanessences.com/gembig/Pyrite.jpg
14/09/2017
3
Molecular Crystals
5
Formed from C60 or molecules,
Known as “buckyballs”
A molecular lattice of 1·KClO4.
Liquid Crystals & Polymers
6
Some properties of liquid,
some of solid
Polymer long chain of atoms
14/09/2017
4
7
Bonds between atoms: contents
• bonding in general, attractive and repulsive forces,
cohesive energy
• ionic bonding
• covalent bonding
• metallic bonding
• hydrogen bonding and van der Waals bonding
• relationship between bonding type and some physical
properties of a solid (in particular melting point)
at the end of this lecture you should understand....
8
Bonding in solids: the general idea
• valence electrons (of the outer shell) achieve bonding (like
in chemistry)
• decrease in total energy stabilises the solid (the solid’s
energy is lower than that of sum of atoms it is made of)
• so the energy gain by the bonding must be higher than the
energy it costs to promote electrons from the atomic orbitals
to the electronic states of the solid.
• this energy difference is a measure for the strength of the
bond. It is called the cohesive energy.
cohesive en.
these slides will help you learn all the basic about chemical bonding. concept of valancy, concept of electronic configuration, types of chemical bonds, and how do atoms form bonds.
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.
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.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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.
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.
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.
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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.
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.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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.
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Thesis Statement for students diagnonsed withADHD.ppt
Solid
1. The Solid State
Crystalline and Amorphous Solids
Ionic Crystals
“The universe consists only of atoms and the void: all else is opinion and
illusion.”–Edward Robert Harrison
2. Exam 3 Post-Mortem—Last time I taught 107
average score=81.2
actually rather high, although 3 points below exams 1&2
high score=98
the biggest problem area:
difficulty completing a calculation with no errors
Exam 3 Post-Mortem—April 2005
average score=82.1
actually rather high, although 3 points below exams 1&2
high score=100
the biggest problem area:
difficulty completing a calculation with no errors
3. A note on attendance policy. The syllabus says:
“The lowest score of the four exams (3 semester, 1 final) will be dropped.
However, you will not be allowed to drop the final exam unless
you attend at least 2/3 of the scheduled class meetings after
exam 3. If you attend less than 2/3 of the scheduled class meetings during
that time, I will drop the lowest of the three in-semester exam scores.”
This means exactly what it says. If you attend at least 2/3 of the
final nine class meetings and skip the final exam, a “0” goes in
your final exam score, and will be the dropped score.
If you attend less than 2/3 of the final nine class meetings and
skip the final exam, a “0” goes in your final exam score. The
dropped score will be the lowest of your 3 exams. The “0” for
the final will kill your grade!
4. Chapter 10
The Solid State
10.1 Crystalline and Amorphous Solids
Crystalline and amorphous are the two major categories into
which solids are divided.
Crystalline solids exhibit long-
range order in their atomic
arrangements.
Silicon crystal surface.
http://www.aip.org/history/einstein/atoms.htm
5. The order in crystals is usually three dimensional, but
lower dimensionality order is possible.
Bonds in crystalline solids are more or less the same in
energy, and crystalline solids have distinct melting
temperature.
mercury
(Don’t try
this at
home!)
6. Amorphous solids exhibit only short-range order in their atomic
arrangements. They lack long-range order.
Their bonds vary in energy and are weaker; they have no
distinct melting temperature.
A good example is B2O3. Here’s figure 10.1b, crystalline B203.
All crystals have atoms which are
“equivalent;” the symmetry of all
equivalent atoms is the same.
For example, these two boron atoms
are equivalent; everything around
them looks the same.
7. Imagine you are a nanohuman. If you build a house on some
boron atom and look out any window in some direction…
You’ll have exactly the same
view if you build your house on
an equivalent boron atom and
look out in an equivalent
direction.
Note that “equivalent direction”
does not mean “the same
direction.”
Here’s how you tell if a solid is crystalline:
8. See Figure 10.1a for amorphous B2O3.
Every B is surrounded by a
“triangle” of 3 O’s, just as for
crystalline B2O3.
