Last modified on 8 May 2014, at 13:46

Introduction to Inorganic Chemistry/Metals and Alloys: Structure, Bonding, Electronic and Magnetic Properties

Chapter 6: Metals and Alloys: Structure, Bonding, Electronic and Magnetic PropertiesEdit

  6.1 Unit cells and crystal structuresEdit

Academ Repetitive patterns by tile groups in a tiling.svg

Crystals can be thought of as repeating patterns, much like wallpaper or bathroom tiles, but in three dimensions. The fundamental repeating unit of the crystal is called the unit cell. It is a three dimensional shape that can be repeated over and over by unit translations to fill space (and leave minimal gaps) in the structure. Some possible unit cells are shown in the tiling pattern at the right, along with arrows that indicate unit translation vectors. In three dimensions, the hexagonal or rhombic unit cells of this pattern would be replaced by three dimensional boxes that would stack together to fill all space. As shown in the figure, the origin of the unit cell is arbitrary. The same set of boxes will fill all space no matter where we define the origin of the lattice. We will see that pure metals typically have very simple crystal structures with cubic or hexagonal unit cells. However the crystal structures of alloys can be quite complicated.

When considering the crystal structures of metals and alloys, it is not sufficient to think of each atom as an isolated system like is done for s- and p-block compounds. Instead, think of the entire metal as a network of atoms surrounded by a sea of delocalized electrons. The electrons are delocalized because there are not enough of them to fill each atoms orbitals; they must be shared by all of the atoms. A transition metal bonds using its d-orbitals, allowing it to make many bonds to adjacent atoms. The atoms in a metal lattice arrange themselves in a certain pattern which can be represented as a 3D box structure known as the unit cell which repeats across the entire metal.

Simple Cubic Body Centered Cubic Face Centered Cubic
Simple cubic.svg
Lattice body centered cubic.svg
Lattice face centered cubic.svg
1 atom/cell 2 atoms/cell 4 atoms/cell

Metal atoms can be thought of as spheres, and therefore will not be 100 % efficient in packing, the same way a stack of cannonballs has some empty spaces between the balls. Different unit cells have different packing efficiencies. The number of atoms that is included in the unit cell only includes the fractions of atoms inside of the box. Atoms on the corners of the unit cell count as ⅛ of an atom, atoms on a face count as ½, an atom in the center counts as a full atom. Using this, let's calculate the number of atoms in a simple cubic unit cell, a face centered cubic (fcc) unit cell, and a body centered cubic (bcc) unit cell.

Simple Cubic:

8 corner atoms × ⅛ = 1 atom/cell

Body Centered Cubic:

(8 corner atoms × ⅛) + (1 center atom × 1)= 2 atoms/cell

Face Centered Cubic:

(8 corner atoms × ⅛)+ (6 face atoms × ½)= 4 atoms/cell

The most efficient packing of metal atoms (fcc) in a metal fills about 74% of space. A bcc metal fills 68% of space. A simple cubic cell fills 52% of space. These can be calculated by dividing the volume of the atoms in the cell by the volume of the cell itself. The volume of the atoms in the cell is equal to the number of atoms/cell times the volume of a sphere, (4/3)πr3. The volume of the cubic cells is found by cubing the side length. As an example, let's calculate the packing efficiency of a simple cubic unit cell. As we saw earlier in the section, a simple cubic unit cell contains one atom. The side length of the simple cubic unit cell is 2r, since the centers of each atom occupy the corners of the unit cell.

Packing efficiency=[(1 atom)×(4/3)πr3]/[(2r)3]=0.523

The same method can be applied to bcc and fcc structures for homonuclear structures.

  6.2 Bravais latticesEdit

Crystal lattices can be classified by their translational and rotational symmetry. In three-dimensional crytals, these symmetry operations yield 14 distinct lattice types which are called Bravais lattices. In these lattice diagrams (shown below) the dots represent lattice points, which are places where the whole structure repeats by translation. For example, in the body-centered cubic (bcc) structure of sodium metal, which is discussed below, we put one atom at the corner lattice points and another in the center of the unit cell. In the NaCl structure, which is discussed in Chapter 8, we place one NaCl formula unit on each lattice point in the face-centered cubic (fcc) lattice. That is, one atom (Na or Cl) would be placed on the lattice point and the other one would be placed halfway between. Similarly, in the cubic diamond structure, we place one C2 unit around each lattice point in the fcc lattice.


