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This tutorial is divided into sections by crystal class
This tutorial is divided into sections by crystal class
In the triclinic system, a
In the triclinic system, a
Now, I’ll produce the point (x, 1-y, z) from my original “blue” point:
Now, I’ll produce the point (x, 1-y, z) from my original “blue” point:
To reiterate, for each space group, as we did for P1, we’ll derive the
To reiterate, for each space group, as we did for P1, we’ll derive the
To reiterate, for each space group, as we did for P1, we’ll derive the
To reiterate, for each space group, as we did for P1, we’ll derive the
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
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Crystal density is conveniently measured by the neutral buoyancy
Bonjour, Professeur
Bonjour, Professeur
Bonjour, Professeur
Bonjour, Professeur
Ah, Professeur
Ah, Professeur
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Рисуем космос 4 класс

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1Knowledge of space groups and the 15that the identity operation is a trivial
implications of space group symmetry on operation of making no change, e.g., a
the physical and chemical properties of 360° rotation. 15.
solids are pivotal factors in all areas of 16The term "Special Position"
structural science. As we met to bring our (not yet encountered...) is defined as
ideas in teaching this subject to life, we follows: A set of symmetrically equivalent
both felt that teaching the concepts with points...is said to be in 'special
repetitive textual and visual position' if each of its points is mapped
reinforcement, “early and often”, will onto itself by at least one further
provide a sound basis for students in this symmetry operation of the space group.
subject. We chose PowerPoint as the Later we'll see that a special position
delivery vehicle for the tutorial, owing always corresponds to the location of a
to the facility of combining narrative point group symmetry element (inversion ?
pedagogy with animations of the buildup of ), rotation (n = 2, 3, 4 or 6 ? rotations
space groups, and the ease of making of 360/n degrees), reflection (m)). A
subtle changes and/or corrections. And, we molecule located on a special position is
realize that others may wish to alter the expected to possess the point group
approach, e.g., by rewriting the text in symmetry of the special position. 16.
another language, or altering the approach 17Our first example is space group No.
(….hopefully, the desired changes will be 1, P1 – the symbol informs us that we have
small!). The student may also wish to a primitive lattice, and only “1”
consider M. E. Kastner's Crystallographic symmetry. A number, n, as a symmetry
Courseware, which provides an alternative element in Hermann-Mauguin notation (used
approach to the teaching of Space Group throughout the tutorial) refers to a
Symmetry. 1. rotation by 360/n degrees. Thus 1 refers
2We trust that the current and to a 360° rotation. This element is also
subsequent versions of our tutorial will called the identity in group theory, as it
become ein lebendiges Buch (a living book) represents a trivial operation of making
to which all can contribute and enjoy, and no change. We will consider the effect of
from which all can learn. 2. symmetry on a molecule or group located at
3The tutorial is based in part upon an a “general position” in the unit cell,
approach to teaching space groups with coordinates (+x, +y, +z). Let’s try
developed in a course in X-ray it now. 17.
Crystallography at Brandeis University. 18a. c. 18. +.
However, through vigorous discussions, the 19So, we’ve got one molecule per unit
ideas have been continuously reworked. 3. cell (Z = 1). Let’s consider some
4Acknowledgements We are particularly properties of space groups that we’d like
grateful to the National Science to track as we take this tour… Which two
Foundation Summer Research Program in phrases describe the properties of
Solid State and Materials Chemistry, handedness and centrosymmetry for P1? 19.
directed by Prof. Shiou-Jyh Hwu, Clemson 20Enantiomorphous. Non-centrosymmetric.
University. The Program provides an 20. +.
opportunity for a faculty member and 21Now, let's review some of the jargon
student to carry out research in a host introduced earlier: In this area we'll
laboratory; Prof. Jasinski and student list the set of equivalent points
Lisa Bennett worked on solid-state belonging to the general position. Number
reactivity during Summer 2003 at Brandeis. of equivalent positions contained within
For the faculty member to return to the one unit cell. 21. Reference atom. An
program in subsequent years, a joint equivalent position. Prefix. +.
