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Perfect. Can we expect that all consequences of band theory, such as the Fermi surface and

energy gaps, will no longer exist? Will insulators become conductors?

Experiment and theory! agree that the consequences of the destruction of the

translational symmetry are much wea"er than one expects at first sight. #f the impurity belongs to

the same column of the periodic table as the host element in replaces, then the effects areparticularly small, in part because the a$erage number of $alance electrons remains constant.

%ne measure of the effect of alloying is gi$en by the residual resisti$ity. %ne atomic

percent of copper dissol$ed in sil$er increases the residual electrical resisti$ity by &.&' (ohm

cm, wich corresponds to a scattering cross sections of perhaps &.&! of the area of the impurity

atom. )imilary, the mobilility of electrons in )i *e alloys is much higher than one would expect

from the simple geometrical argument that a germanium atom +! electrons- is $ery unli"e

aslicon atom + electrons- and therefore a )i atom in *e or a *e atom in si should act as an

efficient scattering center for change carries. /he theory shows us that the effecti$e scattering

potential of impurity may be $ery small.

/here is no experimental e$idence for an intrinsic reduction in band gap due to the

random aspect of alloying. For example, silicon and germanium form solid solutions o$er the

entire0 the energies of the band edges in the alloys $ary continuously with composition +fig.1-.

#t is belie$ed that the desinty of orbitals near the band edges are somewhat smudged out by

alloying.

We now discuss substitutional solid solutions of one metal 2 in another 3 of different

$alance. We suppose that atoms 2 and 3 occupy equi$alent lattice positions at random. /he

distinct effects which accur when the accupancies are regular and not random are consideretunder the heading of the order discorder transformation.

4ume 5othery has discussed empirical requirements for solid solutions to occur. %ne

requirement is si6e. #t is difficult to form solid solutions if the atomic diameters of 2 and 3

differ by more that 7 percent. /he si6es are fa$orable in the Cu +,77 2&- 8n +,11 2 &- system 9

6inc dissol$es in copper as a fcc solid solutions up to !: atomic percent 6inc, the si6es are

somewhat unfa$orable in the Cu +,77 2&- Cd +,;' 2 &- system9 only ,' atomic percent.

!/hese questions can be discussed using the method of orthogonali6ed plane wa$es and effecti$e potentials

described in chapter &. 2 low concentrations of impurity atoms cannot ha$e much effect on the fourier components


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Figure 16+a- calculated $ariation of the ma=or band edges in *e si alloys as a functions of the silicon

concentration band edge in *e. +Compare with fig. .7- /he calculations are by F. 3assani and >. 3rust, phys.

5e$.!, 7 +;1!-, who gi$e extensi$e references to the experimental data. +b- /he calculated $ariations as a

function of pressure for pure *e. ualitati$ely the $ariation of the bands with pressure is similar to the $ariations

with alloying, as we see by comparing energy $alues at equal $alues of the lattice constant.

Cadmium is solube in copper. /he atomic diameters referred to copper are .& for 6inc and

.17 for cadmium.

2lthough the si6es may be fa$orable , solid solutions will not from if there is strong

tendency for 2 and 3 to form stable compounds of defnited chemical proportions. #f 2 is stronglyelectronegati$e and 3 strongly electropositi$e, it is li"ely that compounds such as 23 and 23

will precipitate out of solution. 2lthough the atomic diameter ratio is fa$orable +.&- for 2s in

Cu, only 1 percent 2s is soluble. /he ratio is also fa$orable +.&;- for )b in @g, yet the solubility

of )b in @g is $ery small.

)a$eral aspects of the electronic structure of alloys can be discussed in terms of the

a$erage number7of conduction electrons per atom, denoted by n. /he $alue of n in the alloy 7&

percent Cu 7& percent 8n is .7&0 in 7& Cu 7& 2l, nA.&&. )ome of the principal effects of

alloying elements of different $alence come from the change in electron concentrations. 1

4ume

5othery first drew attention to the importance of the a$erage electron concentration asdetermining structural change in certain alloy systems.

/his number is often called the electron concentration.

1B. friedel, 2d$ances in Physics !, 1 +;7-.