But you can’t find two B atoms
that give you an identical “view.”
Eventually the short-range order
breaks down, and every “view” is
different.
“Amorphous” means “formless” or “shapeless” but amorphous
materials still have short-range order. The appropriate definition
of “amorphous” for us is “lacking a distinct crystalline structure.”
(www.dictionary.com)
9. Defects in Crystals
Image “borrowed” from http://comp.uark.edu/~pjansma/geol3513_25_defmechs1_04.ppt who got
it from “Davis & Reynolds 1996.”
10. It's easy to think of real crystals as having these ideal
structures.
In fact, no crystals are perfect; all crystals have defects.
Crystals can have:
• The "right" atoms in "wrong" places.
• "Wrong" atoms in "right" or "wrong" places.
• Missing atoms.
• Etc.
One type of defect is the point defect.
11. There are three basic kinds of point defects.
• (1) A vacancy.
Actually, a vacancy would probably look more like this:
12. • (2) An interstitial.
There is likely a vacancy somewhere else in the crystal, which
supplied the interstitial atom.
13. • (3) An impurity,
substitutional interstitial
• (3) An impurity, which could be either substitutional• (3) An impurity, which could be either substitutional or
interstitial.
14. Point defects makes diffusion in solids possible.
Either vacancies or interstitial atoms can migrate
through a crystal.
Diffusion is strongly temperature dependent.
Higher dimensional defects include edge and screw
dislocations. A dislocation occurs when a line of atoms is in the
wrong place.
Dislocations are important but more difficult to deal with
than point defects.
15. Edge dislocation are relatively easy to draw and visualize. See
Figure 10.3.
http://uet.edu.pk/dmems/edge_dislocation.htm dead link spring 2005
16. Atoms in crystals are not static, and neither are dislocations.
Screw dislocation are more difficult to draw. See Figures 10.4
and 10.5.
20. Work hardening (you have probably tried this at home).
Work hardening occurs when so many dislocations are
formed in a material that they impede each others'
motion.
Hard materials are usually brittle.
Annealing.
Heating (annealing) a crystal can remove
dislocations. The edge dislocation animation
from Germany we saw showed an edge
dislocation being annealed out.
Annealing makes metals more ductile, and can be used
to remove secondary phases in crystals.
21. 10.2 Ionic Crystals
An atom with a low ionization energy can give up an electron to
another atom with a high electron affinity. The result is an ionic
crystal.
To calculate the stability of an ionic crystal, we need to consider
all of the energies involved in its formation.
Positive energy is required to ionize an atom.
Energy is released when a highly electronegative atom
gains an electron (energy becomes more -).
There are ± contributions to the energy from the
Coulomb force between charged ions.
There is a + contribution to the energy from the overlap
of core atomic electrons (the Pauli exclusion principle at
work).
+
+
±
-
22. Remember that negative energies mean stable systems. If you
add up all the above energies and get a more negative energy
than for the separate, isolated atoms, then the ionic crystal is
stable.
We’ll go through that exercise soon.
Ionic crystals are generally close-packed, because nature
"wants" as many ions of different charge squeezed together as
possible.
Like-charged ions never come in contact.
23. There are two primary structures for ionic crystals.
Face-centered cubic (fcc).
25. An example of the fcc structure is sodium chloride (NaCl).
NaCl NaCl
a bit of the
front sliced off
http://sbchem.sunysb.edu/msl/nacl.html
26. The other main ionic crystal structure is body-centered cubic
(bcc).
27.
28. An example of the bcc structure is cesium chloride (CsCl). Click
here for a model you can manipulate.
It is not too difficult to calculate the energies involved in ionic
bonding, so let’s do it.
We begin by defining the cohesive energy of an ionic crystal
as “the reduction in the energy per ion of the ionic crystal
relative to the neutral atoms,” or “the energy per ion needed to
break the crystal up into individual atoms.”
Later Beiser implicitly generalizes this definition to all
crystals.
