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Knowledge of space groups and the implications of space group symmetry
Knowledge of space groups and the implications of space group symmetry
We trust that the current and subsequent versions of our tutorial will
We trust that the current and subsequent versions of our tutorial will
The tutorial is based in part upon an approach to teaching space
The tutorial is based in part upon an approach to teaching space
Acknowledgements We are particularly grateful to the National Science
Acknowledgements We are particularly grateful to the National Science
Finally, we wish to thank Dr
Finally, we wish to thank Dr
COLORS No doubt you have already noticed that were using underlined
COLORS No doubt you have already noticed that were using underlined
Where should you be, in your knowledge of X-ray structure
Where should you be, in your knowledge of X-ray structure
This tutorial is divided into sections by crystal class
This tutorial is divided into sections by crystal class
In all of the tutorial presentations which show a unit cell diagram,
In all of the tutorial presentations which show a unit cell diagram,
In the triclinic system, a
In the triclinic system, a
On each slide you will see the ac projection of a triclinic or
On each slide you will see the ac projection of a triclinic or
We need to define the coordinate system we'll be working in
We need to define the coordinate system we'll be working in
What is our reference molecule or atom
What is our reference molecule or atom
Well start by adding the translations, and recording the unique set
Well start by adding the translations, and recording the unique set
In the International tables for X-ray Crystallography, Volume A
In the International tables for X-ray Crystallography, Volume A
The term "Special Position" (not yet encountered
The term "Special Position" (not yet encountered
Our first example is space group No
Our first example is space group No
a
a
So, weve got one molecule per unit cell (Z = 1)
So, weve got one molecule per unit cell (Z = 1)
Enantiomorphous
Enantiomorphous
Now, let's review some of the jargon introduced earlier:
Now, let's review some of the jargon introduced earlier:
A final point or reminder regarding P (primitive lattices): The
A final point or reminder regarding P (primitive lattices): The
Now, Ill produce the point (x, 1-y, z) from my original blue point:
Now, Ill produce the point (x, 1-y, z) from my original blue point:
OK! Now lets take the same diagram, that is, the one we just finished
OK! Now lets take the same diagram, that is, the one we just finished
To reiterate, for each space group, as we did for P1, well derive the
To reiterate, for each space group, as we did for P1, well derive the
,
,
Z=2;
Z=2;
Note that there are two molecules per unit cell, and that other
Note that there are two molecules per unit cell, and that other
Above is a perspective view of a unit cell in P1bar, with all centers
Above is a perspective view of a unit cell in P1bar, with all centers
We say that the eight centers of symmetry are independent, since any
We say that the eight centers of symmetry are independent, since any
Each of the eight centers of symmetry corresponds to a special
Each of the eight centers of symmetry corresponds to a special
x
x
So
So
We've completed the triclinic space groups, but there are a few final
We've completed the triclinic space groups, but there are a few final
Can we actually define the density of a crystal in terms of the
Can we actually define the density of a crystal in terms of the
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Often, it is most useful to measure the density by the neutral
Often, it is most useful to measure the density by the neutral
Bonjour, Professeur
Bonjour, Professeur
Ah, Professeur
Ah, Professeur
End of Section 1, Introduction & Pointgroups 1 and 1-bar
End of Section 1, Introduction & Pointgroups 1 and 1-bar

: 4 . : Referee. : 4 .ppt. zip-: 1580 .

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4 .ppt
1 Knowledge of space groups and the implications of space group symmetry

Knowledge of space groups and the implications of space group symmetry

on the physical and chemical properties of solids are pivotal factors in all areas of structural science. As we met to bring our ideas in teaching this subject to life, we both felt that teaching the concepts with repetitive textual and visual reinforcement, early and often, will provide a sound basis for students in this subject.

We chose PowerPoint as the delivery vehicle for the tutorial, owing to the facility of combining narrative pedagogy with animations of the buildup of space groups, and the ease of making subtle changes and/or corrections. And, we realize that others may wish to alter the approach, e.g., by rewriting the text in another language, or altering the approach (.hopefully, the desired changes will be small!).

The student may also wish to consider M. E. Kastner's Crystallographic Courseware, which provides an alternative approach to the teaching of Space Group Symmetry.

1

2 We trust that the current and subsequent versions of our tutorial will

We trust that the current and subsequent versions of our tutorial will

become ein lebendiges Buch (a living book) to which all can contribute and enjoy, and from which all can learn.

