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Anandh Subramaniam & Kantesh Balani
MATERIALS SCIENCE & ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of A Learner’s Guide
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SYMMETRY Symmetry Operators Translation Rotation Inversion Mirror Roto-inversion Roto-reflection Glide reflection Screw axis Point Groups, Space Groups Wyckoff Positions Advanced Reading Elementary Crystallography M.J. Buerger John Wiley & Sons Inc., New York (1956) Fantastic book with good illustrations Crystallography and Crystal Chemistry F. Donald Bloss Holt Rinehart and Winston, Inc., New York (1971) From the ‘heart’ of crystallography
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Pathway to understanding symmetry & crystal structures
If this page makes you a little queasy- skip it! A Individual (simple) Symmetry Operators These are the ‘tools’ you would require for understanding lattices and crystal structures B Compound & Combination of Symmetry Operators How symmetry operators without translation combine C Point Groups 7 Crystal Systems 32 point groups in 3D Lattices have only 7 distinct point groups How symmetry operators with translation combine D Space Groups 14 Bravais Lattices How to repeat? 230 space groups in 3D Lattices have only 14 distinct space groups E Asymmetric Unit Part of the geometry which cannot come from symmetry (what to repeat!) F Wyckoff Positions Where to put your stuff (say atoms) to get a crystal from space groups
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Why study symmetry? Crystals are an important class of materials.
Crystals (and in fact quasicrystals also) are defined based on symmetry. The symmetry being referred to in this context is geometrical symmetry. Symmetry helps reduce the ‘infinite’ amount of information required to describe a crystal into a finite (preferably small) amount of information.* Any property of a crystal will have at least the symmetry of the crystal the Neumann Principle ( A property can have higher symmetry than the crystal). One obvious manifestation of the symmetry inherent in a crystal, is the external shape of the crystal. Symmetry (in conjunction with other elements) helps us define an infinite crystal in a succinct manner. (Infinite information is reduced to finite tangible information- will return to this soon). Note the facets KDP crystals grown from solution * More about how this ‘happens’ soon
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Symmetry of What? Lattice Motif Symmetry of the Crystal Unit Cell
In crystallography (the language of describing crystals) when we talk of Symmetry; the natural question which arises is: Symmetry of What? The symmetry under consideration could be of one the following entities: Lattice Crystal Motif Unit cell (these are distinct and should not be confused with one another!) When the symmetry is normally used, it is the symmetry of the crystal being referred to. Lattice Note the facets Motif Or the molecule Symmetry of the Crystal Unit Cell KDP crystals grown from solution Eumorphic crystal (equilibrium shape and growth shape of the crystal) The external shape of the crystal corresponds to the point group symmetry of the crystal
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SYMMETRY SYMMETRY OPERATOR
Though we all have an intuitive feel for symmetry it needs to be defined formally in terms of symmetry operators If an object is brought into self-coincidence after some operation the object is said to possess symmetry with respect to that operation. SYMMETRY OPERATOR Given a general point a symmetry operator leaves a finite set of points in space. A symmetry operator closes space onto itself. Symmetry operators act on entire space and all its contents. These contents could be Geometrical entities (e.g. atoms), Physical properties, Other symmetry operators (like mirrors, rotation axes) etc.
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Some of the concepts in this slide are ‘pretty’ advanced and can be learnt later via hyperlinks
NOTES: Presence of symmetry enables us to consider only a part of a object (or other entity which could even be infinite) in conjunction with the symmetry operators (see coming slide for explanations) All symmetry operators may not be required to understand/analyze/generate a structure ( but a few basic ones are) The effects of many symmetry operators may be identical (especially in lower dimensions or when mirror symmetric objects are not involved) Symmetry operators may act: (i) ‘alone’ or (ii) as ‘compound’ or (iii) ‘combination’ with other symmetry elements Certain combination* of symmetry operators (without a translational component) can also leave a finite set of points$ and these are called the Point Groups** We can have point groups in 1D, 2D, 3D or nD in general. Certain combination* of symmetry operators involving translations can leave a periodic array of (finite set within unit cell) objects in space the Space Groups We can have space groups in 1D, 2D, 3D or nD in general. * Only certain specific combinations are allowed which possess this property. $ Given a general point. ** “A combination of symmetry elements that can compatibly pass through a common point”.