Let's calculate the contribution to the cohesive energy from the
Coulomb potential energy. Let's do it for an fcc structure, NaCl
for example.
29. Let’s add up all contributions to the Coulomb energy from ion-
ion interactions. Let's take a Na+
ion as the reference* ion (NaCl
is made of Na+
Cl-
ions). We get the same result if we take a Cl-
as the reference.
Here’s the reference ion.
Each Na+ ion has six Cl-
nearest
neighbors a distance r away. Here are
four of them.
Where are the other two Cl’s?
r
showing these 4 Na’s just to help
you get your bearings
here
and
here
*Remember, the energy is
calculated per ion.
30. The contribution to the Coulomb potential from these six nearest
neighbors is
( ) ( )
π π
2
1
0 0
6 +e -e 6 e
U = = - .
4ε r 4 ε r
This represents a negative (more stable) contribution to the total
energy.
r 2
Each Na+
has 12 Na+
next-nearest
neighbors at a distance of 21/2
r.
Each Na+
has 12 Na+
next-nearest
neighbors at a distance of 21/2
r. These
are + ions, so the interaction is repulsive,
and the contribution to the total energy is
positive. This figure shows four of the
next-nearest neighbors.
31. There are four more (red) in the plane aboveThere are four more (red) in the plane above and four more in
the plane below.
“original” plane
“above”
“below”
32. The contribution from the twelve next-nearest neighbors is
π
2
2
0
12 e
U =+ .
4ε r 2
This represents a positive (less stable) contribution to the total
energy.
You could keep on like this for shell after shell. Pretty soon a
pattern would emerge. After “many” shells, you get
π π
2 2
0 0
e 12 e
U= - 6 - +... = -1.748
4ε r 4 ε r2
α
π
2
0
e
U= -
4ε r
33. The constant α (which is constant only for a given type of
structure) is known as the Madelung constant. Beiser gives
values of α for a couple of other structure types.
( )
α
12
= 6 - +... = 6 - 8.485281+...
2
The convergence of this series is very poor. You have to find a
clever way to do this series if you want to calculate α with a
reasonable amount of effort.
We've accounted for the Coulomb attraction. Beiser calls this
Ucoulomb (using U instead of V as in the previous edition) so let’s
make it official:
α
π
2
coulomb
0
e
U = -
4ε r
34. This isn’t the whole story!
α
π
2
coulomb
0
e
U = -
4ε r
Note the - sign in the equation for U. The net coulomb
interaction is attractive.
In fact, the closer the ions, the more negative the energy. The
more negative, the more stable. What is the logical conclusion
from this observation?
As r gets smaller and smaller, electron shells start to overlap,
and electrons from different atoms share the same potential.
What does Pauli have to say about that?
35. Electrons would have to be promoted to higher energies to
allow atoms to come closer together. The result is a more
positive energy, i.e., a less stable crystal.
So we need to account for the repulsive forces that take over
when electron shells start to overlap, and different electrons
share the same set of quantum numbers.
We model this repulsive force with a potential of the form
repulsive n
B
U = ,
r
where n is some exponent. The exact value of n isn't too critical
(see the figure on the next page).
36. A simple scale factor change could make 1/r9
and 1/r10
look
nearly the same.
37. With the electron overlap repulsive energy accounted for, the
potential energy of interaction of our Na+
reference ion with all
the other ions is
α
π
2
total coulomb repulsive n
0
e B
U = U + U = - + .
4ε r r
Even though this says Utotal, we are not done! Do not use this
equation in a test/quiz problem!
For one thing, we have accounted for the total energy of
interaction of the ions (that’s what “total” means), but we haven’t
accounted for the energy required to ionize the neutral atoms.
For another thing, we have this adjustable parameter, B. (You
do remember adjustable parameters?)
Ucoulomb was called Vlattice in the previous text edition, so you’ll see that terminology in
homework solutions and prior exams.
38. At equilibrium the energy is minimized, which allows us to find B
in terms of α and the equilibrium near-neighbor separation r0.
