Crystal Structure 12

Crystal Structure Having introduced a number of important ideas in one dimension, we 1 Warning: Some books (Ashcroft and must now deal with the fact that our world is actually spatially three- Mermin in particular) refer to this as a dimensional. While this adds a bit12 of complication, really the important Bravais lattice.Thisenablesthemto concepts are no harder in three dimensions than they were in one di- use the term lattice to describe other things that we would not call a lattice mension. Some of the most important ideas we have already met in one (e.g., the honeycomb). However, the dimension, but we will reintroduce them more generally here. Having introduced a number of important ideas in one dimension, we definition we use here is more common There are two things1Warning: that might Some be diffi bookscult here. (Ashcroft First, we and do need to among crystallographers, and more cor- wrestle with a bit of geometry. Hopefully most will not find this too hard. rect mathematically as well. must now deal with the fact that our world is actually spatially three- Mermin in particular) refer to this as a Secondly we will also need to establish a language in order to describe 2 Bravais lattice Very frequently “primitive lattice vec- dimensional. While this adds a bit of complication, really the importantstructures in two and three dimensions.Thisenablesthemto intelligently. As such, much of tors” are called “primitive basis vec- concepts are no harder in three dimensions than they were inthis one chapter di- is justuse a list the of definitions term lattice to beto learned, describe but unfortunately other tors” (not the same use of the word this is necessary inthings order to that be able we to wouldCrystal continue not further call Structure a at lattice this point. “basis” as in Section 10.1) or “primi- tive translation vectors”. mension. Some of the most important ideas we have already met in one (e.g., the honeycomb). However, the 12 dimension, but we will reintroduce them more generally here. definition we use here is more common Having introduced a number of important ideas in one dimension, we 12.1 Lattices and Unit Cells 1Warning: Some books (Ashcroft and There are two things that might be difficult here. First, we do need to among crystallographers,must now deal and with more the fact thatcor- our world is actually spatially three- Mermin in particular) refer to this as a [1, 2] = a1 +2a2 rectlattice mathematically1 dimensional. as well. While this adds a bit of complication, really the important Bravais lattice.Thisenablesthemto Definition 12.1 A is an infiniteconcepts set of are points no harder defined in three by dimensions integer than they were in one di- use the term lattice to describe other wrestle with a bit of geometry. Hopefully most will not find this too hard. 2 sums of a set of linearly independent primitive lattice vectors. a2 things that we would not call a lattice mension. Some of the most important ideas we have already met in one (e.g., the honeycomb). However, the Secondly we will also need to establish a language in order to describe 2 dimension, but we will reintroduce them more generally here. Very frequently “primitive lattice vec- a definition we use here is more common For example, in two dimensions, as shownThere are in two Fig. things 12.1 that might the be lattice difficult here. First,1 we do need to among crystallographers, and more cor- rect mathematically as well. structures in two and three dimensions intelligently. As such,points much are described of tors” as are calledwrestle “primitive with a bit of basis geometry. vec- Hopefully most will not find this too hard. 114 Crystal Structure this chapter is just a list of definitions to be learned, but unfortunately tors” (not the sameSecondly use we will of also the need toword establish a language in order to describe 2Very frequently “primitive lattice vec- structures in two and three dimensions intelligently.Fig. 12.1 As such,Alatticeisdefinedasinteger much of tors” are called “primitive basis vec- R = n a + n a thisn chapter,n isZ just a list of definitions(2d) to be learned, but unfortunately tors” (not the same use of the word [n1 n2] 1“basis”1 2 2 as in Section1 2 10.1) or “primi- sums of of primitive lattice vectors. this is necessary in order to be able to continue further at this point.PHYS601 –SSP1 this is necessary∈ in order to be able to continue further at this point. “basis” as in Section 10.1) or “primi- tive translation vectors”. Crystal Structuretive translation vectors”. It turns out that there are several definitions that are entirely equiv- with a1 and a2 being the primitive lattice vectors and n1 and n2 being 12 alent to the one we have just given: integers. In three dimensions points of12.1 a lattice Lattices are analogously and Unit indexed Cells [1, 2] = a1 +2a2 by three integers: Definition 12.1 A lattice1 is an infinite set of points defined by integer Equivalent Definition 12.1.1 A lattice is an infinite set of vectors Having introduced2 a number of important ideas in one dimension, we 1 12.1 Lattices and Unit Cells sums of a set of linearly independent primitive lattice vectors. a2 Warning: Some books (Ashcroft and where addition of any two vectors in the set gives a third vector in the must now deal with the fact that our world is actually spatially three- Mermin in particular) refer to this as a R = n a1 + n a2 + n a3 n ,n ,n Z (3 d). [n1 n2 n3] 1 2 3 For example,1 2 in3 two dimensions,dimensional. as shown Whilein Fig. this 12.1 adds the a lattice bit of complication,a1 really the important Bravais lattice.Thisenablesthemto set. [1, 2] = a∈1 +2a2 use the term lattice to describe other 1 points are described as concepts(12.1) are no harder in three dimensions than they were in one di- Definition 12.1 A lattice is an infinite set of points defined by integer mension. Some of the most important ideas we have already met in one things that we would not call a lattice Note that in one dimension this definition of a lattice fits with our pre- Fig. 12.1 Alatticeisdefinedasinteger (e.g., the honeycomb). However, the It is easy to see that our first definition 12.1 implies the second one 12.1.1. 2 R[n n ] = n1a1 + n2a2 dimension,n1,n2 butZ we will reintroduce(2d) them more generally here. definition we use here is more common a2 1 2 ∈ sums of of primitive lattice vectors. sums of a set of linearly independent primitive lattice vectors.vious description of a lattice as being the points R = na with Theren an are two things that might be difficult here. First, we do need to among crystallographers, and more cor- Here is a less crisply defined, but sometimes more useful definition. integer. with a1 and a2 being the primitivewrestle lattice with vectors a bit and of geometry.n1 and n2 Hopefullybeing most will not find this too hard. rect mathematically as well. integers. In three dimensions points of a lattice are analogously indexed a1 Secondly we will also need to establish a language in order to describe 2Very frequently “primitive lattice vec- It is important to point out that inby two three and integers: three dimensions, the Equivalent Definition 12.1.2 A lattice is a set of points where the For example, in two dimensions, as shown in Fig. 12.1 the lattice 3 structures in two and three dimensions intelligently. As such, much of tors” are called “primitive basis vec- choice of primitive lattice vectors is not unique, as shown in Fig.this 12.2. chapter is just a list of definitions to be learned, but unfortunately tors” (not the same use of the word environment of any given point is equivalent to the environment of any R = n1a1 + n2a2 + n3a3 n1,n2,n3 Z (3 d). “basis” as in Section 10.1) or “primi- points are described as (In one dimension, the single primitive lattice[n1 n2 n3] vector is unique upthis to is the necessary in∈ order to be able to continue further at this point. (12.1) tive translation vectors”. other given point. sign, or direction, of a.) Note that in one dimension this definition of a latticeFig. 12.2 fits withThe our choice pre- of primitive lat- Z Fig. 12.1 Alatticeisdefinedasintegervious description of a lattice as being12.1 the LatticespointsticeR vectors= na andwith for Unit an latticean Cells is not unique. It turns out that any periodic structure can be expressed as a lattice of R[n1 n2] = n1a1 + n2a2 n1,n2 (2d) integer. sums of of primitive lattice vectors. (Four possible sets of primitive lattice [1, 2] = a1 +2a2 ∈ 3 It is important to point out that in two and three dimensions,1 the repeating motifs. A cartoon of this statement is shown in Fig. 12.3. One Given a set of primitive lattice vectors ai anewsetofprimitivelatticevectorsmayDefinition 12.1vectorsA lattice are shown,is an infinite but there set of are points an defined in- by integer 3 2 choice of primitive lattice vectors issums not of unique, a set ofas linearly shown independent in Fig. 12.2. primitive lattice vectors. a2 be constructed as bi = j mij aj so long as mij is an invertible matrix with integer finite number of possibilities!) should be cautious however, that not all periodic arrangements of points and −1 (In one dimension, the single primitive lattice vector is unique up to the 4 with a1 and a2 being the primitive lattice vectors and n1 andentriesn2 beingthe inverse matrix! [m ]ij also has integer entries. For example, in two dimensions, as shown in Fig. 12.1 the lattice a1 are lattices. The honeycomb shown in Fig. 12.4 is not a lattice. This sign, or direction, of a.) Fig. 12.2 The choice of primitive lat- integers. In three dimensions points of a lattice are analogously indexed points are described as tice vectors for a lattice is not unique. is obvious from the third definition 12.1.2: The environment of point (Four possible sets of primitive lattice 3 Given a set of primitive lattice vectors ai anewsetofprimitivelatticevectorsmayR = n a + n a vectorsn ,n areZ shown, but there(2d) are an in- Fig. 12.1 Alatticeisdefinedasinteger P and point R are actually different—point P has a neighbor directly [n1 n2] 1 1 2 2 1 2 sums of of primitive lattice vectors. by three integers: be constructed as bi = j mij aj so long as mij is an invertible matrix with integer finite number∈ of possibilities!) −1 entries and the inverse matrix! [m ]ij also has integer entries. above it (the point R), whereas point R has no neighbor directly above. with a1 and a2 being the primitive lattice vectors and n1 and n2 being integers. In three dimensions points of a lattice are analogously indexed In order to describe a honeycomb (or other more complicated arrange- R = n a1 + n a2 + n a3 n ,n ,n Z (3 d). by three integers: [n1 n2 n3] 1 2 3 1 2 3 ∈ Fig. 12.3 Any periodic structure can ments of points) we have the idea of a unit cell, which we have met before R = n a + n a + n a n ,n ,n Z (3 d). be represented as a lattice of repeating (12.1) [n1 n2 n3] 1 1 2 2 3 3 1 2 3 in Section 10.1. Generally we have ∈ (12.1) motifs. Note that in one dimension this definition of a lattice fits with our pre- Note that in one dimension this definition of a lattice fits with our pre- Definition 12.2 A unit cell is a region of space such that when many vious description of a lattice as being the points R = na with n an vious description of a lattice as being the points R = na with n an integer. identical units are stacked together it tiles (completely fills) all of space It is important to point out that in two and three dimensions, the and reconstructs the full structure. integer. choice of primitive lattice vectors is not unique,3 as shown in Fig. 12.2. (In one dimension, the single primitive lattice vector is unique up to the It is important to point out that in two and three dimensions, the R An equivalent (but less rigorous) definition is sign, or direction, of a.) Fig. 12.2 The choice of primitive lat- 3 tice vectors for a lattice is not unique. 1 choice of primitive lattice vectors is not unique, as shown in Fig. 12.2.VESTA is an important tool for this class . Free to download and install (Four possible sets of primitive lattice P Q 3 Equivalent Definition 12.2.1 A unit cell is the repeated motif which Given a set of primitive lattice vectors ai anewsetofprimitivelatticevectorsmay vectors are shown, but there are an in- be constructed as bi = j mij aj so long as mij is an invertible matrix with integer finite number of possibilities!) (In one dimension, the single primitive lattice vector is unique up to the −1 is the elementary building block of the periodic structure. entries and the inverse matrix! [m ]ij also has integer entries. sign, or direction, of a.) Fig. 12.2 The choice of primitive lat- To be more specific we frequently want to work with the smallest possible tice vectors for a lattice is not unique. unit cell: (Four possible sets of primitive lattice 3 Definition 12.3 A primitive unit cell for a periodic crystal is a unit Given a set of primitive lattice vectors ai anewsetofprimitivelatticevectorsmay vectors are shown, but there are an in- cell containing exactly one lattice point. be constructed as bi = j mij aj so long as mij is an invertible matrix with integer finite number of possibilities!) entries and the inverse matrix [m−1] also has integer entries. Fig. 12.4 The honeycomb is not a lat- ! ij tice. Points P and R are inequivalent As mentioned in Section 10.1 the definition of the unit cell is never (points P and Q are equivalent). unique. This is shown, for example, in Fig. 12.5. Sometimes it is useful to define a unit cell which is not primitive in 4One should be very careful not to order to make it simpler to work with. This is known as a conventional call the honeycomb a hexagonal lattice. unit cell. Almost always these conventional unit cells are chosen so as First of all, by our definition it is not to have orthogonal axes. alatticeatallsinceallpointsdonot Some examples of possible unit cells are shown for the triangular lat- have the same environment. Secondly, some people (perhaps confusingly) use tice in Fig. 12.6. In this figure the conventional unit cell (upper left) is the term “hexagonal” to mean what chosen to have orthogonal axes—which is often easier to work with than the rest of us call a triangular lattice: axes which are non-orthogonal. alatticeoftriangleswhereeachpoint has six nearest neighbor points (see A note about counting the number of lattice points in the unit cell. It Fig. 12.6). is frequently the case that we will work with unit cells where the lattice Primitive lattice vectors