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Why do I need to talk about symmetry and symmetry operators?
If the object, collection of objects, crystal etc. (which is under consideration) has some symmetry then the whole need not be described, but only a part can be described along with the symmetry operators. For example consider a square (as below). An infinite tiling of squares can be thought of as a single square repeated in x and y directions Further one half of the square with a mirror plane (mirror line in 2D) can give the whole square. Or a quarter of a square with two mirror planes or a diagonal half of the quarter with three mirror planes. (note: mirror planes in 2D are lines) Consider an infinite pattern made of squares m m This can be thought of as a single square repeated in x and y directions m Else one could have considered a quarter of the object along with a 4-fold rotation operation (with symbol and which rotates space by 90). Now imagine that the 1/8th triangle had a 1000 atoms in it- we will have to give the coordinates of a 7000 atoms less! Hence, we use the language of symmetry to be terse in our description (i.e. to supply minimum information)
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Classification of Symmetry Operators
Dimension of the Operator Takes an object to its mirror form or not Based on If the operator acts at a point or moves a point (i.e. outside a unit cell) If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic)
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Based on Dimension of the Operator
Classification of Symmetry Operators Dimension of the Operator Based on Classification based on the dimension invariant entity of the symmetry operator Operator Dimension Inversion 0D Rotation 1D Mirror 2D Lower dimensional space
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Translation Rotation Symmetries Based on Type I Roto- reflection
Classification of Symmetry Operators Takes an object to its mirror form or not Based on Translation Takes object to same form → Proper Type I Rotation Symmetries Roto- reflection Mirror Type II Inversion Takes object to enantiomorphic form → improper Roto- inversion (Mirror image form)
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Symmetries Based on Rotation Mirror Macroscopic Inversion Screw Axes
Classification of Symmetry Operators If it plays a role in the shape of a crystal or not (Macroscopic/Microscopic) Based on Rotation Mirror Influence the external shape of the crystal Macroscopic Inversion Symmetries Screw Axes Microscopic Glide Reflection Do not Influence the external shape of the crystal
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Symmetries Based on If the operator acts at a point or moves a point
Classification of Symmetry Operators If the operator acts at a point or moves a point Based on Points remain ‘localized’ and we land up with a finite number of points Rotation Roto-reflection Mirror Roto-inversion Does not move a point Inversion Symmetries (i.e. outside a unit cell) Translation Moves a point Screw Axes Points ‘move’ to create an infinite array Glide Reflection
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Notation for representing left and right handed objects
To start with we use the notation as described below. (Occasionally deviating from this as well!). Ultimately, we will turn to ‘International Tables of Crystallography’ symbols in b/w. +R R +L L
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Translation The translation symmetry operator (t) moves an point or an object by a displacement t or a distance t. A periodic array of points or objects is said to posses translational symmetry. Translational symmetry could be in 2D or 3D (or in general nD). If we have translational symmetry in a pattern then instead of describing the entire pattern we can describe the ‘repeat unit’ and the translation vector(s). t t
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Mirror and Inversion With note on left and right handed objects
Note all mirrors used are ‘double sided’ mirrors (reflect from both sides) The left hand of a human being cannot be superimposed on the right hand by mere translations and rotations The left hand is related to the right hand by a mirror symmetry operation (m) (or the inversion as below). The right hand is called the enantiomorphic form of the left hand Another operator which takes objects to enantiomorphic forms is the inversion operator (i) (in the figure to the right below- between the two hands (in the mid-plane) at the centre is an inversion operator)` m Inversion operator Vertical Mirror Horizontal Mirror