∂ α
∂ π 0
2
2 n+1
0 0 0r =r
U e nB
0 = = -
r 4ε r r
α
π
2
2 n+1
0 0 0
e nB
=
4ε r r
α
π
2
n-1
0
0
e
B= r
4ε n
In a sense, we have “adjusted” B to get the right answer
(energy minimum at equilibrium).
39. Plugging B back into the expression for U gives the total
potential energy at equilibrium separation. Beiser used to call it
the lattice energy, Vlat. Now he calls it U0.
α π
2
0
0 0
e 1
U = - 1- .
4ε r n
U0 is the reduction in the energy of the ionic crystal relative to
the ions at infinite separation.
We set out to calculate the cohesive energy, so we are still not
done yet.
The cohesive energy, Ecohesive, takes into account the energy
needed to create the ions.
40. Beiser doesn’t give an equation for the cohesive energy, so I’ll
make one up. The energy of the ions is U0. The energy to
create a Na+
ion is the ionization energy of Na. I’ll call that Uion.
When a Cl atom combines with an electron to form Cl-
, energy
is actually released. Chemists call this energy the electron
affinity. I’ll write that Uea.
coh, pair 0 ion eaE = U + U + U .
You’ll see what “pair” means in a minute. Also, the numerical
value of Ecoh had better be negative, right?
Uion is positive, because it is energy put into the crystal. Uea is
negative, because it is energy released from the crystal.
41. Part of your homework/quiz/exam responsibility is to get the
right signs on these energies!
Oh, and one more thing. We’ve calculated the coulomb energy
for our reference Na+
with all other ions in the crystal, one pair
at a time. Thus, U0 is the energy per ion pair.
Also, Uion is the energy to make Na+
and Uea is the energy
released on making Cl-
. Thus Uion + Uea represents the energy
needed to make a Na+
Cl-
pair.
Beiser defines the cohesive energy as “per ion.” Thus finally
(and officially):
0 ion ea
coh
U + U + U
E = .
2
42. Beiser on page 342 calculates the cohesive energy for
NaCl.
A “typical” value for n is 9. The equilibrium distance r0 between
Na+
and Cl-
ions is 0.281 nm.
Beiser calculates numerical values for the different energies,
and then combines them to get the cohesive energy.
We know it’s better to do the problem algebraically first, and get
numerical results only at the very end.
Let’s do it Beiser’s way here.
43.
α π
2
0
0 0
e 1
U = - 1-
4ε r n
( )
( )
( )( )
×
π × ×
2
-19
0 -12 -9
1.6 10 1
U = - 1.748 1-
94 8.85 10 0.281 10
× -18
0U = -1.27 10 joules = -7.96 eV .
only for NaCl
structure!
The ionization energy of Na is 5.14 eV and the electron affinity
of Cl is 3.61 eV. With the correct signs, the energies are +5.14
eV and -3.61 eV. Make sure you understand why! Many of you
will lose several points because of sign errors.
44. 0 ion ea
coh
U + U + U
E =
2
coh
-7.96 eV + 5.14 eV -3.61 eV
E =
2
cohE = -3.21 eV
Great news! We got a – sign for the cohesive energy. You can
sleep well at night knowing that NaCl is stable.
Wonder how it compares with experiment? The measured
value is -3.28 eV. That’s a 2% error. For this kind of
calculation, the agreement is good.
45. Why did I bother with this rather lengthy calculation to show that
NaCl is stable, when we already know it is stable?
Stupid?
Physicists may do stupid things, but we are not stupid.
Designer molecules! If you have an application that needs a
compound with special properties, do you spend millions in the
lab and hope you stumble across it…
Designer molecules! If you have an application that needs a
compound with special properties, do you spend millions in the
lab and hope you stumble across it… or do you spend a few
hundred thousand on computational modeling that tells you
what kind of compound to make?
The calculation we did is your first step towards making
designer molecules.
46. Some properties of ionic crystals, which Beiser mentions briefly:
• They have moderately high
melting points.
• They are brittle due to
charge ordering in planes.
• They are soluble in polar
fluids.