The fourteen Bravais lattices fall into seven crystal systems that are defined by their rotational symmetry. In the lowest symmetry system (triclinic), there is no rotational symmetry. This results in a unit cell in which none of the edges are constrained to have equal lengths, and none of the angles are 90º. In the monoclinic system, there is one two-fold rotation axis (by convention, the b-axis), which constrains two of the angles to be 90º. In the orthorhombic system, there are three mutually perpendicular two-fold axes along the three unit cell directions. Orthorhombic unit cells have three unequal unit cell edges that are mutually perpendicular. Tetragonal unit cells have a four-fold rotation axis which constrains all the angles to be 90º and makes the a and b axes equivalent. The rhombohedral system has a three-fold axis, which constrains all the unit cell edges and angles to be equal, and the hexagonal system has a six-fold axis, which constrains the a and b lattice dimensions to be equal and the angle between them to be 120º. The cubic system has a three-fold axis along the body diagonal of the cube, as well as two-fold axes along the three perpendicular unit cell directions. In the cubic system, all unit cell edges are equal and the angles between them are 90º.


The translational symmetry of the Bravais lattices (the lattice centerings) are classified as follows:

  • Primitive (P): lattice points on the cell corners only.
  • Body (I): one additional lattice point at the center of the cell.
  • Face (F): one additional lattice point at the center of each of the faces of the cell.
  • Base (A, B or C): one additional lattice point at the center of each of one pair of the cell faces.


Not all combinations of the crystal systems and lattice centerings are unique. There are in total 7 × 6 = 42 combinations, but it can be shown that several of these are in fact equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A- or B-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, shown in the table below.


When the fourteen Bravais lattices are combined with the 32 crystallographic point groups, we obtain the 230 space groups. These space groups describe all the combinations of symmetry operations that can exist in unit cells in three dimensions. For two-dimensional lattices there are only 17 possible plane groups, which are also known as wallpaper groups.

The 7 lattice systems The 14 Bravais lattices
Triclinic P
Triclinic
Monoclinic P C
Monoclinic, simple Monoclinic, centred
Orthorhombic P C I F
Orthohombic, simple Orthohombic, base-centred Orthohombic, body-centred Orthohombic, face-centred
Tetragonal P I
Tetragonal, simple Tetragonal, body-centred
Rhombohedral P
Rhombohedral
Hexagonal P
Hexagonal
Cubic P (pcc) I (bcc) F (fcc)
Cubic, simple Cubic, body-centred Cubic, face-centred


  6.3 Crystal structures of metalsEdit

Body-centered cubic
hcp (left) and fcc (right) close-packing of spheres
Closepacked layer.jpg

The Crystalline Nature of Metals. All metallic elements (except Cs, Ga, and Hg) are crystalline solids at room temperature. Like ionic solids, metals and alloys have a very strong tendency to crystallize, whether they are made by thermal processing or by other techniques such as solution reduction or electroplating. Metals crystallize readily and it is difficult to form a glassy metal even with very rapid cooling. Molten metals have low viscosity, and the identical (essentially spherical) atoms can pack into a crystal very easily. Glassy metals can be made, however, by rapidly cooling alloys, particularly if the constituent atoms have different sizes. The different atoms cannot pack in a simple unit cell, sometimes making crystallization slow enough to form a glass.


Structure and bonding in metals. Most metals and alloys crystallize in one of three very common structures: body-centered cubic (bcc), hexagonal close packed (hcp), or cubic close packed (ccp, also called face centered cubic, fcc). In all three structures the coordination number of the metal atoms (i.e., the number of equidistant nearest neighbors) is rather high: 8 for bcc, and 12 for hcp and ccp. We can contrast this with the low coordination numbers (i.e., low valences - like 2 for O, 3 for N, or 4 for C) found in nonmetals. In the bcc structure, the nearest neighbors are at the corners of a cube surrounding the metal atom in the center. In the hcp and ccp structures, the atoms pack like stacked cannonballs or billiard balls, in layers with a six-coordinate arrangement. Each atom also has six more nearest neighbors from layers above and below. The stacking sequence is ABCABC... in the ccp lattice and ABAB... in hcp. In both cases, it can be shown that the spheres fill 74% of the volume of the lattice. This is the highest volume fraction that can be filled with a lattice of equal spheres.


