UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
THE ENERGY BAND STRUCTURE OF FERROMAGNETIC NICKEL By JOHN WILLIAM DOMVILLE CONNOLLY A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1966 DEDICATED TO MY WIFE ACKNOWLEDGMENTS I would like to acknowledge the help afforded by ny super visory committee, and in particular by Professor J.C.Slater, who suggested the problem upon which this dissertation is based. Thanks are due to many members of the Quantum Theory Project at the University of Florida,including Drs. S.O.Goscinski and K.H. Johnson for many piquant discussions, and especially to Dr. J.B.Conklin, Jr. for his assistance with the computer programs and constructive criticism of the manuscript. I would also like to express my appreciation to Prof. J.H. Wood of M.I.T. for his generosity in providing several computer pro grams, and to the University of Florida Computing Center for the use of their facilities. Finally, my continuing gratitude goes to my wife, Charlotte, whose patience and understanding were exceeded only by the excellence of her typing of the manuscript. iii TABLE OF CONTENTS ACKNOWLEDGMENTS . .... LIST OF TABLES . .. ..... LIST OF FIGURES . .. Page . iii S. .. vi . vii Chapter I. INTRODUCTION . .. 1.1. Purpose . . 1.2. Outline of Calculation . II. METHODS OF CALCULATION . . 2.1. The Unrestricted HartreeFock Method. 2.2. Exchange Approximations .. 2.3. Superimposedatom Potential . 2.4. The Energy Band Calculation . 2.5. The SelfConsistent Calculation . III. RESULTS OF THE CALCULATION . . e . . a . 1 1 3 8 8 12 18 23 31 35 3.1. The SelfConsistent SpinDependent Energy Bands in Nickel. . 35 3.2. Comparison with Previous Calculations 44 3.3. Comparison with Experimental Data . 53 IV. CONCLUSIONS. . . 63 Appendix I. APW MATRIX ELEMENTS . ... 70 II. RECIPE FOR THE CONSTRUCTION OF A LINEARLY INDEPENDENT BASIS SET . ... 75 III. APW CHARGE DENSITY. . 84 IV. DERIVATION OF THE COULOMB POTENTIAL FROM A GIVEN CHARGE DENSITY.. . .. 88 V. DETERMINATION OF THE DENSITY OF STATES 94 VI. NOTATION .................... 97 BIBLIOGRAPHY .................. .100 BIOGRAPHICAL SKETCH . . .103 LIST OF TABLES Table Page 2.1. HartreeFock oneelectron energy parameters for Fe (3dd)5 (3dJ)3 (4s)2 ... 17 2.2. HartreeFock oneelectron energy parameters for Ni+ (3dd)5 (3dp)3 . . ... 19 3.1. Coordinate grids in reciprocal space . 37 3.2. Comparison of energy differences for various potentials. 43 3.3. Detail at an Lpoint . .. 46 3.4. Results of Snow, Waber and Switendick for paramagnetic nickel . . .. 48 3.5. Charge distributions . . 52 3.6. Comparison of calculated and experimental quantities 62 A2.1. Frequency numbers for the fcc structure . 81 A5.1. A list of Kubic harmonics . .. 95 LIST OF FIGURES ure 1.1. Schematic diagram of energy levels in nickel . 1.2. Flowchart for the selfconsistent calculation . 2.1. Core polarization for the Ni4+ ion .... 2.2. Extrapolation of a new starting potential . 3.1. Convergence of the Lpoint energy values . 3.2. Selfconsistent energy bands for V3 (Vx = x) . 3.3. Selfconsistent energy bands for V4 (Vx ,V) . 3.4. Detail of the energy bands at an Lpoint for the two selfconsistent potentials . 3.5. Comparison of transition element energy band calculations . . 3.6. The density of states curves .. 3.7. Fermi surface crosssections for V4 .. A41. The Ewald problem . . Page . 5 . 7 . 20 . 33 . 38 . 40 . 42 . 45 . 50 . 55 . 58 . 89 vii Fig CHAPTER I INTRODUCTION 1.1. Purpose The properties of ferromagnetic solids have interested physicists for many years, and are of sufficient complexity that a comprehensive theoretical treatment is not yet a reality. The pur pose of this dissertation is to apply the unrestricted (or spin polarized) HartreeFock method to such a solid. This scheme has had a certain amount of success in the explanation of atomic properties, and, as will be shown here, appears to be an acceptable model for a ferromagnet. Experimental information on the metallic properties of ferromagnets, which are dependent on the electronic configuration, has recently become available, anrd with it some analyses to explain these properties on the basis of the energy band model. Of the ferromagnetic materials, nickel is perhaps the best understood. Energy band:,models have been presented, that explain empirically the available data on the electronic and optical properties of nickel. Also within the last five years, several papers have appear ed in the literature,510 which give energy band structures calcu lated from basic considerations, i.e., they present solutions for Schrodinger' s equation in a crystal using some form of oneelectron potential. This potential, in all cases, has been derived from an atomic calculation, corrected for the effects of placing the atoms on a crystalline lattice. Although these previous calculations are qualitatively quite similar, the arbitrariness of the potential used is enough to create differences between them which are large with respect to the experimental effects which are to be explained. The calcula tion described in this dissertation attempts to eliminate this dependence on an arbitrary potential by solving the equations self consistently, in the same way as the HartreeFock method used in atomic calculations. In this way, it is possible to examine the validity of the approximations which must be made in order to solve the equations. The case of ferromagnetic nickel turns out to be extremely sensitive to slight changes in these approximations. In particular, the form in which the exchange effects responsible for the ferro magnetic structure are inserted into the theory can radically change the final results. This effect was not pointed out in previous calculations on nickel or other materials, either because the solu tions were not carried to selfconsistency or because the particu lar case was not sensitive enough to show a definite discrepancy with experiment. In a sense, then, a calculation on nickel is a test case, which examines the validity of an approximation which has been used in many calculations of this type. The procedures used in this work are outlined in the following section, and described in more detail in Chapter II. The results and comparison with the 'experimental data are presented in Chapter III followed by the conclusions and discussion in Chapter IV. 1.2. Outline of Calculation The oneelectron model for a crystal consists of the assump tion that the electronic wave functions satisfy a Schrodinger equa tion of the form ( v' VU ) *C/(f^U c = k(c^ (1.1) where V(r) is a potential identical for all electrons, and includes; (i) an attractive term due to the nuclei situated on the lattice sites, (ii) a repulsive term due to the Coulomb interaction with the other electrons, and (iii) an attractive term which similutates the exchange effects of other electrons. As a first approximation, V(7) can be generated from the assumption that the electronic wave functions are unchanged from their atomic values. The atomic potentials on the appropriate crystalline lattice sites are then overlapped to form what has been called a "superimposedatom potential". This, of course, makes V(?) dependent on the particular atomic configuration chosen. Although the most logical choice would be the ground state configuration of the atom, this is not always the best choice. It is true that the wave functions do not change greatly on going from a free atom to a crystalline environment, but the effective occupation number of each type of orbital (s,p,d, etc.) may change. The reason for this is that the discrete atomic levels which the electrons occupy in a free atom broad en into bands when these atoms come together to form a solid. If these bands happen to overlap, as they do in many metals, then electrons in one band will "spill over" into the other, thus reducing the effective number of the first type in favor of the second. This occurs in the case of nickel (figure 1.1.), in which the overlap of the 3d and 4s bands alters the configuration such that the effective number of "dlike" electrons is increased from the atomic ground state value of 8, to approximately 9 in the solid. However, in a selfconsistent calculation, where the potential is regenerated after each iteration, the arbitrariness in the choice of V(?) disappears. The superimposedatom potential is used only as a starting point for the selfconsistent procedure. Once the initial potential is chosen, the energy bands are calculated for a selected number of vectors in reciprocal space by means of the augmented plane wave (APW) method.11l13 This method has beensufficiently developed and checked out against other methods 1416 so that it gives a solution (for a given potential) as accurate as desired. In its simplest form, the APW method solves equation (1.1) for a V(f) of the "muffintin" type, i.e., spherically symmetric in spheres about each lattice siteand constant between these spheres. This is not a necessary restriction, 17 but it can be removed only at cost of much greater complexity, without much improvement in the final results. The energy eigenvalues obtained in this way are then used to find the associated wave functions. The total charge density is Cf I z d) c 4. 