Q: How can we describe these lattice vectors (there are an infinite number of them)? A: Using primitive lattice vectors (there are only d of them in a d-dimensional space). Primitive lattice vectors For a 3D lattice, we can find three primitive lattice vectors (primitive translation vectors), such that any translation vector can be written as ⃗ Primitive lattice vectors! = &()⃗( + &+)⃗+ + &,)⃗, where &(, &+ and &, are three integers.

For a 2D lattice, we can find two primitive lattice vectors (primitive translation vectors), such that any translation vector can be written asRed (shorter) vectors: )⃗( and )⃗+ !⃗ = &()⃗( + &+)⃗+ Blue (longer) vectors: -( and -+ where &( and &+ are two integers. )⃗ )⃗ For a 1D lattice, we can find one primitive ( and lattice + are primitive lattice vectorsvector (primitive translation vector), such that any translation vector can be written as-( and -+ are NOT primitive lattice vectors ⃗ 1! = &()⃗( -&( = 2)⃗( + 0 )⃗+ )⃗( = -( + 0-+ where ( is an integer. 2

Integer coefficients noninteger coefficients 1D example 1D example 1D crystal 3 atoms/periodicity 1D crystal 3 atoms/periodicity Choice I: Choice I:

Choice II: Choice II:

Choice III: Choice III: 12.1 Lattices and Unit Cells 115

points live at the corners (or edges) of the cells. When a lattice point is on the boundary of the unit cell, it should only be counted fractionally depending on what fraction of the point is actually in the cell. So for example in the conventional unit cell shown in Fig. 12.6, there are two 114 Crystal Structure lattice points within this cell. There is one point in the center, then four points at the corners—each of which is one quarter inside the cell, so we 1 It turns out that there are several definitions that are entirely equiv- obtain 2 = 1 + 4( 4 ) pointsalent in to the the cell. one we (Since have just there given: are two lattice points in this cell, it is by definition not primitive.) Similarly for the primitive cell shown in Fig. 12.6Equivalent (upper right), Definition the two 12.1.1 latticeA lattice pointsis an at infinite the far set of vectors where additiono of any two vectors in the set gives a third vector in the left and the far right haveset. a 60 degree slice (which is 1/6 of a circle) inside the cell. The other two lattice points each have 1/3 of the lattice point inside the unit cell.It is easyThus to thissee that unit our cell first definition contains 12.1 2( 1 implies)+2( the1 )=1 second one 12.1.1. Here is a less crisply defined, but sometimes3 more useful6 definition. point, and is thus primitive. Note however, that we can just imagine shifting the unit cell a tinyEquivalent amount Definition in almost 12.1.2 anyA directionlattice is such a set that of points a where the single lattice point is completelyenvironment inside of any given the unit point cell is equivalent and the to others the environment are of any other given point. completely outside the unit cell. This sometimes makes counting much easier. It turns out that any periodic structure can be expressed as a lattice of Also shown in Fig. 12.6repeating is a so-called motifs. A cartoonWigner–Seitz of this statement unit cell is shown in Fig. 12.3. One should be cautious however, that not all periodic arrangements of points are lattices. The honeycomb4 shown in Fig. 12.4 is not a lattice. This Definition 12.4 Givenis a obvious lattice from point, the third the definition set of all 12.1.2: points The in environment space of point which are closer to thatP givenand point latticeR are point actually than di tofferent—point any other latticeP has a point neighborFig. directly 12.5 The choice of a unit cell is constitute the Wigner–Seitzabove it (the cell pointof theR), given whereas lattice point R point.has no5 neighbor directlynot above. unique. All of these unit cells can be used as “tiles” to perfectly recon- In order to describe a honeycomb (or other more complicated arrange- struct the full crystal. Fig. 12.3ThereAny periodic is a rather structure simple can ments scheme of points) for we constructing have the idea of such a unit a cell, Wigner–Seitz which we have met before be represented as a lattice of repeating in Section 10.1. Generally we have motifs. cell: choose a lattice point and draw lines to all of its possible near neighbors (not just itsDefinition nearest neighbors). 12.212.1ALatticesunit and cell Then Unitis Cells a regiondraw115 of perpendicular space such that when many bisectors of all of theseidentical lines. units The are perpendicular stacked together bisectors it tiles (completely bound fills) the all of space points live at the corners (or edges) of the cells. Whenand a lattice reconstructs point is the full structure. on the boundaryWigner–Seitz of the unit cell, itcell. should Itonly is be counted always fractionally true that the Wigner–Seitz construction Aconventional Aprimitive depending on what fraction of the point is actually in the cell. So for unit cell unit cell examplefor inR the a conventional lattice unit gives cell shown a in primitive Fig.An 12.6, equivalent there are unit two (but cell. less In rigorous) Fig. 12.7 definition we show is another lattice points within this cell. There is one point in the center, then four points atexample the corners—eachP of of the whichQ Wigner–Seitz is one quarter inside the cell, construction so we for a two-dimensional lattice. 1 Equivalent Definition 12.2.1 A unit cell is the repeated motif which obtain 2 = 1 + 4( 4 ) points in the cell. (Since there are two lattice points in this cell, it is by definition not primitive.) Similarlyisthe for the elementary primitive building block of the periodic structure. cell shown in Fig. 12.6 (upper right), the two lattice points at the far left and the far right have a 60o degree slice (whichTo is be 1/6 more of a circle) specific we frequently want to work with the smallest possible inside the cell. The other two lattice points each have 1/3 of the lattice 1 1 point inside the unit cell. Thus this unit cell containsunit 2( 3 cell:)+2(6 )=1 Wigner–Seitz point, and is thus primitive. Note however, that we can just imagine shifting the unit cell a tiny amount in almost any directionDefinition such that 12.3 a A primitive unit cell for a periodic crystal is a unit unit cell single lattice point is completely inside the unit cell and the others are completely outside the unit cell. This sometimes makescell counting containing much exactly one lattice point. Fig.easier. 