research program on educational aspects of 22A final point or reminder regarding P
solid-state chemistry is required. With (primitive lattices): The primitive
the encouragement provided by the program translation operators, expressed in
requirements and additional NSF support fractional coordinates, are (1,0,0),
through grants DMR-0089257 and DMR-0504000 (0,1,0), (0,0,1). These operations apply
to BMF, we designed and wrote this to any and all lattice points and may be
tutorial during Summer 2004 and 2005. 4. applied once or many times. Thus, lattice
5Finally, we wish to thank Dr. Eugene points such as : (x,1+y,z) (x-1,y,z)
Cheung (Cardiff University) and Professors (2+x,y-3,z+4) are all “symmetry-related”
Jen Swift (Georgetown) and Mike Ward to the point (x,y,z). A common error made
(Minnesota) for reviewing the tutorial and in the early stages of understanding is to
providing oodles of helpful comments. suppose that, e.g., (x,1-y,z) is “the same
Local comments and stimulating discussions as”, or is translation-related to
at Brandeis University, from Professors (x,y-1,z). Note the difference! In
Dan Oprian, Chris Miller and undergraduate general, adding or subtracting an integer
students Jeremy Heyman and Stephen C. to a fractional number is NOT the same as
Wilson were of immeasurable help. We first taking the negative of that number,
invite your comment and constructive and then adding or subtracting integers!
criticism. The design of this tutorial 22.
will allow us to incorporate changes on a 23Now, I’ll produce the point (x, 1-y,
regular basis. Bruce M. Foxman Brandeis z) from my original “blue” point: (0.1,
University foxman1@brandeis.edu 0.8, 0.3). Ouch! This differs from the
http://www.xray.chem.brandeis.edu Jerry P. original by (0, 0.6, 0), and the brown
Jasinski Keene State College point by (0, 1.6, 0)….it isn’t related to
jjasinski@keene.edu either point by a primitive translation! I
http://academics.keene.edu/chemistry/facul see the key to it now: we can add positive
y.htm September 2006. 5. or negative integers to a point, but a
6COLORS No doubt you have already translation will NEVER change the sign of
noticed that we’re using underlined green the coordinate. Hmmm…I don’t see what you
text to indicate a hyperlinked page or mean! Let me think about it. So…. I need
pages. In order to minimize issues to satisfy myself that (x, 1-y, z) and (x,
involving broken links, all links are y-1, z) are different, that is, they are
included in the tutorial, and the exact not related by a simple translation. OK.
link references may be found in the I’ll choose a point ….uh….(0.1, 0.2, 0.3).
Credits section (Chapter 7) of this I’ll calculate (x, y-1, z) = (0.1, -0.8,
PowerPoint presentation. Blue is often 0.3). This is clearly related to my “blue”
used for emphasis. Red is usually employed point by the translation (0, 1, 0). Sue N.
when we are referring to symmetry elements Smart. 23.
or unit cell parameters a, b, c, ?, ?, ?. 24OK! Now let’s take the same diagram,
Other colors, used less often, are that is, the one we just finished, and add
generally employed to emphasize particular a center of symmetry at the origin. The
points, or to improve the readability of a symbol for this is a small circle. After
particular presentation. 6. we add the center, where is the next
7Where should you be, in your knowledge molecule? Will it be in front of the
of X-ray structure determination, or screen (as for the (x, y, z) molecule), or
crystallography before beginning this behind? Think about its sign (the sign for
tutorial? We’re starting with an assumed the y-coordinate) before you push the
knowledge of point groups, unit cells, button. Yes, its coordinates are
lattices and space group symmetry elements (-x,-y,-z), and thus we place a “-” next
(there will be some minor review of these to it. If the new molecule that is
topics). In the future we'll be adding a generated has the opposite chirality to
special section on unit cells and lattices its symmetry-related mate, we’ll put a
recently added to the tutorial. We comma inside to indicate that. The other
recommend reviewing or understanding the thing to look for is the appearance of
equivalent of Chapters 1-4 in Sands, other centers of symmetry as you add
Introduction to Crystallography (Dover, molecules: these are the ones that are
1975) or Sections 2.1 and 3.1-3.4 in Stout generated by the interaction (i.e., the
and Jensen, X-Ray Structure Determination: multiplication) of the operations of this
A Practical Guide (Wiley, 1989). Finally, group. 24.
if you have around 3 minutes, listen to a 25To reiterate, for each space group, as
humorous musical revue of Bravais lattices we did for P1, we’ll derive the group from
at : the Hermann-Mauguin symbol. New figures
http://www.haverford.edu/physics-astro/son will "pop up", and you should
s/bravais.htm. 7. continually pause and consider the
8This tutorial is divided into sections figure's HCE : Height, Chirality, and
by crystal class. There are 32 which new Elements have been generated.
crystallographic point groups, or crystal Carl Hermann. Charles-Victor Mauguin. 25.
classes. The 32 crystal classes correspond 26, Z=2; Is this space group
to the external shapes of crystals enantiomorphous or non-enantiomorphous?
actually observed. The crystal class may Non-enantiomorphous. 26. -.