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Figure 17. Equilibrium diagram of phases in the copper 6inc alloy system . the phases is fcc0 D and D

are bcc0 is a complex structure0 G and are both hcp, but which G has a cHa ratio near .71 and +for pure

8n- has cHaA.:1. /he Dphase is ordered bcc, by which we mean that most of the Cu atoms occupy sites on

one sc sublattice and most of the 8n atoms occupy sites on a second sc sublattice which interpenetrates the

first sublattice. /he D phase is disordered bcc9 any site is equally li"ely to be occupied by a Cu or a 8n

atom, almost irrespecti$e of what atoms are in the neighboring sites.

/he phase diagram of the copper 6inc system 'is shown in fig.'. /he fcc structure of

pure copper +nA- persists on the addition of 6inc +nA- until the electron concentration reaches

.!:. 2 bcc structure occurs at a minimum electron concentration of about .:. /he phase

exists for the approxi mate range of n between .7: and .11, and the hcp phase G occurs near

.'7.

/he term electron compound denotes an intermediate phase +such as the D phase of Cu

8n- whose crystal structure is determined by a fairly welldefined electron to atom ratio. /he

$alues of the ratio are called the hume 5othery rules0 they are .7& for the D phase, .1 for the

phase, and .'7 for the G phase. 5epresentati$e experimental $alues are collected in table !,based on the usual chemical $alence of for Cu and 2g0 for 8n and Cd0! for al and *a0 for

)i,*e, and )n.

'/he phases of interest are usually denoted by metallurgist by gree" characters0 in the Cu 8n system we ha$e

+fcc-,D +bcc-, +complex cubic cell of 7 atoms-,G +hcp- and +hcp-0 G and differ considerably in cHa ratio. /he

meaning of the character depends on the alloy system.


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/he 4ume 5othery rules find a simple expression in terms of the band theory of nearly

free alectrons. /he obser$ed limit of the fcc phase occurs close to the electron concentration of

.!1 at which an inscribed Fermi sphere ma"es contact with the 3rillouin 6one boundary for the

fcc lattice. /he obser$ed electron concentration of the bcc phase is close to the concentration

.: at which an inscribed Fermi sphere ma"es contact with the 6one boundary for the bcc

lattice. Contact of the Fermi sphere with the 6one boundary for the phase is at the concentration

.7. Contact for the hcp phase is at the concentration .1; for the idea cHa ratio.

Why is there a connection between the electron concentration at which a new phase

appears and the electron concentration at which the Fermi surface ma"es contact with the

3rillouin 6one boundary? #t is costly in energy to add further electrons to an alloy once the filled

states reach the 6one boundary. 2dditional electrons can be accommodated only in states abo$e

the energy gap which accurs at the boundary or in the states of high energy near the corners of

the lower 6one. #t may therefore be energetically fa$orable for the crystal structure to change to

one which can contain a larger Fermi surface before contact. #n this way the sequence fcc, bcc,

,hcp was made plausible by 4.Bones

.


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Figure :. Iumber of free electron orbitals Figure ; 3ohr magneton numbers of

per unit energy rangr for the first 3rillouin ferromagnetic nic"el copper alloys.

6one of the fcc lattices, as a function of energy.

/he transformation from fcc to bcc is illustrated by fig. :0 this shows the number of

orbital per unit energy range as a function of energy, for the fcc and bcc structures. 2s the

number of electrons is increased, a point is reached where it is easier to accommodateadditional electrons in the 3rillouin 6one of the bcc lattice rather than in the 3rillouin 6one of the

fcc lattice. /he figure is drawn for free electrons, and it may perhaps be ob=ected that the actual

desinty of orbitals in pure fcc metals is somewhat different see the Fermi surface of Cu infigs.&.1, &.!7, and &.!:.

We consider the relationship of the s and d bands in pure nic"el at & &J as suggested by

fig. 1.'b. /here is a certain arbitrariness in the proposed distribution9 we can transfer electronsfrom the d sub band than from the other d sub band. E$idence that our particular choice may correspond to reality is pro$ided by fig.; which shows the effect on the magneton number of

adding copper to nic"el. We add one extra electron with each copper atom because the atomicnumber of copper is larger by one than the atomic number of nic"el. /he desinty of electron

orbitals in the d band is o$er ten times greater than in the s band, so that the extra electron goes at

least ;& percent into the d band and less than & percent into the s band. /he ferromagnetic

magneton number is obser$ed to go to 6ero at about 1& atomic percent copper.