2

3 The tutorial is based in part upon an approach to teaching space

The tutorial is based in part upon an approach to teaching space

groups developed in a course in X-ray Crystallography at Brandeis University. However, through vigorous discussions, the ideas have been continuously reworked.

3

4 Acknowledgements We are particularly grateful to the National Science

Acknowledgements We are particularly grateful to the National Science

Foundation Summer Research Program in Solid State and Materials Chemistry, directed by Prof. Shiou-Jyh Hwu, Clemson University. The Program provides an opportunity for a faculty member and student to carry out research in a host laboratory; Prof. Jasinski and student Lisa Bennett worked on solid-state reactivity during Summer 2003 at Brandeis. For the faculty member to return to the program in subsequent years, a joint research program on educational aspects of solid-state chemistry is required. With the encouragement provided by the program requirements and additional NSF support through grants DMR-0089257 and DMR-0504000 to BMF, we designed and wrote this tutorial during Summer 2004 and 2005.

4

5 Finally, we wish to thank Dr

Finally, we wish to thank Dr

Eugene Cheung (Cardiff University) and Professors Jen Swift (Georgetown) and Mike Ward (Minnesota) for reviewing the tutorial and providing oodles of helpful comments. Local comments and stimulating discussions at Brandeis University, from Professors Dan Oprian, Chris Miller and undergraduate students Jeremy Heyman and Stephen C. Wilson were of immeasurable help. We invite your comment and constructive criticism. The design of this tutorial will allow us to incorporate changes on a regular basis. Bruce M. Foxman Brandeis University foxman1@brandeis.edu http://www.xray.chem.brandeis.edu Jerry P. Jasinski Keene State College jjasinski@keene.edu http://academics.keene.edu/chemistry/faculty.htm September 2006

5

6 COLORS No doubt you have already noticed that were using underlined

COLORS No doubt you have already noticed that were using underlined

green text to indicate a hyperlinked page or pages. In order to minimize issues involving broken links, all links are included in the tutorial, and the exact link references may be found in the Credits section (Chapter 7) of this PowerPoint presentation. Blue is often used for emphasis. Red is usually employed when we are referring to symmetry elements or unit cell parameters a, b, c, ?, ?, ?. Other colors, used less often, are generally employed to emphasize particular points, or to improve the readability of a particular presentation.

6

7 Where should you be, in your knowledge of X-ray structure

Where should you be, in your knowledge of X-ray structure

determination, or crystallography before beginning this tutorial? Were starting with an assumed knowledge of point groups, unit cells, lattices and space group symmetry elements (there will be some minor review of these topics). In the future we'll be adding a special section on unit cells and lattices recently added to the tutorial.

We recommend reviewing or understanding the equivalent of Chapters 1-4 in Sands, Introduction to Crystallography (Dover, 1975) or Sections 2.1 and 3.1-3.4 in Stout and Jensen, X-Ray Structure Determination: A Practical Guide (Wiley, 1989).

Finally, if you have around 3 minutes, listen to a humorous musical revue of Bravais lattices at : http://www.haverford.edu/physics-astro/songs/bravais.htm

7

8 This tutorial is divided into sections by crystal class

This tutorial is divided into sections by crystal class

There are 32 crystallographic point groups, or crystal classes. The 32 crystal classes correspond to the external shapes of crystals actually observed. The crystal class may be obtained from the space group symbol by removing the translations from the symbolmore about that later! In this section we consider the two triclinic space groups, P1 and P-1bar, which belong to crystal classes 1 and 1-bar, respectively.

Paul von Groth

8

9 In all of the tutorial presentations which show a unit cell diagram,

In all of the tutorial presentations which show a unit cell diagram,

we have followed the style of the International Tables for Crystallography, Volume A, Space Group Symmetry, which contains diagrams of the 230 space groups. We will use the Hermann-Mauguin symmetry notation1 throughout, and will be deriving each space group from the Hermann-Mauguin symbol.

Our diagrams are not exactly identical to the diagrams in Volume A, but rather are a composite overlay of the unit cell diagram, symmetry elements, and molecules which reside in the Equivalent General Positions.

9

1There will be an opportunity to learn more of the history of this notation on an upcoming slide.

