MATE 220 EXAM 1

Unit cell

The smallest repeating structural unit of a crystal lattice.

Lattice parameter $a$

The edge length (side length) of a cubic unit cell.

Number of atoms in a simple cubic (SC) unit cell

$$1\text{ atom per unit cell}$$

Number of atoms in a body-centered cubic (BCC) unit cell

$$2\text{ atoms per unit cell}$$

Number of atoms in a face-centered cubic (FCC) unit cell

$$4\text{ atoms per unit cell}$$

Number of atoms in a conventional hexagonal close-packed (HCP) unit cell

$$6\text{ atoms per conventional unit cell}$$

Atom contact direction in simple cubic (SC)

Along the cube edge.

Atom contact direction in body-centered cubic (BCC)

Along the body diagonal.

Atom contact direction in face-centered cubic (FCC)

Along the face diagonal.

Relationship between $a$ and $R$ in simple cubic (SC)

$$a = 2R$$

Relationship between $a$ and $R$ in BCC

$$a = \frac{4R}{\sqrt{3}}$$

Relationship between $a$ and $R$ in FCC

$$a = 2\sqrt{2}R$$

Formula to find $R$ from $a$ for BCC

$$R = \frac{a\sqrt{3}}{4}$$

Formula to find $R$ from $a$ for FCC

$$R = \frac{a}{2\sqrt{2}}$Metric

Atomic packing factor (APF) of simple cubic (SC)

$$0.52$$

Atomic packing factor (APF) of BCC

$$0.68$$

Atomic packing factor (APF) of FCC

$$0.74$$

Atomic packing factor (APF) of HCP

$$0.74$$

APF abbreviation

Atomic Packing Factor

Physical quantity measured by APF

The fraction of the unit-cell volume occupied by atoms.

General APF formula

$$\text{APF} = \frac{\text{volume occupied by atoms}}{\text{total unit-cell volume}}$$

Volume of a cubic unit cell ($V_c$)

$$V_c = a^3$$

Theoretical density equation ($\rho$)

$$\rho = \frac{nA}{V_c N_A}$$

Meaning of $n$ in the theoretical density equation

Number of atoms per unit cell.

Meaning of $A$ in the theoretical density equation

Atomic weight (molar mass), usually expressed in $g\,mol^{-1}$.

Meaning of $V_c$ in the theoretical density equation

Volume of the unit cell.

Avogadro's number ($N_A$)

$$N_A = 6.022 \times 10^{23}\,\text{atoms}\,mol^{-1}$$

Calculation path from atomic radius $R$ to theoretical density ($\rho$)

$$R \rightarrow a \rightarrow a^3 \rightarrow \rho$$

Calculation path from theoretical density ($\rho$) to atomic radius $R$

$$\rho \rightarrow a^3 \rightarrow a \rightarrow R$$

Formula for $a^3$ rearranged from the theoretical density equation

$$a^3 = \frac{nA}{\rho N_A}$$

Conversion factor from $\text{nm}$ to $\text{cm}$

$$1\,\text{nm} = 10^{-7}\,\text{cm}$$

Conversion factor from $\text{cm}$ to $\text{nm}$

$$1\,\text{cm} = 10^7\,\text{nm}$$

Cation

A positively charged ion formed when an atom loses electrons.

Anion

A negatively charged ion formed when an atom gains electrons.

Typical ionic charge of Group 1 elements

$$+1$$

Typical ionic charge of Group 2 elements

$$+2$$

Typical ionic charge of Group 13 elements

$$+3$$

Typical ionic charge of Group 15 elements

$$-3$$

Typical ionic charge of Group 16 elements

$$-2$$

Typical ionic charge of Group 17 elements

$$-1$$

Typical ionic charge of Group 18 elements

$$0$$

Electron change during cation formation

Electrons are lost.

Electron change during anion formation

Electrons are gained.

Electron configuration of argon ($\text{Ar}$)

$$1s^2\,2s^2\,2p^6\,3s^2\,3p^6$$

Electron configuration of $K^+$

$$1s^2\,2s^2\,2p^6\,3s^2\,3p^6 = [\text{Ar}]$$

Electron configuration of $Cl^-$

$$1s^2\,2s^2\,2p^6\,3s^2\,3p^6 = [\text{Ar}]$$

Reason $K^+$ and $Cl^-$ share the same electron configuration

Both species have $18\text{ electrons}$ and are isoelectronic with argon ($\text{Ar}$).

Ionic bonding

Electrostatic attraction between oppositely charged ions.

Covalent bonding

Bonding in which atoms share pairs of electrons.

Metallic bonding

Positive ion cores held together by delocalized valence electrons.

Equilibrium spacing ($r_0$)

The interatomic distance where attractive and repulsive forces balance.

Net force at equilibrium spacing ($r_0$)

$0$ (zero).

Status of individual attractive and repulsive forces at $r_0$

They are non-zero; they are equal in magnitude and opposite in direction.

Significance of a deeper interatomic potential-energy well

Stronger bonding and greater bond energy.

Vacancy defect

A normally occupied lattice site that is missing an atom.

Vacancy equation

$$N_v = N \exp\left(-\frac{Q_v}{kT}\right)$$

Meaning of $N_v$ in the vacancy equation

Number of vacancies per unit volume.

Meaning of $N$ in the vacancy equation

Number of atomic sites per unit volume.

Meaning of $Q_v$ in the vacancy equation

Energy required to form one vacancy.

Meaning of $k$ in the vacancy equation

Boltzmann's constant.

Temperature scale required for the vacancy equation

Kelvin ($\text{K}$).

Formula to convert Celsius to Kelvin

$$T(\text{K}) = T(^\circ\text{C}) + 273$$

Value of Boltzmann's constant $k$ when $Q_v$ is in $\text{eV}/\text{atom}$

$$k = 8.617 \times 10^{-5}\,\text{eV}/(\text{atom}\cdot\text{K})$$

Formula for atomic sites per unit volume ($N$)

$$N = \frac{\rho N_A}{A}$$

Effect of increasing temperature on vacancy concentration

Vacancy concentration increases exponentially.

Substitutional impurity

An impurity atom that replaces a host atom at a regular lattice site.

Interstitial impurity

A small impurity atom that occupies a space between host atoms.

Weight percent ($\text{wt}\%$) formula

$$\text{wt}\% = \left(\frac{\text{mass of component}}{\text{total mass of alloy}}\right) \times 100\%$$

Convenient assumed sample size for $\text{wt}\%$ problems

$$100\,\text{g}\text{ of alloy}$$

Masses in a $100\,\text{g}$ sample of $15\,\text{wt}\%\,\text{Ge}$ and $85\,\text{wt}\%\,\text{Si}$

$15\,\text{g}\,\text{Ge}$ and $85\,\text{g}\,\text{Si}$

Formula to calculate volume from mass and density

$$V = \frac{m}{\rho}$$

Formula to convert grams of an element to moles

$$\text{moles} = \frac{\text{mass}}{\text{atomic weight}}$Metric

Formula to convert moles to number of atoms

$$\text{atoms} = \text{moles} \times 6.022 \times 10^{23}$$

Octahedral interstitial site

An interstitial site coordinated by $6\text{ neighboring atoms}$.

Tetrahedral interstitial site

An interstitial site coordinated by $4\text{ neighboring atoms}$.