Solid State Chemistry and its Applications

ANTHONY R. WEST

Chapter 1 Crystal Structures, Crystal Chemistry, Symmetry and Space Groups - all with Video Answers

Educators

Chapter Questions

Problem 1

What symmetry elements do the following tetrahedral-shaped molecules possess: (a) $\mathrm{CH}_3 \mathrm{Cl}$, (b) $\mathrm{CH}_2 \mathrm{Cl}_2$, (c) $\mathrm{CH}_2 \mathrm{ClBr}$, (d) $\mathrm{CH}_4$ ?

Nicole Mabante

Numerade Educator

Problem 2

What symmetry element do the following everyday objects have in common: (a) a teapot, (b) a pair of trousers, (c) a tricycle?

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03:42

Problem 3

What symmetry elements are present in a cube (use a kiddies' building block for a model)?

Dominador Tan

Numerade Educator

Problem 4

If one pair of opposite cube faces in 1.3 is of different colour to the rest, what symmetries are present?

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02:27

Problem 5

What symmetries does this book have (assume all pages and covers are blank)?

Debasish Das

Numerade Educator

01:35

Problem 6

The hypothetical octahedral molecule $\mathrm{MX}_2 \mathrm{Y}_4$ has two geometrical isomers. One has a centre of symmetry, the other does not. Draw the two isomers and show which one has a centre of symmetry. What other symmetries do the molecules possess?

Arun Bana

Numerade Educator

03:38

Problem 7

How many lattice points are there in (a) a primitive lattice, (b) a body centred lattice, (c) a C -centred lattice, (d) a face centred lattice?

Prabhat Tyagi

Numerade Educator

01:11

Problem 8

What is the Bravais lattice of the calcium carbide structure, Fig. 1.10(a)?

Carlene Jimenez

Numerade Educator

02:06

Problem 9

In the table of crystal systems, Table 1.1, only certain unit celllattice type combinations (i.e. Bravais lattices) are possible. Suggest reasons why the following are not included: (a) C-centred cubic, (b) F -centred tetragonal and (c) C -centred tetragonal.

Kartik Indoliya

Numerade Educator

Problem 10

What are the indices for (a) planes that pass through opposite $a b$ faces of a cubic unit cell, (b) planes parallel to $b$ and $c$ which cut $a$ at $0,1 / 2,1,3 / 2$, etc., (c) planes perpendicular to a unit cell body diagonal, that cut $a, b$ and $c$ at $1 / 21 / 21 / 2,111,3 / 23 / 23 / 2$, etc.?

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01:19

Problem 11

What are the indices for crystallographic directions that are parallel to the unit cell $b$ edge (a) in the positive $y$ direction and (b) in the negative $y$ direction?

Hubert Agamasu

Numerade Educator

16:00

Problem 12

Using the $d$-spacing formula for a cubic substance, equation (1.2), what are the $h k l$ values that correspond to the five largest $d$ spacings? Calculate these $d$ values if $a=5.00 \AA$.

Paul A.

California State Polytechnic University, Pomona

07:32

Problem 13

has the rock salt structure with $a=6.2931 \AA$. Calculate its density.

Shalini Tyagi

Numerade Educator

06:11

Problem 14

TlBr has a cubic unit cell, $a=3.97 \AA, D=7.458 \mathrm{~g} \mathrm{~cm}^{-3}$. How many formula units, $Z$, are in the unit cell? Suggest a possible structure for TlBr .

Pronoy Sinha

Numerade Educator

02:45

Problem 15

In an $f c c / c c p$ metal structure, (a) what are the indices of the $c p$ layers of metal atoms, (b) what are the indices of the $c p$ directions of metal atoms (i.e. in which directions are atoms in contact)?; (c) evaluate the atom radius, $r$, in terms of the unit cell edge, $a$.

Anand Jangid

Numerade Educator

01:12

Problem 16

Metallic Au and Pt have $f c c$ unit cells, $a=4.08$ and $3.91 \AA$. Calculate the metal atom radii.

Dr. Satish Ingale

Numerade Educator

02:30

Problem 17

In a $b c c$ metal structure, (a) what is the atom coordination number, (b) is the structure close packed, (c) what are the indices of the $c p$ directions?; (d) evaluate the atom radius in terms of the unit cell edge, $a$.

