Experiment 5
Solids II - Ionic lattices
Solids II - Ionic lattices
"Well, have you solved it?" I asked.
"Yes. It was the bisulphate of baryta."
"No, No, the mystery!" I cried.
Sir Arthur Conan-Doyle, "A CASE OF IDENTITY"
Introduction
Ionic solids are an important class of chemical compound. Some are bulk chemical
commodities, such as sodium chloride and calcium chloride. They are found in living systems
(NaCl and KCl) or are used as medications (LiCl for the treatment of manic-depressives). They
have many applications in technology, either as the pure compounds, or more commonly as
"doped" lattices containing small amounts of impurities that do not disturb the crystal lattice.
This type of analysis can be used to understand the closely related structures of common minerals
such as rubies and sapphires. A recent important discovery in the field of ceramics, a branch of
solids related to ionic solids, is the class of sub-stoichiometric Y-Ba-Cu oxides that have a crystal
structure closely related to the stoichiometric Perovskite SrTiO3. They are interesting because
they remain superconducting at liquid nitrogen temperatures, something which no other materials
are known to do. (See Shriver, Atkins, Langford, pp.587-590 for a discussion of these so-called
1-2-3 superconductors.)
From an instructional viewpoint the bonding in ionic compounds can be discussed in a fairly
rigorous manner without a great deal of quantum mechanics. This lab is designed to familiarize
you with the common structures of ionic solids. These lattices will be used in calculating lattice
energies.
Instructional goals:
(1)
(2)
(3)
(4)
(5)
Learn to analyze the unit cells of the common ionic lattice types.
Analyze the radius ratio rules for the common ionic lattice types.
Learn to apply the close-packing of anions analogy to ionic structures.
Learn to use a sophisticated computer-modeling package for crystalline solids called
Ca.R.Ine Crystallography (release 3.1).
Calculate sequential terms of the Madelung constant using the “shells” tool in Carine.
Ionic crystal structures
The common ionic structures that we will consider in detail are depicted in the following picture.
In each case, the smaller circles represent the cations, while the larger circles represent the
anions. This kind of open-lattice unit cell picture shows the ions at 20% or less of their true
radii. This gives the advantage that it is possible to see inside the cell and observe the relative
orientation of the ions and their chemical connectivity. It should be borne in mind that in reality
in all ionic lattices the cations and the anions ought to be in contact. The anions may also contact
the other anions, but more normally they are somewhat pulled apart. An anion-anion contact is
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Experiment 5
Solids II - Ionic lattices
electrostatically repulsive, while the cation-anions contacts are attractive. Some alternate
graphical representations are provided later on for some of these structures. Note that some of
these will be stereoviews. These can simulate a 3-dimensional look. To see the perspective in
such structures, one relaxes the eyes and allows then to cross, such that the stereo image forms
mid-way between the two flat images. Alternatively, separate the two views from the other eye
by holding a piece of white cardboard between the images, extending from the surface of the
paper to the observer’s eyes.
Nickel arsenide
Chemistry 2810 Laboratory Manual
Perovskite
Page 5 - 2
Experiment 5
a)
Solids II - Ionic lattices
NaCl lattice
AB stoichiometry
Cation : anion ratio is 1:1
Each Na ion and Cl ion has six nearest neighbours, so that the
coordination number, CN = 6.
Na-Cl-Na angles are 180 and 90°.
Coordinates: Cl– (0, 0, 0) (½,½, 0) (½, 0, ½) (0, ½, ½)
Na+ (0, ½, ½) (½, 0, ½) (½, ½, 0) (½, ½, ½)
Consider the "sliced" view at the right. Note the strong similarity
to the FCC metallic lattice. In fact, the Cl– ions are arranged just
the same as the metals in FCC. This leaves the larger Oh holes for the Na+ to occupy, which
leads to a single sodium ion at the center of the unit cell, and 12 in total along the centres of each
unit cell edge. These edge atoms are on ¼ within the volume of the cubic cell. The next image
is known as a stereoview, and gives 3-D insight into the structure. Note that in this graphic, the
positions of the sodium and chloride ions have been reversed, and this serves to emphasize the
important fact that the NaCl structure is interchangeable between cation and anion.
b)
CsCl lattice
AB stoichiometry
Eight nearest neighbours. CN = 8.