10 In the triclinic system, a

In the triclinic system, a

b ? c; ? ? ? ? ? ? 90?. Triclinic crystals either have only 1 symmetry (a 360 rotation, crystal class 1) or possess a center of inversion (crystal class ). For convenience, well often write 1-bar instead of . 1http://phycomp.technion.ac.il/~sshaharr/intro.html

Primitive Triclinic (click to rotate)1

10

11 On each slide you will see the ac projection of a triclinic or

On each slide you will see the ac projection of a triclinic or

monoclinic unit cell. The b axis, which points toward the viewer, is either inclined to the a and c axes (triclinic) or perpendicular to the page (monoclinic). In the orthorhombic case, we'll use an ab projection.

Each slide opens with the ac (orthorhombic, ab) projection and a reminder about the axial directions. Upon tapping the advance button, various events will occur, depending upon the space group under consideration. Well describe these precisely as we approach each example.

Each atom or group of atoms will be displayed by an open circle. An open circle with a large comma inside will be used to indicate opposite chirality to the reference molecule.

11

12 We need to define the coordinate system we'll be working in

We need to define the coordinate system we'll be working in

Of course, we'll need to specify the positions of atoms or molecules within the unit cells under consideration. Let's imagine that a particular atom is located (in ?) at (X, Y, Z). A preferred method to specify the location of this atom would be to use a formalism that is independent of the size of the unit cell. To do this we use fractional coordinates (x, y, z), where x = X/a y = Y/b and z = Z/c Within the unit cell, values of x, y and z are thus constrained to decimal values between 0 and 1. And, for example, a location in a unit cell adjacent to the reference cell but displaced along a would thus be (1 + x, y, z).

12

13 What is our reference molecule or atom

What is our reference molecule or atom

This is the first atom or group that appears on the screen as an open circle. It will always have the coordinates (x, y, z), and we'll draw an ac projection of the unit cell, with axis b coming out of the page.

Finally, we must specify the location of the atom or group along the third dimension (its "height" in/out of the screen). Well do this by placing the prefix of the y-coordinate next to the open circle. For the reference molecule, this will simply be a +, as it is located at (x, y, z) ? +y. An indicator such as ?+ signifies ?+y; - is just -y. Remember that x, y and z are fractional coordinates, and for molecules within the unit cell, each has a value between 0 and 1.

As you push the advance button, the operations of the group are applied . in the following case, space group P1, we have only unit translations to apply, in sequence.

13

14 Well start by adding the translations, and recording the unique set

Well start by adding the translations, and recording the unique set

of symmetry position(s) generated for a single atom. As the atoms are added, think about how many are actually within the unit cell; well call this number Z. Beside the number of symmetry related atoms, well list their positions. See if you can assign coordinates to each molecule added; some answers are given as you proceed.

At a time after all the unique atoms within the cell have been generated well draw a box around the General Position, the set within the unit cell equivalent by symmetry. The General Position thus contains a certain number of equivalent points per cell; the number is referred to as Z. We will also say that the multiplicity (the number) of the General Position is Z.

14

15 In the International tables for X-ray Crystallography, Volume A

In the International tables for X-ray Crystallography, Volume A

section 2.11, the term "General Position" is defined as follows: A set of symmetrically equivalent points...is said to be in 'general position' if each of its points is left invariant only by the identity operation but by no other symmetry operation of the space group. Each space group has only one general position. We'll see, time and time again, that, in the general position, no symmetry requirements are imposed upon the molecule. Thus, general positions always have 'site-symmetry' 1 (Hermann-Mauguin notation) or alternatively C1 (Schoenflies notation). More info about Hermann-Mauguin notation appears on slides 17, 25 and throughout the tutorial. We hope you will recall from your knowledge of point groups that the identity operation is a trivial operation of making no change, e.g., a 360 rotation.

15

16 The term "Special Position" (not yet encountered

The term "Special Position" (not yet encountered

..) is defined as follows: A set of symmetrically equivalent points...is said to be in 'special position' if each of its points is mapped onto itself by at least one further symmetry operation of the space group. Later we'll see that a special position always corresponds to the location of a point group symmetry element (inversion ? ), rotation (n = 2, 3, 4 or 6 ? rotations of 360/n degrees), reflection (m)). A molecule located on a special position is expected to possess the point group symmetry of the special position.

16

17 Our first example is space group No

Our first example is space group No

1, P1 the symbol informs us that we have a primitive lattice, and only 1 symmetry. A number, n, as a symmetry element in Hermann-Mauguin notation (used throughout the tutorial) refers to a rotation by 360/n degrees. Thus 1 refers to a 360 rotation. This element is also called the identity in group theory, as it represents a trivial operation of making no change. We will consider the effect of symmetry on a molecule or group located at a general position in the unit cell, with coordinates (+x, +y, +z). Lets try it now.