Chai Santi

Numerade Educator

07:23

Problem 18

At $20^{\circ} \mathrm{C}, \alpha-\mathrm{Fe}$ is $b c c, a=2.866 \AA$. At $950^{\circ} \mathrm{C}, \gamma-\mathrm{Fe}$ is $f c c, a= 3.656 \AA$. At $1425^{\circ} \mathrm{C}, \delta$-Fe is again $b c c, a=2.940 \AA$. At each temperature, calculate (a) the density and (b) the atomic radius of Fe .

Sanat Mukherjee

Numerade Educator

03:23

Problem 19

Show that the packing density of a $b c c$ arrangement of spheres is 0.6802 .

Vysakh M

Numerade Educator

03:33

Problem 20

Similarly, show that the packing density in $f c c$ is 0.740 .

Suzanne W.

Numerade Educator

01:44

Problem 21

Starting with a $c c p$ array of anions, what structure types are generated by (a) filling all tetrahedral sites with cations, (b) filling half of the tetrahedral sites, e.g. $\mathrm{T}_{+}$, with cations, (c) filling all octahedral sites with cations and (d) filling alternate layers of octahedral sites with cations?

Manik Pulyani

Numerade Educator

02:07

Problem 22

Repeat the above question but with an $h c p$ array of anions. Comment on the absence of any known structure type in one of these four categories.

Arpit Gupta

Numerade Educator

03:28

Problem 23

Identify the following cubic structure types from the information on atomic coordinates:
i. MX:
M: $1 / 200,01 / 20,001 / 2,1 / 21 / 21 / 2$;
X: $000,1 / 21 / 20,1 / 201 / 2,01 / 21 / 2$
ii. MX:
M: $1 / 41 / 41 / 4,3 / 43 / 41 / 4,1 / 43 / 43 / 4,3 / 41 / 43 / 4$;
X: $000,1 / 21 / 20,1 / 201 / 2,01 / 21 / 2$
iii. MX:
M: $1 / 21 / 21 / 2$;
X: 000
iv. $\mathrm{MX}_2$ :
M: $000,1 / 21 / 20,1 / 201 / 2,01 / 21 / 2$;
X: $1 / 41 / 41 / 4,1 / 41 / 43 / 4,1 / 43 / 41 / 4,3 / 41 / 41 / 4$, 1/4 3/4 3/4, 3/4 1/4 3/4, 3/4 3/4 1/4, 3/4 3/4 3/4
v. $\mathrm{MX}_3$ :
M: 000 ;
X: $1 / 200,01 / 20,001 / 2$
vi. $\mathrm{AMX}_3$ :
A: $1 / 21 / 21 / 2$;
M: 000 ;
X: $1 / 200,01 / 20,001 / 2$

Shalini Tyagi

Numerade Educator

01:34

Problem 24

Starting from a rock salt structure, what structures are generated by the following imaginary steps:
i. removal of all atoms or ions of one type;
ii. removal of alternate layers of cations;
iii. replacement of all cations in O sites by an equal number of cations in one set of T sites?

Matthew Hurlock

Numerade Educator

02:10

Problem 25

Explain why the NiAs structure is commonly found with metallic compounds but not with ionic compounds.
in different orientations; also try hiding certain features to highlight other parts.

Arpit Gupta

Numerade Educator

02:33

Problem 26

MgO has the rock salt structure, $a=4.213 \AA$. Calculate the $\mathrm{Mg}-$ O bond length. Assuming the oxide ion radius is $1.26 \AA$, what is the radius of $\mathrm{Mg}^{2+}$ ? Are the oxide ions in contact?

Madi Sousa

Numerade Educator

07:24

Problem 27

Zinc blende, ZnS , has $a=5.406 \AA$. Calculate the $\mathrm{Zn}-\mathrm{S}$ bond length.
for X-ray Crystallography, Vol 1, Kynoch Press, to get complete listings of atomic coordinates.
Compare your projections with those given in the book. If you are using software, rotate the structures to see them in different orientations; also try hiding certain features to highlight other parts.

Sanat Mukherjee

Numerade Educator

02:55

Problem 28

$ \mathrm{Li}_2 \mathrm{O}$ has the antifluorite structure, $a=4.611 \AA$. Calculate the (a) $\mathrm{Li}-\mathrm{O}$ distance, (b) $\mathrm{O}-\mathrm{O}$ distance, (c) $\mathrm{Li}-\mathrm{Li}$ distance. (d) Are the oxide ions in contact ( $r_0=1.26 \AA$ )?