Periodic trends. Remember where we find the metallic elements in the periodic table - everywhere except the upper right corner. This means that as we go down a group in the p-block (let's say, group IVA, the carbon group, or group VA, the nitrogen group), the properties of the elements gradually change from nonmetals to metalloids to metals. The carbon group nicely illustrates the transition. Starting at the top, the element carbon has two stable allotropes - graphite and diamond. In each one, the valence of carbon atoms is exactly satisfied by making four electron pair bonds to neighboring atoms. In graphite, each carbon has three nearest neighbors, and so there are two single bonds and one double bond. In diamond, there are four nearest neighbors situated at the vertices of a tetrahedron, and so there is a single bond to each one.


The two elements right under carbon (silicon and germanium) in the periodic table also have the diamond structure (recall that these elements cannot make double bonds to themselves easily, so there is no graphite allotrope for Si or Ge). While diamond is a good insulator, both silicon and germanium are semiconductors (i.e., metalloids). Mechanically, they are hard like diamond. Like carbon, each atom of Si and Ge satisfies its valence of four by making single bonds to four nearest neighbors.


The next element under germanium is tin (Sn). Tin has two allotropes, one with the diamond structure, and one with a slightly distorted bcc structure. The latter has metallic properties (metallic luster, malleability), and conductivity about 109 times higher than Si. Finally, lead (Pb), the element under Sn, has the ccp structure, and also is metallic. Note the trends in coordination number and conducting properties:

Carbon bonding.jpg
Element Structure Coord. no. Conductivity
C graphite, diamond 3, 4 semimetal, insulator
Si diamond 4 semiconductor
Ge diamond 4 semiconductor
Sn diamond, distorted bcc 4, 8 semiconductor, metal
Pb ccp 12 metal

The elements C, Si, and Ge obey the octet rule, and we can easily identify the electron pair bonds in their structures. Sn and Pb, on the other hand, adopt structures with high coordination numbers. They do not have enough valence electrons to make electron pair bonds to each neighbor (this is a common feature of metals). What happens in this case is that the valence electrons become "smeared out" or delocalized over all the atoms in the crystal. It is best to think of the bonding in metals as a crystalline arrangement of positively charged cores with a "sea" of shared valence electrons gluing the structure together. Because the electrons are not localized in any particular bond between atoms, they can move in an electric field, which is why metals conduct electricity well. Another way to describe the bonding in metals is nondirectional. That is, an atom's nearest neighbors surround it in every direction, rather than in a few particular directions (like at the corners of a tetrahedron, as we found for diamond). Nonmetals (insulators and semiconductors), on the other hand, have directional bonding. Because the bonding is non-directional and coordination numbers are high, it is relatively easy to deform the coordination sphere (i.e., break or stretch bonds) than it is in the case of a nonmetal. This is why elements like Pb are much more malleable than C, Si, or Ge.

  6.4 Bonding in metalsEdit

Due to the electronic nature of the metal atoms in one unit cell, it is not feasible to form typical two-electron bonds. Sodium, for example, has only one valence electron. There are two atoms per unit cell in the body-centered cubic crystal structure of sodium metal. Each centered sodium atom forms a bond with each of the eight corners of the unit cell, resulting in a bond order of one-eighth.

The bonds made by the central atom in a bcc unit cell are clearly shown in this configuration. A metal exhibiting this crystal structure has a bond order of one-eighth.

A more accurate way to describe the bonding in metals is through band theory. In general, X number of atoms will generate X number of molecular orbitals with X-1 possible nodes. Previously, we showed that the energy versus internuclear distance graph for a two hydrogen atom system has a low energy level and a high energy level corresponding to the bonding molecular orbital and antibonding molecular orbital, respectively. These two energy levels were very distinct from each other, and the two electrons of this two atom system will energetically prefer the lower energy level. If more atoms are introduced to the system, there will be a number of additional levels between the lowest and highest energy levels. In band theory, the atom chain is extrapolated to an indefinitely large number so that the different combinations of bonding and anti-bonding orbitals create "bands" of possible energy states for the metal: this approach is called tight-binding approximation. The number of atoms in these situations is so large, that the energies can be thought of as a continuum rather than a series of distinct levels. A metal will only partially fill this band, as there are far fewer valence electrons than there are energy states to fill.