Angles 180, 70.5 & 109.5°
Coordinates:
Cs+ at (½, ½, ½)
Cl– at (0, 0, 0)
Here again there is a "sliced" view that allows us to accurately count
the unit cell contents. The eight chloride ions at the corners each
contribute only 1/8th of their volume to the cell, while the central
cesium ion is completely within the cell. It is a common mistake to
call this structure "body-centred cubic" - this is false; in fact the lattice type is primitive cubic,
with the cental cesium ion
being intimately related to
the anion at the origin. The
two make a pair. There is
one net CsCl formula per
unit cell.
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Experiment 5
Solids II - Ionic lattices
c)
Zinc blende (cubic ZnS)
AB stoichiometry
CN = 4 for cation and for anion.
Angles 109.5 . Tetrahedra set on one edge.
S: (0, 0, 0) (½ , ½, 0) (½, 0, ½) (0, ½, ½)
Zn: (¼, ¼, ¼) (¼, ¼, ¾) (¾, ¼, ¾) ( ¾ , ¾, ¼)
This structure is closely related
to the diamond structure, but
the arrangement of the anions
is FCC. Thus every second
tetrahedral hole in the lattice is
occupied. There is a close
geometrical affinity to the
diamond structure, and in fact
it is like diamond with every
second carbon atom replaced
by a sulfur atom.
d)
Wurtzite (hexagonal ZnS)
AB stoichiometry
CN = 4 for both ions
Angles 109.5 . Tetrahedra sitting on a
face.
Coordinates:
Zn2+ : (0, 0, 0)
(¼, ¼, ¼)
(¼, ¾, ¾)
(¾, ¼, ¾)
Face atoms:
(½ , ½, 0)
(½, 0, ½)
(0, ½, ½)
Interior atoms
The local geometry is almost identical to that of zinc blende, but the overall symmetry of the
lattice is hexagonal, not cubic.
e)
Fluorite (CaF2)
AB2 stoichiometry
CN = 8 for cation; CN = 4 for anion.
Ca-F-Ca angles are 109.5 ; F-Ca-F
angles are 180, 109.5 and 70.5°.
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Experiment 5
Solids II - Ionic lattices
f)
Antifluorite (e.g. K2O)
(same pictures as fluorite)
A2B stoichiometry
CN = 4 for cation; CN = 8 for anion.
Ca-F-Ca angles are 109.5 ; F-Ca-F angles are 180, 109.5 and 70.5°.
O-K-O angles are 109.5 ; K-O-K angles are 180, 109.5 and 70.5 .
g)
Rutile (TiO2)
AB2 stoichiometry.
CN = 6 for cation; CN = 3 for anion.
Ti-O-Ti angles are 120 ; O-Ti-O angles are 180 and 90 .
h)
Nickel arsenide (NiAs)
AB stoichiometry.
CN = 6 for cation; CN = 6 for anion.
As-Ni-As angles are 90 and 180°; Ni-As-Ni angles are 90 and 180°.
i)
Perovskite (CaTiO3)
ABC3 stoichiometry.
CN = 6 for Ti; CN = 2 for O; CN = 12 for Ca
O-Ti-O angles 90 and 180°; Ti-O-Ti angle 180°; O-Ca-O angles 60, 120 and 180°
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Experiment 5
Solids II - Ionic lattices
Using the close-packed lattice analogy to describe ionic structures
In applying the close packed model, we imagine just the larger of the ions forming a closepacked lattice, in which the anions exist as spheres just touching each other (for anti-lattices, the
cations.) These anions are held in place, however, not by a metallic bond, but by electrostatic
attraction for cations, which occupy the Td and Oh holes.
Now, having said this, considerable expansion of the "close-packed" structure almost always
does occur. I.e. the cations may be larger than the basic size of the holes, and the anions may be
in the same position as in a close-packed structure, but no longer touching. Later we will
consider what kind of size criteria cause alterations from the close-packed structures.
Not all ionic compounds can be understood by this model. Those which can include the
following:
NaCl: ccp array of Cl-; Na+ in all the Oh holes.