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Rotation Axis Rotation axis rotates a general point (and hence entire space) around the axis by a certain angle On repeated operation (rotation) the ‘starting’ point leaves a set of ‘identity-points*’ before coming into coincidence with itself. As we are interested mainly with crystals, we are interested in those rotations axes which are compatible with translational symmetry → these are the (1), 2, 3, 4, 6 – fold axis. If an object come into self-coincidence through smallest non-zero rotation angle of then it is said to have an n-fold rotation axis where: The rotations compatible with translational symmetry are (1, 2, 3, 4, 6) Click here for proof Crystals can only have 1, 2, 3, 4 or 6 fold symmetry * explained in an upcoming slide
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=180 n=2 2-fold rotation axis =120 n=3 3-fold rotation axis
Symbol for 2-fold axis =180 n=2 2-fold rotation axis Then the operation of the 2-fold leaves two points Symbol for 3-fold axis =120 n=3 3-fold rotation axis
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=90 n=4 4-fold rotation axis =60 n=6 6-fold rotation axis
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‘Putting together’ two symmetry operators
Symmetry operators can be put together in two ways: (i) in combination, (ii) as a compound. In a combination the ‘individuality’ of the symmetry operators is not lost. In a compound two symmetry operators perform ‘as-if’ they are a single operator (their ‘individuality’ is not fully expressed). The component operations express themselves and should be compatible with lattice translation Combination How can symmetry operators be put together? Compound Two symmetry operations performed in sequence as a single event → identity of the individual operators lost Ways of ‘putting together’ symmetry operators
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Compound Symmetry Operators
We have so far considered: Translation, mirror, inversion & rotation operators. These are simple symmetry operators. Roto-inversion and Roto-reflection are compound symmetry operators which do not involve translation → both these take left handed objects to right handed forms For generating point groups (to be considered later) one of the two operators is sufficient and hence we will consider roto-inversion only in future considerations. Glide reflection and Screw are compound symmetry operators involving translation → (Only) Glide reflection takes left handed objects to right handed form It is important to note in these operations the ‘complete’ compound operator acts before leaving a identity-point (i.e. Roto-inversion is NOT rotation followed by a inversion). In some cases these compound operators can be broken down into a combination of two operators. In a combination (unlike a compound) the individual operators express themselves fully – i.e. the first operator acts first and then the second acts on the result of the first operation.
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Roto-inversion Compound symmetry operator A roto-inversion operator rotates a point/object and then inverts it (inversion operation) in one go. A left handed object will be taken to its right handed form by the operation. We will only consider 1,2,3,4,6 - fold rotations (crystallographic) as a part of the roto-inversion operation. Roto-inversion operations Compatible with translational symmetry
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Screw Axis Compound symmetry operator A screw (axis) operator rotates a point/object and then moves it a fraction of the repeat distance in one go. The faction which the screw axes move is called the Pitch of the screw. We will only consider (1, 2, 3, 4, 6) - fold rotations (crystallographic) as a part of the screw axes. The screw axes to be considered are: 21 31 , 32 41 , 42 , 43 61 , 62 , 63 , 64 , 65 The normal and screw axis both give the same effect on the external symmetry of the crystal. All identity points have the same enantiomorphic form (i.e. all objects created by the screw operator are all either left-handed or all are right-handed)
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The 32 axis produces a rotation of 120 along with a translation of 2/3.
The set of points generated are: (0,0) (120,2/3) (240,4/31/3) (360,6/32)… This is equivalent to a left handed screw (LHS) of pitch 1/3 Note: these are not fraction calculations! m
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The 43 axis is a RHS with a pitch of 3/4
The set of points generated are: (0,0) (90,3/4) (180,6/42/41/2) (270,9/41/4)… The effect of 43 axis can be thought of as a LHS with a pitch of 1/4 Note: these are not fraction calculations!
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The 42 axis generates the following set of points: (0,0) (90,1/2) (180,2/21) (270,3/21/2) (360,4/22) The grey arrowhead maps the (270,3/2) point to (270,1/2) → to keep points within unit cell Note: these are not fraction calculations!