0  I LJ *o 0> SJ lit) generated from these wave functions, by summing over the occupied levels. The procedure for forming the new muffintin potential to 17, 18 be used for the next iteration is straightforward, 1718 with a certain amount of care necessary in the choice of the intersphere constant. The iterations are continued, going through the above manipulations each time, until there is no significant difference between the energies obtained from two sucessive iterations. Each of these steps is described in detail in Chapter II, and the entire calculation represented by the flow chart in figure 1.2. ASSUMED ATOMIC CONFIGURATION UNRESTRICTED HARTREEFOCK PROGRAM FOR ATOMS ATOMIC ORBITALS SUPERIMPOSEDATOM POTENTIAL (MUFFINTIN FORM) NONREIATIVIS' APW PROGRAM ENERGY BANDS RECIPROCAL LATTICE VECTORS (DERIVED FROM GROUP THEORY) & WAVE FUNCTIONS SPHERICALLY AVERAGED CHARGE DENSITY I NEW MUFFINTIN POTENTIAL (FROM EWALD METHOD) SELF CONSISTENCY Figure 1.2. Flowchart for the selfconsistent calculation 1 CHAPTER II METHOD OF CALCULATION 2.1. The Unrestricted HartreeFock Method The HartreeFock formalism consists of approximating the wave function for an Nelectron system by a single antisymmetrized product of one electron functions, ui of space and spin coordinates xi = ('r, si) (2.1) For a Hamiltonian consisting only of electrostatic interac tion and kinetic energy terms, the variational principle leads to N coupled integrodifferential equations of the fbrm 19 L4. J i1i' .a=l J l/ sps] i(, ,,12. 11] [I/ Sps [ i +, 2, .E. =1 J (2.2) where is the oneelectron kinetic and nuclear potential operator. The second summation is restricted to oneelectron states (j) with spin parallel to state (i), whereas the first is over all the elec trons including the ith state itself. 1E _== 4 at(x,) UL(X ) ... tLI(Xy . These equations can also be written in the form [ V ,() + Ve ()] U i) (2.3) where Vn ( i) = (atom) 2? Z, (solid) is the nuclear potential term, v ( F! d is the Coulomb potential, and S(S;,, ) A I v/ 1 /"] [spin sJ is the "exchange potential". Note, that Vn and V are "local" poten c tials, i.e., they appear in the equations as a multiplicative ope rator acting on ui, but Vx(s) is not. It is this peculiarity of the equations which causes the greater part of the difficulty in their solution. The exchange potential can be expressed as a local operator, but with the result that it will have a different form for each orbital: [II spins] The equations are solved by an indefinite number of iterative steps. Some initial set of ui's is chosen, Vc and Vx are generated from them, and the resulting differential equations are solved for a new set of ui's. The procedure is then repeated for this new set until the final ui's are the same as the initial set (i.e., "self consistency" is achieved). Fortunately, for an initial set of ui's close enough to the selfconsistent solution, this process is conver gent. Because of the difficulties involved in solving the three dimensional integrodifferential equations (2.3), their solution involves a major computational effort, and calculations involving heavy atoms are few. In practice, the HartreeFock formalism is often modified by assuming the potentials in equations (2.3) are spherically symmetric. (Here we are referring to the case of a free atom. The equations for a solid will be discussed in subsequent sections of this chapter.) This allows us to separate the wave function in the form x = P .^ (2.5) J. where Y is a spherical harmonic with angular momentum quantum numbers t and m and (s) is a spin function with spin quantum number, s = 1 This one assumption (known as the central field approximation) effects enormous simplification in the equations, in that all integrals and differential equations become onedimensional. In particular (2.3) reduces to S + P) = (2.6) ;, I^ Ci 9 L L where VC 2Z 4 fl p() dk + 2 J,2 (rq' j .? 7 o k The exchange potential Vx(r) is slightly more complicated than Vc(r) involving an extra summation over integrals which contain Pi(r) as well as P (r). (See reference 19, page 17.) In general, Pi will depend on 1, m, and s, as well as the principal quantum number n which labels the eigenvalues of (2.6) in increasing order. In the socalled "conventional" HartreeFock scheme, Pi is assumed independent of m and s, resulting in a degene racy of 2 (21+1) for each energy eigenvalue, thus reducing the number of equations to be solved. This has the advantage of giving a final wave function (2.1) which is an eigenfunction of the total angular momentum L2 and total spin operator S2. However, this wave function may give entirely false results in interpreting results in which spin effects are important. For example, consider a transition element, for which the ground state has two 4s electrons and an unfilled 3d shell. The conventional HartreeFock solution in this case would have a zero spin density(i.e., the difference between the charge densities of opposite spins) near the nucleus. But the inter pretation of the hyperfine structure of an atom with nonzero nuclear spin demands that the spin density be nonzero. This sort of effect is the justification for the use of "unrestricted" HartreeFock sbheme.20'21 The term "unrestricted" refers to the relaxation of the restriction that Pi be independent of s. The number of equations to be solved is doubled, giving a (21+1) fold degeneracy for each ns. For a configuration which has an unequal number of up and down spins, such as an unfilled 3d shell, the relaxation of the constraint will result in a nonzero spin density. This desira ble feature of the wave function is gained at the sacrifice of the symmetry requirement that (2.1) be an eigenfunction of S2. However, 22 it has been shown that the unrestricted wave functions are in substantially better agreement with experimental data. It also should be noted here that the unrestricted solution can be made an eigen function of S2 by the application of projection operators, 23 but such a complication is beyond the scope of the present calculation. Both the conventional and the restricted forms of the HartreeFock must be regarded as approximations whose error is impossible to esti mate without doing the actual exact calculations. Since these are intractable at the present time for atomic systems with more than a few electrons, appeal must be made to experiment to judge the use fulness of the different forms. In particular, as will be seen, the conventional scheme is much less suitable than the unrestricted for the case of a ferromagnetic solid. 2.2. Exchange Approximations The exchange potential of equation (2.4) for a particular orbital ui involves a sum over all electrons uj of spin parallel to that of ui, of integrals involving both ui and uj. Looking at the situation from a oneelectron point of view, this means that each electron moves in a different effective potential. For a large number of electrons, this results in formidable computational difficulties even when the central field approximation is used. These difficulties are not insurmountable for a free atom, but are virtually impossible to overcome for the case of a solid. Also, we are interested in solving the problem within the framework of the oneelectron model, in which all electrons move in the same effective potential. This simplification has enjoyed a great deal of success despite its obvious limitations, and represents the present limit of complexity to which calculations in solids can be carried. What is clearly indicated, then, is the formation of some kind of average exchange potential which will retain the main fea tures of equation (2.4) One such average has been suggested by Slater, 24i.e., 'C 2' u.(i)Au (2.7) [II sp ,s] i,/ which is simply the sum of Vx weighted by the probability factor, There is one particular case for which the expression (2.7) can be evaluated exactly, viz., the freeelectron gas. If the free electron wave functions ui= exp(dii~ ) are substituted into (2.7), the integral can be easily performed, 19 and the sum over i up to the Fermi level gives X )(2.8) where is the constant electron charge density (in units of the electronic charge). In an atom, of course, the electrons are far from free, and the charge density is not a constant, but if we take the extreme step of assuming the exchange interactions between electrons in an atom ( or a solid) depend only on the local electronic charge density, then a function is obtained which turns out to be remarkably similar to (2.7). Explicitly this function is V) Y 11] (2.9) In at least one case, that of the ion Cu+, a detailed comparison of the approximations (2.7) and (2.9) with the exact HartreeFock expression (2.4), has been worked out. 25 The results show that the Slater average (2.7) and the freeelectron average (2.9) are practically idential. At least, the deviations of (2.7) from (2.9) are much less than the deviations of either from the HartreeFock value (2.4). The point to be drawn here is that there is no advantage in using (2.7) instead of (2.9), and that if one is forced by practical considerations into using a local exchange poten tial, then the errors involved can be ascribed to the averaging pro cedure rather than the use of the freeelectron approximation. Recently, Kohn and Sham 26 by the use of a different averaging scheme have concluded that (2.9) should be multiplied by a factor of 2/3. Calculations on various atoms 27,28 have shown that perhaps this is closer to the truth, in that the resultant wave functions are more similar to the HartreeFock values. Lindgren 29,0 by using a parameterized oneelectron exchange potential and minimizing the correct total