2t 1& atomic percent copper we ha$e added about &.7 electron per atom to the d band

and about &.&1 electron to the s band. 3ut &.7 electron added to the d band of fig. 1.'b will =ustfill both d sub bands and will bring the magneti6ations to 6ero, in excellent agreement with

obser$ations. /he distribution of electrons for 1& Cu Ii is shown in fig. &. 2t low Cu

concentrations the decrease in magneton number from that of pure Ii should be a linear functionof the concentration of copper, in agreement with fig.;.


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Figure >esinty of orbitals in copper !d and s bands. Calculations after C.K.Fong and @.L Cohen

+unpublished-0 Experimental, after W.E. )picer, in international colloquium on optical properties andelectronic structure of metals and alloys, F. 2beles, ed, Iorth 4olland, ;11.

For simplicity the bloc" drawings show the desinty of orbitals as uniform in energy. /he

actual desinty may be quite far from uniform9 fig. gi$es the results of a calculation for copper./he d band is characteri6ed by a high desinty of orbitals. /he desinty of orbitals at the Fermi

surface determines the electronic heat capacity and the pauli paramagnetic suscepbility0 in the

transition metals they ha$e higher $alues than in mono$alent metals.


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Figure @agneti6ation of a free electron gas in neighborhood of a point magnetic moment at the origin rA

&, according to the 5JJK theory. /he hori6ontal axis is "Fr, where "Fis the wa$e$ector on the Fermi

sphere. + after de genes-

Magnetic Alloys and the Kondo Effect

#n dilute solid solutions of a magnetic ion a nonmagnetic metal crystal +such as @n inCu- the exchange coupling between the ion and the conduction electrons has important

consequences. /he conduction electron gas is magneti6ed in the $icinity of the magnetic ion

,with the spatial dependence shown in fig.. /his magneti6ation causes an indirect exchange

interactions: between two magnetic ions, because a second ion percei$es the magneti6ationinduced by the first ion. /he interaction, "now as the 5JJK interaction, also plays a role in the

magnetic spin order of the rare earth metals, where the spins of the f ion cores are coupled

together by the magneti6ation induced in the conduction electron gas.

2 dramatic consequence of the magnetic ion conduction electron interaction is the

Kondo effect. 2 minimum in the resisti$ity temperature cur$e of dilute magnetic alloys at lowtemperature has been obser$ed in alloys of Cu, 2g, 2u, @g, 8n, with Cr, @n, Fe, @o, 5e and %s

as impurities, among others.

:2 re$iew of indirect exchange interactions in metals gi$en by C.Jittel, )olid state physic , +;1:-0 a re$iew of

the "ondo effect is gi$en by B."ondo, M/heory of dilute magnetic alloys, solid state physic !, : +;1;- and 2. B.

4eeger, MLocali6ed moments and nonmoments in metals the Jondo effect, M)olid state physic !, : +;1;-


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Figure ! 9 a Comparison of experimental and theoretical result for the increase of electrical ressti$ity at

low temperature in dilute alloys of iron in gold. /he resistance minimum lies to the right of the figure ,for the

resisti$ity increases at high temperature because of scattering of electrons by thermal phonons. /he experiment aredue to >.J..C. @ac >onald, W. 3. Pearson, and #.@. /empleton , Proc. 5oy.soc. +London- 211,1 +;1-9 the

theory is by B.Jondo,Prog./heo.Physics !, !' +;1-.

/he occurrence of a resistance minimum is connected with the existence of locali6ed magneticmoment on the impurity atoms. Where a resistance minimum is found, there is ine$itably a local

moment. Jondo showed that the anomalously high scattering probability of magnetic ions at lowtemperatures is an esoteric consequence of the dynamic nature of the scattering and of the

sharpness of the Fermi surface at low temperatures. /he temperature region in which the Jondo

effect is important is shown in fig.!. Io simple physical explanations of the effect hasappeared, but the original paper is quite accessible.

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