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9.1 Fundamental Units and Some Physical Constants

SI: LeSystèmac Internation d’Unitès

Seven SI base units

figure 1_9

Derived SI units with a specific name

figure 2_9

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

figure 3_9

aOne twelfth of mass of 12 C. bTemperature 273.15 K, Pressure 101325 Pa(1 atm).

Units frequently used with SI units

figure 4_9

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

figure 8_9
figure 9_9
figure 10_9

Θ: Debye temperature, Unit of density: Mg/m3.

figure 11_9
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Θ: Debye temperature, Unit of density: Mg/m3.

figure 14_9
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Θ: Debye temperature, Unit of density: Mg/m3.

figure 5_9
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Θ: Debye temperature, Unit of density: Mg/m3.

9.3 Atomic Scattering Factors as a Function of sinθ ∕ λ

figure 17_9
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9.4 Quadratic Forms of Miller Indices for Cubic and Hexagonal Systems

figure 20_9

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 : \(\cos \phi= \frac{{d}_{1}{d}_{2}} {{V }^{2}} [{S}_{11}{h}_{1}{h}_{2} + {S}_{22}{k}_{1}{k}_{2} + {S}_{33}{l}_{1}{l}_{2}{73.97733pt} + {S}_{23}({k}_{1}{l}_{2} + {k}_{2}{l}_{1}) + {S}_{13}({l}_{1}{h}_{2} + {l}_{2}{h}_{1}) + {S}_{12}({h}_{1}{k}_{2} + {h}_{2}{k}_{1})]\)

$$\begin{array}{ll} {S}_{11} = {b}^{2}{c{}^{2}\sin }^{2}\alpha&\qquad {S}_{12} = ab{c}^{2}(\cos \alpha \cos \beta-\cos \gamma ) \\ {S}_{22} = {a}^{2}{c{}^{2}\sin }^{2}\beta&\qquad {S}_{23} = {a}^{2}bc(\cos \beta \cos \gamma-\cos \alpha ) \\ {S}_{33} = {a}^{2}{b{}^{2}\sin }^{2}\gamma&\qquad {S}_{13} = a{b}^{2}c(\cos \gamma \cos \alpha-\cos \beta )\end{array}$$

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

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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:

$$\sum y = \sum a + b\sum x$$
(1)
$$\sum xy = a\sum x + b\sum { x}^{2}.$$
(2)

For given n-points, the following four steps are suggested:

  1. (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)
  2. (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)
  3. (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)
  4. (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

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9.9 Crystal Structures of Some Elements and Compounds

figure 24_9

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).