Homework #4
Professor Estrada
ENGR 245: Intro to Materials Science & Engineering
February 9, 2016
In order to receive full credit, all work MUST be shown and appropriate units must
be specified. Working together is encouraged, however, all work that you turn in must be
YOUR OWN work.
Exercise 1. Derive Bragg’s Law. You should find n? = 2d sin(?). In order to receive full
credit, explain the justification for each step.
Exercise 2. Calculate the spacing between the (111) plane in BCC Fe. Cite the source you
used to look up the lattice constant, a.
Exercise 3. Given the following set of experimental data for the number of vacancies (Nv)
in copper as function of temperature, plot Nv on a logarithm scale (or the natural log of Nv)
as a function of T
-1
(i.e., 1/T) for a temperature range of 500K to 1250K.
T(K) Nv(cm-3
)
500 7.23e13
600 1.75e15
700 3.86e16
800 1.094e17
900 8.83e17
1000 2.475e18
1100 6.97e18
Exercise 4. The equilibrium vacancy concentration, Nv, is given by equation 4.1 in the 9th
edition of Callister and Rethwisch. Given the density (?) and atomic weight (ACu) of copper
are 8.4 g/cm3 and 63.5 g/mol, find the activation energy of vacancy formation in copper
(Qv) by fitting equation 4.1 to the experimental data in exercise 3. Show at least three of
your fits on the same plot as the data points.
1

Homework #4
Professor Estrada
ENGR 245: Intro to Materials Science & Engineering
February 9, 2016
In order to receive full credit, all work MUST be shown and appropriate units must
be specified. Working together is encouraged, however, all work that you turn in must be
YOUR OWN work.
Exercise 1. Derive Bragg’s Law. You should find n? = 2d sin(?). In order to receive full
credit, explain the justification for each step.
Exercise 2. Calculate the spacing between the (111) plane in BCC Fe. Cite the source you
used to look up the lattice constant, a.
Exercise 3. Given the following set of experimental data for the number of vacancies (Nv)
in copper as function of temperature, plot Nv on a logarithm scale (or the natural log of Nv)
as a function of T
-1
(i.e., 1/T) for a temperature range of 500K to 1250K.
T(K) Nv(cm-3
)
500 7.23e13
600 1.75e15
700 3.86e16
800 1.094e17
900 8.83e17
1000 2.475e18
1100 6.97e18
Exercise 4. The equilibrium vacancy concentration, Nv, is given by equation 4.1 in the 9th
edition of Callister and Rethwisch. Given the density (?) and atomic weight (ACu) of copper
are 8.4 g/cm3 and 63.5 g/mol, find the activation energy of vacancy formation in copper
(Qv) by fitting equation 4.1 to the experimental data in exercise 3. Show at least three of
your fits on the same plot as the data points.
1

Homework #4
Professor Estrada
ENGR 245: Intro to Materials Science & Engineering
February 9, 2016
In order to receive full credit, all work MUST be shown and appropriate units must
be specified. Working together is encouraged, however, all work that you turn in must be
YOUR OWN work.
Exercise 1. Derive Bragg’s Law. You should find n? = 2d sin(?). In order to receive full
credit, explain the justification for each step.
Exercise 2. Calculate the spacing between the (111) plane in BCC Fe. Cite the source you
used to look up the lattice constant, a.
Exercise 3. Given the following set of experimental data for the number of vacancies (Nv)
in copper as function of temperature, plot Nv on a logarithm scale (or the natural log of Nv)
as a function of T
-1
(i.e., 1/T) for a temperature range of 500K to 1250K.
T(K) Nv(cm-3
)
500 7.23e13
600 1.75e15
700 3.86e16
800 1.094e17
900 8.83e17
1000 2.475e18
1100 6.97e18
Exercise 4. The equilibrium vacancy concentration, Nv, is given by equation 4.1 in the 9th
edition of Callister and Rethwisch. Given the density (?) and atomic weight (ACu) of copper
are 8.4 g/cm3 and 63.5 g/mol, find the activation energy of vacancy formation in copper
(Qv) by fitting equation 4.1 to the experimental data in exercise 3. Show at least three of
your fits on the same plot as the data points.
1