HartreeFock energy with respect to these parameters gets amazingly good agreement with the best HartreeFock values. This "optimum" potential, he finds,31 is always smaller in magnitude than (2.9). The consensus appears to be that, although the free fe electron approximation Ve reproduces the general features of the HartreeFock scheme, it overestimates the exchange effects. The extension of this approximation to the unrestricted case is straightforward. In the course of the derivation of (2.8), half of the electron gas was assumed to have up spin, and the other half down spin. This introduces an extra factor of 2 in equation (2.9) when we replace the total charge density by that of only one spin. The effective exchange potential is then X4 GP) r, )1 (2.10) where the summation now contains the orbitals of one spin only. This is the approximation used in this calculation. Substitution of (2.10) into (2.6) gives the actual equations to be solved: where V () = 2 f(r)d/ 1 2 d 0 ,^ ) = P )/ r7 s = , (2.11) f'(')= ^ <^ r} + t (r) Wnas is the number of electrons in the configuration with quantum numbers n and f, and spin s. These equations are exactly the same, with the exception of the added spin subscript, as those used by Herman and Skillman in their atomic calculations.32 Their program, which was published with their calculations, was modified for the unrestricted case during the course of this work. As a test case, this program was used to find the wave function for the Fe atom in the configuration (3dd)5 (3dl)1 (4sd)1 (4sf)1, in order to compare with the results of Wood and Pratt,21 who did exactly this calculation. A comparison of the oneelectron energies is shown in table 2.1. The energy values in the unrestricted case tend to bracket those of Herman and Skillman and the splitting are larger for the n=3 electrons than for the other shells, as would be expected. Apart from a large discrepancy (almost certainly an error in their paper) at the ls level, the results are comparable to those of Wood and Pratt. The results are not identical in that their calculation was not taken out to as great a degree of selfconsistency. At first glance, the discrepancy between the energies calcu lated using the free electron approximation and the exact Hartree Fock values seems discouragingly large. However, this does not mean that the corresponding wave function is inaccurate to the same degree. As Lindgren30 has pointed out, Koopman's theorem no longer holds when the HartreeFock equations are solved approximately, and therefore the oneelectron energies no longer have the same Table 2.1. HartreeFock one electron energy parameters for Fe (3do)5 (3df)3 (4s)2 Exact Approx. Approx. Apprx. CHFa CHFb UHFc UHF Is 522.7e 515.8 584.5 584.2 515.8 515.8 2s 63.84 60.96 61.08 60.40 61.03 60.75 2p 54.79 53.08 53.16 52.65 53.12 52.91 3s 8.308 7.269 7.463 6.930 7.492 6.929 3p 5.455 4.891 5.061 4.540 5.111 4.558 3d 1.271 0.963 1.122 0.664 1.168 0.664 4s 0.510 0.545 0.532 0.428 0.592 0.491 a Exact conventional HartreeFock values, taken from Watson33. b fe Approximate conventional HartreeFock values (Vx = Vx ) taken from Herman and Skillman32. c Approximate unrestricted HartreeFock values (Vx = V ) taken from Wood and Pratt21. d Approximate unrestricted HartreeFock values (Vx = Vf) calcu lated during the course of this work. e All energies are in Rydbergs. significance of being equal to the binding energies. By applying the appropriate corrections, he has shown that binding energies can be extracted from the approximate solutions which are in much better agreement with the exact values. To further examine the method, a calculation was done for the ion Ni+ in the configuration (3dU)5 (3df)3 for which an exact 20 unrestricted HartreeFock solution is available. The results are shown in table 2.2,both for the standard exchange and for one 2/3 as large. Note that the discrepancies with the exact energy values are larger in the latter case, but that core polarization (figure 2.1) is in better agreement. This serves as an illustration of the fact that inaccuracies in the oneelectron energies are not necessarily indicative of inaccuracies in the wave function itself. 2.3. SuperimposedAtom Potential In the preceding section we have shown how to obtain a oneelectron potential valid for electrons in an atom. We are now faced with the problem of how to extend this method when the atoms come together to form a solid. Suppose we make the rash assumption that the electron charge density in the solid is unchanged from the atomic values. If J (r) is the spherical charge density of spin s in the atom, derived from a selfconsistent calculation, then the charge density in the solid (A ) is just a sum, (2.12) i Table 2.2. HartreeFock oneelectron energy parameters for Ni (3do()5 (3df)3 Exact UHFa Approximate UHFb Vx SO Vx 3 x Is 612.6 612.6 605.3 605.3 597.2 597.2 2s 77.30 77.15 74.20 74.04 72.17 72.08 2p 67.35 67.21 65.47 65.35 63.27 63.20 3s 11.35 10.96 10.19 9.901 9.477 9.294 3p 8.200 7.725 7.445 7.158 6.762 6.583 3d 2.898 2.705 2.782 2.519 2.266 2.111 a Exact unrestricted HartreeFock values, taken from Watson and Freeman.20 b Approximate unrestricted HartreeFock values, calculated during the course of this work. c All energies are in Rydbergs. o oN o o < I I 0 C.. 0 r4 a U) 0~ where ai is the position vector of the ith lattice site. We consider a particular case here, that of only one atom in a unit cell The Coulomb potentials defined in (2.11) will also combine linearally; v^?V C ( (2.13) where Vc(r) is the atomic Coulomb potential and the exchange is =' 4 (2.14) For computational reasons, which will be made clear in the next section, we wish to have a potential which will have the so called muffintin form, i.e., spherically symmetric in nonoverlapping spheres of radius R about each lattice site, and a constant between these spheres. To find the spherical average <(J> of the charge density (2.12), we integrate over all angles to get < > n, = 1 j 2, J si+ (L 2,.S i. 0 ai + (2.15) where ni is the number of neighboring atoms at distance ai. The spherically averaged Coulomb potential the spherically averaged exchange potential T( . The constant intersphere potential V0s is defined by S / .A 0 where 12 = J.I L R3 is that part of the unit cell which 3 lies outside the spheres. This can be evaluated in terms of the spherically averaged quantities by splitting up the integral over V, (r) into separate parts over. and the sphere J, i.e., = 4 T Jf4Ve(r)& + e' ( v'>d . 0 1 Finally we have o f,, (2.18) a 0 6 00 is the charge of sin in the volume J2 and is the fraction of electrons with spin s. This scheme affords a straightforward method of deriving a oneelectron potential suitable for a solid from the results of a selfconsistent atomic calculation. The sum over neighbors in equa tion (2.15) typically converges after two or three terms, and the integrals involved, being all onedimensional, present no difficul ties. The sphere radius R is arbitrary, but better convergence in the energy values is obtained in the solution of the energy band problem if R is taken as large as possible, i.e., equal to onehalf the nearest neighbor distance. For the face centered cubic structure with one atom per unit cell, R= a/242, where a is the "lattice parameter", the distance between second nearest neighbors. There are two basic difficulties with the potential descri bed here: (1) The neglect of the nonspherical terms in the potential. This can be remedied by adding them in as a perturbation after a solution is found. However, if these terms are not negligible, formi dable complexities are introduced into the solution. (2) The assumption that the charge density is unchanged from its atomic value is not always valid. The orbitals will certainly be distorted to some degree by the influence of neighboring atoms. This would be much more noticeable in a metal, where the wave functions have a substantial overlap, than in an insulator where the charge density is more confined. It may be possible to choose an atomic configuration other than the ground state which will generate a charge density close to the actual charge density in the solid, but this is impossible to determine nonempirically. However, this problem will not concern us in this work, since we will be finding a selfconsistent solution. The superimposed / atom potential as described here is used only as a logical starting point for the first iteration. 