be obtained from the space group symbol by 27Z=2; Is this space group
“removing” the translations from the centrosymmetric or non-centrosymmetric?
symbol…more about that later! In this Centrosymmetric. 27.
section we consider the two triclinic 28Note that there are two molecules per
space groups, P1 and P-1bar, which belong unit cell, and that other centers of
to crystal classes 1 and 1-bar, symmetry (at a/2, b/2, c/2 and
respectively. Paul von Groth. 8. combinations thereof) were generated as we
9In all of the tutorial presentations added atoms or groups. It turns out that
which show a unit cell diagram, we have there are eight centers of symmetry: where
followed the style of the International are they? (0, 0, 0) ; (?, 0, 0); (0, 0,
Tables for Crystallography, Volume A, ?); (?, 0, ?); (0, ?, 0); (?, ?, 0); (0,
Space Group Symmetry, which contains ?, ?); (?, ?, ?). 28.
diagrams of the 230 space groups. We will 29Above is a perspective view of a unit
use the Hermann-Mauguin symmetry notation1 cell in P1bar, with all centers of
throughout, and will be deriving each symmetry shown. The eight independent
space group from the Hermann-Mauguin centers of symmetry (i. e., those NOT
symbol. Our diagrams are “not exactly” related by any symmetry operation) are
identical to the diagrams in Volume A, but shown in blue. 29.
rather are a composite overlay of the unit 30We say that the eight centers of
cell diagram, symmetry elements, and symmetry are independent, since any single
molecules which reside in the Equivalent center of symmetry is unrelated to another
General Positions. 9. 1There will be an by the operations of the group : It’s easy
opportunity to learn more of the history to see that the centers of symmetry are
of this notation on an upcoming slide. unrelated; plug any value into the above
10In the triclinic system, a ? b ? c; ? set of equivalent positions, and we get
? ? ? ? ? 90?. Triclinic crystals either the same value back: try (0,0,0) – this is
have only 1 symmetry (a 360° rotation, obvious. If we try (?,0,0) we generate
crystal class 1) or possess a center of (-?,0,0). Do these two points represent
inversion (crystal class ). For the same position? Yes, of course, since
convenience, we’ll often write “1-bar” we can always add a positive or negative
instead of . integer (unit cell translation) to any
1http://phycomp.technion.ac.il/~sshaharr/i point. Formally, the two have different
tro.html. Primitive Triclinic (click to locations, but they are identical upon
rotate)1. 10. unit translation along the a axis. 30.
11On each slide you will see the ac 31Each of the eight centers of symmetry
projection of a triclinic or monoclinic corresponds to a special position in .
unit cell. The b axis, which points toward Note that each point will be mapped onto
the viewer, is either inclined to the a itself by an operation of the group (cf.
and c axes (triclinic) or perpendicular to slides 16 & 30). Special positions
the page (monoclinic). In the orthorhombic always correspond to a point group
case, we'll use an ab projection. Each symmetry element, i. e. a rotation axis,
slide opens with the ac (orthorhombic, ab) reflection, inversion or rotary inversion
projection and a reminder about the axial axis. A special position always has
directions. Upon tapping the advance reduced multiplicity compared to the
button, various events will occur, general position. 31.
depending upon the space group under 32x. y. z. Why is the multiplicity for
consideration. We’ll describe these the special positions in equal to 1?
precisely as we approach each example. Recall that the equivalent positions for
Each atom or group of atoms will be are: If we "run" any of the
displayed by an open circle. An open special position coordinates through these
circle with a large comma inside will be general positions, we only get one value
used to indicate opposite chirality to the in return: (0, 0, 0) gives identically (0,
reference molecule. 11. 0, 0), and, (?, ?, ?) gives (-?, -?, -?).
12We need to define the coordinate Placing these two points within the same
system we'll be working in. Of course, unit cell by unit translations along a, b
we'll need to specify the positions of and c renders them identical. 32.
atoms or molecules within the unit cells 33So. It is easy to see that P1-bar must
under consideration. Let's imagine that a contain racemic molecules….but could P1
particular atom is located (in ?) at (X, contain a racemate? Of course. In P1, the
Y, Z). A preferred method to specify the left- and right-handed molecules are
location of this atom would be to use a simply unrelated by any symmetry
formalism that is independent of the size operation, while in P1-bar, they are
of the unit cell. To do this we use related by the crystallographic inversion
fractional coordinates (x, y, z), where x center. 33.