Lower energy bands fill with electrons first. The conductivity of metals is related to the band gap between the valence band and conduction band: the smaller the gap between bands, the better conductive properties the metal displays.

The degree to which bands fill also affects the conductive property of metals. The lowest filled energy band in an energy band diagram is referred to as the valence band, which contains the valence electrons of the atoms in the metal. Above that band is the conduction band, which is the band that excited electrons will begin to fill if the appropriate amount of energy is applied. These electrons are charge-carriers, and are capable of traveling freely through the metal. The region between the top of the valence band and the bottom of the conduction band is referred to as the band gap, and the energy between the two bands is typically referred to as Eg. The value of Eg varies from species to species and is directly affected by orbital overlap. A strong overlap of s and p orbitals will reduce the band gap and vice versa. A small band gap between the valence and conduction bands will allow electrons to much more easily leave their orbit about the nucleus and freely move through the metal. Therefore, a metal with these properties is a conductor. A small band gap typically corresponds to a semiconductor, and a large band gap typically corresponds to an insulator, where the region between the two bands is designated as the forbidden region for electrons.

The ability of metals to conduct electricity can be understood by looking at the density of states, which describes the number of orbitals per unit energy. When no electric field is applied, the density of states (D.O.S.) is the same in any direction and the band is filled halfway to the Fermi energy level. When an electric field is applied, the D.O.S. increases in energy in one direction and is lowered in the other. As a result, the electrons fill the orbitals in the direction opposite the electric field (since electrons are attracted to the positive field) more than the orbitals in the direction of the electric field. They still only fill up to the Fermi level. As a result, the electrons begin to move in the direction opposite the electric field, causing a current to flow in the direction of the E-field (remember that current is the movement of positive charge, so it is opposite the direction of the movement of electrons). When an electric field is applied to a conductor, electrons are moving extremely fast due to the force applied to them by the field. A typical value for the speed of electrons in a metal is approximately 3.0 x 106 m/s, but the drift speed is approximately 0.01 m/s. The slow drift speed across the surface of a metal is attributable to the lack of a free path for the electrons. Other particles in the metal "scatter" electrons, causing a change in direction. This cumulative effect eventually results in electrons traveling against the direction of the electric field across the metal.

Why don't insulators conduct electricity? The band is filled for an insulator, so even if the energy of the density of states in increased in one direction and lowered the energy in the other, the electrons would not move to the lower energy states because all of them are already full.

  6.5 Conduction in metalsEdit

MetalEvsDOS.png

In metals, the valence electrons are in molecular orbitals that extend over the entire crystal lattice. As we have learned, metallic crystals are typically micron size, meaning that the spatial extent of the orbitals is very large compared to the size of the atoms or the unit cell. The diagram at the left shows a generic plot of electron energy vs. density of states for a metal such as Na, Cu, or Ag. In these cases, there are N orbitals for N electrons, and each orbital can accomodate two electrons. Therefore the Fermi level, which corresponds to the energy of the highest occupied MO at zero temperature, is somewhere in the middle of the band of orbitals. The energy level spacing between orbitals is very small compared to the thermal energy kT, so we can think of the orbitals as forming a continuous band.


Classically, if the electrons in this band are free to be thermally excited, we would expect them to have a specific heat of 3R per mole of electrons. However, experimentally we observe that Cp ≈ 0.02 R per mole. This suggests that only about 1% of the electrons in the metal can be thermally excited at room temperature. However, essentially all of the valence electrons are free to move in the crystal and contribute to electrical conduction. To understand this apparent paradox, we need to recall that the electrons exist in quantized energy levels.


Because of quantization, electrons in metals have a Fermi-Dirac distribution of energies. In this distribution, most of the electrons are spin-paired, although the individual electrons in these pairs can be quite far apart since the orbitals extend over the entire crystal. A relatively small number of electrons at the top of the Fermi sea are unpaired by thermal excitation. This is the origin of the Pauli paramagnetism of metals.