2.4. The Energy Band Calculation Within the framework of the oneelectron model for a solid containing N atoms and ZN electrons, the total electronic wave func tion is an ZN by ZN determinant of functions 4i, each of which satisfy an equation of the form [ V2 + V ( )] VL = E 4 (2.19) where Ei are the allowed energy levels. The potential V(P) includes the interactions with the N nuclei, which are assumed to be in fixed positions on the lattice sites, and simulates the interactions with all the other electrons. Effectively, the problem of solving the Schr8dinger equation for the (Z+1)N particles has been replaced by an easier one, in which the ZN electrons are pictured as noninter acting particles moving in a fixed potential. In this picture, the Fermi statistics hold, so that the ground state of the system is obtained by choosing the solutions of (2.19) corresponding to the lowest ZN values of Ei. For a solid with a ferromagnetic structure, the problem is generalized slightly, in that the electrons are divided in two separate systems, according to their spin, which interact differently with their environment. In the oneelectron model, this means solving two equations: [V V )]E (2.20) The lowest ZN eigenvalues of the combined set of solutions will represent the ground state. Since the potentials V. and Vg are unequal, the two sets Ei and Ej will be different. This will, in general, imply an excess of one spin over the other in the ground state, which is found experimentally in a ferromagnetic material. Once this model has been chosen, we are left with the pro blem of solving the equations (2.20)..This can be made feasible by the application of the symmetry properties of the Hamiltonian de =  V+ Vs(r). First, the translational symmetry of RL allows us to choose the eigenfunctions i*(r) to be Bloch functions, i.e., to satisfy the relation l( Jt ) = e ') (2.21) where aj is a lattice vector, and k is a real vector which takes on N values corresponding to the N onedimensional representations of the translation group. In the language of group theory, a Bloch function is a basis function for the kth representation of this translation group T. The reciprocal lattice vectors g are defined by the rela tion lw n (2.22) where the nij are integers. If Ik> is a Bloch function belonging the kth representation of T, then it follows from (2.21) and (2.22) that iZ+ i> also belongs tothe same representation, and further that I and Ik') belong to the same representation if and only if kk' is a reciprocal lattice vector. We define the Brillouin zone as that set of N vectors ko in kspace satisfying the relation: Sj T V Tr (2.23) Every vector in kspace is equal to the sum of a "reduced" vector to (in the Brillouin zone) and a reciprocal lattice vector. Suppose, we have available a complete set of Bloch functions ( > corresponding to all possible vectors in kspace, then 4i can be expressed as a linear combination of these functions. But, if 4L. is to satisfy (2.21), then only those functions which belong to the th representation will contribute to it. Therefore lt. can be expres sed as a linear combination of that subset of the complete set which has the same reduced vector, i.e., SCo (2.24) is the most general eigenfunction corresponding to the irreducible representation associated with the reduced vector ko. A secular equation for a particular ko can now be set up by substituting (2.24) into (2.20): ^ < i?,r le E_!,T) > C j/ = 0 < kb+ + E( o0)I 0i, '> C.+ o (2.25) and the energies E(ko) can be found by the solution of the determi nantal equation I h E( o) Aj  0 (2.26) where = < and A = o The order of the determinant in (2.26) will be equal to the number of lattice vectors needed to give convergent values for E(ko). This, of course, will be dependent on the basis set chosen. For example, the plane waves, ei"' form a complete set of Bloch functions, but are quite unsuitable for a problem in which the potential is atomiclike. For a "muffintin" potential, as described in Section 2.3, suitable basis functions are the augmented plane waves 1 (APW'S) defined by I e outside the spheres, Z mt u E) Ylm(g9)? (2.27) inside the spheres, where Ymare spherical harmonics, iU satisfy a radial equation corresponding to energy E, and the a1 are constants defined so that Ik)is continuous at the sphere radius r = R, (see Appendix I). These satisfy the requisite Bloch condition (2.21), and the matrix elements required for the secular equation (2.26) are easily derived. For reference, these elements are listed in Appendix I, equations (AI16,17). The translational symmetry used to derive the secular equa tion (2.26) is only part of the total symmetry of a crystal. In gene ral, the Hamiltonian is invariant under a space group, whose elements are combinations of translations and rotations (either proper or improper). The translation group is always a subgroup and in certain cases, called symmorphic, the space group can be expressed as a semidirect product of the translational subgroup and another subgroup called the point group, consisting only of rotations. This is the case for the facecentered cubic structure, for which the point group is the full cubic group Oh. The representations for a symmorphic space group are parti cularly simple, and it can be shown that a basis function ikoT of the translation group can also be chosen to be a basis function for that subgroup of the point group which consists of elements transforming the vector ko into an equivalent vector. This subgroup, known as the group of the reduced vector ko, can be defined mathemati cally as (0 =I^ (2.28) where are the rotational elements of the space group. A basis function for this group can be manufactured from (Tk by the application of the standard "projection" operator oiI = T ^^z^ (2.29) where p F) is the (I,J) matrix element of the pth representation of f(. The projected function, li? r > O (2.30) will transform as the Ith row of rp, i.e., 3tlEPt> = ^ 00^t) ( I^pIW>. (2.31) It can be shown 3536 that the effect of operating on a Bloch function I1o such as an APW, with the rotation a is to transform it into another Bloch function corresponding to ko Therefore the projected basis function is S(2.32) The projection operators (2.29) satisfy the orthogonality relation P Q P QTL 0MN In P LM TN ? (2.33) where G is the order of 1(2) and n is the dimension of r . 0 p p Because of the relation (2.33), any operator A which is invariant under all operations of(fk) has zero matrix elements between projected functions belonging to different representations, or to different rows of the same representation. It follows directly from (2.33) that < 'po P I3" A I AP Y i 'pp"< i' (2.34) The requisite overlap and energy integrals are found by substituting A=I (the identity operator) and A=~Prespectively. The explicit form of these matrix elements between projected AFW's is shown in Appendix I. If the projected functions Iko PIJ are chosen instead of I k then the secular equation (2.26) is considerably simplified at symmetry points, i.e., where (k)~ contains more than one element. The orthogonality of equation (2.34) has the effect of "splitting" up the secular equation into blocks, one for each combi nation of P and I, which can be solved separately. Also, since the right hand side of (2.34) is independent of the row I, then we need solve for only one row of each representation. Thus for each P we have a smaller secular equation: P P Ole 7.;J' 3 1J'l= (2.35) where O= K P I IE > P 0 a ' < o l cer  The projected functions (2.30) at a symmetry point are not all independent. In Appendix II is given a recipe for selecting out of all the functions Iko + P I J >, a linearally independent set. In practice, the energies E(ko) are calculated at a few symmetry points, because of the saving in time afforded by the use of projected functions, and some kind of interpolation scheme is used to find the energy bands over the whose Brillouin zone. It is a straightforward matter for a computer to calculate the matrix elements (equations AI16 to 22) and diagonalize the secular equa tion. There are two infinite sums involved (over the values, and the lattice vectors 1), but empirically it is found 14 that, for a metal, 61 values and 50 lattice vectors (at a nonsymmetry point) are enough to give convergence in the energy values to less than .001 Rydberg. This means that one never has to diagonalize a matrix larger than 50 by 50, and at a symmetry point, or if less accuracy is desired, considerably less. 2.5. The SelfConsistent Calculation In the preceding section, it has been shown how the energy eigenvalues and eigenfunctions can be found from a given potential, ( or two potentials as in the ferromagnetic case). If a selfconsis tent solution is desired, there remains the problem of generating, from the calculated wave functions, a new potential to be used on the subsequent iteration. When an approximate exchange of the type given in equation (2.8) isused, the oneelectron potential is a functional of the charge density (t) V,(?) = VC [] + V [j] (2.36) where Vx is the particular exchange approximation to be used, and Vc is the Coulomb potential satisfying Poisson's equation, V Vc j oJ (atomic units). In the computations reported here, the charge density is assumed to be spherically symmetric in a sphere about each lattice site, and constant between. The departures of the actual charge density from this simplified model can be handled as perturbations on the final solution, but only at a cost of considerable complication. In Appendix III it is shown how to obtain an averaged charge density of this type from the occupied wave functions. The quantities to be calculated are closely related to those already found in the expression for the overlap integral used in the secular equation. Taking advantage of this fact simplifies the work to be done. The formation of the Coulomb part of the muffintin poten tial from the averaged charge density can be carried out, using Ewald's technique for handling the lattice summations. This is out lined in Appendix IV, the result being v, 2"(rlOrd t +i(t) dt Sv (2.38) \ t= 2 c'/ , where Qo= Jp(r) dr is that part of the charge density found outside the spheres, a= 22 R is the lattice parameter and c'2.41583 is a constant characteristic of the fcc structure. The superscripts on Vc indicate the potential inside and outside the spheres. Equations (2.38) give us a straightforward way of generating a new potential, once the spherical average of the charge density has been computed using the formulas of.Appendix III. This can now be used as the input to the energy band calculation. However this pro cedure very often leads to a divergence, i.e., the difference between the potentials of successive iterations increases. As in atomic calcu lations, it is found that convergence is usually obtained by the use of an extrapolation procedure. The simplest of these is the arithmetic average. If (Vin, Vout) is the initial (muffintin) potential, and (VjC Vout) is the iterated potential, then we take the new potential to be (V i V ) defined by v.