Here we will focus on how to add figures, tables, and equations into your document. Here are the complete source file and results in PDF format: Figures To insert a figure in a LaTeX document, you write lines like this: \begin{figure} \centering \includegraphics {imagefile1} \caption{Caption for figure} \label{fig_sample} \end{figure} The whole block is enclosed between \begin{figure} and \end{figure}. The command \includegraphics does the actual insertion of the image. Here we insert a file named imagefile1.eps (or imagefile1.pdf when using PdfLaTeX). LaTeX assumes a .eps file extension (and PdfLaTeX assumes .pdf). You don't need to write it. You can also specify the width of the image. Give it as a parameter (enclosed in brackets) to the \includegraphics command. Acceptable measurement units are for example in, mm, and cm. Also the height of the figure is scaled proportionally so the image doesn't get distorted. The \caption command gives a caption for the figure. We have also added the \label which is useful when you want to refer to the equation in your paragraph text (see References ). Additionally, we have used a \centering command to center the figure in the column. If you don't yet know how to create EPS images for LaTeX documents, read the Creating figures tutorial. Subfigures If you want to divide a figure into many smaller parts, use the \subfigure command. First, you have to add this in the beginning of your .tex file: \usepackage{graphicx,subfigure} You probably already have the graphicx package loaded so add only the word subfigure here. Let's add three small figures in place of one normal figure. Use the \subfigure command: \begin{figure} \centering \subfigure { \includegraphics {imagefile2} \label{fig_firstsub} } \ \ \subfigure { \includegraphics {imagefile2} \label{fig_secondsub} } \subfigure { \includegraphics {imagefile2} \label{fig_thirdsub} } \caption{Common figure caption.} \label{fig_subfigures} \end{figure} The result is: Write as many \subfigure commands as you have figures. \subfigure takes an argument (enclosed between ) which specifies the caption for that subfigure. Then put the \includegraphics and \label commands between { and } of the subfigure. Here we use an image file named imagefile2.eps. We have also specified a width for each image using the optional width parameter of the \includegraphics command. Note the \ \ command after the first subfigure. This command creates a line break. In this case, it separates the three subfigures into two rows. Without the \ \ all the three subfigures may end up in just one row. You can try the \ also in other places and see its effect. In the end, we put one more \caption and \label. These are for the whole three-part figure element. Tables A table in LaTeX may look a bit scary bunch of code at first. But you can copy and paste the basic lines that are needed. Then inserting your own text into the table is a piece of cake. Here we go: \begin{table} \renewcommand{\arraystretch}{1.3} \caption{Simple table} \label{table_example} \centering \begin{tabular}{c|c} \hline Heading One Heading Two\ \ \hline \hline Three Four\ \ \hline Five Six\ \ \hline \end{tabular} \end{table} The result will look like this: Hence it's a table with two columns and two rows. Here is how you organize the text in a table: Horizontal lines are separated by \ in the end of line. That is, \ begins a new row. Then write \hline to insert a horizontal line (one or more). Write an where you want a vertical line. The number of columns is specified like this: Here we used a line like \begin{tabular}{c|c}. The | represents a vertical line and c makes the text of a column centered. Thus, c|c creates two columns with centered text. Text can also be left and right aligned if you use l or r instead of c. More columns can be added by using many | symbols. For example, this produces four columns: l|c|c|c . Now the leftmost column is left-aligned and the others are centered. You may wonder about the strange line \renewcommand{\arraystretch}{1.3}. This is needed for adjusting the white space around text in the table cells. The value 1.3 produces quite a pleasing look. Double column figures and tables If you are writing a two column document and you would like to insert a wide figure or table that spans the whole page width, use the starred versions of the figure and table constructs. Like this: \begin{figure*}...