17

18 a

a

c

18

+

19 So, weve got one molecule per unit cell (Z = 1)

So, weve got one molecule per unit cell (Z = 1)

Lets consider some properties of space groups that wed like to track as we take this tour

Which two phrases describe the properties of handedness and centrosymmetry for P1?

19

20 Enantiomorphous

Enantiomorphous

Non-centrosymmetric

20

+

21 Now, let's review some of the jargon introduced earlier:

Now, let's review some of the jargon introduced earlier:

In this area we'll list the set of equivalent points belonging to the general position

Number of equivalent positions contained within one unit cell

21

Reference atom

An equivalent position

Prefix

+

22 A final point or reminder regarding P (primitive lattices): The

A final point or reminder regarding P (primitive lattices): The

primitive translation operators, expressed in fractional coordinates, are (1,0,0), (0,1,0), (0,0,1). These operations apply to any and all lattice points and may be applied once or many times. Thus, lattice points such as : (x,1+y,z) (x-1,y,z) (2+x,y-3,z+4) are all symmetry-related to the point (x,y,z). A common error made in the early stages of understanding is to suppose that, e.g., (x,1-y,z) is the same as, or is translation-related to (x,y-1,z). Note the difference! In general, adding or subtracting an integer to a fractional number is NOT the same as first taking the negative of that number, and then adding or subtracting integers!

22

23 Now, Ill produce the point (x, 1-y, z) from my original blue point:

Now, Ill produce the point (x, 1-y, z) from my original blue point:

(0.1, 0.8, 0.3). Ouch! This differs from the original by (0, 0.6, 0), and the brown point by (0, 1.6, 0).it isnt related to either point by a primitive translation! I see the key to it now: we can add positive or negative integers to a point, but a translation will NEVER change the sign of the coordinate.

HmmmI dont see what you mean! Let me think about it. So. I need to satisfy myself that (x, 1-y, z) and (x, y-1, z) are different, that is, they are not related by a simple translation.

OK. Ill choose a point .uh.(0.1, 0.2, 0.3). Ill calculate (x, y-1, z) = (0.1, -0.8, 0.3). This is clearly related to my blue point by the translation (0, 1, 0).

Sue N. Smart

23

24 OK! Now lets take the same diagram, that is, the one we just finished

OK! Now lets take the same diagram, that is, the one we just finished

and add a center of symmetry at the origin. The symbol for this is a small circle.

After we add the center, where is the next molecule? Will it be in front of the screen (as for the (x, y, z) molecule), or behind? Think about its sign (the sign for the y-coordinate) before you push the button. Yes, its coordinates are (-x,-y,-z), and thus we place a - next to it. If the new molecule that is generated has the opposite chirality to its symmetry-related mate, well put a comma inside to indicate that.

The other thing to look for is the appearance of other centers of symmetry as you add molecules: these are the ones that are generated by the interaction (i.e., the multiplication) of the operations of this group.

24

25 To reiterate, for each space group, as we did for P1, well derive the

To reiterate, for each space group, as we did for P1, well derive the

group from the Hermann-Mauguin symbol. New figures will "pop up", and you should continually pause and consider the figure's HCE : Height, Chirality, and which new Elements have been generated.

Carl Hermann

Charles-Victor Mauguin

25

26 ,

,

Z=2;

Is this space group enantiomorphous or non-enantiomorphous?

Non-enantiomorphous

26

-

27 Z=2;

Z=2;

Is this space group centrosymmetric or non-centrosymmetric?

Centrosymmetric

27

28 Note that there are two molecules per unit cell, and that other

Note that there are two molecules per unit cell, and that other

centers of symmetry (at a/2, b/2, c/2 and combinations thereof) were generated as we added atoms or groups.

It turns out that there are eight centers of symmetry: where are they?

(0, 0, 0) ; (?, 0, 0); (0, 0, ?); (?, 0, ?); (0, ?, 0); (?, ?, 0); (0, ?, ?); (?, ?, ?)

28

29 Above is a perspective view of a unit cell in P1bar, with all centers

Above is a perspective view of a unit cell in P1bar, with all centers

of symmetry shown. The eight independent centers of symmetry (i. e., those NOT related by any symmetry operation) are shown in blue.

