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SI: LeSystèmac Internation d’Unitès
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9.1 Fundamental Units and Some Physical Constants
SI: LeSystèmac Internation d’Unitès
Seven SI base units
Derived SI units with a specific name
Time: minute and hour, Plane angle: degree, minute, second, Volume: liter and Mass: metric ton.
These units are non-SI units, but they are accepted for use with the SI units.
Physical constants
aOne twelfth of mass of 12 C. bTemperature 273.15 K, Pressure 101325 Pa(1 atm).
Units frequently used with SI units
aÅ is also used in comparison to the electric current A
9.2 Atomic Weight, Density, Debye Temperature and Mass Absorption Coefficients ({ cm}2 ∕ { g}) for Elements
Θ: Debye temperature, Unit of density: Mg/m3.
Θ: Debye temperature, Unit of density: Mg/m3.
Θ: Debye temperature, Unit of density: Mg/m3.
Θ: Debye temperature, Unit of density: Mg/m3.
9.3 Atomic Scattering Factors as a Function of sinθ ∕ λ
9.4 Quadratic Forms of Miller Indices for Cubic and Hexagonal Systems
9.5 Volume and Interplanar Angles of a Unit Cell
Cubic : V = a 3
Tetragonal :V = a 2 c
Hexagonal : \(V = \frac{\sqrt{3}{a}^{2}c} {2} = 0.866{a}^{2}c\)
Trigonal : \(V = {a}^{3}\sqrt{1 - {3\cos }^{2 } \alpha+ {2\cos }^{3 } \alpha }\)
Orthorhombic : V = abc
Monoclinic : V = abcsinβ
Triclinic : \(V = abc\sqrt{1 {-\cos }^{2 } \alpha{-\cos }^{2 } \beta{-\cos }^{2 } \gamma+ 2\cos \alpha \cos \beta \cos \gamma }\)
Interplanar angles
The angle ϕ between the plane (h 1 k 1 l 1) of spacing d 1 and the plane (h 2 k 2 l 2) of d 2 is estimated from the following equation, where V is the volume of a unit cell:
Cubic : \(\cos \phi= \frac{{h}_{1}{h}_{2} + {k}_{1}{k}_{2} + {l}_{1}{l}_{2}} {\sqrt{({{h}_{1 } }^{2 } +{ {k}_{1 } }^{2 } +{ {l}_{1 } }^{2 } )({{h}_{2 } }^{2 } +{ {k}_{2 } }^{2 } +{ {l}_{2 } }^{2}})}\)
Tetragonal : \(\cos \phi= \frac{\frac{{h}_{1}{h}_{2}+{k}_{1}{k}_{2}} {{a}^{2}} + \frac{{l}_{1}{l}_{2}} {{c}^{2}} } {\sqrt{\left (\frac{{{h}_{1 } }^{2 } +{{k}_{1 } }^{2 } } {{a}^{2}} + \frac{{{l}_{1}}^{2}} {{c}^{2}} \right )\left (\frac{{{h}_{2}}^{2}+{{k}_{2}}^{2}} {{a}^{2}} + \frac{{{l}_{2}}^{2}} {{c}^{2}} \right )}}\)
Hexagonal : \(\cos \phi= \frac{{h}_{1}{h}_{2} + {k}_{1}{k}_{2} + \frac{1} {2}({h}_{1}{k}_{2} + {h}_{2}{k}_{1}) + \frac{3{a}^{2}} {4{c}^{2}} {l}_{1}{l}_{2}} {\sqrt{({{h}_{1 } }^{2 } +{ {k}_{1 } }^{2 } + {h}_{1 } {k}_{1 } + \frac{3{a}^{2 } } {4{c}^{2}}{ {l}_{1}}^{2})({{h}_{2}}^{2}+{{k}_{2}}^{2} + {h}_{2}{k}_{2} + \frac{3{a}^{2}} {4{c}^{2}}{ {l}_{2}}^{2})}}\)
Trigonal : \(\cos \phi= \frac{{a}^{4}{d}_{1}{d}_{2}} {{V }^{2}} {[\sin }^{2}\alpha ({h}_{ 1}{h}_{2} + {k}_{1}{k}_{2} + {l}_{1}{l}_{2}){73.97733pt} + {(\cos }^{2}\alpha-\cos \alpha )({k}_{ 1}{l}_{2} + {k}_{2}{l}_{1} + {l}_{1}{h}_{2} + {l}_{2}{h}_{1} + {h}_{1}{k}_{2} + {h}_{2}{k}_{1})]\)
Orthorhombic : \(\cos \phi= \frac{\frac{{h}_{1}{h}_{2}} {{a}^{2}} + \frac{{k}_{1}{k}_{2}} {{b}^{2}} + \frac{{l}_{1}{l}_{2}} {{c}^{2}} } {\sqrt{\left (\frac{{{h}_{1 } }^{2 } } {{a}^{2}} + \frac{{{k}_{1}}^{2}} {{b}^{2}} + \frac{{{l}_{1}}^{2}} {{c}^{2}} \right )\left (\frac{{{h}_{2}}^{2}} {{a}^{2}} + \frac{{{k}_{2}}^{2}} {{b}^{2}} + \frac{{{l}_{2}}^{2}} {{c}^{2}} \right )}}\)
