Introduction

The previous article dealt with a uniform and error-free structure of the metals. However, such ideal crystals do not exist in reality or can only be achieved approximately on a very small scale under extreme expense (for example in so-called whiskers).

Real metals do not have a perfect lattice structure but show so-called crystallographic defects. At these defects, the real lattice deviates from the idealized perfect structure.

The figure above shows examples of different crystallographic defects, which can be subdivided depending on their impact on the surrounding structure. The following sections discuss these types of impurities more closely:

• 0-dimensional crystallographic defects (punctiform defects)
• 1-dimensional crystallographic defects (linear defects)
• 2-dimensional crystallographic defects (area-shaped defects)
• 3-dimensional crystallographic defects (volume-shaped defects)

Point defects

Point defects are also referred to as 0-dimensional defects and disturb the lattice only within a very limited (punctiform) region. The following sections discuss these types of point defects:

• vacancies
• Substitutional atoms
• Interstitial atoms

Vacancies

The vacancies are sites in the structure that are not occupied by an atom and thus remain vacant. The vacancy density usually increases exponentially with increasing temperature. The reason for this is the increased lattice vibration, which “tears” some atoms from their regular sites. If the atoms migrate by self-diffusion to the metal surface, they leave a vacancy defect in the material. However, this also means that vacancies in a metal can not be avoided in principle, since they are in a thermodynamic equilibrium. These types of vacancies are also called thermal vacancies.

Vacancies are unoccupied sites in the lattice structure. They are usually in thermodynamic equilibrium and can not be avoided!

At 1 quadrillion (10¹⁵) atoms comes at room temperature about one vacancy. Immediately below the melting temperature already one vacancy comes at 10,000 atoms (10⁴).

Vacancies can also be “trapped” in the material by rapid cooling from the hot state. However, these are not in thermodynamic equilibrium and partially heal over time due to diffusion processes (athermal vacancies).

Substitutional atom

However, there are not only vacancies interfering with a lattice structure. In addition, a metal is not free of foreign atoms. In the lattice structure, these foreign atoms can displace the actual metal atoms and thus occupy their lattice sites. Since the actual metal atom has been substituted by the foreign atom, it is also called a substitutional atom.

Substitutional atoms are foreign atoms which are occupying regular sites!

Interstitial atoms

A further possibility of the arrangement of foreign atoms consists in the storage on an interstitial space, e.g. in the cube center of a face-centered cubic unit cell (also referred to as octahedral site). Atoms that are not on regular sites but have interposed therebetween, are referred to as interstitial atoms.

Interstitial atoms do not occupy regular sites but interstitial sites.

Foreign atoms can migrate (diffuse) particularly well through the material due to lattice vibrations if there are many vacancies. There is plenty of room to get from one vacancy to another. Therefore, vacancies play an important role in diffusion processes of foreign atoms!

Line defects

Line defects are also called 1-dimensional crystallographic defects. They disturb the crystalline structure within a larger area compared to point defects. This only includes the so-called edge dislocations and the screw dislocations:

• edge dislocations
• screw dislocations

A edge dislocation is a lattice plane that ends in the metal structure without further connection. It can be thought of as an inserted atomic plane in the already existing structure. The “edge” of this inserted atomic plane is also referred to as a dislocation line or dislocation core and often symbolized in drawings with a “T”. Along these dislocation lines, the lattice structure is heavily distorted and has stresses. The dislocation line either forms a closed ring or exits the surface of the crystal (grain) or terminates at other defects.

A edge dislocation is an inserted atomic plane that is not continued in the lattice structure!

In contrast to a edge dislocation, a screw dislocation winds an atomic plane along the dislocation line through the crystal like the thread of a screw. In a crystal always combinations of both types of dislocation occur.

Dislocations occur during solidification of melts or due to stresses in the metal. But they are also introduced by plastic deformation (work hardening or strain hardening). Dislocations play a central role in deformation processes, since they are largely responsible for the good ductility of the metals.

Dislocations play a central role in the ductility of metals!

The dislocation density in a crystal is given as the total length of all dislocation lines per volume. Dislocation lines with a total length of about 1 km are found per square millimeter in a metal. Cold working increases the dislocation length per square millimeter to around 1 million kilometers! From a dislocation density of about 100 million kilometers per square millimeter, however, the material is so damaged that it is destroyed in principle.