DOS for an insulator or semiconductor

The number of electrons in an element or compound can also determine their conductivity. The energy to DOS picture to the left shows how an element or compound with filled orbitals will have no change when an electric field is applied. The splitting of the orbitals does not stabilize the electrons. This is the case with semiconductors and insulators. However, when an electric field is applied to an element with a half filled orbital shell, the shift in the DOS stabilizes the unfilled electrons. This property makes the element or compound metallic and an conductor of electricity. This explains the conductivity of Na (3s1), but what about Mg(3s2)? The promotional energy (+267 kJ) is more than offset by the bonding energy (-410kJ). The ΔEvaporization, or the cohesive energy of metals, is determined by the bonding energy - the promotion energy.

The cohesive energy diagram of Mg

The situation is more complex for Re, Os, Ir, and Pt. These can create d-"acid-base" alloys. For instance, ZrC (a very stable carbide) and Pt form an even more stable alloys (Zrpt3) by donating their d electrons in a very exothermic reaction. HF and Pt also undergo this reaction. Hf is the "acid" with an electron configuration of 6s25d2, while Pt is the "base" with the electron configuration of 6s15d9 to create a stable "salt" product with a filled 5d electron configuration. Some d-acid/base alloys are of particular interest for use in catalysis. For instance Sc-Pt alloys are good catalysts for oxygen reduction in fuel cells. Even though Sc is an active metal, it is stabilized by the aqueous acid environment of the fuel cell.






How fast are electrons traveling in a typical metal? Because of the bell shape of the E vs. DOS curve, most of the electrons have E ≈ EF. At the midpoint energy (EF) of the band, the MO's have one node for every two atoms. We can calculate the de Broglie wavelength as twice the distance between nodes and thus:

λ = 4a

where a is the interatomic spacing. Since a typical value of a is about 2 Å, we obtain the de Broglie wavelength λ ≈ 8 Å.

Using the de Broglie relation h = λ/p, we can write:

p = λ/h = mevF,

where me is the mass of the electron and vF is the velocity of electrons with energy EF.

Solving for vF = h/meλ we obtain vF = (6.62 x 10-34 J s)/(9.1 x 10-31 kg)(8 x 10-10 m) = 1.0 x 106 m/s

Experimental values of vF are 1.07 x 106 and 1.39 x 106 m/s for Na and Ag, respectively, so our approximations are pretty good.

How fast are electrons moving in metals? Really, really fast!! 1,000,000 meters per second! This is about 1/300 the speed of light, and about 3000 times the speed of sound (3 x 102 m/s).


However, the drift velocity of electrons in metals is quite slow, on the order of 0.0001 m/s, or .01 cm/s. You can easily outrun an electron drifting in a metal, even if you have been drinking all night and have been personally reduced to a very slow crawl.

Escattering.png

In order to understand the great disparity between the Fermi velocity and the drift velocity of electrons in metals, we need to consider a picture for the scattering of electrons, and their acceleration in an electric field, as shown at the left. If we apply a voltage across a metal (e.g., a metal wire), the electrons are subjected to an electric field E, which is the voltage divided by the length of the wire. This electric field exerts a force on the electron, causing it to accelerate. However, the electron is frequently scattered, mostly by phonons (lattice vibrations). Each time the electron is scattered its acceleration starts all over again. The time between scattering events is τ and the distance the electrons travel between scattering events is the mean free path, λ. (Note that this is NOT the same λ as the de Broglie wavelength, they just unfortunately have the same symbol!)


We can write the force on the electron as:

F = eE = mea = mevdrift
Beavermeanfreepath.png

In this equation, a is the acceleration in the electric field, me is the mass, and vdrift is the drift velocity of the electron.


Experimentally, the mean free path is typically obtained by measuring the scattering time. For an electron in Cu metal at 300 K, the scattering time τ is about 2 x 10-14 s. From this we can calculate the mean free path as:

λ = vavgτ ≈ vFτ = (1 x 106 m/s)(2 x 10-14 s) = 40 nm

The mean free path (40 nm = 400 Å) is quite long compared to the interatomic spacing (2 Å). To put it in perspective, if the interatomic spacing was scaled to the length of a football (0.3 m), the mean free path would be over half the length of the football field (60 m). Thus an electron travels a fairly long way between scattering events and scarcely notices the atomic structure of the metal in which it is traveling.

