6

Adsorption on metal electrodes: principles

6.1 Adsorption phenomena

Whenever the concentration of a species at the interface is greater than canbe accounted for by electrostatic interactions, we speak of specific adsorption.It is usually caused by chemical interactions between the adsorbate and theelectrode, and is then denoted as chemisorption. In some cases adsorption iscaused by weaker interactions such as van der Waals forces; we then speak ofphysisorption. Of course, the solvent is always present at the interface; so theinteraction of a species with the electrode has to be greater than that of thesolvent if it is to be adsorbed on the electrode surface. Adsorption involvesat least a partial desolvation. Cations tend to have a firmer solvation sheaththan anions, and are therefore less likely to be adsorbed.

The amount of adsorbed species is usually given in terms of the coverageθ, which is the fraction of the electrode surface covered with the adsorbate.When the adsorbate can form a complete monolayer, θ equals the ratio of theamount of adsorbate present to the maximum amount that can be adsorbed.In a few systems the area covered by a single adsorbed molecule changes withcoverage; for example, some organic molecules lie flat at low coverage andstand up at higher coverages. In this case one must specify to which situationthe coverage refers. Another definition of the coverage, often used in surfacescience, is the following: θ is the ratio of the number of adsorbed species tothe number of surface atoms of the substrate. Fortunately, most authors statewhich definition they use.

The chemisorption of species occurs at specific sites on the electrode, forexample on top of certain atoms, or in the bridge position between two atoms.Therefore, most adsorption studies are performed on well-defined surfaces,which means either on the surface of a liquid electrode or on a particular sur-face plane of a single crystal. Nowadays, most work is done on single crystals,and mercury, which was extensively used before, has almost dropped out ofuse.

52 6 Adsorption on metal electrodes: principles

Nevertheless, liquid electrodes are not only easier to prepare than singlecrystal surfaces of solid electrodes, they also have another advantage: Adsorp-tion can be studied by measuring the variation of the surface tension. Wedefer the thermodynamics of interfaces till Chapter 8; here we merely statethat such measurements yield the total surface excess Γi of a species. Roughlyspeaking, Γi is the amount of species i per unit area in excess over the amountthat would be present if its concentration were the same at the interface asin the bulk. Γi can be positive or negative – cations, for example, can beexcluded from the surface region near a positively charged electrode. Untilthe advent of modern spectroscopic methods for studying the electrochemicalinterface, thermodynamic measurements were the only reliable way of deter-mining specific adsorption. While such measurements are easier to performon liquid electrodes, they have been extended to solid surfaces.

The surface excesses Γi not only include the atoms or molecules of speciesi that are adsorbed on the metal surface, but also those that are in the space-charge region considered in Chapter 5. The latter is also known as the diffusepart of the double layer, or simply as the diffuse double layer; in contrast,adsorbed particles are said to be part of the compact part of the double layer.Usually we are only interested in the amount of particles that is specificallyadsorbed. The excess of species i in the diffuse double layer can be minimizedby working with a high concentration of nonadsorbing, inert (also called sup-porting) electrolyte. Suppose we want to study the adsorption of an anionA−; if we add a large excess of nonadsorbing ions B

+ and C− to the solution,

keeping the electrode charge constant, the amount of ions A− in the diffuse

layer will be greatly reduced. This can be seen from the following argument.Let Q be the total surface charge at the interface, that is, Q contains both theexcess charge on the metal and the charge of any adsorbed species. This chargemust be balanced by a charge -Q in the space-charge region. The concentra-tion ni(x) of a species i in this region is proportional to its bulk concentrationni,o (see Eq. 5.3): ni(x) = ni,o exp[zie0φ(x)/kT ]. Adding an excess of an inertelectrolyte will therefore drastically reduce the amount of ions A

− in this re-gion. Since the charge number enters into the exponent, highly charged ionsrequire a higher concentration of supporting electrolyte. In practice it maynot be easy to find an inert electrolyte that is not specifically adsorbed; obvi-ously, coadsorption of other ions should be avoided since it drastically changesthe conditions at the electrode surface, and makes the interpretation of theexperimental data difficult.