12.4 The honeycomb is not a lat- Also shown in Fig. 12.6 is a so-called Wigner–Seitz unit cell tice. Points P and R are inequivalent As mentioned in Section 10.1 the definition of the unit cell is never (pointsDefinitionP and 12.4QGivenare equivalent). a lattice point, the set ofunique. all points in This space is shown, for example, in Fig. 12.5. which are closer to that given lattice point than to any other lattice point Fig. 12.5 The choice of a unit cell is Fig. 12.6 Some unit cells for the trian- constitute the Wigner–Seitz cell of the given lattice point.Sometimes5 itnot is unique. useful All of to these define unit cells a can unit cell which is not primitive in be used as “tiles” to perfectly recon- 4 gular lattice. OneThere should is a rather be simple very scheme careful for constructingnot to suchorder a Wigner–Seitz to make itstruct simpler the full crystal. to work with. This is known as a conventional callcell: the choose honeycomb a lattice a point hexagonal and draw lattice. lines to allunit of its possible cell. Almost near always these conventional unit cells are chosen so as neighbors (not just its nearest neighbors). Then draw perpendicular First of all, by our definition it is not to have orthogonal axes. 5 alatticeatallsinceallpointsdonotbisectors of all of these lines. The perpendicular bisectors bound the AconstructionanalogoustoWigner– Wigner–Seitz cell. It is always true that the Wigner–SeitzSome construction examplesAconventional of possibleAprimitive unit cells are shown for the triangular lat- have the same environment. Secondly, unit cell unit cell Seitz can be performed on an irregular for a lattice gives a primitive unit cell. In Fig. 12.7tice we in show Fig. another 12.6. In this figure the conventional unit cell (upper left) is someexample people of the (perhaps Wigner–Seitz confusingly) construction for use a two-dimensional lattice. collection of points as well as on a peri- the term “hexagonal” to mean what chosen to have orthogonal axes—which is often easier to work with than odic lattice. For such an irregular set of the rest of us call a triangular lattice: axes which are non-orthogonal. Fig. 12.7 The Wigner–Seitz construction for a lattice in two dimensions. On the left point the region closer to one particular alatticeoftriangleswhereeachpoint A note about counting the numberWigner–Seitz of lattice points in the unit cell. It has sixperpendicular nearest neighbor bisectors points (see are added between the darker pointunit and cell each of its neighbors. point than to any other of the points is is frequently the case that we will work with unit cells where the lattice Fig. 12.6).The area bounded defines the Wigner–Seitz cell. On the right it is shown that the known as a Voronoi cell. Wigner–Seitz cell is a primitive unit cell. (The cells on the right are exactly the same Fig. 12.6 Some unit cells for the trian- shape as the bounded area on the left!) gular lattice.

5AconstructionanalogoustoWigner– Seitz can be performed on an irregular collection of points as well as on a peri- odic lattice. For such an irregular set of Fig. 12.7 The Wigner–Seitz construction for a lattice in two dimensions. On the left point the region closer to one particular perpendicular bisectors are added between the darker point and each of its neighbors. point than to any other of the points is The area bounded defines the Wigner–Seitz cell. On the right it is shown that the known as a Voronoi cell. Wigner–Seitz cell is a primitive unit cell. (The cells on the right are exactly the same shape as the bounded area on the left!) 2D Bravais lattices 5 Bravaus lattices in 2D

http://en.wikipedia.org/wiki/Bravais_lattice 12.1 Lattices and Unit Cells 115 points live at the corners (or edges) of the cells. When a lattice point is on the boundary of the unit cell, it should only be counted fractionally depending on what fraction of the point is actually in the cell. So for example in the conventional unit cell shown in Fig. 12.6, there are two lattice points within this cell. There is one point in the center, then four points at the corners—each of which is one quarter inside the cell, so we 1 obtain 2 = 1 + 4( 4 ) points in the cell. (Since there are two lattice points in this cell, it is by definition not primitive.) Similarly for the primitive cell shown in Fig. 12.6 (upper right), the two lattice points at the far left and the far right have a 60o degree slice (which is 1/6 of a circle) inside the cell. The other two lattice points each have 1/3 of the lattice 1 1 point inside the unit cell. Thus this unit cell contains 2( 3 )+2(6 )=1 point, and is thus primitive. Note however, that we can just imagine shifting the unit cell a tiny amount in almost any direction such that a single lattice point is completely inside the unit cell and the others are completely outside the unit cell. This sometimes makes counting much easier. Also shown in Fig. 12.6 is a so-called Wigner–Seitz unit cell Definition 12.4 Given a lattice point, the set of all points in space which are closer to that given lattice point than to any other lattice point Fig. 12.5 The choice of a unit cell is constitute the Wigner–Seitz cell of the given lattice point.5 not unique. All of these unit cells can be used as “tiles” to perfectly recon- There is a rather simple scheme for constructing such a Wigner–Seitz struct the full crystal. cell: choose a lattice point and draw lines to all of its possible near neighbors (not just its nearest neighbors). Then draw perpendicular bisectors of all of these lines. The perpendicular bisectors bound the Wigner–Seitz cell. It is always true that the Wigner–Seitz construction Aconventional Aprimitive for a lattice givesWigner Seitz construction a primitive unit cell. In Fig. 12.7 we show another unit cell unit cell example of the Wigner–Seitz construction for a two-dimensional lattice.

Wigner–Seitz unit cell

Fig. 12.6 Some unit cells for the trian- gular lattice.

5AconstructionanalogoustoWigner– Seitz can be performed on an irregular collection of points as well as on a peri- odic lattice. For such an irregular set of Fig. 12.7 The Wigner–Seitz construction for a lattice in two dimensions. On the left point the region closer to one particular perpendicular bisectors are added between the darker point and each of its neighbors. point than to any other of the points is The area bounded defines the Wigner–Seitz cell. On the right it is shown that the known as a Voronoi cell. Wigner–Seitz cell is a primitive unit cell. (The cells on the right are exactly the same shape as the bounded area on the left!) 116 Crystal Structure

6Eugene Wigner was yet another Nobel A similar construction can be performed in three dimensions in which laureate who was one of the truly great case one must construct perpendicular-bisecting planes to bound the minds of the last century of physics. Wigner–Seitz cell.6 See for example, Figs. 12.13 and 12.16. Perhaps as important to physics was the fact that his sister, Margit, mar- ried Dirac. It was often said that Dirac Definition 12.5 The description of objects in the unit cell with respect could be a physicist only because Mar- to the reference lattice point in the unit cell is known as a basis. 12.2 Lattices in Three Dimensions 117 git handled everything else. Fredrick Seitz was far less famous, but gained This is the same definition of “basis” that we used in Section 10.1. In notoriety in his later years by being other words, we think of reconstructing(triangular) the entirelattice, wherecrystal we by can associating write the primitive lattice vectors as aconsultantforthetobaccoindustry, with each lattice point a basis of atoms. astrongproponentoftheRegan-era a1 = a xˆ Star Wars missile defense system, and In Fig. 12.8 (top) we show a periodic structure in two dimension made a2 =(a/2) xˆ +(a√3/2) yˆ. (12.3) aprominentscepticofglobalwarming. of two types of atoms. On the bottom we show a primitive unit cell He passed away in 2007. (expanded) with the position ofIn the terms atoms of the given reference with points respect of the lattice, to the the basis for the primitive reference point of the unit cell whichunit cell, is taken i.e., the to coordinates be the lower of the left-hand two larger circles with respect to the reference point, are given by 1 (a + a )and 2 (a + a ). corner. We can describe the basis of this crystal as follows: 3 1 2 3 1 2 Basis and location of atoms in unit cell