^ yrvi. + V 1 i[ de ie d(2.39) Vo.t= y[ v.,, vo.] Even better, empirically, is a scheme invented by Pratt 37 and used to advantage by Herman & Skillman 32 in which the results of two iterations are used to extrapolate a new starting potential. This is best illustrated by a diagram (figure 2.2) ITERATED POTENTIAL / II I V I I I I I I I I I I I INITIAL I POTENTIAL Figure 2.2. Extrapolation of a new starting potential In the figure, the abcissa represents the starting potential, and the ordinate, the potential obtained after the iteration by equation (2.38). Each calculation is therefore represented by a single point on the diagram. The next starting potential in this procedure is chosen at the intersection of the line joining the points repre senting two successive iterations with the 450 line. If (V1 VI) and (V2 +V2) are the initial and iterated potentials for the two iterations, then the new potential V3 is defined by the equations; V ut potV t / out 3Jui P I J L Poot 1 Pout = [1V '< o /[ t V.j] /n of 0 0 u Pin [V ] V1 Vo t [V ~V t] Vi VI ou (2.40) (Since the zeropoint of the potential is arbitrary, the extrapola tion procedure uses the potentials with respect to their constant value between the spheres.) It was found, during this calculation, as Herman and Skillman 32 had for atoms, that quick convergence is obtained if the two schemes (2.39) and 2.40) are alternated. CHAPTER III RESULTS OF THE CALCULATION 3.1. The SelfConsistent SpinDependent Energy Bands in Nickel: The selfconsistent energy eigenvalues were computed for two values of the exchange potential: (i) the averaged free electron value a= 6(6g/ 8r ) and (ii) the value suggested by Kohn and Sham 26 = 2/3 *t The initial potential chosen was a superimposed atom potential, as defined in Section 2.3, generated from selfconsis tent atomic orbitals corresponding to the configuration 3d= 5.0, 3dP= 4.0, 4sg= 4sf= 0.5 . The value of the lattice parameter used in the calculation 38 is the one quoted by Wyckoff, i.e., a = 6.6586 atomic units. The radius of the sphere about each lattice site corres ponding to this value is R = 2.3544 atomic units. The summations over 1 in the matrix elements (AI 26) were taken up to a value of 1 = 6 and the maximum reciprocal vector magnitude ( which determines the size of the secular equation) was K = 61T/ a. By calculating the max energy eigenvalues for a few points for larger values, it was found that the chosen values, of emax and Kmax were sufficient to insure a convergence in the energy values of less than .005 Ry. The. "dlike" states tend to converge more slowly than the "s and p like" states, because of the large negative values of the logarithmic derivatives for i= 2 This implies that the APW basis functions corresponding to the dstates will have a large discontinuity in their derivatives, and more basis functions are necessary to "smooth out" the wave function. The eigenvalues and the corresponding wave functions were calculated at the 32 points of a cubic mesh in reciprocal space (grid #1 of table 3.1) during the course of each iteration. Because of symmetry, actually only 6 points were calculated, and the corresponding charge densities were multiplied with the appropriate weights to derive the total charge density. For the last two itera tions of each calculation, the mesh was reduced to half the former size, for an equivalent of 256 points (grid #2 of table 3.1). It was found that 6 or 7 iterations were sufficient to achieve a convergence of the energy levels to within .01 Rydbergs. More accuracy than this was deemed unnecessary, because of the uncertainties in the oneelectron potential and the approximations inherent in the muffintin form of this potential. The superimposed atom potential used for the first iteration turned out to be quite far from the selfconsistent solution, with the result that the first two iterations appeared to be diverging (See figure 3.1). However, the extrapolation procedure described in Section 2.5 was sufficient to give eventually the selfconsistent solution. In each iteration, after the energies were calculated, the Fermi energy E was estimated by counting the states of both spins Table 3.1. Coordinate grids in reciprocal space. Grid #1 k*(a/7r) 000 001 002 011 012 111 (32 equivalent points) Symmetrya Weight Grid #2 (256 Symmetry Weight equivalent points) k*(a/2TT) Symmetry Weight a The symmetry symbols are those of Bouckaert, Wigner.39 Smoluchowski and k*(a/2wr) 000 001 002 003 004 011 012 013 014 022   SI I p it I ..>  0 n:1 N II ICn .t o o 0 m  0 d o CA 0O 0o f 103d'S3 HIM I I I I I I I I I I I o (AU) 0 d A9O3N3  0 o o  > N in order of increasing energy until 10 bands were filled, corres ponding to the 10 valence electrons of.the nickel atom. For example, if the energies had been computed at the 32 points of grid #1, then the lowest 320 states were chosen as the occupied levels for the generation of the next potential. Because of this procedure, the configuration (i.e., the effective number of s,p, and dlike states) can change from iteration to iteration, unlike the usual atomic self consistent calculation, in which the configuration is held fixed. This might be expected to lead to a "collapse" of the energy bands, i.e., the number of occupied levels of the minority spin might tend to increase with each iteration until finally it has been become equal to that of the majority spin. In the course of the calculation, it was found that the difference between the number of electrons of either spin did decrease slightly, but levelled off at a constant value. This may be a fortuitous result of using a finite number of points in the Brillouin zone to calculate the density but the fact that this constant value is close to the experimental is encouraging. The calculation was carried out on an IBM 709 computer. The time taken per iteration was approximately one hour for grid #1 (32 points in the Brillouin zone) and three hours for grid #2 (256 points in the Brillouin zone). Figure 3.2 shows the final bands for the first calculation (denoted by V3) in which the exchange potential was chosen to have the averaged free electron value, the averaged free electron value, x* 40 ,1 . mll I % !/ / 1 '' 1 a Ia S' 0 .r    ^ c ^ II / ., 1 I I i m in o So o o S   I JII Ip p 0 d d d 0. N I I ID U) 0i 0 0 6"A 01 .L dS3HI HLIM (ANL) AkO3N3 n Here, the energy values are plotted along the three main symmetry directions in reciprocal space: A: r o,oo) X x(, o) A: P L( Figure 3.3 shows the same information for the second calculation (denoted by V4), for which Vx = 2/3 V e Both figures have the same general shape, which is essen tially determined by the symmetry of the facecentered cubic lattice. Both show the feature, typical of the transition metals, of a narrow ( .3 Ry.) "dband" imbedded in a much broader (^.8 Ry.) "sp" or "conduction" band. The splitting between the two spin bands is not uniform throughout the zone, but is approximately .07 Ry for the dlike states and zero in the conduction states. The principal difference between the two sets of results is in the width and position of the dbands. These are considerably wider (30%) and higher ( 1.15 Ry.), with respect to the conduction band, for V4 than for V3 (See table 3.2). The reason for this is that reducing the exchange effects elevates the potential by an amount proportional to ./3. Therefore, the energies of the dtype electrons, which are mostly confined to the higherdensity interior of the atom, are raised more than the energies of the conduction band electrons, which are found further out where the density is less. This elevation of the dbands tends to spread out the wave functions thus increasing a, Of o a s I I I  II I I I I I O (0 oU) 1 N j o o o d do d d o d DOA 01 1J03dS3U HIM C(A) AOU3N3 rf 0) ,.. ct C.. Table 3.2.Comparison of energy differences for various potentials. Potential V1o(b V113 V2qc V2P d V3 V4e V4 vB v54 V5fl ds separation .481a .627 .537 .687 .437 .491 .322 .403 .478 .542 .488 .535 .571 .633 .441 .531 .625 .698 .639 .695 d width XFX, .320 .349 .299 .325 .251 .282 .330 .362 .347 .372 sp width X/ _pr  .837 .838 .836 .837 .833 .834 .841 .842 .865 .866 .070 .070 .070 .076 splitting .016 .013 .005 .060 .004 .067 .018 a The energy differences are given inRydbergs. b V =a superimposedatom potential with Vx = V, corresponding to the atomic configuration (3dc = 5.0, 3d = 4.4, 4sq = 4s = 0.3). c V = a superimposedatom potential with Vx V=x, corresponding to the atomic configuration (3do = 5.0, 3d = 4.0, 4so(= 4s, = 0.5). V3 = the selfconsistent potential with Vx = Vx. e V4 = the selfconsistent potential with Vx = (2/3) Ve. V5 = Wakoh's potential of reference 10. the dband width. Although the general shape of the bands is quite similar, the change in the position of the dbands with respect to the con duction band is sufficient to alter some of the details. For example, at the L point (fT/a, '/a, T/a) there are two doubly degenerate dlike levels Lo, and two singly degenerate levels: L ( a mixture of s and dlike states ) and L2 (which has plike symmetry). Figure 3.4 shows the bands in the region around this point. A is the line from r (0,0,0) to L, and Q is the line perpendicular toAon the face of the zone, going from L to W (ir/a,2ir/a, O ). It can be seen from this figure that the energy bands are substantially different in the two cases, the plike level I4 being much lower with respect to the dbands for Vq than for V3. This is particularly significant in that the position of the Fermi level with respect to the conduction band, and consequently the Fermi surfaces, are considerably altered by the shifting of the 12 level. Table 3.3 lists the ordering and the resultant Fermi surfaces, and shows that the reduction of the exchange effects has qualitatively changed these surfaces. The experimental data (See Section 3.3) favor the second calculation (VN), because of the multiplyconnected Fermi surface in the sixth Oband. This surface is similar to the famous copper Fermi surface, having "necks" along the (1,1,1) direction. 