6.2 Adsorption isotherms

Consider the adsorption of a species A with concentration cA in the bulk of thesolution. The variation of the coverage θ with cA, keeping all other variablesfixed, is known as the adsorption isotherm.

6.2 Adsorption isotherms 53

–4 –2 0 2 40.0

0.4

0.8

ln(y)

g=1

g=-5 g=0g=-4

Fig. 6.1. Frumkin isotherms for various values of the adsorbate interaction pa-rameter g; the Langmuir isotherm corresponds to g = 0. The abscissa is: ln y =ln(c/c0) + (µsol − µad)/kT .

We first consider the simplest possible model, in which adsorption occursat fixed sites, and the interaction between adsorbed particles can be neglected.Let the surface contain N adsorption sites, of which M are occupied, and let�ad be the adsorption energy per particle. The internal energy of the adsor-bate is simply M�ad. To obtain the free energy, we need the entropy, whichaccording to the Boltzmann formula is: S = k lnW , where W is the numberof realizations of the system, or the number of ways of selecting M sites outof N . Therefore:

F = M�ad − kT lnN !

M !(N −M)!(6.1)

Using Stirling’s formula: lnn! ≈ n lnn− n for large n gives:

F = M�ad +�M ln

M

N+ (N −M) ln

N −M

N

�(6.2)

At equilibrium, the chemical potential of the adsorbate must equal the chemi-cal potential of the same particle in the solution. For the adsorbate we obtain:

µad =∂F

∂M= �ad + kT ln

θ

1− θ(6.3)

where θ = M/N is the coverage. The chemical potential for an ideal solutehas the form:

µsol = µ0 + kT lnc

c0(6.4)

where c is the concentration, and the unit concentration c0 makes the argu-ment of the logarithm dimensionless. Setting the chemical potentials equal

54 6 Adsorption on metal electrodes: principles

results in the Langmuir isotherm:

θ

1− θ=

c

c0exp

�µsol − µad

kT

�(6.5)

The model from which it has been derived is the two-dimensional lattice-gasmodel . We shall meet a three-dimensional version in Chapter 18.

So far, we have ignored interactions between the adsorbates completely.In a simple, phenomenological way one can account for such interactions byassuming that ∆µad is proportional to θ: µad = µ

0ad + γθ, where the constant

γ is positive if the adsorbed particles repel, and negative if they attract eachother. The resulting isotherm:

θ

1− θ=

c

c0exp

�µsol − µad

kT

�e−gθ (6.6)

with g = γ/RT , is known as the Frumkin isotherm. At present there is no gen-eral satisfactory theory for adsorbate-adsorbate interaction at electrochemicalinterfaces, and consequently none for adsorption isotherms. Besides Eqs.(6.5)and (6.6), various other isotherms have been proposed based on rather sim-ple models, but none of them is really satisfactory. Figure 6.1 shows Frumkinisotherms for a few different values of the interaction parameter g. Positivevalues of g broaden the isotherm because the adsorbed particles repel eachother; for negative values of g the isotherms are narrow because adsorption isthen a cooperative effect. For g < −4, a phase transition occurs, in which theadsorbate condenses on the electrode. This gives rise to a vertical slope in theisotherm. The case g = 0 corresponds to the Langmuir isotherm.