To summarize, electrons are traveling in metals at the Fermi velocity vF, which is very large (106 m/s), but the flux of electrons is the same in all directions. That is, they are going nowhere fast. In an electric field, a very small but directional drift velocity is superimposed on this fast random motion.


An important consequence of this calculation is Ohm's Law, V = iR. From the equations above, we can see that the drift velocity increases linearly with the applied electric field:

vdrift = aτ = eEτ/me

The drift velocity is proportional to the current, and the electric field is proportional to the voltage.

The conductivity (σ) of a metal is proportional to τ (and λ):

σ = ne2τ/me

where n is the density of electrons (8.5 x 1028 m-3 for Cu) and e is the electronic charge. Putting in the numbers and converting to cm units we obtain σ = 7 x 105 Ω-1 cm-1 for Cu, in good agreement with the measured value (6 x 105 Ω-1cm-1).

  6.6 Atomic orbitals and magnetismEdit

Metal heats of vaporization.png




There are two main types of periodic trends that are observed relating to orbital size. As one moves across the periodic table ( Sc - Ti - V - Cr - Fe ), orbitals contract, increasing the nuclear charge of the ions. When moving down the periodic table ( V - Nb - Ta ), orbitals expand, increasing the principal quantum number. Other key trends involving 3d, 4d, and 5d elements are observed as well. 3d elements are small because of their contracted orbitals. The contracted orbitals effect the ability for the electrons to bond, as one moves past V and across the periodic table, 3d electrons become much less effective in bonding because they are contracted. Also, 4d and 5d elements are similar to each other in size and have similar chemical properties. Poor overlap of 3d orbitals gives narrow d-bands. Good overlap of 4s, 4p orbitals gives a wide s,p band.

Orbital diagrams of Cr and W



Magnetism, like heat of vaporization, is also related to orbitals and bonding. Materials can be diamagnetic, paramagnetic, ferromagnetic, ferrimagnetic, and antiferromagnetic. If a material has no unpaired electrons, it will be diamagnetic, and if it has one or more unpaired electron it can be characterized by any of the other four types of magnetic conditions. When a material is diamagnetic, it is repelled by magnetic fields. A material that is diamagnetic will not have its own magnetic field. This is because all electrons are paired. In order for a material to have a magnetic field or magnetic susceptibility, it needs to have unpaired electrons. Magnetic fields are generated by electron spins. When electrons are paired, their spins cancel, cancelling their induced magnetic fields.

Using orbitals and bonding, the strength of the magnetic properties of a material can be predicted by determining the number of unpaired electrons. The magnetic moment of the atom, μ, can be calculated in terms of the Bohr magneton, μb. μ and μb are related by the number of unpaired electrons in the atom: μ=2√(S(S+1))μb where S=n/2 and n is the number of unpaired electrons. The stronger the magnetic moment per atom, the stronger the magnetic properties of the material.

The magnetic properties of the material also depend on the alignment of orbitals in the structure of the material. The way bonds form and align in metals, alloys, and various magnetic crystalline compounds can determine whether the compound will be ferromagnetic, ferrimagnetic, or antiferromagnetic. For example, in magnetite, Fe3O4, the oxygen between iron causes the 3d orbitals of iron atoms to be opposite its nearest iron neighbor.

  6.7 Ferro-, ferri- and antiferromagnetismEdit

A rough description of Ferrimagnetism, Ferromagnetism, Antiferromagnetism, and Paramagnetism

The magnetism of metals and other materials are determined by the orbital and spin motions of the electrons and the way in which electrons line up to one another and interact. Whether a material exhibits magnetism or not, it is affected by the interaction of atomic magnetic moments or in some cases, the lack of. Ferromagnetism, anti-ferromagnetism and ferrimagnetism, in each of these cases, long-range magnetic order takes places below a critical temperature, the Curie temperature. Magnetic coupling is observed in systems that have two or more metal atoms bridged by ligands. These types of ordering are primarily seen among 3d and 4f elements and their compounds.