Fig. 12.9 Left: The honeycomb from Basis for crystal in Fig. 12.8 = Fig. 12.4 is shown with the two inequiv- 2 (a + a ) alent points of the unit cell given dif- a1 3 1 2 ferent shades. The unit cell is out- lined dotted and the corners of the Large Light Gray Atom Position= [a/2,a/2] unit cell are marked with small black 1 dots (which form a triangular lattice). 3 (a1 + a2) Right: The unit cell is expanded and Small Dark Gray Atoms Position= [a/4,a/4] coordinates are given with respect to the reference point at the lower left cor- [a/4, 3a/4] a2 ner. [3a/4,a/4] [3a/4, 3a/4] a 12.2 Lattices in Three Dimensions

The simplest lattice in three dimensions is the simple cubic lattice shown a 3a 3a 3a The reference points (the small blackin Fig. 12.10 dots (sometimes in the figure) known forming as cubic “P” the or cubic-primitive lattice). [ 4 , 4 ] [ 4 , 4 ] square lattice have positions The primitive unit cell in this case can most conveniently be taken to be a single cube—which includes 1/8 of each of its eight corners (see R =[an ,anFig.]= 12.11).an xˆ+ an yˆ (12.2) [n1 n2] 1 Only2 slightly1 more complicated2 than the simple cubic lattice are the a a [ 2 , 2 ] tetragonal and orthorhombic lattices where the axes remain perpendicu- with n1,n2 integers so that the largelar, but light the gray primitive atoms lattice have vectors positions may be of different lengths (shown in Fig. 12.11). The orthorhombic unit cell has three different lengths of light−gray R =[anits1 perpendicular,an2]+[a/2 primitive,a/2] lattice vectors, whereas the tetragonal unit [ a , a ] [ 3a , a ] [n1 n2] 4 4 4 4 cell has two lengths the same and one different. Fig. 12.10 Acubiclattice,otherwise known as cubic “P” or cubic primitive. [0, 0] whereas the small dark gray atoms have positions a, b, c dark−gray1 c = a all different R =[an1,an2]+[a/4,a/4] ̸ a [n1 n2] Rdark−gray2 =[an ,an ]+[a/4, 3a/4] [n1 n2] 1 2 Fig. 12.8 Top: Aperiodicstructurein dark−gray3 c c To remember: R CRYSTAL = LATTICE + BASIS=[an ,an ] + [3a/a 4,a/4] two dimensions. A unit cell is marked [n1 n2] 1 2 with the dotted lines. Bottom: A dark−gray4 Fig. 12.11 Unit cells for cubic, tetrag- blow-up of the unit cell with the coor- R =[an1a,an2] + [3a/4, 3a/4]a . b onal, and orthorhombic lattices. [n1 n2] a a a dinates of the objects in the unit cell with respect to the reference point in Cubic Tetragonal Orthorhombic In this way you can say that the positionsunit cell of the atoms inunit the cell crystal unit cell the lower left-hand corner. The basis is are “the lattice plus the basis”. the description of the atoms along with these positions. We can now return to the case of the honeycomb shown in Fig. 12.4. The same honeycomb is shown in Fig. 12.9 with the lattice and the basis explicitly shown. Here, the reference points (small black dots) form a 122 Crystal Structure

122 Crystal Structure the spheres than packing the spheres in a simple cubic lattice10 (see also 10In fact it is impossible to pack spheres Exercisethe spheres 12.4). than packing Correspondingly, the spheres in a simple bcc and cubic fcc lattice lattices10 (see alsoare realized much more densely10In factthan it is you impossible would to pack get spheres by placing themore spheres densely at than the you vertices would get of by moreExercise frequently 12.4). Correspondingly, in nature thanbcc and simple fcc lattices cubic are realized (at least much in the case of a an fcc lattice.placing This the resultspheres (knownat the vertices em- of singlemore frequently atom basis). in nature For than example, simple cubic the (at elements least in the Al, case Ca, of Au, a Pb, Ni, Cu, pirically toan people fcc lattice. who This have result tried (known to em- single atom basis). For example, the elements Al, Ca, Au, Pb, Ni, Cu, pirically to people who have tried to AgAg (and (and many many others) others) are fcc are whereas fcc whereas the elements the Li, elements Na, K, Fe, Li, Mo, Na, K, Fe, Mo, pack orangespack in oranges a crate) in a crate)was first was first offi o-ffi- cially conjecturedcially conjectured by Johannes by Johannes Kepler Kepler CsCs (and (and many many others) others) are bcc. are bcc. in 1611, butin 1611, was but not was mathematically not mathematically proven untilproven 1998! until Note 1998! however Note however that that there is another lattice, the hexago- 12.2.4 Other Lattices in Three Dimensions there is anothernal close lattice,packed lattice the whichhexago- achieves 12.2.4 Other Lattices in Three Dimensions nal close packedpreciselylattice the same which packing achieves density for precisely thespheres same as the packing fcc lattice. density for 14 spheresBravaus as the fcc lattice. lattices in 3D

Fig. 12.19 Conventional unit cells for the fourteen Bravais lattice types. Note that if you tried to construct a “face- centered tetragonal” lattice, you would find that by turning the axes at 45 de- Fig. 12.19greesConventional it would actually unit be cells equivalent for the fourteento Bravais a body-centered lattice tetragonal types. Note lattice. that if youHence tried face-centered to construct tetragonal a “face- is not listed as a Bravais lattice type (nor is centered tetragonal”base-centered lattice, tetragonal you for would a similar find that byreason, turning etc.). the axes at 45 de- grees it would actually be equivalent to a body-centered tetragonal lattice. Hence face-centered tetragonal is not listed as a Bravais lattice type (nor is base-centered tetragonal for a similar reason, etc.).

11Named after Auguste Bravais who classified all the three-dimensional lat- tices in 1848. Actually they should be In addition to the simple cubic, orthorhombic, tetragonal, fcc, and named after Moritz Frankenheim who bcc lattices, there are nine other types of lattices in three dimensions. studied the same thing over ten years 11 earlier—although he made a minor er- These are known as the fourteen Bravais lattice types. Although the ror in his studies, and therefore missed study of all of these lattice types is beyond the scope of this book, it is getting his name associated with them. probably a good idea to know that they exist. Figure 12.19 shows the full variety of Bravais lattice types in three di-

11Named after Auguste Bravais who classified all the three-dimensional lat- tices in 1848. Actually they should be In addition to the simple cubic, orthorhombic, tetragonal, fcc, and named after Moritz Frankenheim who bcc lattices, there are nine other types of lattices in three dimensions. studied the same thing over ten years 11 earlier—although he made a minor er- These are known as the fourteen Bravais lattice types. Although the ror in his studies, and therefore missed study of all of these lattice types is beyond the scope of this book, it is getting his name associated with them. probably a good idea to know that they exist. Figure 12.19 shows the full variety of Bravais lattice types in three di- 12.2 Lattices in Three Dimensions 117

(triangular) lattice, where we can write the primitive lattice vectors as

a1 = a xˆ

a2 =(a/2) xˆ +(a√3/2) yˆ. (12.3)

In terms of the reference points of the lattice, the basis for the primitive unit cell, i.e., the coordinates of the two larger circles with respect to 1 2 the reference point, are given by 3 (a1 + a2)and 3 (a1 + a2). 12.2 Lattices in Three Dimensions 117

(triangular) lattice, where we can write the primitive lattice vectors as Fig. 12.9 Left: The honeycomb from a1 = a xˆ Fig. 12.4 is shown with the two inequiv- √ 2 alent points of the unit cell given dif- a2 =(a/2) xˆ +(a 3/2) yˆ. (12.3) (a + a ) a1 3 1 2 ferent shades. The unit cell is out- In terms of the reference points of the lattice, the basis for the primitive lined dotted and the corners of the unit cell, i.e., the coordinates of the two larger circles with respect to 1 2 unit cell are marked with small black the reference point, are given by (a1 + a2)and (a1 + a2). 3 3 1 dots (which form a triangular lattice). 3 (a1 + a2) Right: The unit cell is expanded and