The difference ∆µ = µsol − µad depends on the electrode potential φ.This dependence will be different for anions, cations, and neutral species. Thesimplest possible case is the adsorption and total discharge of an ion accordingto the equation:

Az+ + ze− � Aad (6.7)

obeying the Langmuir isotherm with a potential dependence of the form:

∆µ(φ) = ∆µ(φ0) + ze0(φ− φ0) (6.8)

where φ0 is a suitably chosen reference potential. The choice of φ0 is notimportant since it just determines the zero of the potential scale. A convenientchoice may be one for which the coverage is θ = 1/2 for a given electrolyteconcentration. The resulting isotherm takes the form:

θ

1− θ=

c

c0K exp

�−ze0(φ− φ0)

kT

�(6.9)

which is illustrated in Fig.6.2. However, the assumptions on which this equa-tion is based rarely hold in practice. When ions are adsorbed, the interaction

6.2 Adsorption isotherms 55

-­0.15 -­0.10 -­0.05 0.00 0.05 0.10 0.15

Charg

eCurrent

! " !#  /  V

Qo s

Fig. 6.2. Current and charge versus potential for a Langmuir isotherm whose poten-tial dependence is given by Eq. (6.9). Such curves are obtained by a slow potentialsweep. The absolute value of the current depends on the sweep rate (see text).

of the adsorbates is quite important; in addition, the adsorbed ions need notbe totally discharged – a point which we will discuss further – and Eq. (6.8)need not hold. The simple adsorption isotherm Eq. (6.9) should rather beviewed as an ideal reference situation.

A simple way to study the potential dependence of an adsorption reactionis a potential sweep. In this procedure the electrode potential is first kept in aregion where the adsorption is negligible; then the potential is varied slowlyand with a constant rate vs = dφ/dt, and the resulting current i is measured.The sweep rate vs must be chosen with care. It must be so slow that thereaction is in equilibrium, and that the current due to the charging of thedouble layer is negligible or small. On the other hand, vs must be so largethat a sizable current flows. Sweep rates of the order of a few mV s−1 orsomewhat faster are common. In simple cases the current is proportional tothe rate of change of the coverage:

I = Q0dθ

dt(6.10)

where Q0 is the charge required to form a monolayer of the adsorbate. Thisequation only holds if the charge required to adsorb one particle is independentof the coverage. This need not be the case; at small coverages the adsorbatemay still be charged, while at high coverages Coulomb repulsion prevents theaccumulation of a sizable charge on the interface, and the adsorbates will bedischarged. Figure 6.2 shows the form of the current-potential curve if bothEqs. (6.9) and (6.10) hold; the absolute value of the current depends on thesweep rate and on Q0. Again, this curve should be viewed as an ideal reference

56 6 Adsorption on metal electrodes: principles

case, and real curves will differ significantly. A repulsive adsorbate interaction,for example, will broaden the peak in Fig.6.2, while an attractive interactionwill lead to narrow peaks. If the charge per adsorbed particle is constant, thecoverage at a given potential can be determined by measuring the charge thathas flowed:

θ(φ) =Q(φ)Q0

=1

Q0

� φ

φ1

II

vsdφ (6.11)

where the potential φ1 has to be in the region where the species is not ad-sorbed.

Other phenomena such as phase formation or phase transitions will alsoshow up in such current-potential curves. It will be apparent by now thatadsorption is a complicated process; only a few systems are well understood.We will consider a few illustrative examples later in this chapter, and deferthe more complicated theoretical aspects to later chapters.

6.3 The dipole moment of an adsorbed ion

In general a polar bond is formed when an ion is specifically adsorbed ona metal electrode; this results in an uneven distribution of charges betweenthe adsorbate and the metal and hence in the formation of a surface dipolemoment. So the adsorption of an ion gives rise to a dipole potential dropacross the interface in addition to that which exists at the bare metal surface.

Electrode Ion µ× 10−30 CmHg Rb+ 4.07Ga Rb+ 0.90Hg Cs+ 4.65Ga Cs+ 0.90Hg Cl− -3.84Hg Br− -3.17Hg I− -2.64

Table 6.1. Dipole moments of a few ions adsorbed from an aqueous solution at lowcoverage.