[Note it is a special property of the ccp array alone, that the Oh holes describe a
ccp array of their own. Thus NaCl is often spoken of as two interpenetrating ccp
arrays, one of Cl-, the other of Na+, displaced from each other by a/2.]
Zinc blende: ccp array of S2-; Zn2+ in every second Td
hole.
22+
Wurtzite: hcp array of S ; Zn in every second Td hole.
Fluorite (CaF2): ccp array of Ca2+ (i.e. this is an anti-lattice), F- in all Td holes.
Antifluorite: the reverse of CaF2, where a 2- anion is ccp; 1+ cations in Td holes
Rutile: cannot be so described (distorted close packing)
CsCl: is not a close packed array.
The radius ratio of the larger and smaller ions
A cation and anion will approach till they
gently nudge; closer than this will set up
repulsion between the outer electrons in each
electron:
In the lattice, we want to maximize cations and
anions just touching, for this is the greatest
coulombic attraction. But we want to avoid
anion-anion touching (or cation-cation, should
the cations happen to be larger than the anions,
as sometimes happens), since these are purely
repulsive interactions. So the ideal packing of
an ionic crystal is ion pairs touching, but
cations well separated.
Attractive forces
+
−
Approaching
Attractive forces
-
-
e
-
e
e
+
−
-
e
Just touching
-
-
e
e
-
e
+
-
e
-
−
e
Interpenetrating
-
e
Repulsive forces
This is where the idea of coordination number comes in, because what the most favourable
interactions will be depends on relative ion sizes. The radius ratio expresses the relative size of
the cation and the anion. For each type of lattice, we can calculate the ideal radius ratio for
perfect packing using solid geometry. This is the minimum radius ratio that this type of structure
will tolerate; any thing less and the anions will remain touching, but the ion-pairs are no longer
touching. When this occurs, the lattice will switch to one with lower coordination number.
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Experiment 5
Solids II - Ionic lattices
In practice, the range of radius ratio values which still lead to the same structure are given in the
following table. The first number in each range is the limiting ratio. Drop below this number,
and the lattice should jump back to the structure with lower coordination number. The upper
number is the limit of the range in which lattice expansion, or loss of anion-anion contact, occurs
without altering the lattice type. Ideal radius ratios are highlighted in blue.
Stoichiometry
1:1
r+/r_
Lattice type
0.225-0.414
C.N.
3
4
6
8
C.N. of smaller ion
4
0.414-0.732
0.732-1.00
6
8
0.00-0.155
0.225-0.414
0.414-0.732
0.732-1.00
1:2
No examples
Wurtzite and zinc blende
NaCl
CsCl
β-quartz (not dealt with
previously)
Rutile
Fluorite (note r(Ca2+) / r(F-)
Procedure
You will work individually in this tutorial using your workstation running CaRIne. Physical
models of the lattices will also be available for study. As you work through the exercises, fill out
the reporting sheets provided for this lab (a copy of the sheets is provided at the end of this
tutorial). Report sheets must be submitted individually.
Instructions for basic operation of CaRIne were provided as part of the instructions for
Experiment 4. Instructions for additional operations will be provided as they are needed (blue
text). All the data files that are needed to complete these exercises are provided (see the
filenames below). Any students who wishes to (a) use the models supplied by the software
authors, or (b) create their own models from scratch may do so, and it is hoped that many of you
will find this tool to be applicable to further study or research.
Data files are provided for each exercise; once introduced, it is assumed that you will feel free to
go back and consult files already treated, and this may be necessary for the subsequent exercises.
For each exercise, respond to the questions on the reporting sheets provided (this may help
explain what it is you are supposed to be doing!)