Monoclinic : \(\cos \phi={ \frac{{d}_{1}{d}_{2}} {\sin }^{2}\beta } \left [\frac{{h}_{1}{h}_{2}} {{a}^{2}} +\frac{{k}_{1}{k{}_{2}\sin }^{2}\beta } {{b}^{2}} +\frac{{l}_{1}{l}_{2}} {{c}^{2}} -\frac{({l}_{1}{h}_{2} + {l}_{2}{h}_{1})\cos \beta } {ac} \right ]\)
Triclinic : \(\cos \phi= \frac{{d}_{1}{d}_{2}} {{V }^{2}} [{S}_{11}{h}_{1}{h}_{2} + {S}_{22}{k}_{1}{k}_{2} + {S}_{33}{l}_{1}{l}_{2}{73.97733pt} + {S}_{23}({k}_{1}{l}_{2} + {k}_{2}{l}_{1}) + {S}_{13}({l}_{1}{h}_{2} + {l}_{2}{h}_{1}) + {S}_{12}({h}_{1}{k}_{2} + {h}_{2}{k}_{1})]\)
9.6 Numerical Values for Calculation of the Temperature Factor
Values of \(\phi (x) = \frac{1} {x}{\int\nolimits \nolimits }_{0}^{x} \frac{\xi } {{\mathrm{e}}^{\xi } - 1}\mathrm{d}\xi \) \(\quad x = \frac{\Theta } {T}\), Θ: Debye temperature
For x lager than 7, ϕ(x) values are approximated by (1. 642 ∕ x). Debye temperatures are compiled in Appendix A.2 using the following reference: (C.Kittel: Introduction to Solid State Physics, 6th Edition, John Wiley & Sons, New York (1986), p.110.)
9.7 Fundamentals of Least-Squares Analysis
Let us consider that the number of n-points have coordinates(x 1, y 1), (x 2, y 2) ⋯ (x n , y n ), and the x and y are related by a straight line with the form of y = a + bx. Our problem is to find the best value of a and b which makes the sum of the squared errors a minimum by using the least-squares method. In this case, we use the following two normal equations:
For given n-points, the following four steps are suggested:
-
(i)
Substitute the experimental data into y = a + bx for obtaining n-equations.
$$\left.\begin{array}{l} {y}_{1} = a + b{x}_{1}\\ \\ \\ {y}_{2} = a + b{x}_{2}\\ \\ \\ \qquad \vdots\\ \\ \\ {y}_{n} = a + b{x}_{n}\\ \end{array} \right \}.$$(3) -
(ii)
Multiply each equation by the coefficient of a (1 in the present case) and add for obtaining the first normal equation.
$$\begin{array}{rcl} & {y}_{1} = a + b{x}_{1}, & \\ & {y}_{2} = a + b{x}_{2}, & \\ & \qquad \vdots & \\ & {y}_{n} = a + b{x}_{n}, & \\ & \overline{{\sum\nolimits }^{n}y = \sum\nolimits a + b\sum\nolimits x}.4& \end{array}$$(4) -
(iii)
Multiply each equation by the coefficient b and add for obtaining the second normal equation.
$$\begin{array}{rcl} & {x}_{1}{y}_{1} = {x}_{1}a + b{x{}_{1}}^{2}, & \\ & {x}_{2}{y}_{2} = {x}_{2}a + b{x{}_{2}}^{2}, & \\ & \qquad \vdots & \\ & {x}_{n}{y}_{n} = {x}_{n}a + b{x{}_{n}}^{2} & \\ & \overline{{ \sum\nolimits }^{n}xy = a\sum\nolimits x + b\sum\nolimits {x}^{2}}5& \end{array}$$(5) -
(iv)
Simultaneous solution of the two equations of (4) and (5) yields the value of a and b.
9.8 Prefixes to Unit and Greek Alphabet
9.9 Crystal Structures of Some Elements and Compounds
These data are taken from the following references. B.D. Cullity: Elements of X-ray Diffraction (2nd Edition), Addison-Wesley (1978). F.S. Galasso: Structure and Properties of Inorganic Solids, PergamonPress (1970).
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Waseda, Y., Matsubara, E., Shinoda, K. (2011). Appendix A. In: X-Ray Diffraction Crystallography. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16635-8_9
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DOI: https://doi.org/10.1007/978-3-642-16635-8_9
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