\end{figure*} or \begin{table*}...\end{table*}. Write the contents in the usual way. You can use also subfigures inside figure*. Note that double column figures and tables have some limitations. They can't be placed at the bottom of pages. Additionally, they will not appear on the same page where they are defined. So you have to define them prior to the page on which they should appear. Equations Short mathematical expressions can be inserted within paragraph text by putting the math between $ and $. For example: ... angle frequency $\omega = 2\pi f$ ... This is called an inline equation. The result is: . In equations the normal text symbols are written as such, for example 2 and f. Greek symbols are named for example \alpha, \beta and so on. You don't need to remember these because in WinEdt (and TeXnicCenter) you can use the symbol toolbar which has buttons for all the Greek letters and other math symbols. Numbered equations are separate from paragraph text and they are automatically numbered. The contents of the equation are written using the same ideas as inline equations but now we write \begin{equation} and \end{equation} instead of $s. \begin{equation} \label{capacitor_impedance} X_{C} = \frac{ 1 }{ \omega C } \end{equation} The result is: Here we learn another structure which is often used in equations: the \frac command inserts a fraction whose numerator and denominator are enclosed in braces. http://blog.sina.com.cn/s/blog_4a582a1f0100cyiw.html
http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list 下面是从其他网站转的，读过后，对wyckoff position 可以有大致的了解了。 Space Groups The International Tables for Crystallography From examination of a space group in “The International Tables for Crystallography” Vol. A, you should be able to ascertain the following information: · Herman-Mauguin (HM) Symbol (Long, Short) · Point Group (HM, Schoenflies) · Locate and identify symmetry elements · Understand Wyckoff site multiplicity and symmetry · Distinguish general and special positions · Extinction conditions · Identify possible subgroups and supergroups Understanding the Herman-Mauguin Space Group Symbol Space groups are typically identified by their short Herman-Mauguin symbol (i.e. Pnma, I4/mmm, etc.). The symmetry elements contained in the short symbol are the minimum number needed to generate all of the remaining symmetry elements. This symbolism is very efficient, condensed form of noting all of the symmetry present in a given space group. We won’t go into all of the details of the space group symbol, but I will expect you to be able to determine the Crystal system, Bravais Lattice and Point group from the short H-M symbol. You should also be able to determine the presence and orientation of certain symmetry elements from the short H-M symbol and vice versa. The HM space group symbol can be derived from the symmetry elements present using the following logic. The first letter identifies the centering of the lattice, I will hereafter refer to this as the lattice descriptor : · P Primitive · I Body centered · F Face centered · C C-centered · B B-centered · A A-centered The next three symbols denote symmetry elements present in certain directions, those directions are as follows: Crystal System Symmetry Direction Primary Secondary Tertiary Triclinic None Monoclinic Orthorhombic Tetragonal / Hexagonal/ Trigonal / / Cubic / / – Axis parallel or plane perpendicular to the x-axis. – Axis parallel or plane perpendicular to the y-axis. – Axis parallel or plane perpendicular to the z-axis. – Axis parallel or plane perpendicular