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Glide Reflection Compound symmetry operator A glide (reflection) operator move a point/object by a fraction of the repeat distance and reflects the object in one go. Kinds of ‘Glides’ are considered in crystallography: Axial Glide (a, b, c) → Diagonal Glide (n) → Diamond Glide (d) →
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Different type of glides
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Complete set of symmetry operators
3 numbers each 3 numbers each
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Point Groups and Space Groups
We have so far considered various types of symmetry operators- those with translation and those without (keeping our focus on those related to crystals). The symmetry operators without translation (rotation, inversion, mirror, roto-inversion, roto-reflection) leave a finite number of identity-points and even those involving translation (glide and screw) leave a finite number of identity-points within the unit cell. Symmetry operators which do not involve translation can combine with one another in certain specific ways so as to leave a finite number of identity-points (i.e. arbitrary combinations are not possible). The number of such possible combinations (along with ‘single’ symmetry operators) is 32 and these are called the 32 Point Groups. One such combination is 4mm* An example of a disallowed combination is 22 (with an included angle of (say) 15)*. There are 7 distinct point group symmetries of lattices (14 Bravais Lattices) which correspond to the 7 Crystal Systems. When all symmetry elements are allowed to combine- including those with translation- then we end up with 230 space groups. There are 14 distinct space group symmetries of Lattices → the 14 Bravais Lattices Leave at least one point unmoved * Considered in upcoming slides We shall not formally derive the 32 point groups or the 230 space groups- interested readers may consult Elementary Crystallography by M.J. Buerger
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Allowed combinations As mentioned before only some combinations of symmetry operators are allowed. 4mm is an allowed combination (as below) provided that the two mirrors are at 45 and the line of intersection of the mirror is the line of the 4-fold axis. When ever we write a symbol for a combination (according to the Hermann–Mauguin notation)- the symbol has a precise meaning w.r.t to the relative orientation of the component operators. As shown below 2-fold axes with an included angle of 30 is an allowed combination → leading to point group 622 → Starting with just the two 2-fold axes- by repeated action of the two folds twelve 2-fold axes are created → which automatically implies that a 6-fold is perpendicular to the two 2-folds! Active 2-fold is in red 622 Last bullet is unclear. Please show example, or elaborate. The 2-folds have been coloured differently to understand the origin of the 6-fold 622 4mm Right Handed Left Handed 32
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Disallowed combinations
Most of the possible combinations of symmetry elements are actually disallowed! If we randomly chose two rotation axes and put them at some random angle- more likely than not that would be a disallowed combination (note that there are only 22 allowed combinations → along with the single operators (10 in number) we get the 32 point groups) As shown below two 2-fold axis with an included angle of (say) 15 is a disallowed combination → this is because the presence of two 2-folds with an included angle of 15 implies the presence of a 12 fold perpendicular to the plane of the 2-folds → which is a disallowed rotational symmetry in crystallography. Another example of a combination which is disallowed is (say) two 2-fold axes with an included angle of 7 (360 is not divisible by 7!). In this case: the action of one two fold on the other repeatedly, would lead to an infinite number of two folds on the plane and hence an infinite number of points (if we start with one point) (i.e. space would not close on itself!).
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R Rotation 32 Point Groups R Roto-inversion
Minimum set of symmetry operators required to create point groups Though there are a number of symmetry operators, we have already seen a redundancy in their combined action. E.g. 6 (six bar)= 3/m. This raises the question: what is the minimum number of symmetry operators required to create all possible combinations (i.e. the 32 point groups)? The answer is we can create all the 32 point groups using Only Rotations and Roto-inversions (pure and combinations)! (As in the table below). R Rotation 32 Point Groups R Roto-inversion
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Symmetries acting at a point
Point group symmetry of Lattices → 7 The 7 crystal systems Symmetries acting at a point R R 32 point groups Along with symmetries having a translation G + S 230 space groups Space group symmetry of Lattices →14 The 14 Bravais lattices R + R → rotations compatible with translational symmetry (1, 2, 3, 4, 6)
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The 32 Point Groups Highest symmetry class is in blue The possible combinations of crystallographic symmetry operators Point groups on Blue are Holohedral symmetry classes (highest symmetry for a crystal system) → these 7 point groups are the only possible symmetries of lattices
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Identity Points/Objects
If we start with a general point, then the operation of symmetry operator(s) will leave a (finite) set of points. These symmetrically related set of points are called identity points. An extension of the concept of Identity points is to use identity objects which can show left or right handedness. (Some examples are shown below). The number of identity points is the Order of the symmetry operator or the point group and is a ‘measure of the symmetry’ of the point group (/operator). Alternate diagram 4-fold leaves 4 identity points 4mm Right Handed 4mm Left Handed 4mm point group leaves 8 identity points: 4 left handed (orange circle) and 4 right handed (green circle) Right Handed Left Handed
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Concept of Sub-group A point group of higher symmetry may contain within it the operations of one of more point group(s) of lower symmetry. The lower symmetry point group(s) are called Subgroup(s) of the higher symmetry point group. E.g. the 4 point group contains the operations of 2 point group the 4mm point group contains the operations of the point groups 4, 2, m 4mm 4-fold contains 2-fold 4 m 2 Click here to know more
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Why space groups at all? Why not work with Lattice + Motif picture?