The same effect exists for adsorption on a metal surface from the gas phase.In this case the adsorbate-induced dipole potential changes the work functionby an amount ∆Φ. If nad is the number of adsorbed molecules per unit area,the component µx of the dipole moment of single adsorbed molecule can beinferred from the relation:

∆Φ =nadµx

�0(6.12)

6.3 The dipole moment of an adsorbed ion 57

As before, the x direction has been taken normal to the metal surface. Inelectrochemistry, the dipole moment µx associated with an adsorbate bondcan be defined in the following way: For simplicity suppose that the electrodehas unit area. At the beginning the electrode surface is bare and kept atthe pzc. Then a number nad of ions with charge number z are adsorbed;simultaneously a counter charge −ze0nad is allowed to flow onto the metalsurface. The change ∆φ in the electrode potential is related to the dipolemoment through:

e0 ∆φ =nadµx

�0(6.13)

Note that both before and after the experiment the sum of the charges on themetal surface and in the adsorbate layer is zero, and hence there is no excesscharge in the diffuse part of the double layer. However, after the adsorptionhas occurred, the electrode surface is no longer at the pzc, since it has takenup charge in the process.

!

Fig. 6.3. Two alternative ways of viewing the charge distribution in an adsorptionbond. The upper part of this figure shows the dipole moment; the lower part showsa partially charged adsorbate and its image charge. The dipole moments of thesurrounding solvent molecules are oriented in the direction opposite to the adsorbatedipole.

In the gas phase the dipole moment determined through Eq. (6.12) refers toan individual adsorbed particle. This is not so in the electrochemical situation.The dipole moment of an adsorbed species will tend to align neighboringsolvent molecules in the opposite direction, thereby reducing the total dipolepotential drop (see Fig. 6.3). Only the total change in dipole potential canbe measured, and there is no way of dividing this into separate contributionsfrom the adsorbate bond and the reorientation of the solvent. A few values ofsuch electrochemical dipole moments are given in Table 6.1. For comparisonwe note that the dipole moments of alkali ions adsorbed from the vacuum are

58 6 Adsorption on metal electrodes: principles

!

"#$%&!

'#(") &#*#&

#+#(,-!

%./0(1%$#!

.#+/)$-!02!

/$%$#/!

Fig. 6.4. Density of states of an adsorbed cation (schematic).

usually of the order of the order of 10−29 Cm. Because of the screening bythe solvent, the apparent dipole moment of an ion adsorbed from a solutionon a particular metal is often substantially smaller than that of the same ionadsorbed in the vacuum.

If the adsorbate bond has a strong ionic character, as is the case for thealkali and the halide ions, the concept of a partial charge is useful. One thinksof the adsorbed ion as carrying a charge zade0, which is generally fractional,i.e. zad is not an integer. The excess charge on the ion induces an imagecharge of equal and opposite magnitude on the metal surface (see Fig. 6.3),resulting in a surface dipole moment. The concomitant potential reorientsthe neighboring polar solvent molecules in the opposite direction so that thedipole potential induced by the adsorbate is reduced by the dipole momentsof the solvent molecules. A related concept is that of a partial charge transfercoefficient l, which is defined as l = zion − zad.

In principle the partial charge on an adsorbate is ill defined, since one hasto introduce a plane separating the electronic density into a part belonging tothe adsorbate and one belonging to the metal; obviously, it is not measurable.However, the notion of partial charge can be understood in terms of quantum-mechanical considerations. To be specific, let us consider the adsorption of aCs+ ion from an aqueous solution, and assume that the electrode potential is inthe range where no reactions occur. When the ion is in the bulk of the solutionthe valence orbital has a well-defined energy lying above the Fermi level of theelectrode; hence the valence orbital is empty. When this ion is adsorbed onthe metal electrode, its valence orbital overlaps with the metal orbitals. If weput an electron into the valence orbital of the adsorbed Cs atom, it has only afinite lifetime τ in this state before it is transferred to the metal; the strongerthe interaction, the shorter is τ . According to the Heisenberg uncertaintyprinciple, a finite lifetime τ entails an energy uncertainty ∆ = �/τ . Hencethe valence orbital is broadened and acquires a density of states ρ(�) of width