Lottie Adams

Numerade Educator

02:25

Problem 29

$\mathrm{SrTiO}_3$ has the cubic perovskite structure, $a=3.91 \AA$, with atomic coordinates: Sr: $1 / 21 / 21 / 2, \mathrm{Ti}: 000, \mathrm{O}: 1 / 200,01 / 20,00 1 / 2$.
a. Draw a projection of the structure on the $x-z$ plane.
b. What is the coordination environment of (i) Sr , (ii) Ti , (iii) O ?
c. Calculate the $\mathrm{Sr}-\mathrm{O}$ and $\mathrm{Ti}-\mathrm{O}$ bond lengths. Calculate the density of $\mathrm{SrTiO}_3$.
d. Is the structure close packed? If so, describe it. What is the lattice type?
e. What compositional modifications may be made to $\mathrm{SrTiO}_3$ in an attempt to induce (i) ferroelectricity, (ii) superconductivity, (iii) ionic conductivity.

Wilson Ma

The University of Alabama

03:46

Problem 30

Si has the diamond structure, $a=5.4307 \AA$. Calculate the radius of Si .

Crystal Wang

Numerade Educator

01:41

Problem 31

Silver oxide, $\mathrm{Ag}_2 \mathrm{O}$, has a cubic unit cell, $Z=2, a=4.726 \AA$, with atomic coordinates:
Ag: $1 / 41 / 41 / 4,3 / 43 / 41 / 4,3 / 41 / 43 / 4,1 / 43 / 43 / 4 ;$
O: $000,1 / 21 / 21 / 2$.
What are the atomic coordinates if the unit cell is displaced so that an Ag atom is at the origin? Draw a projection of the structure onto the $a-b$ face, using the new set of atomic coordinates. What is the lattice type? What are the coordination numbers of Ag and O ? Calculate the $\mathrm{Ag}-\mathrm{O}$ distance? Does the structure possess a centre of symmetry?

Aadit Sharma

Numerade Educator

00:45

Problem 32

In an ideal rock salt structure, MX , in which the anions are in contact, calculate the radius ratio, $r_{\mathrm{M}} / r_x$.

Hast Aggarwal

Numerade Educator

01:16

Problem 33

Repeat question 1.32 for an ideal zinc blende structure.

Manik Pulyani

Numerade Educator

Problem 34

Repeat question 1.32 for an ideal CsCl structure.

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00:45

Problem 35

By consideration of the radius ratio rules, derived in questions 1.32-1.34, and assuming that the calculated radius ratios represented the lowest values possible for a given structure, what structure type would you expect for an MX compound with:
(a) $r_{\mathrm{M}} / r_x=0.3$, (b) $r_{\mathrm{M}} / r_x=0.6$, (c) $r_{\mathrm{M}} / r_x=0.8$ ?

Hast Aggarwal

Numerade Educator

03:16

Problem 36

Compare the packing density of the NaCl and CsCl structures for which, in both cases, anion-anion and anion-cation direct contacts occur.

Khalida Dawar

Numerade Educator

03:11

Problem 37

What kind of complex anion do you expect in the following:
(a) $\mathrm{Ca}_2 \mathrm{SiO}_4$; (b) $\mathrm{NaAlSiO}_4$ (tetrahedral Al ); (c) $\mathrm{BaTiSi}_3 \mathrm{O}_9$; (d) melilite, $\mathrm{Ca}_2 \mathrm{MgSi}_2 \mathrm{O}_7$; (e) diopside, $\mathrm{CaMgSi}_2 \mathrm{O}_6$; (f) tremolite, $\mathrm{Ca}_2 \mathrm{Mg}_5 \mathrm{Si}_8 \mathrm{O}_{22}(\mathrm{OH}, \mathrm{F})_2$; in this amphibole, the $\mathrm{OH}, \mathrm{F}$ are not bonded to Si ; (g) margarite, $\mathrm{CaAl}_2(\mathrm{OH})_2\left(\mathrm{Si}_2 \mathrm{Al}_2\right) \mathrm{O}_{10}$; in this mica, two Al are tetrahedral and two are octahedral; (h) kaolinite, $\mathrm{Al}_2(\mathrm{OH})_4 \mathrm{Si}_2 \mathrm{O}_5$; Al is octahedral and OH is not bonded to Si ?