How do we go from a space group to a crystal? Why space groups at all? Why not work with Lattice + Motif picture? Click here The Space Group gives us a distribution of symmetry elements in space. (Given this distribution some points in space have a higher symmetry than others.) If the Asymmetric Unit is used as a tile, then this tile in conjunction with the space group can fill entire space. Like unit cell (as a tile) in conjunction with basis vectors can fill entire space. Wyckoff Positions for atomic species distribute (put) the atomic entities with respect to the symmetry operators. Wyckoff positions specify Site Symmetry and Occupancy by entities (usually atomic species) Further values for variables in Wyckoff table (x,y,z) have to be specified. Obviously there is no Scale in ‘symmetry related stuff’→ scale has to be added in via the Lattice Parameters (Unit Cell Parameters → Lengths and Angles consistent with the space group). Making a Crystal Space Group + Asymmetric Unit + Wyckoff Positions + Lattice Parameters Consistent with the crystal system Site symmetry, Values for variables & Occupancy Asymmetric Unit is that part of the crystal which cannot be generated using symmetry operators → “Crystal Symmetry = Asymmetric Unit”
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Positioning a object with respect to the symmetry elements
In this part we briefly consider the effect of positioning an object with respect to the distribution of symmetry elements As seen in the example of 4mm point group- placing an object in special positions reduces the number of identity-points/’objects’ produced by the point group. General site 8 identity-points (4R, 4L) On mirror plane (m1) 4 identity-points On mirror plane (m2) 4 identity-points Site symmetry 4mm 1 identity-point Object has to have mirror symmetry (bilateral symmetry) Note: this is for a point group and not for a lattice the black lines are not unit cells
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Positioning a object with respect to the symmetry elements
So it is clear that merely specifying the symmetry operations (along with their distribution) is not sufficient to generate a pattern. We have to know where the entity (say atoms) are positioned with respect to the symmetry operators in the unit cell. It is also clear that if an object is placed on a (or a combination of) symmetry operator(s), it has to be compatible with the symmetry operator(s). E.g. a left handed object cannot be placed on a mirror plane or a rectangular object cannot be placed on a 4 fold axis or a square can be placed on a 4mm site only if the diagonal of the square coincides with a mirror This positioning also determines the number of such entities within the unit cell → called the Multiplicity. Higher the site symmetry → lower will be the number of entities in the unit cell Wyckoff had developed a notation to label the symmetry positions Each site (with a certain symmetry) is labeled with a alphabet. The labeling starts with the highest symmetry sites (‘a’ for highest symmetry, then ‘b’ …) Many different sites with differing symmetry may have the same multiplicity → but they will have different ‘Wyckoff’ labels.
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Positioning of a motif w. r
Positioning of a motif w.r.t to the symmetry elements of a lattice Wyckoff positions A 2D lattice with symmetry elements
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g e b d a c 4mm 8 g Area 1 4 f Lines ..m e .m. d 2 c Points 2mm. b 4mm
Multi-plicity Wyckoff letter Site symmetry Coordinates 8 g Area 1 (x,y) (-x,-y) (-y,x) (y,-x) (-x,y) (x,-y) (y,x) ((-y,-x) 4 f Lines ..m (x,x) (-x,-x) (x,-x) (-x,x) e .m. (x,½) (-x, ½) (½,x) (½,-x) d (x,0) (-x,0) (0,x) (0,-x) 2 c Points 2mm. (½,0) (0,½) b 4mm (½,½) a (0,0) f g e b d a c Number of Identi-points Any site of lower symmetry should exclude site(s) of higher symmetry [e.g. (x,x) in site f cannot take values (0,0) or (½, ½)]
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Within a line or a region special points/lines of higher symmetry have to be excluded.