Fig. 12.9 Left: The honeycomb from coordinates are given with respect to Fig. 12.4 is shown with the two inequiv- the reference point at the lower left cor- a2 2 (a + a ) alent points of the unit cell given dif- ner. a1 3 1 2 ferent shades. The unit cell is out- lined dotted and the corners of the unit cell are marked with small black 1 dots (which form a triangular lattice). 3 (a1 + a2) Right: The unit cell is expanded and 12.2coordinates are given Lattices with respect to in Three Dimensions the reference point at the lower left cor- a2 Thener. simplest lattice in three dimensions is the simple cubic lattice shown in Fig. 12.10 (sometimes known as cubic “P” or cubic-primitive lattice). 12.2 Lattices in Three Dimensions The primitive unit cell in this case can most conveniently be taken to The simplest lattice in three dimensions is the simple cubic lattice shown be a single cube—which includes 1/8 of each of its eight corners (see in Fig. 12.10 (sometimes known as cubic “P” or cubic-primitive lattice). The primitive unit cell in this case can most conveniently be taken to Fig. 12.11). be a single cube—which includes 1/8 of each of its eight corners (see Only slightly more complicated than the simple cubic lattice are the Fig. 12.11). Only slightly more complicated than the simple cubic lattice are the tetragonal and orthorhombic lattices where the axes remain perpendicu- tetragonal and orthorhombic lattices where the axes remain perpendicu- lar, but the primitive lattice vectors may be of different lengths (shown lar, but the primitive lattice vectors may be of different lengths (shown in Fig. 12.11). The orthorhombic unit cell has three different lengths of in Fig. 12.11). The orthorhombic unit cell has three different lengths of its perpendicular primitive lattice vectors, whereas the tetragonal unit its perpendicular primitive lattice vectors, whereas the tetragonal unit cell has two lengths the same and one different. Fig. 12.10 Acubiclattice,otherwise Fig. 12.10 Acubiclattice,otherwise cellknown has as cubic two “P” orlengths cubic primitive. the same and one different. known as cubic “P” or cubic primitive. a, b, c c = a all different ̸ a, b, c c = a all different ̸ c c a Fig. 12.11 Unit cells for cubic, tetrag- a a b onal, and orthorhombic lattices. a a a c c Cubic Tetragonal Orthorhombic a unit cell unit cell unit cell Fig. 12.11 Unit cells for cubic, tetrag- a a b onal, and orthorhombic lattices. a a a Cubic Tetragonal Orthorhombic unit cell unit cell unit cell 118 Crystal Structure

Conventionally, to represent a given vector amongst the infinite num- ber of possible lattice vectors in a lattice, one writes

[uvw]=ua1 + va2 + wa3 (12.4) where u,v,andw are integers. For cases where the lattice vectors are orthogonal, the basis vectors a1, a2,anda3 are assumed to be in the xˆ, 7This notation is also sometimes yˆ,andzˆ directions. We have seen this notation before,7 for example, in abused, as in Eq. 12.2 or Fig. 12.8, the subscripts of the equations after definition 12.1. where the brackets enclose not integers, Lattices in three dimensions also exist where axes are not orthogonal. but distances. The notation118 Crystal can Structure also be abused to specify points which are We will not cover all of these more complicated lattices in detail in 12.2 Lattices in Three Dimensions 119 not members of the lattice, by choos- this book.Conventionally, (In Section to represent a 12.2.4 given vector we amongst will the briefly infinite num- look through these other ber of possible lattice vectors in a lattice, one writes Packing together these unit cells to fill space, we see that the lattice ing, u, v,orw to be non-integers. We cases, but only at a very cursory level.) The principles we learn inpoints the of a full bcc lattice can be described as being points having co- will sometimes engage in such abuse. ordinates [x, y, z] where either all three coordinates are integers [uvw] more simple cases (with[uvw]=u orthogonala1 + va2 + wa3 axes) generalize(12.4) fairly easily, and justtimes the lattice constant a, or all three are half-odd-integers times the add further geometric and algebraic complexity without illuminatinglattice the constant a. where u,v,andw are integers. For cases where the lattice vectors are It is often convenient to think of the bcc lattice as a simple cubic lattice with a basis of two atoms per conventional cell. The simple cubic lattice physicsorthogonal, much the further. basis vectors a1, a2,anda3 are assumed to be in the xˆ, 7This notation is also sometimes yˆ,andzˆ directions. We have seen this notation before,7 for example, in contains points [x, y, z] where all three coordinates are integers in units abused, as in Eq. 12.2 or Fig. 12.8,Twothe particularsubscripts of the equations lattices after (withdefinition 12.1. orthogonal axes) which we will coverof the lattice constant. Within the conventional simple-cubic unit cell where the brackets enclose not integers, Lattices in three dimensions also exist where axes are not orthogonal. we put one point at position [0, 0, 0] and another point at the position but distances. The notation canin also some detail are body-centered cubic(bcc)latticesandface-centered[ 1 , 1 , 1 ] in units of the lattice constant. Thus the points of the bcc be abused to specify points which are We will not cover all of these more complicated lattices in detail in 2 2 2 bcc 12.2 Lattices in Three Dimensions 119 not members of the lattice, bycubic choos- this (fcc) book. lattices. (In Section 12.2.4 we will briefly look through these other lattice are written in units of the lattice constant as ing, u, v,orw to be non-integers. We cases, but only at a very cursory level.) The principles we learn in the will sometimes engage in such abuse. Packing togetherR thesecorner unit=[ cellsn1 to,n2 fill,n space,3] we see that the lattice more simple cases (with orthogonal axes) generalize fairly easily, and just 1 1 1 add further geometric and algebraic complexity without illuminating the points of a full bccR latticecenter can=[ ben described1,n2,n3]+[ as2 being, 2 , 2 ] points having co- ordinates [x, y, z] where either all three coordinates are integers [uvw] physics much further. as if the two different types of points were two different types of atoms, times the lattice constant a, or all three are half-odd-integersLattice sites times the Two particular lattices (with orthogonal axes) which we will cover although all points in this lattice should be considered equivalent (they lattice constant a. ( ( ( in some detail are body-centered cubic(bcc)latticesandface-centered only look inequivalent because we have chosen a conventional unit cell It is often convenient to think of the bcc lattice as a simple)( cubic? @A lattice+ B CAFig.+ n 12.13D̂) and The Wigner–Seitz)[ ? cell+ of @A + B + CA + & + D̂] cubic (fcc) lattices. with two lattice points in it). From this representation we see that we can + + + 12.2.1 The Body-Centered Cubic (bcc) Lattice with a basis of two atoms per conventional cell. The simple cubic lattice the bcc lattice (this shape is a “trun- also think of the bcc lattice as being two interpenetrating simple cubic contains points [x, y, z] where all three coordinates are integers in units cated octahedron”). The hexago- lattices displaced from each other by [ 1 , 1 , 1 ]. (See also Fig. 12.14.) nal face is the perpendicular bisecting of the lattice constant. Within the conventional2 2 2 simple-cubic unit cell We may ask why it is that this set of points forms a lattice. In terms of plane between the lattice point (shown ( we put one point at position [0, 0, 0] and another point at the position as a sphere) in the center and the lattice 12.2.1 The Body-Centered Cubic (bcc) Lattice our first definition of a lattice (definition 12.1) we can writeLattice point per conventional cell: the primitive 2 = 8× + 1 [ 1 , 1 , 1 ] in units of the lattice constant. Thus the points of the bcc point (also a sphere) on the corner. The G 2lattice2 2 vectors of the bcc lattice as square face is the perpendicular bisect- lattice are written in units of the lattice constant as , Volume (conventional cell): ing plane between the lattice point) in a1 = [1, 0, 0] the center of the unit cell and a lattice Rcorner =[n1,n2,n3] point in the center of the neighboring, a2 = [0, 1, 0] 1 1 1 Volume (primitive cell) : ) /2 Rcenter =[n1,n2,n3]+[ , , ] unit cell. a =[1 , 1 , 1 ] 2 2 2 Fig. 12.12 ConventionalFig. unit 12.12 Conventional cell for unit cell for a 3 2 2 2 Number of nearest neighbors: 8 the body-centered cubic (I) lattice. a a/2 as if the two different types of points were two different types of atoms, the body-centered cubicLeft: (I)3D view. lattice.Right: Aplanview althoughin units ofall the points lattice in this constant. lattice It should is easy be to considered check that equivalent any combination (they of the conventional unit cell. Unlabeled c c c , a a/2 only look inequivalent because we have chosen a conventional unit cell + + + Left: 3D view. Right:pointsAplanview are both at heights 0 and a. a R = n1a1 + n2a2 + n3a3 Nearest neighbor distance: (12.5) ( ) +( ) +( ) = ) ≈ 0.866) a with two lattice points in it). From this representation we see that we can Fig. 12.13 The Wigner–Seitz cell of+ + + + Body-centered cubic the bcc lattice (this shape is a “trun- of the conventional unit cell. Unlabeled a Plan view alsowith thinkn1,n of2,and the bccn3 integers lattice as gives being a point two withininterpenetrating our definition simple of the cubic bcc cated octahedron”). The hexago- unit cell lattice (that the three coordinates are1 either1 1 all integers or all half-odd nal face is the perpendicular bisecting points are both at heights 0 and a. a lattices displaced from each other by [ 2 , 2 , 2 ]. (See alsoNumber of second neighbors: 6 Fig. 12.14.) integersWe may times ask why the it lattice is that constant). this set of points Further, forms one a lattice.can check In terms that any of plane between the lattice point (shown a as a sphere) in the center and the lattice ourpoint first satisfying definition the of a conditions lattice (definition for the bcc 12.1) lattice we can can write beSecond neighbor distance: writtenthe primitive in the ) The body-centeredBody-centered cubic (bcc) cubic lattice is a simple cubic lattice where point (also a sphere) on the corner. The there is an additional lattice point in the very center of the cube (this Plan view latticeform of vectors Eq. 12.5. of the bcc lattice as square face is the perpendicular bisect- 8Cubic-I comes from “Innenzentriert” is sometimes known8 as cubic-I.) The unit cell is shown in the left of We can also check that our description of a bcc lattice satisfies our ing plane between the lattice point in unit cell a = [1, 0, 0] (inner-centered). This notation was in- Fig. 12.12. Another way to show this unit cell, which does not rely on second description of a lattice1 (definition 12.1.1) that addition of any the center of the, unit cell and a lattice Simple cubic point in the center of the neighboring troduced by Bravais in his 1848 trea- showing a three-dimensional picture, is to use a so-called plan view of the two points of the lattice (givena2 = by[0 Eq., 1, 12.5)0] gives another point of the 9 ≈ 0.680 tise (Interestingly, Europe was burning Packing fraction: unit cell. unit cell, shown in the right of Fig. 12.12. A plan view (a term used in lattice. a =[1 , 1 , 1 ] G in 1848, but obviously that didn’t stop )(? @A + B CA D̂ 3 2 2 2 Fig. 12.14 The Wigner–Seitz cells of science from progressing.) engineering and architecture) isLattice sites: a two-dimensional projection+ n from) the More qualitatively we can consider definition 12.1.2 of the lattice— the bcc lattice pack together to tile all The body-centered cubic (bcc) lattice is a simple cubic( lattice wherein units of the lattice constant. It is easy to check that any combination top of an object where heights areLattice point per conventional cell: labeled to show the third dimension. 1 = 8× that the local environment of every point in the latticeCoordinates of the sites: should be the of space. Note that the structure(?, & of, theB) Simple cubicIn the picture of the bcc unit cell, there are eight lattice points on the G same. Examining the point in the center of the unit cell, we see that bcc lattice is that of two interpenetrat- there is an additional lattice point in the very, center of the cube (this R = n1a1 + n2a2 + n3a3 (12.5) ing simple cubic lattices. corners of the cell (each of whichVolume (conventional cell): is 1/8 inside of the conventional) unit it has precisely eight nearest neighbors in each of the possibleFor the site diagonal 0,0,0 , 8 8 , directions. Similarly, any of the points in the corners of the unit cells will Cubic-I comes from “Innenzentriert” is sometimesLattice sites: cell) and one point known)(? in@A + theB center CAas+ n Volume (primitive cell) : of cubic-I.)D thê) cell. Thus theThe conventional unit) unitcell is shown in the leftwith ofn1,n2,andn3 integers gives a point within our definition of the bcc ( ( ( latticehave eight (that nearest the three neighbors coordinates corresponding are either all to theintegers points or in all the half-odd center (inner-centered). This notation was in- cell contains exactly two (= 8 1Number of nearest neighbors: 6/8+1)latticepoints.( 8 nearest neighbors: ± , ± , ± Fig. 12.12.Lattice point per conventional cell: Another way× to show1 = 8× this unit cell, which does not relyintegersof on the eight times adjacent the lattice unit constant). cells. Further, one can check that any + + + Nearest neighbor distance: G ) troduced by Bravais in his 1848 trea- , point satisfying the conditions for the bcc lattice can be written in the showingVolume (conventional cell): a three-dimensionalNumber of second neighbors: 12) picture, is to use a so-called plan view ofform the of Eq. 12.5. 6 nest nearest neighbors: ±1,0,0 , 0, ±1,0 and (0,0, ±1) tise (Interestingly, Europe was burning , Volume (primitive cell) : ) We can also check that our description of a bcc lattice satisfies our in 1848, but obviously that didn’t stop unit cell, shown in the rightSecond neighbor distance: of Fig. 12.12. A2) plan view (a term used in Number of nearest neighbors: 6 second description of a lattice (definition 12.1.1) that addition of any science from progressing.) engineering and architecture) is a two-dimensionalI projection fromtwo the points of the lattice (given by Eq. 12.5) gives another point of the Nearest neighbor distance: )Packing fraction: ≈ 0.524 lattice. J Fig. 12.14 The Wigner–Seitz cells of top ofNumber of second neighbors: 12 an object where heights are labeled to show the third dimension.More qualitatively we can consider definition 12.1.2 of the lattice— the bcc lattice pack together to tile all Second neighbor distance: 2) that the local environment of every point in the lattice should be the of space. Note that the structure of the In the picture of the bcc unit cell, there are eight lattice points onsame. the Examining the point in the center of the unit cell, we see that bcc lattice is that of two interpenetrat- it has precisely eight nearest neighbors in each of the possible diagonal ing simple cubic lattices. cornersCoordinates of the sites: of the cellI (each(?, &, B of) which is 1/8 inside of the conventional unit Packing fraction: ≈ 0.524 directions. Similarly, any of the points in the corners of the unit cells will cell)For the site and one0 point,0,0 ,J in the center of the cell. Thus the conventionalhave unit eight nearest neighbors corresponding to the points in the center cell contains6 nearest neighbors: exactly±1 two,0,0 , (=0, ± 81,0 and 1/8+1)latticepoints.0,0, ±1 of the eight adjacent unit cells. 12 nest nearest neighbors: ±1, ±1,0 ×, 0, ±1, ±1 and (±1,0, ±1) Coordinates of the sites: (?, &, B) For the site 0,0,0 , 6 nearest neighbors: ±1,0,0 , 0, ±1,0 and 0,0, ±1 12 nest nearest neighbors: ±1, ±1,0 , 0, ±1, ±1 and (±1,0, ±1) Packing fraction Packing fraction Packing fraction: We try to pack N spheres (hard, cannot deform). OP The total volume of the spheres is N4 9 , OP The volume these spheres occupy V > N4 9 (there are spacing) , Packing fraction=total volume of the spheres/total volume these spheres occupy