Planar defects

Compared to linear crystallographic defects, planar defects disturb the lattice structure in a spatially larger area. The following defects fall into this categorization of the so-called 2-dimensional defects:

• high angle grain boundaries
• low angle grain boundaries
• phase boundaries
• stacking fault

Grain boundary

Grain boundaries delimit areas in a crystal within the lattice structure shows a uniform spatial orientation. These boundaries are structureless regions of thickness on the order of only 2 to 4 atomic distances. The uniformly aligned areas themselves are referred to as grains or as crystallites. The unit cells are identical for each grain, they only have a different spatial orientation (rotated, mirrored, etc.).

The grain structure is formed during the solidification of the molten metal, since a melt usually does not solidify starting from a single point but at many points at the same time (exception: single crystals or monocrystals!). At each of these solidification points (so-called nuclei), the lattice structure is formed with its own orientation. The growing grains collide after complete solidification of the melt and thus form the grain boundaries.

Grain refers to a crystal region with a uniform spatial lattice orientation. Different grains are separated by grain boundaries!

Small angle grain boundary

A deviation of the crystal orientation can also arise if several dislocations are superimposed. Since these errors change the lattice orientation by only a few degrees (<15 °), one also speaks of a low angle grain boundary. Due to the better delimitation, the grain boundaries described above are often referred to as high angle grain boundaries.

Twin grain boundary

A special type of grain boundary is the so-called twin boundary. In this case, the opposing lattice structures are just ordered in mirror image. Such a twin grain boundary has a high symmetry and thus low energy.

Twin boundaries are often seen under the microscope as straight lines, while “normal” grain boundaries are characterized by more curvy lines. Twin grain boundaries are very often formed in metals like brass, copper and austenite (γ-Fe).

While grain boundaries generally form incoherent interfaces, twin grain boundaries show a fully coherent interface (for the concept of coherence, see the next section).

Phase boundary

The so-called phase boundary forms another type of planar crystallographic defects.

A phase is a space with a uniform chemical structure!

A phase may be an accumulation of alloying elements in the host lattice of the metal. Thus a phase boundary spatially delimits two different chemical structures. Depending on how the structures of the different phases merge, a distinction is made between a coherent, partially coherent or incoherent phase boundary.

With a coherent phase boundary, the two structures merge into one another without any gaps. This is true if the two phases have a consistent structure and similar chemical properties.

However, if the phases differ somewhat in their properties, the lattice structures no longer completely merge into one another. Therefore dislocations must be present at regular intervals. One then speaks of a partially coherent phase boundary..

By contrast, with an incoherent phase boundary, neither the lattice structures nor the chemical properties of the two phases match. The structure is similar to a high angle grain boundary, but consists of two distinct phases. The phase boundaries are not distorted to the extent that is the case with high angle grain boundaries.

Stacking fault

Another planar defect is the so-called stacking fault. It is a locally different stacking sequence of otherwise periodically arranged planes. For example, the stacking sequence of the closest packed planes in the face-centered cubic lattice with normally ABCABC may locally have the sequence ABACAB. Such stacking faults can arise when a dislocation is split into two smaller dislocations.

Stacking faults are local deviations from the actual stacking sequence of the lattice structure!

Bulk defects

Bulk defects are also referred to as 3-dimensional defects and interfere with the lattice structure to a greater extent than the planar defects do:

• precipitations
• pores
• inclusions

The so-called precipitations are an accumulation of chemical compounds (phases) in the metal. In addition to precipitations, pores or other inclusions are among the 3-dimensional defects.

Definition of the packing density

The packing density is the ratio of atomic volume \(V_A\) within a unit cell to the total volume of the unit cell \(V_{U}\):

\begin{align}
\boxed{\text{PD}=\frac{V_A}{V_{U}} } \\[5px]
\end{align}

Depending on the grid structure, there is a certain packing density. The packing factors of the most important lattice types are to be derived in this article.

Body-centered cubic lattice

In order to determine the packing density for the body-centered cubic crystalline structure, the spatial diagonal \(e\) of the cube-shaped unit cell is considered. The three atoms lying on this diagonal are just touching each other. Thus, the spatial diagonal corresponds to 4 times the atomic radius \(r\). In a cube, the spatial diagonal is larger by a factor √3 than the edge of the cube \(a\). Thus, the atomic radius \(r\) depending on the cube edge \(a\) as follows:

\begin{align}
e=\sqrt{3} \cdot a = 4 \cdot r ~\Rightarrow ~ \underline{r= \frac{\sqrt{3}}{4} \cdot a}
\end{align}

