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% Document Info
\author{marrgotthewise (marrgotthewise)}
\pdfinfo{
  /Title (bonds-and-crystal-structures-of-metals.pdf)
  /Creator (Cheatography)
  /Author (marrgotthewise (marrgotthewise))
  /Subject (Bonds and Crystal Structures of Metals Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Bonds and Crystal Structures of Metals Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{marrgotthewise (marrgotthewise)} via \textcolor{DarkBackground}{\uline{cheatography.com/113347/cs/21691/}}}
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\noindent
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}marrgotthewise (marrgotthewise) \\
  \uline{cheatography.com/marrgotthewise} \\
  \end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 2nd February, 2020.\\
   Page {\thepage} of \pageref{LastPage}.
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.32733 cm} x{1.09848 cm} x{2.15119 cm} }
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\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Types of Bonds}}  \tn
% Row 0
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\mymulticolumn{3}{x{5.377cm}}{{\bf{Primary}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
Ionic & Electron {\emph{transfer}}. & Non-directional \tn 
% Row Count 4 (+ 3)
% Row 2
\SetRowColor{LightBackground}
Covalent & Electron {\emph{sharing}}. & Directional, strong, brittle, high melting temps, less conductive. \tn 
% Row Count 8 (+ 4)
% Row 3
\SetRowColor{white}
Metallic & "Sea of \seqsplit{electrons}." & Ductile, conductive. \tn 
% Row Count 11 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\{\{nobreak\}\}{\bf{Secondary}} & {\bf{(van der Waals)}} &  \tn 
% Row Count 14 (+ 3)
% Row 5
\SetRowColor{white}
 & {\emph{atom 1}} & {\emph{atom 2}} \tn 
% Row Count 15 (+ 1)
% Row 6
\SetRowColor{LightBackground}
London & induced dipole & induced dipole \tn 
% Row Count 17 (+ 2)
% Row 7
\SetRowColor{white}
Debye & \seqsplit{permanent} dipole & neutral \{\{nl\}\}(non-polarized) \tn 
% Row Count 19 (+ 2)
% Row 8
\SetRowColor{LightBackground}
Keesom & \seqsplit{permanent} dipole & permanent dipole \tn 
% Row Count 21 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Lattice}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580593708_lattice.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{A collection of points arranged on a periodic pattern so that the surroundings of each lattice point are identical.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{0.86963 cm} x{1.8308 cm} x{1.87657 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{3}{x{5.377cm}}{\bf\textcolor{white}{Atomic Radii and Crystal Structures for 16 Metals}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Metal}} & {\bf{Crystal Structure}} & {\bf{Atomic Radius (nm)}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\seqsplit{Aluminum} & FCC & 0.1431 \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{Cadmium} & HCP & 0.1490 \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
\seqsplit{Chromium} & BCC & 0.1249 \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Cobalt & HCP & 0.1253 \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
Copper & FCC & 0.1278 \tn 
% Row Count 9 (+ 1)
% Row 6
\SetRowColor{LightBackground}
Gold & FCC & 0.1442 \tn 
% Row Count 10 (+ 1)
% Row 7
\SetRowColor{white}
Iron & BCC & 0.1241 \tn 
% Row Count 11 (+ 1)
% Row 8
\SetRowColor{LightBackground}
Lead & FCC & 0.1750 \tn 
% Row Count 12 (+ 1)
% Row 9
\SetRowColor{white}
\seqsplit{Molybdenum} & BCC & 0.1363 \tn 
% Row Count 14 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Nickel & FCC & 0.1246 \tn 
% Row Count 15 (+ 1)
% Row 11
\SetRowColor{white}
\seqsplit{Platinum} & FCC & 0.1387 \tn 
% Row Count 17 (+ 2)
% Row 12
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Silver & FCC & 0.1445 \tn 
% Row Count 18 (+ 1)
% Row 13
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\seqsplit{Tantalum} & BCC & 0.1430 \tn 
% Row Count 20 (+ 2)
% Row 14
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\seqsplit{Titanium} & HCP & 0.1445 \tn 
% Row Count 22 (+ 2)
% Row 15
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\seqsplit{Tungsten} & BCC & 0.1371 \tn 
% Row Count 24 (+ 2)
% Row 16
\SetRowColor{LightBackground}
Zinc & HCP & 0.1332 \tn 
% Row Count 25 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}---}
\SetRowColor{LightBackground}
\mymulticolumn{3}{x{5.377cm}}{{[}1, Tab 3.1{]}}  \tn 
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\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{E of Various Crystallographic Directions}}  \tn
% Row 0
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 & {\bf{Young's Modulus (GPA)}} &  &  \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
{\bf{Metal}} & {\bf{{[}100{]}}} & {\bf{{[}110{]}}} & {\bf{{[}111{]}}} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{Aluminum} & 63.7 & 72.6 & 76.1 \tn 
% Row Count 6 (+ 2)
% Row 3
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\seqsplit{Copper} & 66.7 & 130.3 & 191.1 \tn 
% Row Count 7 (+ 1)
% Row 4
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Iron & 125.0 & 210.5 & 272.7 \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
\seqsplit{Tungsten} & 384.6 & 384.6 & 384.6 \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Plane From Miller Index}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1. Look at the plane, determine where the plane intersects the various axes (X, Y, and Z intercepts).} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{2. Verify that the origin does not intersect with the plane.} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{3. Take reciprocals.} \tn 
% Row Count 6 (+ 1)
% Row 3
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\mymulticolumn{1}{x{5.377cm}}{4. Clear fractions (if needed).} \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{5. Any negative numbers? Use the overbar.} \tn 
% Row Count 8 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{6. Enclose final result in parentheses.} \tn 
% Row Count 9 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Miller Index From Direction}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1. Using a right-handed coordinate system, find the coordinates of TWO points that lie in the direction.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{2. Subtract the "tail" point from the "head" point.} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{3. Clear fractions and/or reduce the results from step 2 to the lowest integers.} \tn 
