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Lattice Lecture 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.
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