In the unit cell, there is a whole atom in the middle and eight others on the cube corner, but only with one eighth each. In total the volume \(V_A\) of two atomic spheres is in the unit cell:

\begin{align}
\underline{V_A} =2 \cdot V_{sphere}
=2 \cdot \frac{4}{3} \pi \cdot r^3
=\frac{8}{3} \pi \cdot \left( \frac{\sqrt{3}}{4} \cdot a \right)^3
= \underline{ \frac{\sqrt{3}}{8} \pi \cdot a^3}
\end{align}

This atomic volume \(V_A\) can now put into relation to the unit cell volume \(V_{U}=a^3\) in order to obtain the packing density \(\text{PD}\) of the body-centered cubic lattice:

\begin{align}
\underline{\underline{\text{PD}}}= \frac{V_A}{V_{U}} =\frac{\frac{\sqrt{3}}{8} \pi \cdot a^3}{a^3}=\frac{\sqrt{3}}{8} \pi \approx \underline{\underline{0,68}}
\end{align}

Thus, the bcc-lattice has a packing facotr of 68 %.

Face-centered cubic and hexagonal closest packed lattice (fcc, hcp)

The packing density of the face-centered cubic lattice (fcc) can be determined in an analogous manner as for thebody-centered cubic structure. Three atomic spheres touch each other on the surface diagonal of the unit cell. This diagonal \(f\) thus corresponds to 4 times the atomic radius and equals to the value \(\sqrt{2} \cdot a\) (where \(a\) is the cube edge). Thus, the atomic radius \(r\) depends on the cube edge as follows:

\begin{align}
f=\sqrt{2} \cdot a = 4 \cdot r ~\Rightarrow ~ \underline{r= \frac{\sqrt{2}}{4} \cdot a}
\end{align}

In the fcc unit cell, there are six atoms on the cube surfaces, but only with one half of their sphere volume (in total 3 whole atomic volumes). In addition, there are eight other atoms on the cube corner, but only with one eighth each (1 whole atomic volume). In total the volume of four atomic spheres is in the unit cell with the atomic volume \(V_A\):

\begin{align}
\underline{V_A} =4 \cdot V_{sphere}
=4 \cdot \frac{4}{3} \pi \cdot r^3
= \frac{16}{3} \pi \cdot \left( \frac{\sqrt{2}}{4} \cdot a \right)^3
= \underline{ \frac{\sqrt{2}}{6} \pi \cdot a^3}
\end{align}

This atomic volume \(V_A\) can now put into relation to the unit cell volume \(V_{U}=a^3\). The packing density \(\text{PD}\) of the face-centered cubic grid is calculated as follows:

\begin{align}
\underline{\underline{\text{PD}}}= \frac{V_A}{V_{U}} =\frac{\frac{\sqrt{2}}{6} \pi \cdot a^3}{a^3}=\frac{\sqrt{2}}{6} \pi \approx \underline{\underline{0,74}}
\end{align}

The fcc-lattice thus has an packing factor of 74 %. However, there is no need to differentiate between the fcc-structure and the hexagonal closest packed crystal (hcp), since in both cases they built up by densest packed atomic planes (for further information see post on Important lattice types). The packing density in the hcp-lattice thus also has the maximum possible value of 74 %.

Introduction

The article on lattice structure of metals explains why the atoms in a metal are arranged with a certain regularity and thus form a lattice structure. Only in rare cases does a simple cubic crystalline structure appear, as shown in the animation below.

Of much greater technical importance are variations of this simple cubic lattice structure. These include the body-centered cubic lattice (bcc) and the face-centered cubic lattice (fcc), as well as the hexagonal closest packed lattice (hcp).

The difference in the lattice structure has in particular strong effects on the ductility of the respective metals! While the fcc-lattice shows the highest ductility and the hcp-lattice the lowest deformability, the bcc-lattice has a malleability that lies between the other structures.

In order to be able to understand these differences in ductility, it is first necessary to understand the structure of the different types of lattice. For this reason, this article discusses the realization of the three most important lattice types.

Body-centered cubic lattice structure

A variation of the primitive cubic lattice is the so-called body-centered cubic lattice structure (bcc). In this atomic structure, the individual planes of atoms (lattice plane) lie in the gaps of the respective lower planes of atoms. As with the primitive cubic lattice, the unit cell also has the basic shape of a cube. In addition to the corner atoms, which each sit in the corners of the cube-shaped unit cell, there is now another atom centered in the middle of the cube (“body”).

