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<p>Solid materials are classified according to the uniformity with which atoms or ions are arranged with respect to one another.</p>
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<p>A crystalline materialis one in which the atoms are positioned in a recurring arrangement over large atomic distances. That means a long-range order exists.</p>
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<p>When solidification, the atoms will position themselves in a repetitive three-dimensional pattern. There, each atom is bonded to its nearest neighbor atoms. All metals, most ceramic materials, and certain polymers form crystalline structures under normal solidification conditions.</p>
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<p>This long-range atomic order is absent for materials that do not crystallize. Those are called <em>non-crystalline </em>or <em>amorphous </em>materials<strong>. </strong>Examples of such materials are glass and some complex metallic materials.</p>
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<p>Properties of crystalline solids depend on the Crystal Structureof the material.</p>
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<p>.</p>
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<h2 class="wp-block-heading"><strong>Lattice Space and Unit Cell</strong></h2>
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<p>When describing crystalline structures, atoms (or ions) are considered solid spheres with well-defined diameters. This is expressed as the <em>atomic hard sphere model. </em>In this model, spheres representing nearest-neighbor atoms touch one another.</p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-9.png" alt="" class="wp-image-2208"/></figure>
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<p>An example of the hard sphere model for the atomic arrangement found in some of the common elemental metals is displayed in the above Figure.</p>
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<p>In this particular case, all the atoms are identical.</p>
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<p>Sometimes the term <strong>lattice </strong>is used in the context of crystal structures. In that sense lattice implies a three-dimensional array of points coinciding with atom positions or sphere centers.</p>
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<h2 class="wp-block-heading"><strong>The unit cell</strong></h2>
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<p>The atomic order in crystalline solids shows that small groups of atoms form a repetitive pattern. Therefore, when describing crystal structures, it is easier to subdivide the structure into small repeat entities called unit cells.</p>
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<p>The unit cell is the basic structural unit or building block of the crystal structure. It establishes the crystal structure by geometry and the atom positions within the unit cell.</p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-10.png" alt="" class="wp-image-2209"/></figure>
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<h2 class="wp-block-heading"><strong>Types of Crystal structures of materials</strong></h2>
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<p>There are 14 types of crystal structures in various engineering materials as shown in the figure above.</p>
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<p>The main crystal structures among all of the above are: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>
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<ul><!-- wp:list-item -->
<li>The body-centered cubic structure (BCC).</li>
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<li>The face-centered cubic structure (FCC).</li>
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<li>Hexagonal close-packed structure (HCP).</li>
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<h3 class="wp-block-heading"><strong>The Face-Centered Cubic Crystal Structure</strong></h3>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-11.png" alt="" class="wp-image-2210"/></figure>
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<p>The crystal structure found in many metals has a unit cell of cubic geometry. In these unit cells, atoms are located at each corner and the centers of all the cube faces. It is termed as the <strong>face-centered cubic (FCC) </strong>crystal structure.</p>
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<p>Some of the common metals having this crystal structure are copper, aluminum, silver, and gold.</p>
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<p>The mechanical properties of FCC are:</p>
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<ul><!-- wp:list-item -->
<li>Low Young’s modulus,</li>
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<li>Low yield strength,</li>
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<li>Low hardness,</li>
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<li>Good ductility,</li>
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<li>High ability for forming.</li>
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<p>Figure <strong>(</strong><em>a</em><strong>) </strong>illustrates the hard sphere model for the FCC unit cell.</p>
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<p>In Figure <strong>(</strong><em>b</em><strong>) </strong>the atom centers are represented by small circles to provide a better perspective of atom positions.</p>
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<p>The collection of atoms in Figure <strong>(</strong><em>c</em><strong>) </strong>represents a section of crystal consisting of many FCC unit cells. These spheres or ion cores touch one another along a face diagonal.</p>
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<p>The cube edge length <strong><em>a </em></strong>and the atomic radius <strong><em>R </em></strong>are related through;                                                               </p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-12.png" alt="" class="wp-image-2211"/></figure>
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<p>In the FCC crystal structure, each corner atom is shared between eight unit cells. However, a face-centered atom belongs to only two. Therefore, 1/8<sup>th</sup> of each of the 8 corner atoms and 1/2 of each of the 6 face atoms, or a total of 4 whole atoms, is assigned to a given unit cell.</p>
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<p>Two other important characteristics of a crystal structure are the <strong>Coordination number </strong>and the <strong>Atomic Packing Factor (APF).</strong></p>