3.2. Comparison with Previous Calculations: Within the last five years, several calculations or estimates of the energy band structure of nickel have appeared in the literature. 45 V3 v3 0.6 L2  A, 0.5 Ef L3 Q  0 0.4  Q a 0.3 L SI I I I I Li L3 I V4 a V4 0.7  SCDL __ __ iL w0.6 A3 La A3 SL QL3, Q . 0.5 Q 0.4  Figure 3.4. Detail of the energy bands at an Lpoint for the two aelfconsiatent potentials Table 3.3. Detail at an Lpoint. Potential Ordering at Resultant shape of Fermi the Lpoint surfaces near the Lpoint V3o L3 < Ef < Closed surface in the 6th band. V3 3 Ef 5th and 6th bands, holes in the 4th band. V4 L3 ace in the 6th band with a "neck" at the Lpoints. V4J3 L The augmented plane wave method has been applied to the paramagnetic case by (i) Hanus, 5 who used a potential generated from renormalized atomic orbitals of the configuration (3d)8(4s)2, (ii) Mattheiss,6 who used a superimposedatom potential corresponding to the atomic configuration (3d)9(4s)1, (iii) Snow, Waber and Switendick, 7 who tried several different superimposedatom potentials with configu rations (3d)10(4s)x, 0 < x < 2. The three calculations are qualitatively similar to each other, differing only in the position of the dbands with respect to the conduction band. In reference 7 it is shown, as confirmed by the results presented in Section 3.1, that this dband shift has a profound effect on the topology of the Fermi surface, caused mainly by the altered ordering of the L point levels. (See table 3.4.) Yamashita and Wakoh, 810 have also done several calculations for nickel, both in the paramagnetic and ferromagnetic states, using the Green's function method. Since this method is formally equivalent to the APW method, 40 their results are also very similar, showing the same sensitivity of the dbands to changes in the potential. In order to compare all of these results, we note that the results of each calculation can be fairly well described by only two parameters: (i) the position of the dband with respect to the conduction band, which can be characterized by the energy difference between the states r2 and P, and (ii) the dband width, a measure of which is the energy difference between X5 and X1. Table 3.4. Results of Snow, Waber and Switendick7 for paramagnetic nickel Atomic configuration Ordering of Fermi surface in used for the super energies at the 6th band imposedatom potential the Lpoint (3d)8"0 (4s)2.0 1e Ef <( Closed surface with pro trusions toward the X points. (3d)8.5 (48)1.5 3 (3d)9.0 (4s)10 4< T < Ef Multiplyconnected surf ace, with "necks" at the Lpoints. (3d)9'5 (4s)05 L4 and Lpoints. (3d)10.0 IL Using these two parameters as coordinates, each calculation is plotted on the graph in figure 3.5. For comparison, a few calcu lations for other elements (Cu,Co,Fe) with an fcc structure are included. We note that no matter what the potential, all the calcu lations tend to lie on a straight line, which suggest that the ener gy band structure of an fee transition metal could be described by a single parameter, say, the position of the dbands. This diagram shows clearly the considerable variation between the results of previous work. The principal differences can be traced back to the atomic configuration used. Those calculations which assumed a smaller number of delectrons are found toward the lower left hand corner of the diagram. Although all of the calculations listed here used the averaged freeelectron exchange potential V , the effect of reducing the exchange moves the results in the opposite direction.(See Section 3.1.) In short, anything which makes the potential more attractive for the delectrons narrows the dband and lowers it with respect to the sp band. Of the previous calculations on transition elements, only two have been carried to selfconsistency, viz., that of Wakoh 10 on copper and ferromagnetic nickel, and that of Snow and Waber 41 on copper, both of which used Vx We can see by looking at figure 3.5, that there is a considerable difference between the two results for Cu (references 10 and 41) although both lie near the straight line. Wakoh's dbands are displaced upward from those of Snow and Waber by approximately .15 Ry. This is almost exactly the same as the 50 ITERATION OF THIS CALCULATION 0 PREVIOUS CALCULATIONS m I 0 a/ INCREASING " NUMBER OF SD ELECTRONS/ INCREASING/ I* EXCHANGE / 1i EFFECTS / N /  0.7 n INCREASING/ z ATOMIC / < NUMBER / m / 90.6  . Ni. I/ NIP WAKOH (10) o NIpv, I / COMATTHEISS(6) uw NI MATTHEISS(6) K od 0D N NNIP4 FE(fcc)WOOD (14) . 0 .5 2 o f w NIaV1'V 0 NNi WAKOH (10) w: NNl*_V4 S CUMATTHEISS(6)D Ni V4 z 0 NIHANUS (5) a CUARLINGHAUS (4) NNIV2 NI YAMASHITA & WAKOH (9) A 0.4 CUBURDICK o (16) NISNOW ET AL(?) z CUWAKOH (10) NIBJV3 o 3 o8 CUSEGALL (15) 0.3  S0.3 NlaV3 I 0.2 0 C1 U SNOW a WABER (41) 0.2 0.3 0.4 0.5 0.6 DBAND WIDTH E(x5)E(x) RYDBERGS Figure 3.5. Comparison of transition element energy band calculations displacement between Wakoh's bands for ferromagnetic Ni and those of V3 (the selfconsistent bands for Ve ). The reason for this discrepancy lies in the simplified version of selfconsistency used by Wakoh. The wave functions used to generate a new potential after each iteration were not chosen uniformly over the Brillouin zone. His procedure was to pick out 5 functions representative of the dlike electrons (corresponding to local maxima in the density of states curve) and another function representing a conduction electron, corresponding to energy E( ?) + (3/5)EI. These functions were then weighted to give 5.0 doelectrons, 4.4 d3 electrons and 0.3 conduc tion electrons of both spins. For the copper calculation, the corres ponding weights were 10 delectrons and one conduction electron. Since the "conduction electron" function chosen here is a mixture of s and dlike functions, this procedure tends to fix the amount of dlike character at too high value, thus displacing the dbands upward. His final result is very close to that of V4 (the selfconsistent bands for 2/3 Vxe ), since the effect of overestimating the exchange tends to balance the effect of using too many delectrons. For comparison, the effective number of dlike electrons calculated for Ni from the selfconsistent results, according to the formulas of Appendix III, are listed in table 3.5. We note that, although the effective number of delectrons is about the same ( 9.4) for the superimposedatom potential V and the selfconsistent poten tial V3, the resulting bands are displaced 1.l tRy. from each other. This is a reflection of the distortion of the dfunctions on going Table 3.5. Charge distributions. Inside charge .14 .16 .24 .26 .44 .50 Total charge 4.72 4.02 4.47 3.79 9.45 9.33 5.33 4.67 5.31 4.69 10.97 10.95 Effective dcharge I a %d 4.97 4.38 4.79 4.14 9.96 9.90 a = Qd(QtQtQo)). The dix III. quantities Qo etc. are defined in Appen b The results for Cu are taken from Arlinghaus2 who calculated the bands for a superimposedatom potential (CuSAP) and for the poten tial used by Burdicklo (CuB). Potential Outside charge V3c V40o .20 .20 .24 .24 .50 .51 CuB to selfconsistency.The atomic dfunctions are more localized than the selfconsistent ones, which has the effect of elevating the bands through the Coulomb repulsion term. There is a similar displa cement between the selfconsistent copper bands of Snow and Waber 4 and those of Burdick 16 and Arlinghaus, 2 even though the number of delectrons is close to 10 in both cases. (See table 3.5.) 