The body-centered cubic lattice structure (bcc) contains another atom in the middle of the cube-shaped unit cell next to the corner atoms!

Typical representatives of suchbody-centered cubic structures include the metals iron, chromium, molybdenum, vanadium and tungsten.

In the body-centered cubic crystal, one atom is in direct contact with eight surrounding atoms. This number of direct neighbor atoms is also called coordination number. The coordination number in the bcc lattice is consequently 8. In the simple cubic lattice, on the other hand, a coordination number of 6 is obtained.

The coordination number describes the number of directly adjacent atoms!

For the representation of unit cells, it makes sense to draw the atoms downsized, even if they “touch” in reality. In order to clarify the spatial arrangement of the atoms, it is also common to indicate the basic form of the unit cell by dashes. Note, that these lines do not represent binding forces!

If the atoms are realistically regarded as touching spheres, the so-called packing density of an atomic structure can be determined (also referred to as packing factor). The packing density indicates what percentage of the unit cell is filled with atoms. For the body-centered cubic lattice, a packing density of 0.68 can be determined in this way. This means that 68 % of the total lattice volume (or unit cell) is occupied by atoms. The remaining 32 % are attributable to the “gaps” between the atoms. The derivation of this packing density is shown in a separate post.

The packing density (or packing factor) is the ratio of the atomic volume to the total volume of the unit cell!

Hexagonal closest packed lattice structure

In the body-centered cubic crystalline structure, the superposed atomic planes have gaps so that the respectively underlying plane fits exactly into the gaps of the upper one. The atomic levels are thus not maximally packed. The situation is different with the hexagonal lattice structure.

In the hexagonal structure, the atoms of a lattice plane are densely packed together. A single atom in this lattice plane A (shown in red) is always touched by a total of six neighboring atoms. The overlying atomic plane B (shown in blue) is constructed identically in principle. However, it is just so shifted that the atoms of this plane fit exactly in the indentation of the underlying plane. An atom of the upper lattice plane thus sits in the gap formed by three atoms of the lower layer. This lattice plane sequence AB (also called stacking sequence) is now repeated continuously (ABAB …).

Stacking sequence is the order of the permanently repeating atom layers!

The metals titanium, cobalt, zinc and magnesium typically occur in such a hexagonal lattice structure with maximum packing density. An atom in the hexagonal closest packed lattice is surrounded by 12 direct neighboring atoms. The coordination number in this type of lattice is therefore 12.

The unit cell of a hexagonal closest packed lattice can be reduced to a hexagonal base area. In the middle of the unit cell are three other atoms that sit in the resulting atomic gaps of the base and top surfaces. Since the individual atomic levels that make up the structure are maximally packed, we also speak of closest packed atomic planes. Consequently, the lattice is referred to as a hexagonal closest packed lattice structure (hcp).

In the hexagonal closest packed lattice, two atomic layers with a hexagonal base form the shape of the unit cell. In the middle of the base areas is another atom, as well as a total of three other atoms in the middle of the unit cell!

The packing density of the hcp-lattice corresponds to the maximum possible packing density of 74 %. This maximum packing density is generally applicable to any spherical body.

For spherical bodies a packing density of 74 % can be reached at maximum!

The addition of “closest packed” in the name of the hexagonal lattice already implies that there are also other hexagonal lattice structures. For example, graphite also has a hexagonal lattice structure (hex), but this is not closest packed as in the hcp-lattice.

While in a atomic plane of the hcp-lattice an atom is directly surrounded by 6 other atoms, in the hexagonal lattice of graphite there are only three neighboring atoms. The individual atomic layers are arranged with an offset and can be relatively easily shifted against each other. The shifted atomic planes can thereby be detached relatively easily. For example, this process uses a graphite pencil when drawing on a sheet of paper.

Face-centered cubic lattice structure

As well as the hcp-lattice, the face-centered cubic lattice structure (fcc) has maximum packed atomic planes. However, the stacking sequence is different. The second lattice plane is initially stacked as in the hcp-lattice and sits in the gaps of the underlying layer. In contrast to the hcp-structure, however, the third atomic layer is located in the “free” gaps. Accordingly, the stacking sequence of the closest packed planes results in the series ABCABC …

Typical metals with a face-centered cubic lattice structure are aluminum, lead, copper and nickel. As in the hcp-lattice, an atom in the fcc-lattice has 12 neighboring atoms. The coordination number is therefore also 12.