6.4 Electrosorption valence 59

∆, a phenomenon known as lifetime broadening and familiar from electronicspectroscopy. This density of states is filled up to the Fermi level of the metal.For an adsorbed Cs atom the center of the density of states lies well abovethe Fermi level EF (see Fig. 6.4), the occupancy n is generally quite small,and the partial charge zad = 1 − n is close to unity. In contrast, halide ionstypically carry a negative excess charge, and the center of the density of statesof their valence orbitals lies below or near the Fermi level of the metal.

6.4 Electrosorption valence

The distribution of charges on an adsorbate is important in several respects:It indicates the nature of the adsorption bond, whether it is mainly ionic orcovalent. Therefore, a fundamental problem of classical electrochemistry is:What does the current associated with an adsorption reaction tell us aboutthe charge distribution in the adsorption bond? Ultimately the answer is alittle disappointing: All the quantities that can be measured do not refer toan individual adsorption bond, but involve also the reorientation of solventmolecules and the distribution of the electrostatic potential at the interface.This is not surprising; after all, the current is a macroscopic quantity, which isdetermined by all rearrangement processes at the interface. An interpretationin terms of microscopic quantities can only be based on a specific model.Therefore, DFT calculations, especially if they include some water besidesthe electrode and the adsorbate, are especially valuable to understand theadsorption bond.

There is a formal similarity between adsorption and reactions such as metaldeposition which gives rise to the concept of electrosorption valence. Considerthe deposition of a metal ion of charge number z on an electrode of the samematerial. If the electrode potential φ is kept constant, the current density jis:

j = −ze0

�∂N

∂t

�

φ

(6.14)

where N is the number of particles deposited per unit area. Likewise, whenan adsorbate (index i) is deposited on a metal electrode, the resulting currentwill be proportional to the adsorption rate:

j = −le0

�∂Γi

∂t

�

φ,Γj �=Γi

(6.15)

where Γi is the amount adsorbed, also called the surface excess the surfaceexcess of species i; this includes any excess present in the diffuse layer –a precise definition will be given in Chapter 8. By using dσ = j dt andrearranging: �

∂σ

∂Γi

�

φ,Γj �=Γi

= le0 (6.16)

60 6 Adsorption on metal electrodes: principles

where σ is the surface charge density on the metal. The coefficient l was givendifferent names by different authors,1 but the term electrosorption valence,coined by Vetter and Schultze [18], has stuck. Equivalently, it can be definedthrough:

le0 =�

∂µsi

∂φ

�

Γi

(6.17)

We defer the proof to Chapter 8, which contains the required thermodynamicrelations.

The definition of the electrosorption valence involves the total surface ex-cess, not only the amount that is specifically adsorbed. It is common to correctthe surface excess Γi for any amount that may be in the diffuse double layerin order to obtain the amount that is specifically adsorbed. This can be doneby calculating the excess in the diffuse layer from the Gouy-Chapman theory.Often this correction is small, particularly if the species is strongly adsorbed,or if an excess of a nonadsorbing electrolyte is used. However, if the correctionterm is large, Eq. (6.17) need not hold for the corrected quantity, since thisequality has been proved only for the total excess.

The interpretation of the electrosorption valence is difficult. The following,somewhat naive argument shows that it involves both the distribution of thepotential and the amount of charge transferred during the adsorption process.Suppose that an ion S

z is adsorbed and takes up λ electrons in the process.λ need not be an integer since there can be partial charge transfer. We canthen write the adsorption reaction formally as:

Sz → S

z+λ + λe− (6.18)

As noted before, the partial charge transfer is not well defined. Nevertheless,let us suppose that we can treat S

z+λ like a normal species. Its electrochemicalpotential is then:

µ̃adi = µ

adi + (z + λ)e0φad (6.19)

where φad denotes the electrostatic potential at the adsorption site. Since thereaction is in equilibrium, the electrochemical potentials must balance. Settingthe electrostatic potential in the solution equal to zero, we obtain:

µsi = µ

adi + (z + λ)e0φad − λe0φm (6.20)

where φm is the potential of the metal. Differentiating with respect to theelectrode potential φ, which differs from φm by a constant, gives:

l = gz − λ(1− g), where g =�

∂φad

∂φ

�

Γi

(6.21)

While this equation is certainly not exact, it can be used for qualitativeinterpretations. In particular, the following limiting cases are of interest:1 Usually the electrosorption valence is denoted by γ, which we use for the surface

tension. The symbol l was used earlier by Lorenz and Salie [17].

6.5 Electrosorption valence and the dipole moment 61

Electrode Ion l

Hg Rb+ 0.15Ga Rb+ 0.20Hg Cs+ 0.18Ga Cs+ 0.20Hg Cl− -0.20Hg Br− -0.34Hg I− -0.45

Table 6.2. Electrosorption valences of a few simple ions at the pzc and at lowcoverage.

1. Total discharge: λ = −z, or l = z

2. Incorporation into the electrode: φad = φm, g = 1, and l = z.3. No charge transfer: λ = 0, l = gz.

In general, the electrosorption valence depends both on the electrode poten-tial and on the amount adsorbed, as may be expected from Eq. (6.21). Table6.4 lists the electrosorption valences of a few simple ions at the pzc and atlow coverages on liquid metals [19], where measurements are easier than onsolids. The low values for the alkali ions Rb+ and Cs+ are generally thoughtto indicate the absence of partial charge transfer. In contrast, the values forthe halide ions may indicate a partial transfer of an electron to the metal.In underpotential deposition of a monolayer of metal ions (see next chapter)the electrosorption valence is generally equal to the charge number of themetal ion, indicating total discharge. A final word of caution: In mixed solu-tions coadsorption may take place and make a proper determination of theelectrosorption valence difficult.

6.5 Electrosorption valence and the dipole moment

The electrosorption valence can be related to the dipole moment of an ad-sorbed species introduced above. For this purpose consider an electrode sur-face that is initially at the pzc and free of adsorbate. When a small excesscharge density σ is placed on the metal, its potential changes by an amount∆φ given by:

∆φ =σ

C=

σ

CH+

σ

CGC(6.22)

where we have split the interfacial capacity C into the Gouy-Chapman partCGC and the Helmholtz part CH. Equation (6.22) is a linear expansion interms of σ. When a small number Ni of a species with charge number z

is adsorbed per unit area at fixed σ, the resulting change in the electrodepotential is proportional to Ni. The total charge density at the interface isnow σ + ze0Ni ≡ σ + σi, and this is balanced by the charge in the diffuselayer. So we have:

62 6 Adsorption on metal electrodes: principles

∆φ = Bσi +σ

CH+

σ + σi

CGC(6.23)

with an unknown coefficient B. We have made no assumption about the actualdistribution of the charge at the interface. The electrosorption valence is:

l̃ = −z

�σ

σi

�

∆φ=0

= zB + 1/CGC

1/CGC + 1/CH(6.24)

The tilde indicates that the value is not corrected for the diffuse layer. Thecorrected value is obtained by eliminating the CGC terms:

l = zBCH (6.25)

To obtain the dipole moment we set σ = −σi in Eq. (6.23) so that the diffusedouble layer is free of excess charge (see Section 4.3).