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<p>For metals, each atom has the same number of nearest-neighbor or touching atoms, which is the coordination number.</p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-13.png" alt="" class="wp-image-2212"/></figure>
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<p>For face-centered cubic, <strong>the coordination number is 12</strong>. This may be confirmed by examination of the Figure. Below the front face atom, there are 4 corner nearest-neighbor atoms surrounding it, 4 face atoms that are in contact from behind, and 4 other equivalent face atoms residing in the next unit cell to the front, which is not shown.</p>
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<p><strong>The APF </strong>is the sum of the sphere volumes of all atoms within a unit cell (assuming the atomic hard sphere model) divided by the unit cell volume.</p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-14.png" alt="" class="wp-image-2213"/></figure>
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<p>For the FCC structure, the<strong> atomic packing factor </strong>is<strong> 0.74</strong>, which is the maximum packing possible for spheres with the same diameter.</p>
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<h3 class="wp-block-heading"><strong>The Body-Centered Cubic Crystal Structure</strong></h3>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-15.png" alt="" class="wp-image-2214"/></figure>
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<p><em>A collection of spheres showing BCC crystal structure is shown in Figure <strong>(</strong>c<strong>)</strong>, whereas Figures (a)and (b)are diagrams of BCC unit cells with the atoms represented by hard sphere and reduced-sphere models, respectively</em></p>
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<p>Another common metallic crystal structure having a cubic unit cell with atoms located at all eight corners and a single atom at the cube center is shown above. This is called a <strong>Body-Centered Cubic (BCC) </strong>crystal structure.</p>
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<p>Chromium, iron, and tungsten are listed as having a BCC structure.</p>
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<p>Their mechanical properties are:</p>
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<ul><!-- wp:list-item -->
<li>High yield strength,</li>
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<li>High young modulus,</li>
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<li>High hardness,</li>
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<li>High tensile strength,</li>
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<li>Limited ability to form.</li>
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<p>Center and corner atoms touch one another along cube diagonals, and unit cell length <strong><em>a </em></strong>and atomic radius <strong><em>R </em></strong>are related through;  </p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-16.png" alt="" class="wp-image-2215"/></figure>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-17.png" alt="" class="wp-image-2216"/></figure>
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<p>There are 2 atoms associated with each BCC unit cell. It is the equivalent of 1 atom from the 8 corners, each shared among 8 unit cells, and the single center atom wholly contained within its cell.</p>
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<p><strong>The coordination number for the BCC crystal structure is 8. </strong>Each center atom has as nearest neighbors its eight corner atoms as shown in the figure above. Since the coordination number is less for BCC than FCC, so also is the <strong>atomic packing factor for BCC is lower, 0.68 </strong>versus 0.74.</p>
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<h3 class="wp-block-heading"><strong>Hexagonal Close-Packed Crystal Structure</strong></h3>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-18.png" alt="" class="wp-image-2217"/></figure>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-19.png" alt="" class="wp-image-2218"/></figure>
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<p>All metals do not have unit cells with cubic symmetry. The final common metallic crystal structure having a unit cell is hexagonal.</p>
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<p>Figure <strong>(</strong><em>a</em><strong>) </strong>shows a reduced-sphere unit cell for this structure. It is termed <strong>Hexagonal Close Packed (HCP). </strong>It is an assemblage of several HCP unit cells presented in Figure <strong>(</strong><em>b</em><strong>)</strong>.</p>
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<p>The top and bottom faces of the unit cell consist of 6 atoms. It forms standard hexagons and encircles a single atom in the center.</p>
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<p>Another plane that provides 3 additional atoms to the unit cell is located between the top and bottom planes. The atoms in this mid-plane have as nearest neighbors atoms in both of the adjacent two planes.</p>
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<p>The equivalent of 6 atoms is contained in each unit cell, 1/6<sup>th</sup> of each of the 12 top and bottom face corner atoms, 1/2 of each of the 2 center face atoms, and all 3 mid-plane interior atoms.</p>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-20.png" alt="" class="wp-image-2219"/></figure>
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<p>The<strong> coordination number </strong>and the<strong> atomic packing factor </strong>for the HCP crystal structure are the same as for FCC: 12 and 0.74, respectively, see figure above.</p>
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<p>Unit cell length <strong><em>a </em></strong>and atomic radius <strong><em>R </em></strong>are related through</p>
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<p><strong>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; a = 2R</strong></p>
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<p>The HCP metals include cadmium, magnesium, titanium, and zinc.</p>
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<p>Their properties are:</p>
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<ul><!-- wp:list-item -->
<li>brittle,</li>
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<li>Low yield strength,</li>
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<li>Inability to form.</li>
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<h2 class="wp-block-heading"><strong>Atomic Radii and Crystal Structures for 16 Metals</strong></h2>
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<figure class="wp-block-image size-full"><img src="https://civilengineeringlectures.com/wp-content/uploads/2024/07/image-21.png" alt="" class="wp-image-2220"/></figure>
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