to the line running at 45° to the x and y axes. – Axis parallel or plane perpendicular to the long face diagonal of the ab face of a hexagonal cell. – Axis parallel or plane perpendicular to the body diagonal. For a better understanding see specific examples from class notes. However, with no knowledge of the symmetry diagram we can identify the crystal system from the space group symbol. · Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m) · Tetragonal – The primary symmetry symbol will always be either 4, (-4), 4 1 , 4 2 or 4 3 (i.e. P4 1 2 1 2, I4/m, P4/mcc) · Hexagonal – The primary symmetry symbol will always be a 6, (-6), 6 1 , 6 2 , 6 3 , 6 4 or 6 5 (i.e. P6mm, P6 3 /mcm) · Trigonal – The primary symmetry symbol will always be a 3, (-3) 3 1 or 3 2 (i.e P31m, R3, R3c, P312) · Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc2 1 , Pnc2) · Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P2 1 /n) · Triclinic – The lattice descriptor will be followed by either a 1 or a (-1). The point group can be determined from the short H-M symbol by converting glide planes to mirror planes and screw axes to rotation axes. For example: Space Group = Pnma Point Group = mmm Space Group = I`4c2 Point Group =`4m2 Space Group = P4 2 /n Point Group = 4/m Wyckoff Sites One of the most useful pieces of information contained in the International Tables are the Wyckoff positions. The Wyckoff positions tell us where the atoms in a crystal can be found. To understand how they work consider the monoclinic space group Pm. This space group has only two symmetry elements, both mirror planes perpendicular to the b-axis. One at y = 0 and one at y = ½ (halfway up the unit cell in the b direction). Now let’s place an atom in the unit cell at an arbitrary position, x,y,z. If we now carry out the symmetry operation associated with this space group a second atom will be generated by the mirror plane at x,-y,z. However, if we were to place the atom on one of the mirror planes (its y coordinate would have to be either 0 or ½) then the reflection operation would not create a second atom. All of the information in the proceeding paragraph is contained in Wyckoff positions section of the International Tables. Pm has three Wyckoff sites as shown in the table below: Multiplicity Wyckoff Letter Site Symmetry Coordinates 2 c 1 (1) x,y,z (2) x,-y,z 1 b m x,½,z 1 a m x,0,z The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. In this case for every atom we insert at an arbitrary position (x,y,z) in the unit cell a second atom will be generated by the mirror plane at x,-y,z. This corresponds to the uppermost Wyckoff position 2c. The letter is simply a label and has no physical meaning. They are assigned alphabetically from the bottom up. The symmetry tells us what symmetry elements the atom resides upon. The uppermost Wyckoff position, corresponding to an atom at an arbitrary position never resides upon any symmetry elements. This Wyckoff position is called the general position. The coordinates column tells us the coordinates of all of the symmetry related atoms (two in this case). All of the remaining Wyckoff positions are called special positions. They correspond to atoms which lie upon one of more symmetry elements, because of this they always have a smaller multiplicity than the general position. Furthermore, one or more of their fractional coordinates must be fixed. In this case the y value must be either 0 or ½ or the atom would no longer lie on the mirror plane. Generating a Crystal Structure from its Crystallographic Description Using the space group information contained in the International Tables we can do many things. One powerful use is to generate an entire crystal structure from a brief description. Consider the following description of the crystal structure of Sr 2 AlTaO 6 . Space Group = Fm`3m a = 7.80 Atomic Positions Atom X Y Z Sr 0.25 0.25 0.25 Al 0.0 0.0 0.0 Ta 0.5 0.5 0.5 O 0.24 0.0 0.0 From the space group tables we see that