3.3. Comparison With Experimental Data Of the wealth of experimental data that has been published on nickel, there are several firmly established facts which should be explained by an energy band calculation, if the model is to have any validity at all: (1) the saturation value of the magnetization, (2) the electronic specific heat at low temperatures, (3) the Fermi surface topology as deduced from de Haasvan Alphen measurements, (4) the saturation of the magnetoresistance, and (5) the negative spin density found from neutron diffraction and positron annihilation data. In order to interpret the first two of these properties, we need to calculate from the energy bands, the density of states, i.e., the number of allowed energy levels per unit energy. Mathematically 44 this is 4 E 3 VI k (3.1) (2I) 70i d f S, t where Es( k ) is the energy of an electron with spin s and reduced wave vector k andI2 is the volume of the unit cell. Since Es is not known analytically from the computation, we approximate (3.1) by Z w(e) (3.2) XE ( ;Rf ,< *46f where the sum is over those I points at which Es ( ) is known, and w ( k ) is the weight for the point !. (See table 3.1.) In this calculation, Es ( k ) was calculated at only 20 nonequivalent points of the Brillouin zone, which is not enough to give an accurate n. (E). In Appendix V is shown a procedure for inter polating Es ( k ) at enough points to give sufficient accuracy. The calculated n. (E) for both spins is shown in figure 3.6 for the two selfconsistent potentials V3 and V4. It can be seen that the narrowing of the dbands in V3 substantially increases the height of the peaks in n. (E). The Fermi level E; is determined by the relation, Z)2 (3.3) where nv is the number of valence electrons, 10 in nickel, and EFti is the minimum energy of the corresponding bands. The difference in the number of electrons of either spin is then .mis] The measured value of corresponding to the saturation value of the magnetization is 0.606 electrons/atom (reference 43 page 317). 55 3 V3 o A t 0 V3 3 0 CO a) H I 3 4 .1 .1 .3 .4 .5 .6 1 I I I I.5 Energy (Rydbergs) with respect to Voc( Figure 3.6. The density of states curves The calculated values ofu. for the superimposedatom potentials using Vx are slightly larger than the experimental value, which tends to substantiate the conjecture that this appro ximation overestimates the exchange effects. Going to selfconsis tenty slightly reduces the value bringing it closer to .6, i.e., ,kA(V3) = .65, t(V4) = .62. The error in the calculation of A is fairly large ,A n (Ef)AEf, since np (Ef) is high ( 40 electrons/ atomRy.) An error of AEf '.001 Ry. therefore gives an error in / of .04 electrons/atom, so that the agreement here is quite satisfac tory. The electronic specific heat, c,, at low temperatures is related to the density of states at the Fermi energy by the equation V J [ 4(^> A (3.5) J (reference 45, page 150). The measured value of cv corresponds to a 46 total density of 3.1 electrons/atomev. The calculated values of n(Ef) are n(V3) 4.5, n(V4) 3.0 electrons/atomev. The discrepancy in the value for V3 is a further indictment of these bands, since the method of calculating n(E) should be expected to give a density lower than the exact result. Both calculations show the Fermi energy occurring at a high peak of the density of states curve which agrees with Slater's criterion, 47 for ferromagnetism. The results are also confirmed by the recent measurements of the magnetic susceptibility, 48 X which satisfies a relationship of the form; I I I ( + (3.6) xa h(W E() 11" y The measurements indicate a very small Xd, which would be obtained only if no (El) is small. We note that in both V3 and V4, since the A dbands are filled, n 0 (Ef) < np (Ef). Although the Fermi surface of nickel is not as completely mapped out as that of Cu and the noble metals, at least one feature has been firmly established. The de Haasvan Alphen measurements,49'50 indicate a small crosssectional area of the Fermi surface in the (1,1,1) direction. This is interpreted as a "neck" in a Fermi surface similar to that found in copper, i.e., roughly spherical with protru sions that make contact with the Brillouin zone at the Lpoints. The neck area is about 1/10 that found in Cu, corresponding to an angle of 6.80 + 0.20 subtended at the V point. 50 If the energy surfaces are assumed to be hyperbolic, M (3 "7) then mt and mt are the transverse and longitudinal effective masses, whose measured values are 50 mt = .26 + .04, mA = .65 + .10 The Fermi surface derived from the energy bands for V4 is shown in two crosssections in figure 3.7. It consists in this case of four sheets for the downspin (P) bands and one in the upspin (oc) bands. The i sheets consist of small hole surfaces at the CLU C) 0 030 5~ C..4 02 03 0) ~d Ed L L U 0 0 X <1 4 04 Sz 14 I2 a1 C' Xpoints in the 3rd and 4th bands, a closed surface with protrusions along the 7 directions in the 5th band, and closed surface with protrusions along the A directions in the 6th band. The sole o( sheet is a multiplyconnected surface with necks at the Lpoints, in qualitative agreement with experiment. The calculated angle sub tended at F is 100, slightly larger than the measured value. The theoretical values of the effective masses at the neck are in much worse agreement; mt .5, m1 = 1.1. This discrepancy is probably due to the fact that the energy surfaces at L2 do not conform to the simple equation (3.7), but is more like the solution of a quadratic equation E  ".'[ (,5 ti^ akc9a k 4(p*k)] (3.8) along a particular direction in rspace, where s = E(L3) + E(L2) and p = E(3) E(I). The principal result of this behavior is that the effective mass mX = ( 8E/ d1k)EE varies strongly with the position of Ef, unlike (3.7) where it is a constant. In fact, shifting the level L2 up with respect to Ef by an amount of .02 Ry would bring the effective masses into approximate agreement with the experimental values. A shift of this magnitude would also bring the neck size into agreement with the measured value. The model of the o( Fermi surface is consistent with the magnetoresistance date of Fawcett and Reed. 51,52 These measurements also show that the magnetoresistance saturates at high magnetic fields. This type of behavior is usually associated with an uncompensated material. Nickel being an evenvalenced metal, would ordinarily be expected to behave like a compensated material. 52 This would be the case in the ordinary energy band model in which the two spins are degenerate. However, the unrestricted case can give rise to the situation where a different number of bands are occupied for either spin, the sum of the occupation numbers being an odd number. Experimentally, 51 the number of electrons per atom must satisfy the relation We teQ..) t4(0d () 10 (3.9) where ne (s) is the number of electrons/atom on the electron Fermi surfaces, and nh (s) is that on the hole Fermi surfaces. Equation (3.9) is satisfied by both of the energy bands V3 and V since in both cases five of the o( bands and four of the bands are filled, so that only one of the ten valence electrons is left to be distri buted over the Fermi surfaces. Recently, it has been discovered through measurements of positron annihilation 5and neutron diffraction 54 in ferromagnetic Ni, that the electron spin density is negative in the outer regions of the unit cell. It is found that for both V3 and V the conduction electrons are polarized opposite to the delectrons, so there is a theoretical negative spindensity outside the spheres, and just inside the spheres. The measured 54 value of 19% of the magneton number /A (.606) is larger than the theoretical ones for both calculations, perhaps because of the neglect of the nonspherical terms of the potential. As pointed out in reference 4, these nonspherical terms are essential for the interpretationof the neutron diffraction data, and will have to be included before quantitative agreement can be obtained. There are also a great deal of optical data available for nickel. There are summarized in references 1 and 2, which analyze the ferromagnetic Kerr effect, the dielectric constant and the conductivity. The structure found in these data (at .3, .6 and 1.4 ev) is interpreted as interband transitions. Because of the errors involved in calculating a density of states curve (See Appendix V), it is difficult to determine whether or not structure on this small a scale is present. It can be seen from figures 3.2 and 3.3 that transitions to empty states above the Fermi level are possible in the regions around the X and L points. These are the most probable since the density of states is high at these points. However, the density of states curve calculated here is not accurate enough to compare the theoretical energy bands with the experimental optical data. A summary of the experimental and calculated data is found in table 3.6. It can be seen that the selfconsistent bands for x = 2/3 Ve (V ) are in much better agreement than those for ,= V; (V3)' Table 3.6. Comparison of calculated and experimental quantities. Calculated Experimental V3 V4 Saturation moment (Bohr magnetons) .65 .62 .606 Density of states at the Fermi level (electrons/atomev) 4.5 3.0 3.1 Neck of the 6d Fermi surface (angle subtend no ed at the P point) neck 10.2 6.804i.2 Effective masses at the mt 0.5 .26+.02 neck (in units of the electron mass) mI 1.1 .65+.10 High field Hall coefficient (electrons) 1.0 1.0 1.0 Negative spindensity (percentage of the saturation moment) 7.8% 9.4% 19% CHAPTER IV CONCLUSIONS The purpose of this work has been to solve as accurately as possible within the limits of present computational techniques, the unrestricted HartreeFock (UHF) equations in a ferromagnetic solid. Whether or not this is a valid model will not be discussed here. To really test this method would involve a determination of the excited state energies, which would permit an analysis of the temperature dependent properties in a way similar to that of the Heisenberg model. However, this calculation has been limited to the ground state of ferromagnetic nickel. This follows naturally from the use of the energyband model, which