Z, Z, Z, N4 9 4 9 4 9 R)STU&V WX)S!UY& = 3 = 3 = 3 [ [/N [Y?\B] ^]X _U!] Z, 4 9 = 3 [Y?\B] YW ) ^XUBU!U`] S]??

High packing fraction means the space is used more efficiently Simple cubic

Lattice sites: )(? @A + B CA+ n D̂) ( Lattice point per conventional cell: 1 = 8× G Volume (conventional cell): ), Volume (primitive cell) : ), Number of nearest neighbors: 6 Nearest neighbor distance: ) Number of second neighbors: 12 Second neighbor distance: 2)

Packing fraction of simple cubic I Packing fraction: ≈ 0.524 J Z, 4 9 R)STU&V WX)S!UY& = 3 Nearest distance= 2 R [Y?\B] YW ) ^XUBU!U`] S]?? R= Nearest distance/2=)/2 Z, 4 9 4 9 Z 4 9 )/2 9 = 3 = ( ),= ( ),= ≈ 0.524 ), 3 ) 3 ) 6

Ø About half (0.524=52.4%) of the space is really used by the sphere. Ø The other half (0.476=47.6%) is empty. 120 Crystal Structure

The coordination number of a lattice (frequently called Z or z) is the number of nearest neighbors any point of the lattice has. For the bcc lattice the coordination number is Z =8. As in two dimensions, a Wigner–Seitz cell can be constructed around each lattice point which encloses all points in space that are closer to that lattice point than to any other point in the lattice. This Wigner–Seitz 120 Crystal Structure unit cell for the bcc lattice is shown in Fig. 12.13. Note that this cell is bounded by the perpendicular bisecting planes between lattice points. These Wigner–Seitz cells, being primitive, can be stacked together to fill all of space as shown in Fig. 12.14. The coordination number of a lattice (frequently called Z or z) is the number of nearest neighbors any point of the lattice has. For the bcc lattice the coordination number is Z =8. 12.2.2 The Face-Centered Cubic (fcc) Lattice As in two dimensions, a Wigner–Seitz cell can be constructed around 120 Crystal Structure each lattice point which encloses all points in space that are closer to that a/2 lattice point than to any other point in the lattice. This Wigner–Seitz The coordination number of a lattice (frequently called Z or z) is the Fig. 12.15 Conventional unit cell for the face-centered cubic (F) lattice. a a/2 a/2 unitnumber cell for of nearest the bcc neighbors lattice any point is shown of the lattice in Fig. has. For 12.13. the bcc Note that this cell is Left: 3D view. Right: Aplanview lattice the coordination number is Z =8. of the conventional unit cell. Unlabeled bounded by the perpendicular bisecting planes between lattice points. points are both at heights 0 and a. a As in two dimensions, a Wigner–Seitz cell can be constructed around a a/2 Theseeach Wigner–Seitz lattice point which encloses cells, all being points inprimit space thative, are can closer be to that stacked together to fill Face-centered cubic lattice point than to any other point in the lattice. This Wigner–Seitz a unit cell all ofunit space cell for as the shown bcc lattice in is shown Fig. in 12.14. Fig. 12.13. Note that this cell is Plan view bounded by the perpendicular bisecting planes between lattice points. fcc These Wigner–Seitz cells, being primitive, can be stacked together to fill The face-centered (fcc) lattice is a simple cubic lattice where there all of space as shown in Fig. 12.14. is an additional lattice point in the center of every face of every cube (this is sometimes known as cubic-F, for “face-centered”). The unit 12.2.2 The Face-Centered Cubic (fcc) Lattice cell is shown in the left of Fig. 12.15. A plan view of the unit cell is shown on the right of Fig. 12.15 with heights labeled to indicate the 12.2.2 The Face-Centered Cubic (fcc)Lattice sites Lattice ( ( third dimension. )(? @A + B CA+ n D̂) )[ ? + @A + B + CA + & D̂] In the picture of the fcc unit cell, there are eight lattice points on the + + corners of the cell (each of which is 1/8 inside of the conventional unit a/2 ( a/2 ( cell) and one point in the center of each of the six faces (each of which )[ ? + @A + B CA + & + D̂] is 1/2 inside the cell). Thus the conventional unit cell contains exactly Fig. 12.15 Conventional unit cell for + + four (= 8 1/8+6 1/2) lattice points. Packing together these unit the face-centered cubic (F) lattice. a a/2 a/2 ( ( cells to fill× space, we× see that the lattice points of a full fcc lattice can Fig. 12.15 ConventionalLeft: 3D view.unitRight: cell forAplanview )[?@A + B + CA + & + D̂] 12.2 Lattices in Three Dimensions 121 be described as being points having coordinates (x, y, z)whereeitherall of the conventional unit cell. Unlabeled a + + the face-centered cubic (F) lattice. a a/2 a/2 three coordinates are integers times the lattice constant a,ortwoofthe points are both at heights 0 and a. a Left: 3D view. Right: Aplanview a/2 three coordinates are half-odd integers times the lattice constant a and at [ 1 , 0, 1 ] and another point at [0, 1 , 1 ]. Thus the lattice points of the of the conventional unit cell. Unlabeled Face-centered cubic a 2 2 ( ( 2 2 Fig. 12.16 The Wigner–Seitz cell of the remaining one coordinate is an integer times the lattice constant a unit cell fcc lattice are written in units of the lattice constant as the fcc lattice (this shape is a “rhombic points are both at heights 0 and a. PlanLattice point per conventional cell: view 4 = 8× + 6× = 1 + 3 a. Analogous to the bcc case, it is sometimes convenient to think of a a/2 G + dodecahedron”). Each face is the per- the fcc lattice as a simple cubic lattice with a basis of four atoms per , Rcorner =[n1,n2,n3] (12.6) pendicular bisector between the central The face-centered (fcc) lattice is a simple cubic lattice where there ) point and one of its 12 nearest neigh- conventional unit cell. The simple cubic lattice contains points [x, y, z] Face-centered cubic Volume (conventional cell): R =[n ,n ,n ]+[1 , 1 , 0] is an additional lattice point in the center of every face of every cube a , face−xy 1 2 3 2 2 bors. where all three coordinates are integers in units of the lattice constant 1 1 a. Within the conventional simple-cubic unit cell we put one point at (this is sometimesunit known cell as cubic-F, for “face-centered”).Volume (primitive cell) : The unit ) /4 Rface−xz =[n1,n2,n3]+[2 , 0, 2 ] Plan view 1 1 position [0, 0, 0] and another point at the position [ 1 , 1 , 0] another point cell is shown in the left of Fig. 12.15. A plan view of the unit cell is Rface−yz =[n1,n2,n3] + [0, 2 , 2 ]. 2 2 shown on the right of Fig. 12.15 with heights labeledNumber of nearest neighbors: 12 to indicate the third dimension. Again, this expresses the points of the lattice as if they were four dif- c c + TheIn face-centered the picture of the fcc (fcc) unit cell, lattice there are is eighta lattice simpleNearest neighbor distance: points cubic on the lattice where there( )++ferent( ) types++( of0 points)+= but they) only≈ 0 look.707) inequivalent because we have is ancorners additional of the cell (eachlattice of which point is 1/8 in inside the of center the conventional of every unit face of every cube + chosen+ a conventional unit+ cell with four lattice points in it. Since the Fig. 12.17 The Wigner–Seitz cells of conventional unit cell has four lattice points in it, we can think of the the fcc lattice pack together to tile all cell) and one point in the center of each of the six faces (each of which of space. Also shown in the picture are (thisis is 1/2 sometimes inside the cell). Thus known the conv asentional cubic-F, unit cellNumber of second neighbors: 6 for contains “face-centered”). exactly The unit fcc lattice as being four interpenetrating simple cubic lattices. two conventional (cubic) unit cells. cell isfour shown (= 8 1/ in8+6 the1/ left2) lattice of points. Fig. 12.15.Packing together A plan these view unit of the unit cell is Again we can check that this set of points forms a lattice. In terms cells to fill× space, we× see that the lattice points of aSecond neighbor distance: full fcc lattice can ) of our first definition of a lattice (definition 12.1) we write the primitive shownbe described on the as beingright points of having Fig. coordinates 12.15 with (x, y, z heights)whereeitherall labeled to indicate the lattice vectors of the fcc lattice as three coordinates are integers times the lattice constant a,ortwoofthe third dimension. 1 1 three coordinates are half-odd integers times the lattice constant a and a1 =[2 , 2 , 0] For the site 0,0,0 , 1 1 Fig. 12.16 The Wigner–Seitz cell ofInthe the remaining picture one of coordinate the fcc is anunit integer cell, times there the lattice are eightconstant lattice points on the a2 =[2 , 0, 2 ] the fcc lattice (this shape is a “rhombic a. Analogous to the bcc case, it is sometimes convenient to think of ( ( ( ( 1 1( ( a3 = [0, , ] dodecahedron”). Each face is thecorners per- the fcc of lattice the as cell a simple (each cubic of lattice which with is a basis 1/812 nearest neighbors: of inside four atoms of per the conventional± unit, ± , 0 , ± , 0, ± and 0, ±2 2 , ± pendicular bisector between the central + + + + + + cell)conventional and one unit point cell. Thein the simple center cubic lattice of each contains of points the [x, six y, z] faces (each of which point and one of its 12 nearest neigh- 6 nest nearest neighbors: ±1,0,0 in, units0, ± of1 the,0 latticeand constant.(0,0 Again, ±1) it is easy to check that any bors. is 1/2where inside all three the coordinates cell). are Thus integers the in units conv ofentional the lattice constant unit cell contains exactly combination a. Within the conventional simple-cubic unit cell we put one point at R = n1a1 + n2a2 + n3a3 1 1 fourposition (= 8 [0, 01, 0]/8+6 and another1 point/2) at lattice the position points. [ 2 , 2 , 0] another Packing point together these unit × × with n1,n2,andn3 integers gives a point within our definition of the cells to fill space, we see that the lattice points of a full fcc lattice can fcc lattice (that the three coordinates are either all integers, or two of be described as being points having coordinates (x, y, z)whereeitherall three are half-odd integers and the remaining is an integer in units of the lattice constant a). three coordinates are integers times the lattice constant a,ortwoofthe We can also similarly check that our description of a fcc lattice satisfies three coordinates are half-odd integers times the lattice constant a and our other two definitions of (definition 12.1.1 and 12.1.2) of a lattice. The Wigner–Seitz unit cell for the fcc lattice is shown in Fig. 12.16. In Fig. 12.16 The Wigner–Seitz cell of the remaining one coordinate is an integer times the lattice constant Fig. 12.17 it is shown how these Wigner–Seitz cells pack together to fill the fcc lattice (this shape is a “rhombic a. Analogous to the bcc case, it is sometimes convenient to think of all of space. dodecahedron”). Each face is the per- the fcc lattice as a simple cubic lattice with a basis of four atoms per pendicular bisector between the central Fig. 12.18 Top: Simple cubic, Mid- conventional unit cell. The simple cubic lattice contains points [x, y, z] 12.2.3 Sphere Packing dle: bcc, Bottom: fcc. The left shows point and one of its 12 nearest neigh- packing of spheres into these lattices. bors. where all three coordinates are integers in units of the lattice constant Although the simple cubic lattice (see Fig. 12.10) is conceptually the The right shows a cutaway of the con- simplest of all lattices, in fact, real crystals of atoms are rarely simple ventional unit cell exposing how the fcc a. Within the conventional simple-cubic unit cell we put one point at and bcc lattices leave much less empty cubic.9 To understand why this is so, think of atoms as small spheres 1 1 space than the simple cubic. position [0, 0, 0] and another point at the position [ 2 , 2 , 0] another point that weakly attract each other and therefore try to pack close together. When you assemble spheres into a simple cubic lattice you find that it 9Of all of the chemical elements, polo- is a very inefficient way to pack the spheres together—you are left with nium is the only one which can form a a lot of empty space in the center of the unit cells, and this turns out simple cubic lattice with a single atom to be energetically unfavorable in most cases. Packings of spheres into basis. (It can also form another crystal simple cubic, bcc, and fcc lattices are shown in Fig. 12.18. It is easy structure depending on how it is pre- pared.) to see that the bcc and fcc lattices leave much less open space between 12.2 Lattices in Three Dimensions 123