the atoms are located on the following Wyckoff sites Sr 8c Al 4a Ta 4b O 24e The number associated with the Wyckoff sites tells us how many atoms of that type there are in the unit cell. In this So there are 40 atoms in the unit cell, with stoichiometry Sr 8 Al 4 Ta 4 O 24 which reduces to the empirical formula Sr 2 AlTaO 6 . Since the number of atoms in the unit cell is four times the number of atoms in the formula unit, we say that Z = 4. Using the face centering generators (0,0,0), (½,½,0), (½,0,½), (0,½,½) together with the coordinates of each Wyckoff site we can generate the fractional coordinates of all atoms in the unit cell: Sr 1:(0.25,0.25,0.25), 2:(0.75,0.75,0.25), 3:(0.75,0.25,0.75), 4:(0.25,0.75,0.75) 5:(0.25,0.25,0.75), 6:(0.75,0.75,0.75), 7:(0.75,0.25,0.25), 8:(0.25,0.75,0.25) Al 1:(0.0,0.0,0.0), 2:(0.5,0.5,0.0), 3:(0.5,0.0,0.5), 4:(0.0,0.5,0.5) Ta 1:(0.5,0.5,0.5), 2:(0.0,0.0,0.5), 3:(0.0,0.5,0.0), 4:(0.5,0.0,0.0) O 1:(0.24,0.0,0.0), 2:(0.74,0.5,0.0), 3:(0.74,0.0,0.5), 4:(0.24,0.5,0.5) 5:(0.76,0.0,0.0), 6:(0.26,0.5,0.0), 7:(0.26,0.0,0.5), 8:(0.76,0.5,0.5) 9:(0.0,0.24,0.0), 10:(0.5,0.74,0.0), 11:(0.5,0.24,0.5), 12:(0.0,0.74,0.5) 13:(0.0,0.76,0.0), 14:(0.5,0.26,0.0), 15:(0.5,0.76,0.5), 16:(0.0,0.26,0.5) 17:(0.0,0.0,0.24), 18:(0.5,0.5,0.24), 19:(0.5,0.0,0.74), 20:(0.0,0.5,0.74) 21:(0.0,0.0,0.76), 22:(0.5,0.5,0.76), 23:(0.5,0.0,0.26), 24:(0.0,0.5,0.26) From these fractional coordinates you can sketch out the structure of Sr 2 AlTaO 6 . With some luck I will provide a link to a picture of the structure here, at some point in the future. We can also work out bond distances from this information. The first Al ion is octahedrally coordinated by six oxygens (1,5,9,13,17,21) and the Al-O distance is : d = 7.80 1/2 = 1.87 while the first Ta ion is also surrounded by 6 oxygens (4,8,11,15,18,22) at a distance of d = 7.80 1/2 = 2.03 and Sr is surrounded by 12 oxygens (1,4,6,7,9,11,14,16,17,18,23,24) at a distance of d = 7.80 1/2 = 2.76 Determining a Crystal Structure from Symmetry Composition Another use is that given the stoichiometry, space group and unit cell size (which can typically be determined from diffraction techniques) and the density of a compound we can often deduce the crystal structure of relatively simple compounds. As an example consider the following information: Stoichiometry = SrTiO 3 Space Group = Pm3m a = 3.90 Density = 5.1 g/cm 3 To derive the crystal structure from this information the first step is to calculate the number of formula units per unit cell : Formula Weight SrTiO 3 = 87.62 + 47.87 + 3(16.00) = 183.49 g/mol Unit Cell Volume = (3.9010 -8 cm) 3 = 5.9310 -23 cm 3 (5.1 g/cm 3 )(5.9310 -23 cm 3 )(mol/183.49 g)(6.02210 23 /mol) = 0.99 Thus there is one formula unit per unit cell (Z=1), and the number of atoms per unit cell is : 1 Sr, 1 Ti and 3 O. Next we compare the number of atoms in the unit cell with the multiplicities of the Wyckoff sites. · From the multiplicities of the special positions in space group Pm3m we see that Sr must occupy either the 1a or 1b positions (otherwise there would be more than one Sr in the unit cell) · By the same reasoning Ti must also reside in either the 1a or 1b position, and, since there are no free positional parameters (x,y or z) in either 1a or 1b, the two ions cannot occupy the same site. · To maintain 3 oxygen ions in the unit cell it must reside at either site 3c or 3d. If we arbitrarily put Ti at the origin (1a), then by default Sr must go to 1b. To evaluate the prospects of putting O at either 3c or 3d we calculate the Ti-O bond distances: D (O @ 3c) = 3.90 1/2 = 2.76 D (O @ 3d) = 3.90 1/2 = 1.95 Of these two the latter (3d) is obviously more appropriate for a Ti-O bond (consult tables of ionic radii to convince yourself of this statement). Thus we obtain the structure of SrTiO 3 to be Space Group = Pm3m a = 3.90 Atomic Positions Atom Site X Y Z Sr 1b 0.5 0.5 0.5 Ti 1a 0.0 0.0 0.0 O 3d 0.5 0.0 0.0
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