Exercise 1
Ionic Lattice Unit Cells
Data files provided: NaCl(unit cell).CRY CsCl(unit cell).CRY ZnBlende(unit cell).CRY
CaF2(unit cell).CRY K2O(unit cell).CRY NiAs(unit cell).CRY Wurtzite(unit cell).CRY
TiO2(unit cell).CRY CaTiO3(unit cell).CRY
NaCl(Clcentre).CRY CsCl(Clcentre).CRY
ZnBlende(Scentre).CRY
CaF2(Zncentre).CRY K2O(Ocentre).CRY NiAs(NiCentre).CRY
NiAs(AsCentre).CRY CaTiO3(CaCentre).CRY
CaTiO3(OCentre).CRY
In this exercise, you start by analyzing each lattice type. Most of the exercise must be done with
reference to the unit cell of the lattice type. Notice that unit cell data files are provided for each
lattice type. The unit cell model in each case allows you to define the cell contents, and hence
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Experiment 5
Solids II - Ionic lattices
arrive at the stoichiometry of the cell (i.e. the chemical identity of the species). From this the
generic formula for the lattice can be determined. As you work through this exercise, fill in the
number ion ions in each structure that are located at corners, faces, edges or are entirely within
the borders of the cells. Apply the appropriate correction factors (for corner atoms, only 1/8th of
the atom is within the cell; for face atoms, ½ of the atom is within the cell; for edge atoms, ¼ is
within, and internal atoms count fully to the unit cell); then calculate the number of each ion type
within the cell, and confirm that this fits the chemical formula for the compound. Then
determine the generic lattice type.
Finally, you are also asked to determine the coordination number (CN) of each type of ion in
your structure. Just as for metallic lattices, CN is defined as the number of ions equidistant to a
given ion, picking those ions that are closest to the ion. In almost all cases, the closest ions will
be of a different charge (i.e. cations to anions or vice verse.) For most of the lattice types, the
unit cell is not a reliable view to determine the CN, since many atoms are located at corners,
edges and faces, where counting is unreliable. Therefore several additional data files are
provided (e.g. for NaCl, CsCl, zinc blende, CaF2, K2O, NiAs (two additional files) and CaTiO3
(two additional files). However, in the case of Wurtzite and TiO2, the full CN can be determined
by judicially chose ions within the unit cell. For the Wurtzite model, use the sulfide ion coloured
green, and the zinc ion coloured red. For TiO2, count the green-coloured Ti ion, and the pink
coloured oxide ion. The CN of all the other ions in the structure are the same as the coloured
examples, but this is not evident because the ions are at the edges.
The data files have been scaled in such a way that all the files indicated for a given lattice type
can be open at the same time. Use the Tile command under the Windows menu to make them all
visible at the same time (e.g. both the NaCl(unit cell) and the NaCl(Clcentre) files can be viewed
simultaneously).
Exercise 2
The radius ratio concept
In this exercise, you will use the Unit Cell data files for each lattice being considered. To help
visualize along which lines in the unit cell the ions touch each other, it may help you to re-scale
the ion radii. As provided, all the ionic lattices are scaled back to 50% of the actual radii. This
lets you see “inside” the structure. Within Carine, on the View menu, the Radii scale can be
adjusted to any value from 0 to 100%. CTRL-X opens this menu as well. Altering the radii scale
does not affect the other attributes of the crystal models. You may also need to remove a surface
ion to see the internal lines of contact. This is most easily achieved using the Hide Atom
command on the Crystal menu. Once activated, any atom that you click on will become
“hidden”. Note that for static models, the effect does not become visible until the model has
been rotated or moved in some fashion. The Recall Atom feature is the reverse procedure (click
on the lattice position where the atom is supposed to be). If many ions have been hidden, the
Recall All Atoms command is more convenient.
Instructions: In each of these lattices, the anion is the larger radius ion (note that we use the
“antifluorite” lattice K2O rather than Fluorite itself, for which this rule does not hold.) The ideal
radius ratio is defined as the radius ratio when the anions touch and the cations just touch anions.
It is a strictly geometric concept.
d(Anion–Anion)/2
Using the Distance between atoms tool in Carine, determine the line in the
lattice along which the anions do (or should) just touch, and measure the distance between the
centres of the ions. Take half this distance as the ideal anion radius.
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Experiment 5
Solids II - Ionic lattices
d(Anion–Cation)
Determine the distance between the centres of the cation and anion along
the line of nearest contact.
Difference
= d(Anion–Cation) – d(Anion–Anion)/2
Ideal rr
= Difference / { d(Anion–Anion)/2}, i.e. the r+ / r– so long as r– > r+.
r(cation)
In the Carine Creation/List window (under the Cell menu), locate the actual radii
the program has assigned to the cation and write it in.
r(anion)
Do the same for the anion radius.