mensions. While it is an extremely deep fact that there are only fourteen 12There is a real subtlety here in clas- lattice types in three dimensions, the precise statement of this theorem, sifying a crystal as having a particu- lar lattice type. There are only these as well of the proof of it, are beyond the scope of this book. The key re- fourteen lattice types, but in principle a sult is that any crystal, no matter how complicated, has a lattice which crystal could have one lattice, but have is one of these fourteen types.12 the symmetry of another lattice. An ex- ample of this would be if the a lattice were cubic, but the unit cell did not 12.2.5 Some Real Crystals look the same from all six sides. Crys- tallographers would not classify this as Once we have discussed lattices we can combine a lattice with a basis to being a cubic material even if the lat- describe any periodic structure—and in particular, we can describe any tice happened to be cubic. The reason crystalline structure. Several examples of real (and reasonably simple) for this is that if the unit cell did not look the same from all six sides, there crystal structures are shown in Figs. 12.20 and 12.21. would be no particular reason that the three primitive lattice vectors should have the same length—it would be an insane coincidence were this to happen, 12.2 Lattices in Three Dimensions 123 and almost certainly in any real mate- Sodium (Na) rial the primitive lattice vector lengths 12 would actually have slightly different mensions. While it is an extremely deep fact that there are only fourteen There is aLattice real subtlety = Cubic-I here in clas- (bcc) values if measured more closely. lattice types in three dimensions, the precise statement of this theorem, sifying a crystalBasis as = having Na at a [000] particu- lar lattice type. There are only these Plan view as well of the proof of it, are beyond the scope of this book. The key re- fourteen lattice types, but in principle a 12.2 Latticesunlabeled in Three points Dimensions at z =0, 1 123 sult is that any crystal, no matter how complicated, has a lattice which crystal could have one lattice, but have is one of these fourteen types.12 the symmetry of another lattice. An ex- mensions. While it is an extremelyample d ofeep this fact would that be there if the a are lattice only fourteen 12There is a real subtlety here in clas- were cubic, but the unit cell did not sifying a crystal1/2 as having a particu- 12.2.5 Some Real Crystals lattice types in three dimensions,look the the precisesame from statement all six sides. of Crys- this theorem, lar lattice type. There are only these as well of the proof of it, are beyondtallographers the scope would notof this classify book. this as The key re- Once we have discussed lattices we can combine a lattice with a basis to being a cubic material even if the lat- fourteen lattice types, but in principle a sult is that any crystal, no mattertice happened how complicated, to be cubic. The has reason a lattice which crystal could have one lattice, but have describe any periodic structure—and in particular, we can describe any 12 is one of these fourteen types. for this is that if the unit cell did not the symmetry of another lattice. An ex- crystalline structure. Several examples of real (and reasonably simple) look the same from all six sides, there ample of this would be if the a lattice crystal structures are shown in Figs. 12.20 and 12.21. Fig. 12.20 Top: Sodium forms a bcc would be no particular reason that the were cubic, but the unit cell did not lattice. Bottom: Caesium chloride three primitive lattice vectors should 12.2.5 Some Real Crystals look the same from all six sides. Crys- forms a cubic lattice with a two atom have the same length—it would be an tallographers would not classify this as basis. Note carefully: CsCl is not bcc! Once we have discussed latticesinsane we can coincidence combine were a this lattice to happen, with a basis to being a cubic material even if the lat- and almost certainly in any real mate- In a bcc lattice all of the points (includ- Sodium (Na) describe any periodic structure—and inCaesium particular, chloride we can describe (CsCl) any tice happened to be cubic. The reason ing the body center) must be identical. rial the primitive lattice vector lengths for this is that if the unit cell did not crystalline structure. Several exampleswould actuallyLattice of real have (and= slightly Cubic-P reasonably different simple) For CsCl, the point in the center is Cl Lattice = Cubic-I (bcc) look the same from all six sides, there crystal structures are shown invalues Figs. if 12.20 measuredBasis and more= 12.21. Cs closely. at [000] whereas the points in the corner are Cs. Basis = Na at [000] Plan view would be no particular reason that the 1 1 1 three primitive lattice vectors should unlabeled points at z =0, 1 and Cl at [ 2 2 2 ] Plan view Atoms inside a unit cell have the same length—it would be an insaneunlabeled coincidence points were at thisz =0 to, happen,1 Sodium (Na) and almost certainly in any real mate- 1/2 rial the primitive lattice vector lengths Lattice = Cubic-I (bcc) would actually have slightly different values if measured1/2 more closely. Basis = Na at [000] Ø Plan view We choose three lattice vectors unlabeled points at z =0, 1 Ø Three lattice vectors form a primitive or a conventional unit cellFig. 12.20 Top: Sodium forms a bcc Ø Length of these vectors are called: lattice.the lattice constantsBottom: Caesium chloride forms a cubic lattice with a two atom basis. Note carefully:1 CsCl/2 is not bcc! In a bcc lattice all of the points (includ- Caesium chloride (CsCl)We can mark any unit cell by three integers: ing the body center)?B& must be identical. Lattice = Cubic-P !⃗ = ?)⃗ + B )⃗For+ CsCl,& the)⃗ point in the center is Cl Basis = Cs at [000] ( whereas+ the points, in the corner are Cs. 1 1 1 and Cl at [ 2 2 2 ] Plan view Fig. 12.20 Top: Sodium forms a bcc unlabeled points at z =0, 1 lattice. Bottom: Caesium chloride forms a cubic lattice with a two atom Coordinates of an atom: basis. Note carefully: CsCl is not bcc! We can mark Caesiumany atom in a unit cell chloride (CsCl) by three real numbers: @CD. In a bcc lattice all of the points (includ- 1/2 ing the body center) must be identical. The location of this atom: Lattice = Cubic-P @ )⃗( + C )⃗+ + D )⃗, For CsCl, the point in the center is Cl whereas the points in the corner are Cs. Notice that Basis0 ≤ =@ Cs< at1 [000]and 0 ≤ C < 1 and 0 ≤ D < 1 1 1 1 and Cl at [ 2 2 2 ] Plan view Q: Why x cannot be 1? unlabeled points at z =0, 1 A: Due to the periodic structure. 1 is just 0 in the next unit cell 1/2 Sodium Chloride structure Sodium Chloride Face-centered cubic lattice Na+ ions form a face-centered cubic lattice Cl- ions are located between each two neighboring Na+ ions Sodium Chloride structure Equivalently, we can say that Cl- ions form a face-centered cubic lattice Na+ ions are located between each two Primitive cells neighboring Na+ ions Cesium chloride structure

Cesium chloride structureSimple cubic lattice Cesium chloride structureCs+ ions form a cubic lattice Cl- ions are located at the center of each cubeCesium Chloride

Simple cubic latticeSimple cubic lattice Equivalently, we can say thatCs+ ions form a cubic latticeCs+ ions form a cubic lattice Cl- ions are located at the center of each cubeCl- ions are located at the center of each cube Cl- ions form a cubic latticeEquivalently, we can say that Cs+ ions are located at the center of each Cl- ions Equivalently, we can say thatform a cubic lattice cube Cs+ ions are located at the center of each cube Cl- ions form a cubic lattice Coordinates:Cs+ ions are located at the center of each cube Coordinates:Cs: 000 ( ( ( Cl: Cs: 000 + +Coordinates:+ ( ( ( Cs: 000 Cl: ( ( ( + + + Notice that this is a simple cubic latticeCl: NOT a body centered cubic lattice+ + + Ø For a bcc lattice, the center site is the same as the corner sites Ø Here, center sites and corner sites are Notice that this is a simple cubic latticedifferentNotice that this is a simple cubic lattice NOT a body centered cubic lattice NOT a body centered cubic latticeØ For a bcc lattice, the center site is the Ø For a bcc lattice, the center site is the same as the corner sites same as the corner sitesØ Here, center sites and corner sites are Ø Here, center sites and corner sites are different different Diamond lattice is Diamond is not a NOT a Bravais Lattice Bravais lattice either

Same story as in graphene: We can distinguish two different type of carbon sites (marked by different color) We need to combine two carbon sites (one black and one white) together as a (primitive) unit cell If we only look at the black (or white) sites, we found the Bravais lattice: fcc