Unpaired electrons vs transition metal or compound

Ferromagnetic materials exhibit parallel alignment of spins, which produces a large magnetization in both the presence and absence of a magnetic field. These types of materials possess a magnetic susceptibility value that is greater than zero. Magnetic susceptibility (X) is defined by how easy it is to align electron spins with the applied magnetic field. The value of magnetic susceptibility is magnetization (M) divided by applied magnetic field (H). The value is typically given in molar units in a cgs system. Elements iron, nickel and cobalt and many of their alloys are typical ferromagnet materials.

Antiferromagnetic materials exhibit spin alignment that is completely opposite of what was seen with ferromagnetism. In antiferromagnets, neighboring spins are locked in an anti-parallel alignment. The antiparallel formation results in the cancellation of magnetic spins and moments, hence producing a low magnetic moment and magnetic susceptibility, with the trend moving towards zero. The characteristic temperature associated with antiferromagnetism is called the Néel temperature. Below this temperature there is antoferromagnetic ordering, above the temperature this type of ordering is not seen. Many d-metal oxides exhibit this magnetic behavior because of the superexchange occurring with oxygen; examples include FeO and MnO. Superexchange is seen in electrons’ having come from the same donor atom and are being coupled with the receiving ions’ spins.

Ferrimagnetism is a net magnetic ordering of ions with different individual magnetic moments and is observed below the Curie temperature. These ions can order with opposed spins, but because the individual spin moments are different, there is an incomplete cancellation and the material has a net overall moment. The magnetic susceptibility of ferrimagnets is greater than zero. A good example of ferromagnetic material is magnetite, Fe3O4 . Two of the iron ions are trivalent and one is divalent, hence the trivalent ions align with opposite moments and cancel one another out and a net moment is produced due to the divalent iron ion.

Whether a material is Ferromagnetic, Ferrimagnetic, or Antiferromagnetic, above the Curie temperature, it will be Paramagnetic. A Paramagnetic material has a small positive magnetic susceptibility, and is weakly attracted to magnetic fields. Paramagnetic materials will align with magnetic fields, but will not retain any magnetic properties once the external magnetic field is removed. This is because in the Paramagnetic state, thermal motion constantly randomizes and disorders the orientation of magnetic domains.

Magnetic susceptibility vs temperature (kelvin) of ferrimagnetic ferromagnetic and anti-ferromagnetic materials

  6.8 Hard and soft magnetsEdit

Linear magnetic susceptibility relationships between temperature in ferromagnets, ferrimagnets, and antiferromagnets.




A graphical representation of how remnant magnetization, saturation magnetization, and coercive field relate to magnetic materials characterized as being hard and soft.

Whether a material is a hard magnet or a soft magnet depends on the ability of the material's magnetic domains to retain magnetic alignment when no field is present. This ability is characterized by the constant Br, or remnant magnetization at zero applied field. If a material has a large remnant magnetization, it is characterized as a hard magnet, and if a material has a small remnant magnetization, it is characterized as a soft magnet.

Typically only Ferromagnetic and Ferrimagnetic materials are capable of having remnant magnetization when no field is present. However, these materials are usually found or produced with their magnetic domains pointed in randomized directions, with their net dipoles cancelling. To achieve a remnant magnetization, these domains need to be aligned. This can be done with an applied magnetic field. The field required to align the magnetic domains in a material is known as the coercive field, Hc. Also, if a remnant magnetization is already present, Hc represents the field needed to bring the net magnetization to zero.

Saturation magnetization, Bsat represents the maximum magnetic field that can be present in a material. This field is stronger than Br, the remnant magnetization of the material, but unlike Br, Bsat is only present when a strong enough field is applied in alignment with the magnetic field of the material.

Materials characterized as hard magnets have large remnant magnetization, Br. A large coercive field Hc is required to bring B of a hard magnet to zero.

Materials characterized as soft magnets have small remnant magnetization, Br, and a smaller coercive field Hc is required to bring B of a soft magnet to zero. Soft magnets also typically have a larger saturation magnetization, Bsat.