Actual rr
Now calculate the actual value for r+ / r– in this particular ionic compound.
Note: for the purpose of this exercise, the calcium and oxygen ions in the Perovskite structure
both act as “anions”. You should take the average of the values calculated from these two ions,
and it should be a weighted average, i.e. 25% of the calcium value, and 75% of the oxygen value.
Exercise 3
The Holes in Close-Packing Analogy
In this exercise, the Unit Cell models for each lattice type will be most helpful. You may also
wish to refresh your memory of the structures of the Cubic Closest Packed (= FCC) and
Hexagonal Closest Packed structures we considered under the Metallic Lattices (Experiment 4),
or any of the metallic lattices we considered there. The most useful file type are:
CCP(Copper)ABCLayers.CRY; HCP(Magnesium)ABLayers.CRY. You may also wish to
consult the holes in these lattices, cf. CCP(Copper)Holes.CRY; HCP(Magnesium)Holes.CRY.
In addition, you may find some or all of the following prepared models helpful in making your
determinations: NaCl(ABC layers with Td holes).CRY; NaCl(FCC lattice).CRY
ZnBlende(FCC).CRY; ZnBlende(ABC layers; Td holes).CRY; CaF2(as FCC lattice).CRY
CaF2(as FCC lattice).CRY; NiAs(AB layers Ni in Oh holes).CRY; Wurtzite(AB layers).CRY
CaTiO3(mixed ABC layers).CRY
First consider the lattice carefully, and analyze the lattice type of the anions assuming the cations
are not present at all. You can of course use Carine Hide Atom tool to temporarily remove from
view all the cations. Then compare the resulting lattices that you generate with the metallic
lattices we studied in Experiment 4. Enter the anion lattice type on the reporting sheet.
Now, only if the anion lattice type is one of the two closest-packed lattices, continue on to
consider the location(s) of the cations in the resulting holes (i.e. are they tetrahedral or octahedral
holes; how many of the available are holes are occupied.)
Using geometry, along with the Distance Between Atoms tool to calculate the radius of the holes
assuming that the anions are truly touching (this is the same calculation as the radius ratio
determination). Use the Carine Creation/List menu to report that actual cation radius of the
cation (again, this was done in the radius ratio exercise.)
If the parent lattice is not closest-packed leave the other entries blank. Enter all your responses
on the reporting sheets, and answer the two questions that are posed.
Exercise 4
Calculation of Madelung Constants
The Madelung Constant is a geometrical factor that counts the number of attractive and
repulsive terms in the geometry of the crystal lattice that affect the net electrostatic attractive
force that holds the crystal lattice together. It is the sole term in the calculation of the lattice
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Experiment 5
Solids II - Ionic lattices
energy that accounts for the type of crystal lattice being considered. All Madelung Constants are
infinite series that only slowly converge to the limit. In this exercise, you will calculate no more
and no less than eight terms of the Madelung series for four of the cubic lattice types.
You will need to open the new datafiles NaCl(2x2).CRY, CsCl(4x4).CRY, ZnBlende(2x2).CRY
and Fluorite(2x2).CRY. These models are large enough to allow the eight terms to be calculated.
The centre atom in each model is coloured red. Pay attention to the sign of each term; an
attractive term is positive (an addition), while a repulsive term is negative (a subtraction.) Not all
lattice types alternate consistently (i.e. two terms in succession may have the same sign.)
Carine has a function called Shells that can be activated from the toolbar icons, or else from the
Specials>Environment>Shells menu. Once activated, the function is initiated by clicking on a
given ion. For the purpose of this exercise, the central atom in each lattice coloured red. This is
the atom you should click on. Then a window opens, displaying the first shell (those atoms that
are closest to the central atom), and gives the distance in Å to those atoms, and the number of
atoms at that distance. Notice that the program can have significant difficulty with internal
rounding of the calculated atom positions. Thus two apparently distinct ion types with almost the
same distance (e.g. 4.1231 and 4.1232 Å) should be combined.