In a face-centered cubic lattice, the basic shape of the unit cell is not hexagonal, as might be suggested by its similar structure to the hexagonal lattice. The unit cell actually has the shape of a cube and is therefore one of the cubic crystal systems! Regarding the horizontally aligned closest packed planes, the unit cell stands on the corner of the cube. The anomation above shows that besides the atoms in the corner of the cube, other atoms are centered on the cube surfaces. Therefore, this type of lattice is called face-centered.

In the face-centered cubic lattice (fcc), there is another atom next to the corner atoms on the surfaces of the cubic unit cell!

Note that the fcc and hcp lattice differ only in the stacking sequence of the closest packed planes (ABCABC … or ABAB …)! The packing density in the face-centered cubic lattice is therefore identical to the maximum possible packing density of 74 %.

However, the seemingly small difference in the stacking sequence between the fcc-lattice and the hcp-lattice has enormous effects on the mechanical properties, especially on the ductility of these structures. The reason for this will be explained in more detail in the ductility.

Introduction

Metals play an important role in mechanical engineering. Compared to other materials, they can withstand relatively high loads, but still have sufficient plasticity (ductility) not to break immediately under stress. In addition, metals have a very good thermal conductivity and electrical conductivity, which gives these materials a wide range of applications. Due to this special position of metals in mechanical engineering, their atomic structure is discussed in more detail below.

Metallic bonding

The structure of metals has already been explained shortly in the chapter on metal bonding. The metal atoms release all their outer electrons and thus reach the noble gas configuration. In this way, positively charged metal ions remain (cations). Due to the released electrons, a kind of “gaseous” state of the free electrons is formed around the positive ions. This is also known as electron gas.

Two forces act on the cations. On the one hand, there are attractive forces between the positively charged ions and the negatively charged electron gas. On the other hand, there are repulsive forces between the cations themselves due to their identical charges. The electron gas thus tries to approach the various cations, while the cations themselves repel each other.

Consequently, there is an state of equilibrium in which both forces compensate each other. In this stable position the cations keep a fixed distance. This ultimately leads to a regular build-up of the atomic structure. Such regularity in atomic structure is also referred to as crystal structure or lattice structure. The substance itself is called crystalline. However, the grid structure can take different forms, which will be discussed in more detail in the article Important types of lattice structures.

A crystalline structure is a typical feature of metals. Substances lacking such a crystalline structure are called amorphous. A typical example of an amorphous material is glass, but many plastics also have an irregular atomic structure.

A regularity in the atomic structure of a substance is called crystal structure or lattice structure. Materials lacking such a crystalline structure are called amorphous!

The animation above shows the forces acting between two ions. At a large distance, the attractive force of the electron gas predominates, so that the ions initially approach each other. If the distance between the two ions is reduced, the repulsive force increases more and more and rises disproportionately compared to the attracting effect. If the approach is too close, however, the repulsive force is greater, so that the distance increases again. Finally an equilibrium state is reached, in which both forces are equal and a resultant force disappears.

Of course, a metal consists not just of two atoms, but of a multitude of particles. The individual particles then perceive not only the repulsive forces of their immediately adjacent particles but also the forces of particles located further away. The whole particles thus influence each other in their electrostatic effect. Therefore, the atomic distance of two neighboring ions will not correspond exactly to the value obtained in the figure above with only two atoms.

The distance between two ions is a characteristic value for the corresponding metal and is called a lattice constant. The lattice constant is on the order of about 250 to 500 pm. A massive iron cube with an edge length of 25 mm contains about a quadrillion (10²⁴) atoms! This corresponds approximately to the number of 1-litre-milk-bags that would be needed to fill the entire volume of the earth with milk!

Unit cell

In a crystalline structure, a smallest unit can always be found, that repeats itself at regular intervals. Such an elementary unit is also called a unit cell.

A unit cell is the smallest repetitive unit of a lattice structure!

In the simplest case, the unit cell has the shape of a cube with the respective atoms in the corners. This structure is repeated over and over again throughout the metal. Such a simple cubic lattice structure is also referred to as a primitive cubic lattice structure.

However, there are only a few substances that have such a simple cubic crystalline structure (e.g. the highly radioactive polonium). From a technical point of view, variations of this type of lattice are much more common. These include the so-called body-centered cubic lattice (bcc) , the face-centered cubic lattice (fcc) , and the hexagonal closest packed lattice (hcp) .

Especially with regard to ductility, metals show different properties depending on the type of lattice structure. In order to understand this, more detailed knowledge of the individual lattice types is required. Therefore, the three lattice types mentioned above will be explained in more detail in the following section.