Some materials can exist as either hard or soft magnets. Iron, for example can exist as both hard and soft magnet. In the case of iron whether it is a hard or soft magnet depends on the crystal grain size. When crystal grains in iron are smaller, the magnetic domains are pinned. When the magnetic domains are pinned a stronger coercive magnetic field needs to be applied to cause them to align. Also a larger remnant magnetism is present as a result of domains being pinned. When iron is annealed crystal grains grow and become more free to align with applied magnetic fields. This decreases the coercive field required to align magnetic domains. This also reduces the remnant magnetism when no applied field is present, as domains are more free to lose alignment when the magnetic field is removed. Also as a result of the magnetic domains being more free to more, the saturated magnetization increases.


Nd2Fe14B is a hard magnet used in disk drives, refrigerator magnets, electric motors and other applications. Drive motors for hybrid and electric vehicles such as the Toyota Prius require 1 kilogram (2.2 pounds) of neodymium.[1]. A high-resolution transmission electron microscope image of Nd2Fe14B is shown below and compared to the crystal structure with the unit cell marked.

  6.9 Discussion questionsEdit

  • Discuss the relationship between the trends in bonding energy of the transition metals and their magnetic properties.
  • Draw the hcp unit cell in sections and show how you would calculate (a) the number of atoms in the unit cell and (b) the fraction of space filled by equal spheres.

  6.10 ProblemsEdit

1. C-centered Bravais lattices exist in the monoclinic and orthorhombic systems but not in the tetragonal system. That is because the C-centered tetragonal lattice is equivalent to one of the other Bravais lattices. Which one is it? Show with a drawing how that lattice is related to the C-centered tetragonal lattice. How many atoms are in the unit cell of that lattice?


2. Which of the following sequences of close packed layers fills space most efficiently? Explain your answer.

(a) ABACCABA...

(b) ABABAB...

(c) AABBAABB...

(d) ABACABAC...


3. Calculate the fraction of space that is occupied by packing spheres in the (a) simple cubic, (b) body-centered cubic structure, (c) face-centered cubic (cubic close packed) structures. Assume that nearest neighbor spheres are in contact with each other.

HCP crystal structure.svg


4. The hexagonal close packed (hcp) structure is shown at the right. If the radius of the spheres is R, what is the vertical distance between layers in units of R? What fraction of space is filled by the spheres in the hcp lattice?


5. Consider a one-dimensional chain of sodium atoms that contains N atoms, where N is a large number. The distance between atoms in the chain is the lattice constant a. In the highest occupied molecular orbital of the chain, what is the distance between nodes (in units of a)?


6. Starting from the left side of the periodic table, the melting and boiling points of the elements first increase, and then decrease. For example, the order of boiling points is Rb < Sr < Y < Zr < Nb < Mo > Tc ≈ Ru > Rh > Pd > Ag. Briefly explain the reason for this trend.


7. Going across the periodic table from left to right starting from potassium, the bonding energies and heats of vaporization increase in the order K < Ca < Sc < Ti < V, but then decrease going from V to Mn (V > Cr > Mn). The heats of vaporization of V, Cr, and Mn are much less than those of Nb, Mo, and Tc, respectively. Explain these trends.


8. Some alloys of early and late transition metals (e.g., ZrPt3) have much higher enthalpies of vaporization (per metal atom) than either pure metal. Why are such alloys unusually stable, relative to the pure metals?


9. Graphite is a semimetal composed of sheets of fused benzene rings. There are no bonds between sheets, only van der Waals interactions. What is the C-C bond order in graphite? Show why the C-C distance (1.42 Å) is different from that of benzene (1.40 Å).


10. Fe, Co, and Ni are ferromagnetic, whereas Ru, Ir, and Pt are diamagnetic. Explain why magnetic metals are found only among the 3d and 4f elements. Why are Sc and Ti, which are also 3d elements, diamagnetic?


11. The Curie temperature of cobalt is 1127•C, which is higher than that of Fe (770•C) or Ni (358•C). Why does Co have the highest Curie temperature?


12. Iron metal comes in magnetically “soft” and “hard” forms. Briefly explain the structural differences between them and draw magnetic hysteresis curves for each, indicating on your curves the coercive field and the remanent magnetization.

  6.11 ReferencesEdit