Madi Sousa

Numerade Educator

02:51

Problem 38

Fullerene, $\mathrm{C}_{60}$, forms compounds in which inert gas atoms enter interstitial sites in the $f c c$ array of $\mathrm{C}_{60}$ molecules; $a=14.17 \AA$. Calculate (a) the van der Waals radius of the $\mathrm{C}_{60}$ molecule and (b) the radii of octahedral and tetrahedral sites. (Assume that the $\mathrm{C}_{60}$ molecules are spherical and are 'touching' in the close packed [110] direction.). Inert gas atoms have van der Waals radii $\mathrm{Ar} 1.91 \AA, \mathrm{Kr} 1.98 \AA$ and Xe $2.05 \AA$. Comment on the probable formulae and structures of inert gas fullerides and on the relative ease of formation of such compounds for $\mathrm{Ar}, \mathrm{Kr}$ and Xe .

Arpit Gupta

Numerade Educator

02:25

Problem 39

The perovskite structure is found in a wide range of materials, both stoichiometric and non-stoichiometric. Depending on composition, a variety of electrical, magnetic and optical properties are exhibited by perovskites. Describe (a) the perovskite structure, (b) typical compounds and their formulae that have this structure, (c) how non-stoichiometry can be accommodated, (d) three properties of your choice exhibited by perovskites, focusing on the links between stoichiometry, structure and properties.

Wilson Ma

The University of Alabama

Problem 40

What point groups result on adding a centre of symmetry to point groups (a) 1 , (b) 2 , (c) 3 , (d) $\overline{4}$, (e) 4 , (f) 222 , (g) $m m 2$, (h) $4 m m$, (i) 6 , (j) $\overline{6}$, (k) $6 m 2$ ?

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06:47

Problem 41

What point groups result from the combination of two mirror planes at (a) $90^{\circ}$ to each other, (b) $60^{\circ}$, (c) $45^{\circ}$, (d) $30^{\circ}$ ?

Susan Hallstrom

Numerade Educator

Problem 42

What point groups result from the combination of two intersecting 2-fold axes at (a) $90^{\circ}$ to each other, (b) $60^{\circ}$, (c) $45^{\circ}$, (d) $30^{\circ}$ ?

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Problem 43

An atom in an orthorhombic unit cell has fractional coordinates $0.1,0.15$ and 0.2 . Give the coordinates of a second atom in the unit cell that is related to the first by each of the following, separately: (a) body centring, (b) a centre of symmetry at the origin, (c) a 2 -axis parallel to $z$ and passing through the origin, (d) a $2_1$ axis parallel to $z$ and passing through the origin, (e) A-centring.

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02:55

Problem 44

$\mathrm{Li}_2 \mathrm{PdO}_2$ has an orthorhombic unit cell, $a=3.74, b=2.98, c= 9.35 \AA, Z=2$, space group $\boldsymbol{\operatorname { I m m m }}$. Atomic coordinates are: Pd : 2(a) 000; Li:4(i) $00 z: z=0.265$; $\mathrm{O}: 4(\mathrm{j}) 0 \frac{1}{2} z: z=0.143$. Draw projections of the unit cell, determine coordination numbers and bond lengths and describe the structure [J. Solid State Chem. 6, 329 (1973).]

Lottie Adams

Numerade Educator

02:19

Problem 45

Prepare crystal structure projections, either as drawings on paper or (if you have access) using Crystal Maker software, for the structures of $\mathrm{CaCu}_3 \mathrm{Ti}_4 \mathrm{O}_{12}$ (Fig_1.42a), brownmillerite (Fig_1.42b), $\mathrm{La}_2 \mathrm{Ni}_2 \mathrm{O}_5$ (Eig_1.42c), spinel (Eig_1.44), corundum (Eig_1.46a),
ilmenite (Fig._1.46b), $\mathrm{LiNbO}_3$ (Fig._1.46c, d), pyrochlore (Fig._1.48), garnet (Fig._1.49), $\mathrm{K}_2 \mathrm{NiF}_4$ (Fig._1.50), $\mathrm{MgB}_2$ (Fig_1.51). For each of these, you need access to International Tables for X-ray Crystallography, Vol 1, Kynoch Press, to get complete listings of atomic coordinates.
Compare your projections with those given in the book. If you are using software, rotate the structures to see them in different orientations; also try hiding certain features to highlight other parts.

Vivek Singh

Numerade Educator