= X/a y = Y/b and z = Z/c Within the unit 34We've completed the triclinic space
cell, values of x, y and z are thus groups, but there are a few final relevant
constrained to decimal values between 0 points worth making from both experimental
and 1. And, for example, a location in a and pedagogic points of view. It is very
unit cell adjacent to the reference cell useful for us to know something about the
but displaced along a would thus be (1 + density of a crystal. If the measured and
x, y, z). 12. calculated densities agree, we may be
13What is our reference molecule or atom confident that we know the stoichiometry
? This is the first atom or group that of the crystal contents well.
appears on the screen as an open circle. Alternatively, if the two values do not
It will always have the coordinates (x, y, match, we can calculate the unit cell's
z), and we'll draw an ac projection of the molecular weight from the measured
unit cell, with axis b coming out of the density, and often deduce the
page. Finally, we must specify the stoichiometry. 34.
location of the atom or group along the 35Can we actually define the density of
third dimension (its "height" a crystal in terms of the language &
in/out of the screen). We’ll do this by ideas we have used thus far? The mass, m,
placing the “prefix” of the y-coordinate of one unit cell is just nM/N0 , where n =
next to the open circle. For the reference the number of molecules per unit cell, M =
molecule, this will simply be a “+”, as it formula weight in grams, and N0 =
is located at (x, y, z) ? +y. An indicator Avogadro's number. V is the unit cell
such as “?+” signifies “?+y”; “-” is just volume. Normally we will use formula
“-y”. Remember that x, y and z are weight in grams and volume in cm3. 35.
fractional coordinates, and for molecules 36Crystal density is conveniently
within the unit cell, each has a value measured by the neutral buoyancy
between 0 and 1. As you push the advance technique. Imagine what would happen if we
button, the operations of the group are placed crystals of density 1.30 g-cm-3 in
applied …. in the following case, space a solution of heptane, ? = 0.68 g-cm-3:
group P1, we have only unit translations Now imagine what would happen if we used
to apply, in sequence. 13. CCl4, ? = 1.59 : 36.
14We’ll start by adding the 37Often, it is most useful to measure
translations, and recording the unique set the density by the neutral buoyancy
of symmetry position(s) generated for a technique, and then calculate n, the
single atom. As the atoms are added, think number of molecules in the unit cell.
about how many are actually within the Thus, a compound C12H12N2O2 crystallizes
unit cell; we’ll call this number Z. in the triclinic system, and has formula
Beside the number of symmetry related weight 216.24, a density of 1.341 g-cm-3
atoms, we’ll list their positions. See if and a unit cell volume of 535.59 ?3. Let's
you can assign coordinates to each calculate n = ?VNo/M = 1.341(535.59 ?
molecule added; some answers are given as 10-24)(6.02 ? 1023)/(216.24) = 2.002! So,
you proceed. At a time after all the there's 2 molecules per unit cell…..can we
unique atoms within the cell have been tell, in general, whether the space group
generated we’ll draw a box around the is P1 or P-1bar from such data? More
“General Position”, the set within the pertinently: can we ever determine the
unit cell equivalent by symmetry. The space group from density information? 37.
General Position thus contains a certain 38Bonjour, Professeur! I have a
number of equivalent points per cell; the crystal…I have measured the density and
number is referred to as Z. We will also determined the crystal system. It’s
say that the multiplicity (the number) of triclinic, and Z = 1. Therefore, it MUST
the General Position is Z. 14. belong to space group P1 !!!!!!!!!! Silly
15In the International tables for X-ray boy! No. You may be correct, but it just
Crystallography, Volume A. section 2.11, as well could be in P1-bar, with a
the term "General Position" is molecule simply occupying a special
defined as follows: A set of symmetrically position (the center of symmetry). You
equivalent points...is said to be in CANNOT obtain the space group from a
'general position' if each of its points density measurement! 38.
is left invariant only by the identity 39Ah, Professeur! I beg to differ! Now I
operation but by no other symmetry have a different crystal. Again I have
operation of the space group. Each space measured its density and determined the
group has only one general position. We'll crystal system. It’s triclinic, and this
see, time and time again, that, in the time Z = 2. It is very clear that the
general position, no symmetry requirements space group is P1-bar !!! No. No. No.
are imposed upon the molecule. Thus, Again, you may be correct, but it just as
general positions always have well could be in P1, with two molecules
'site-symmetry' 1 (Hermann-Mauguin each occupying a general position; the two
notation) or alternatively C1 (Schoenflies are unrelated by symmetry. As I said, you
notation). More info about Hermann-Mauguin CANNOT obtain the space group from a
notation appears on slides 17, 25 and density measurement! 39.
throughout the tutorial. We hope you will 40End of Section 1, Introduction &
recall from your knowledge of point groups Pointgroups 1 and 1-bar. 40.
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