% Row Count 7 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{4. Enclose the numbers with brackets {[} {]}. {\emph{If a negative number is present, use a bar over the number.}}} \tn 
% Row Count 10 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Equalibrium Distance}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580592240_binding energy.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Equilibrium separation occurs when the total energy is minimized and also when the net force is zero.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Coefficient of Thermal Expansion}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{→CTE = α \{\{nl\}\}→∆L/L₀=α(T₂+T₁)\{\{nl\}\}→Metal CTE is higher than ceramic CTE. \{\{nl\}→Heat = elongation \{\{nl\}\}→Cooling = contraction% Row Count 3 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Sources}}  \tn
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{{[}1{]} W.D. Callister, {\emph{Fundamentals of Materials Science and Engineering}}, 5\textasciicircum{}th\textasciicircum{} ed. New York, NY, USA: John Wiley \& Sons, 2001, pp. 33% Row Count 3 (+ 3)
} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.29735 cm} p{0.62655 cm} p{0.62655 cm} p{0.62655 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{4}{x{5.377cm}}{\bf\textcolor{white}{Common Crystal Structures}}  \tn
% Row 0
\SetRowColor{LightBackground}
 & {\bf{FCC}} & {\bf{BCC}} & {\bf{HCP}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
{\bf{C.N.}} & 12 & 8 & 12 \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
{\bf{total atoms/unit cell}} & 4 & 2 & 6 \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
{\bf{APF}} & 0.74 & 0.68 & 0.74 \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
{\bf{a}} & \seqsplit{2R√2} & \seqsplit{4R/√3} & td \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}----}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Atomic Packing Factor}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580595933_apf.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{V\_s = (4/3)(pi)(R\textasciicircum{}3\textasciicircum{})  \{\{nl\}\}V\_c = a\textasciicircum{}3\textasciicircum{}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Theoretical Density of a Metallic Solid}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580669046_materialdensity.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Direction From Miller Index}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1. Look at the index values and if any are larger than one, take out a factor such that none of the integer values are greater than one.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{2. For example, we have a Miller index of {[}2 1 1{]}, we can take out a factor of 2   2* {[}1 1/2 1/2{]}.} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{3. Essentially we have created a new point to plot with coordinates of x = 1, y = 1/2 and z = 1/2.} \tn 
% Row Count 7 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{4. Set up your unit cube and it is fine to place the origin at the lower left hand corner as shown below.} \tn 
% Row Count 10 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{5. Plot the new point (1, 1/2, 1/2) and connect the 'dots'.} \tn 
% Row Count 12 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Terms}}  \tn
% Row 0
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\mymulticolumn{2}{x{5.377cm}}{{\bf{Single crystal}}} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Anisotropic}}} \tn 
% Row Count 2 (+ 1)
% Row 2
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\mymulticolumn{2}{x{5.377cm}}{{\bf{Polycrystal}}} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Grain boundaries}}} \tn 
% Row Count 4 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Isotropic}}} \tn 
% Row Count 5 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{{\bf{Polymorphism}}} \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
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\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Stiffness}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580593369_stiffness.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Stiffness of a material is related to the slope of the F vs atomic separation curve (r). \{\{nl\}\}The stiffness (dF/da) of the material is known as the {\bf{Young's modulus, E.}}}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Dimensions and Angles}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580593996_points.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{{\bf{Lattice parameters}} are the dimensions a,b and c of the unit cell. \{\{nl\}\}{\bf{Interaxial angles}} are α, β, and γ.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{(top to bottom)BCC, FCC, \& HCP}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{5.377cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/marrgotthewise_1580594178_crystalstructures.png}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Miller Index From Plane}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{1. Draw the unit cube and right handed coordinate system \& be sure that the plane does not pass thru the point you've selected to be the origin.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{2. Take reciprocals} \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{3. Plot the plane.} \tn 
% Row Count 5 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{4. DO NOT CLEAR FRACTIONS.} \tn 
% Row Count 6 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Linear and Planar Density}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Linear Density} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Number of atoms per unit cell whose centers are aligned in a specific direction.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{LD=} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}\#atoms/unit length} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Planar Density} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}Number of atoms per unit area that are centered on the area of the plane.} \tn 
% Row Count 8 (+ 3)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{PD=} \tn 
\mymulticolumn{1}{x{5.377cm}}{\hspace*{6 px}\rule{2px}{6px}\hspace*{6 px}\#atoms centered on plane/area of plane} \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}