Exclude these points Exclude these points d e Exclude these points
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Solved Example Identify the 2D space group (plane group) of the crystal below. What is the Wyckoff symbol for each of the species? What is the Stoichiometry of the crystal? A B C Unit cell
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First we overlay the symmetry operators
Looking at the space group table we can see that it matches with p4mm space group. First we overlay the symmetry operators Stoichiometry by inspection and by the Wyckoff symbols is A1 B1 C4 Wyckoff position Site Symmetry x y Occupancy A 1a 4mm 1 B 1b C 4f ..m 0.75 Asymmetric unit Wyckoff position Coordinates Site Symmetry x y Occ. A 1a (0,0) 4mm 1 B 1b (½, ½) C 4f (x,x), (x, x), ( x,x), (x, x) ..m - Can we have fractional numbers for Wyckoff occupancy? Slide 15
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Effect of decoration of a lattice on the symmetry
We briefly consider this aspect here- details can be found in the topics on Geometry of Crystals and Making Crystals. An Infinite Lattice can be represented by a Unit Cell. On decorating the lattice with objects the symmetry of the lattice may be: Maintained lowered A special type of object (/collection of objects), which is repeated identically (in shape, orientation colour etc.) at each lattice point is called a Motif.
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4 points at the vertices of a square
Consider a square (which could also function as a unit cell of a crystal if decorated with a motif) The square shape (and also the collection of four points in the corners of a square) have some basic symmetries as shown below Complete set of symmetries i Square Which can be written as Symmetries 4 m1 m2 i 4-fold Which can be further abbreviated as 4mm mv = m1 md = m2 4 points at the vertices of a square
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Redundant mirrors which need not be drawn
Effect of the decoration a 2D example Two kinds of decoration are shown (i) for an isolated object, (ii) an object which can be an unit cell. 4mm Redundant mirrors which need not be drawn Redundant inversion centre Can be a unit cell for a 2D crystal 4mm Decoration retaining the symmetry This is not a Motif as it is not repeated identically at each point Motif
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mm m m Possible UCs of Crystals Motifs
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4 No symmetry If this is an unit cell of a crystal → then the crystal would still have translational symmetry This is a Crystal This is Amorphous!! Motif: object with no symmetry
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Amorphous Not a Motif (as repeated at random orientations)
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Lattices have the highest symmetry
(Which is allowed for it) Crystals based on the lattice can have lower symmetry
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Amorphous arrangement No unit cell
Unit cell of Triclinic crystal
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Symmetry of lattices → the 7 crystal systems
Funda Check Why 7 crystal systems? As we have seen lattices have the highest symmetry and crystals based on these can have lower symmetry. If we consider lattices in conjunction with point groups, then out of the 32 only 7 point groups survive → these correspond to the 7 crystal systems. (See example below). Another way of looking at this is there are 14 Bravais lattices, these have 7 distinct symmetries → these are the 7 crystal systems. (See example below). Out of the 32 only 7 such point group symmetries will survive Crystal has 4mm symmetry Lattice has 4/m 2/m 2/m symmetry
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The most important of the lot is the symmetry of the crystal!
Funda Check In crystallography when we talk of Symmetry, Symmetry of Which entity are we referring to? The symmetry being referred to could be for one of the following entities Lattice Motif Symmetry of the Crystal Unit Cell Eumorphic crystal (equilibrium shape and growth shape of the crystal) The shape of the crystal corresponds to the point group symmetry of the crystal The most important of the lot is the symmetry of the crystal!
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Funda Check How much information is ‘present’ in the Herman-Mauguin symbol? In the Herman-Mauguin notation, symmetry of structures is represented by a symbol. E.g. 4 (for a ‘pure’ operator), 63 (for a compound operator), 4mm (for a combination of operators/point group symbol), I (space group symbol). It is important to note that like any other symbol (say H2O) the Herman-Mauguin symbols give some information– but not all the information required. In the symbol ‘H2O’ for instance the H-O-H bond angle is not specified. Similarly in the symbol I 4/m –3 2/m, the location of these operators in the lattice/crystal, their relative orientation, etc. are not specified in the symbol and one has to refer to the International Tables of Crystallography understand all aspects of the symbol.
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The order of the group 4/m 2/m 2/m is 48.
Solved Example What is the asymmetric unit for the space group: 4/m 2/m 2/m Asymmetric unit is that part of space, which when operated by the space group symmetry gives entire space (becomes space filling on the operation of space group symmetry) Unique 4-fold implies that this belongs to the tetragonal crystal system. The order of the group 4/m 2/m 2/m is 48. This space group has no operators involving translation (glide or screw). This implies that the asymmetric unit is 1/48 the unit cell (shown in green below)
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