29

30 We say that the eight centers of symmetry are independent, since any

We say that the eight centers of symmetry are independent, since any

single center of symmetry is unrelated to another by the operations of the group :

Its easy to see that the centers of symmetry are unrelated; plug any value into the above set of equivalent positions, and we get the same value back: try (0,0,0) this is obvious. If we try (?,0,0) we generate (-?,0,0). Do these two points represent the same position?

Yes, of course, since we can always add a positive or negative integer (unit cell translation) to any point. Formally, the two have different locations, but they are identical upon unit translation along the a axis.

30

31 Each of the eight centers of symmetry corresponds to a special

Each of the eight centers of symmetry corresponds to a special

position in . Note that each point will be mapped onto itself by an operation of the group (cf. slides 16 & 30). Special positions always correspond to a point group symmetry element, i. e. a rotation axis, reflection, inversion or rotary inversion axis. A special position always has reduced multiplicity compared to the general position.

31

32 x

x

y

z

Why is the multiplicity for the special positions in equal to 1?

Recall that the equivalent positions for are:

If we "run" any of the special position coordinates through these general positions, we only get one value in return: (0, 0, 0) gives identically (0, 0, 0), and, (?, ?, ?) gives (-?, -?, -?). Placing these two points within the same unit cell by unit translations along a, b and c renders them identical.

32

33 So

So

It is easy to see that P1-bar must contain racemic molecules.but could P1 contain a racemate?

Of course. In P1, the left- and right-handed molecules are simply unrelated by any symmetry operation, while in P1-bar, they are related by the crystallographic inversion center.

33

34 We've completed the triclinic space groups, but there are a few final

We've completed the triclinic space groups, but there are a few final

relevant points worth making from both experimental and pedagogic points of view.

It is very useful for us to know something about the density of a crystal. If the measured and calculated densities agree, we may be confident that we know the stoichiometry of the crystal contents well. Alternatively, if the two values do not match, we can calculate the unit cell's molecular weight from the measured density, and often deduce the stoichiometry.

34

35 Can we actually define the density of a crystal in terms of the

Can we actually define the density of a crystal in terms of the

language & ideas we have used thus far?

The mass, m, of one unit cell is just nM/N0 , where n = the number of molecules per unit cell, M = formula weight in grams, and N0 = Avogadro's number. V is the unit cell volume. Normally we will use formula weight in grams and volume in cm3.

35

36 Crystal density is conveniently measured by the neutral buoyancy

Crystal density is conveniently measured by the neutral buoyancy

technique. Imagine what would happen if we placed crystals of density 1.30 g-cm-3 in a solution of heptane, ? = 0.68 g-cm-3:

Now imagine what would happen if we used CCl4, ? = 1.59 :

36

37 Often, it is most useful to measure the density by the neutral

Often, it is most useful to measure the density by the neutral

buoyancy technique, and then calculate n, the number of molecules in the unit cell. Thus, a compound C12H12N2O2 crystallizes in the triclinic system, and has formula weight 216.24, a density of 1.341 g-cm-3 and a unit cell volume of 535.59 ?3. Let's calculate n = ?VNo/M = 1.341(535.59 ? 10-24)(6.02 ? 1023)/(216.24) = 2.002! So, there's 2 molecules per unit cell..can we tell, in general, whether the space group is P1 or P-1bar from such data? More pertinently: can we ever determine the space group from density information?

37

38 Bonjour, Professeur

Bonjour, Professeur

I have a crystalI have measured the density and determined the crystal system. Its triclinic, and Z = 1. Therefore, it MUST belong to space group P1 !!!!!!!!!!

Silly boy! No. You may be correct, but it just as well could be in P1-bar, with a molecule simply occupying a special position (the center of symmetry). You CANNOT obtain the space group from a density measurement!

38

39 Ah, Professeur

Ah, Professeur

I beg to differ! Now I have a different crystal. Again I have measured its density and determined the crystal system. Its triclinic, and this time Z = 2. It is very clear that the space group is P1-bar !!!

No. No. No. Again, you may be correct, but it just as well could be in P1, with two molecules each occupying a general position; the two are unrelated by symmetry. As I said, you CANNOT obtain the space group from a density measurement!

39

40 End of Section 1, Introduction & Pointgroups 1 and 1-bar

End of Section 1, Introduction & Pointgroups 1 and 1-bar

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