∆φσ=−σi = σi

�B − 1

CH

�(6.26)

The dipole moment µi per adsorbate is obtained by dividing this potentialdrop by Ni/�0, and changing the sign, so that a positive charge on the adsor-bate corresponds to a positive dipole moment:

µi = −ze0�0

�B − 1

CH

�=

ze0�0

CH(1− l/z) (6.27)

which is the desired relation. For a different derivation see Ref. [20]. We thinkthat the dipole moment is a more useful quantity than the electrosorption va-lence since it can be interpreted without recourse to the badly defined conceptof partial charge transfer. Even so, µi is not the dipole moment of an individ-ual adsorbed molecule. The solvent molecules in the vicinity of the adsorbatewill be oriented by the dipole moment of the adsorbate, and the resultingchange in the interfacial potential is reflected in µi [21].

6.6 Structures of commensurate overlayers on singlecrystal surfaces

The overlayer structure that result when a species is adsorbed on a singlecrystalline surface depends on various factors, such as the relative sizes of theadsorbate and substrate atoms (or molecules), and the interactions betweenthe particles involved. If the interactions between the adsorbed species arerepulsive, the resulting overlayer often shows a homogeneous structure. Onthe contrary, if attractive forces exist, there is a tendency to form islands orpatches on the surface. Moreover, as we have already discussed in Chapter 2there are also preferential sites for the adsorption of different species (atop,

6.6 Structures of commensurate overlayers on single crystal surfaces 63

Fig. 6.5. Unit cells of the principle planes of an fcc crystal.

bridge, threefold sites, etc.). In this section we shall look at the formation ofcommensurate overlayers, which we shall define below, and the notation torepresent them.

Single crystals have a periodically ordered arrangement of atoms at thesurface. The structure depends on the direction of cutting the crystal as al-ready mentioned in Chapter 2. Just like the structure of three-dimensionalcrystals is defined by three-dimensional unit cells, the structure of a single-crystal surface can be characterized by a two-dimensional unit cell, which isdefined as the simplest periodically repeating unit which can be identified inan ordered two-dimensional array. Thus, the whole surface structure can beconstructed by repeated translation of the unit cell. As examples we show inFig. 6.5 the three principal lattices planes of a fcc crystal and several possiblechoices for the unit cell are indicated. Of course, all legitimate choices of theunit cell give rise to the same lattice.

Fig. 6.6. 2× 2 structures on the principle planes of an fcc crystal.

64 6 Adsorption on metal electrodes: principles

Two vectors a1 and a2 with a common origin are usually selected to definethe unit cell. In the case of ordered overlayers formed by adsorbates we canemploy the same procedure as before and define a unit cell for the overlayerthrough vectors b1 and b2. When the substrate and the adsorbate struc-tures are related by a simple mathematical transformation, one speaks of acommensurate structure. otherwise it is incommensurate. Frequently, Woodsnotation is employed when the vectors form the same angle for the two unitcells that corresponding to the substrate and to the overlayer. The lengths ofthe two vectors b1 and b2 are expressed in terms of a1 and a2, respectively,as (|b1/|a1|× |b2/|a2|). As an example, Fig. 6.6 shows the (2× 2) structureson the three principal lattice planes of a fcc crystal. Here the adsorbate issmaller than the substrate atom and adsorbed on an atop site, but the sameconcepts can be applied for other sites and sizes. Often, the letter p precedesthe description of the structure p(2×2) to indicate that it is a primitive struc-ture, the simplest possible unit, and to distinguish from the closely relatedc(2 × 2) structure. The latter actually does not correspond to a primitivecell because it has an additional species in the centre of the (2 × 2) struc-ture. The corresponding primitive cell for this structure is better described as(√

2 ×√

2)R45, which means that the vectors b1 and b2 are√

2 larger than√a1 and

√a2, respectively, and that the unit cell of the overlayer is rotated

45 degrees with respect to the substrate unit cell (see Fig. 6.7). Another typi-cal overlayer structure which is frequently observed on the (111) surface, alsoshown in Fig. 6.7, is that represented by the primitive cell (

√3×

√3)R30.

c(2x2)-(100) or:

( 2 x 2)R45-(100) ( 3x 3)R30-(111)

a2

a1

a2

a1

Fig. 6.7. Two examples of rotated structures.