MCEN 2024, Spring 2008
The week of Feb 04
HW 4–Final...w/solutions
A Quiz related to HW4 will be posted on CULearn on Friday, Feb 8th\. You must complete the quiz
before 1:30PM on Monday the 11th. You get only one shot to complete the quiz, that is, once you
open it you must complete it. You cannot open it a second time.
References to A&J:
The reading list for this week remains the same as from last week.
Chapter 3: Definitions of stress and strain, and the elastic constants.
Chapter 4: Bonding between atoms.
Chapter 5; Packing of atoms in solids.
Chapter 6.1–6.3: Models for elastic moduli.
Overview:
Last week, and this week, and right up until the first SuperQuiz we shall be learning how the elastic
modulus of solids is related to the nanoscale structure. This process has two elements:
(a) What is the structure at the atomic and the molecular scale?
(b) How is the local bonding between atoms and molecules related to the elastic modulus of the
solids that are constructed from them.
Last week we learnt how to estimate the size of
atoms (and molecules) and the average spacing
between them from handbook information such
as density, Avogadro’s Number and the atomic or
the molecular weight. We learnt how the
stoichiometry of molecular compounds is related
to the periodic table. Also we studied how the
periodic table can guide us to understand the
different kinds of bondings that are possible
among atoms and molecules such as metallic,
covalent, ionic and van der Waal’s bonding. The
directionality of these different type of bonds
was addressed.
This week we shall focus on two topics: (a) the
packing of atoms on the nanoscale such that they
create crystals, and the formal way of describing
the structure of simple crystals, and (b) how the
elastic modulus can be related to the strength of
the bonds among the atoms in solids.
1.
Problems related to the packing of atoms
1. Describe the crystal translation vector along the edge, the face diagonal, and the body
diagonal of a cubic structure.
2. Describe the vector that lies perpendicular to the face plane, the prismatic plane and the
diagonal plane of a cubic structure.
3. Aluminum has a face centered cubic structure. Calculate the length of the cube edge in this
structure. Also calculate the closest spacing between the atoms.
4. Iron has a body centered cubic structure. Calculate the length of the cube edge in this
structure. Also calculate the closest spacing between the atoms.
The packing of atoms (or molecules) in a solid is determined by the following questions:
1) Are the atoms packed randomly, or are they packed such that they form a “periodic
lattice”?
2) Is the bonding directional or non–directional?
3) How does the packing conform to the stoichiometry of the molecules that constitute
the solid?
Please keep these points in mind as we discuss “packing” in class.
Further comments with regard to crystal structures that have a cubic unit cells. These comments apply
to all three cubic structures (simple–cubic, face–centered–cubic and body–centered–cubic):
(a) The lattice parameter, usually written by the symbol “a” is the length of the edge of cube.
3
(b) The volume of the cube is a . The volume is related to the number of atoms in the unit cell,
multiplied by the volume per atom in the solid ( Ω =
MW
), that is,
ρN A
a 3 = (#atoms per unit cell)*Ω .
(c) When the question ask for spacing between atoms, it is asking for the center–to–center
distance between the atoms.
(d) Remember that we discussed the crystal translation vectors, such as [100] and <100>, and
crystal planes, for example (110) and {110}. Crystal planes is described by the vector that is
normal to the crystal plane. The <xxx> and {xxx} refers to the family of vectors and planes
while the other notation refers to a specific vector or plane.
Solution to Problem 1 (Lattice translation vectors):
Solution to Problem 2:
Vectors normal to the face planes belong to the family {100} (there are three of them)
Vectors normal to the prismatic planes belong to the family {110} (there are six such planes)
Vectors normal to the body–diagonal planes belong to the family {111} (four such planes)
Other Problems: Create an Excel File
Calculate lattice parameter
At or Mol
Density
weight
g/mol
27.00
12.00
63.50
56.00
59.00
Volume
per atom
g/cm^3
2.70
3.50
8.96
7.87
8.90
nm^3
0.02
0.01
0.01
0.01
0.01
Which
atom
molecule
Al
C (diamond)
Copper
Fe
Ni
Calculate the lattice Parameter
STRUCT
fcc
dc
fcc
bcc
fcc
#atoms/cel Cell Vol
nm^3
4
0.07
8
0.05
4
0.05
2
0.02
4
0.04
3
Lattice para
nm
0.41
0.36
0.36
0.29
0.35
The volume of the cube is a . The volume is related to the number of atoms in the unit cell, multiplied by
the volume per atom in the solid ( Ω =
MW
), that is, a 3 = (#atoms per unit cell)*Ω .
ρN A
Another type of problems ask you to calculate the size (that is the diameters) of the hard spheres that will
lead to the different kinds of cubic structures, for examples fcc or bcc.
The approach in these problems is to find the direction in the cubic cell along which the atoms are touching,
then calculate the length of the direction in the cell, and then divide by the number of atoms in the unit cell
that are in contact.
For example consider the fcc structure: here the face plane appears as follows:
Therefore the face diagonal, a 2 , where a is the lattice parameter, or the cube edge is equal to two sphere
diameters.
In the case of bcc structure the body diagonal of the cube, which is equal to a 3 , is equal to the sum of two
atom diameters.
I will leave it up to you to draw the positions of the atoms on various planes in simple cubic, face–centered
cubic, and body–centered cubic structures.
In Problem 5.2, ignore questions
on (210) and [211].
Please ignore Mg, c.p.h.
structure in problem 5.4