To identify the kind of ion contained in each shell, you can highlight that shell (or with multiple
clicks, more than one shell – this particularly helpful with ions you expect to be within the same
shell because of rounding off errors), and then click on OK – now all atoms except the central
atom and those ions in the selected shell become hidden. You can now identify the type of ion
involved in that shell. After doing this, you need to recall all the hidden atoms (Crystal>Recall
all atoms Menu) before you can do this for the next shell. In this way you can very quickly build
up the number of each ion at each unique distance and also the type of ion (i.e. repulsive or
attractive interaction.)
Note: In some cases, the program has problems with internal rounding-off; where terms are very
close in distance they should be combined. Such ions should also be identical by symmetry. The
very useful feature of being able to highlight atoms from more than one shell for display
purposes will help in making such decisions.
The Madelung constant A is computed as follows:
6
12
etc
A=
−
+
(continue for 8 terms )
2.823 3.9924 etc
Samples of the reporting sheets
(starting on the next page)
Chemistry 2810 Laboratory Manual
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Experiment 5
Exercise 1
Solids II - Ionic lattices
Ionic Lattice Unit Cells
Ionic structures belonging to the cubic crystal system
Structure
Corner
Face
Edge
Internal
model
NaCl
Anion
(Rock salt)
Cation
CsCl
Anion
(same name)
Cation
ZnS
Anion
(zinc blende)
Cation
CaF2
Cation!
(fluorite)
Anion!
K2O
Anion
(antifluorite)
Cation
Ionic structures belonging to the hexagonal crystal system
NiAs
Anion
(Ni arsenide)
Cation
ZnS
Anion
(wurtzite)
Cation
An ionic structure belonging to the tetragonal crystal system
TiO2
Anion
(rutile)
Cation
A complex (cubic) ionic crystal structure
CaTiO3
Anion
(perovskite)
Cation A
Total ion C.N.
count
AxBy…
Cation B
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Experiment 5
Exercise 2
Solids II - Ionic lattices
The Radius-Ratio (rr) Concept
Lattice type
d(Anion–
Anion)/2
d(Anion Differ–Cation) ence
Ideal rr
r(cation)
r(anion)
Actual rr
NaCl
CsCl
ZnS (zinc blende)
K2O
NiAs
ZnS (Wurtzite)
TiO2
CaTiO3 (Ca = O)
O value
Ca value
Note: for the purpose of this exercise, the calcium and oxygen ions in the Perovskite structure
both act as “anions”. You should take the average of the values calculated from these two ions,
and it should be a weighted average, i.e. 25% of the calcium value, and 75% of the oxygen value.
Question: Discuss the discrepancy, if any, between the ideal and actual radius ratios for the 8
lattice types in this exercise. Is the deviation in the direction you expect? Explain.
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Experiment 5
Exercise 3
Solids II - Ionic lattices
The Holes in Close Packing Analogy
Lattice type
Anion lattice
type
Type of hole
# holes
occupied
Ideal hole
size (radius)
Actual cation
radius
NaCl
CsCl
ZnS (zinc blende)
K2O
NiAs
ZnS (Wurtzite)
TiO2
CaTiO3
(Ca = O)
Question 1: The Perovskite structure, CaTiO3 differs from the simple binary salts in the way the
primary lattice (larger ions) is defined, while the secondary lattice (i.e. the constituents of the
holes) are similar to those of the other examples. Describe the CaTiO3 lattice in detail using the
holes in close packing analogy.
Question 2: Using the concepts above (especially your treatment of K2O), describe fluorite,
CaF2, using the holes in close packing analogy.
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Experiment 5
Exercise 4
Solids II - Ionic lattices
Estimating Madelung Constants
NaCl (2x2 model) Centre ion =
CsCl (4x4 model) Centre ion =
Distance
Distance
No.
Type
No.
Type
ZnBlende (2x2 model) Centre ion =
CaF2 (2x2 model) Centre ion =
Distance
Distance
No.
Type
No.
Type
The centre atom in each model is coloured red. Pay attention to the sign of each term; an
attractive term is positive (an addition), while a repulsive term is negative (a subtraction.) Not all
lattice types alternate consistently (i.e. two terms in succession may have the same sign.)
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