although it can explain certain temperature independent properties, has not yet been extended in a rigorous way. In attempting to solve the UHF equations, we have used the oneelectron approximation. This has had considerable success for atomic systems, and it can be shown, 30 that it can give a solution that is very nearly as good as the exact HartreeFock value. There is no reason to believe that the same should not be true for a solid. The problem here is to find the "best" oneelectron approxi mation, i.e., to find an effective local exchange potential that can accurately reproduce the exact HartreeFock solution. One way to do this would be to calculate the total energy of the solid in the correct way, viz., Et == ( ft', where i is a determinant made up of the approximate selfconsistent oneelectron functions and e is the exact Hamiltonian. and then to minimize Et with respect to the exchange approximation. Lindgren 29 has found that this method works very well for atoms, giving better agreement with the experi mental binding energies than the exact HartreeFock method. Such a scheme is of course much more difficult in a solid, but de Cicco 17 has recently shown that it is feasible to compute the total energy of a solid, so that an investigation of this type might be possible in the future. This has not been done in this work, but is has been shown that slightly changing the oneelectron Hamiltonian critically changes the resulting energy band structure. In particular, it turns out that using the averaged freeelectron exchange approximation e )1/3 Vx = 6(6 /81r)/ gives qualitative disagreement with experiment, whereas reducing the exchange to (2/3)Vx gives agreement for most of the experimental data. The strongest argument for this conclusion is the Fermi surface structure deduced from the calculated energy bands. The existence of a copperlike Fermi surface with a small "neck" in the (1,1,1) direction, which has been firmly established 4952 by de Haasvan Alphen and magnetoresistance experiments, isnot predict ed by the V~bands, but does occur when the exchange is reduced. This tends to confirm the conclusion (See Section 2.2.) that others have reached for atomic systems, i.e., that V overestimates the exchange effects. This sensitivity of the energy bands to changes in the oneelectron approximation was not pointed out in previous calculations of this type, mainly because they were not carried out to self consistency, and therefore no conclusions could be made about the adequacy of the Hamiltonian. There has been one other calculation (that of Snow and Waber 41 on Cu) that was selfconsistent using the Vx exchange. The discrepancy of the Fermi surface did not occur in this case, since the position of the Fermi level with respect to the position of the dbands is different for Cu than for Ni. In Cu, there are 11 valence electrons per atom, 10 of which are "dlike", so that the dbands are full. The Fermi level therefore lies above the dbands in Cu, whereas in Ni, having one less valence electron, has its Fermi level slightly below the top of the dbands. In both cases (this work and reference 41), going to selfconsistency lowers the dbands with respect to the sp bands. For Cu, the position of Ef with respect to the sp bands remains essentially the same, but for Ni, Ef is lowered with respect to the sp bands. As shown in Section 3.1, the position of Ef with respect to the sp bands is what determines the shape of the Fermi surfaces. Therefore, going to self consistency in Cu showed no basic change in the Fermi surface, as is the case for Ni. This sensitivity makes Ni a better test case than 6e Cu for testing the validity of an exchange approximation such as Vx Actually, before we can definitely say whether the discre pancy is due to the inaccuracy of V we should examine the approxi mations made in this computation, in order to see if there could possibly be some other effect which would shift the bands enough to bring the V bands into agreement with experiment. From Figure 3.2, it can be seen that this shift would have to be approximately .1 Ry. in order to bring the dbands up far enough to give the experimentally observed Fermi surface. There are four important approximations which have been used in this computation: (1) the neglect of relativistic effects. They can be handled within the framework of the energy band model, and are certainly important for heavy atoms. Calculations which have been made (e.g., reference 36) show that these effects are on the same order of magni tude in a solid as in a free atom. For the Ni atom, the relativistic splitting of the oneelectron energy values are 32 .01 Ry., so that it does not appear that they could qualitatively change the Fermi surface structure. (2) the neglect of the nonspherical component of the potential. In deriving a potential from a muffintin charge density (See Appendix IV) we used the spherical average instead of the actual Ewald potential. In Reference 18, it is shown how these effects can be taken into account, by expanding the nonmuffintin part of the potential in terms of a Fourier series. For example, it turns out that the L2 level can be well approximated by a single symmetrized plane wave corresponding to k = (ir/a) ( 1,1,1). The Fourier component of the Ewald potential corresponding to i = (2ar/a) (1,1,1) is 18 equal to (Q/a) (.0096). Substituting in the values of Q = .26 and a = 6.65 shows that the energy shift of the L level would be on the order of only .001 Ry. Since this level is the one which determines the neck of the Fermi surface, it appears that the inclusion of these nonmuffintin terms would change the resultant Fermi surface only by a negligible amount. (3) the neglect of the nonspherical components of the charge density. The Ewald potential was derived on the assumption of a muffintin charge density. However, the departures of the actual density from this approximation are certainly significant, as can be seen from the neutron diffraction data. 4 These effects are much harder to estimate, 17 especially the nonspherical parts within the spheres. It can be done by an expansion in terms of spherical harmonics, but if more than a few avalues are important, the number of integrals to be calculated becomes prohibitively large. They should, of course, be investigated if the data are to be accurately interpreted, but it is unlikely that they will seriously affect the energy band structure as presented here. (4) the neglect of the distortion of the core states. In this calculation, the core state wave function (ls,2s,2p,3s and 3p) were assumed to remain unchanged from the values found for the Ni atom. Snow and Waber 41 in their calculation on Cu tried to estimate the error involved in this assumption. They used two values of the core density, one obtained from an ordinary selfconsistent atomic calculation, and the other, from an atomic calculation in which all of the wave functions were constrained to be in the WignerSeitz sphere (i.e., that sphere with volume equal to a unit cell). It was found that constraining the core states had the effect of pushing the dstates Mu with respect to the sp band by approximately .03Ry. During the course of this work, the bands corresponding to the 3s and 3p levels were computed using the actual crystalline potential, and a new potential derived from the resultant wave functions. The changed potential shifted the dbands up by an amount less than .01 Ry. Neither estimate is enough to qualitatively change the Fermi surfaces, although a more accurate calculation should certainly take the distortion of the core states into account. The arguments presented here, although not conclusive, tend to suggest that any discrepancy found can be attributed to the approximation made in the Hamiltonian, rather than in the approxima tion made to find the eigenvalues of this Hamiltonian. To the list of approximations, could possibly be added another, i.e., the neglect of any correlation effects. These are of course not included in the HartreeFock model, but they may be im portant in ferromagnetic nickel. However, the importance of correla tion cannot be determined if the exchange is not known accurately. Indeed, it may be that the averaged freeelectron approximation is a better estimate of the HartreeFock exchange effects than it appears here, and that reducing the magnitude of the exchange merely introdu ces some effective correlation. More study is needed on these effects, if the energy band model is to be successful in the explanation of ferromagnetic effects. 69 In conclusion, the main result of this work is that the unrestricted HartreeFock equations can be solved in a solid, at least in the oneelectron model, and that this scheme forms a reasonable model for a ferromagnetic solid. The accuracy obtained is sufficient to give at least qualitative agreement with experiment, and could probably be improved if better approximations to the Hamiltonian were used. APPENDIX I APW MATRIX ELEMENTS A single APW basis function has the form, for a crystal with one atom per unit cell APW fs = It where E (X) an ( d. ) is Ea step function, is a step function, S, ,I  4.,i.jf,.) ki~ (Al1) tbC ILe( 8 Y^),(J YA: (k) (Al2) The spherical harmonics Jmare chosen according to the phase convention of Condon and Shortley, i.e., y, (i)# Jim 4,j (#)!e I cls'(os W4)Ak so that they have the following properties: (Al5) (P) = S . (Al6) SA #7r A( I A (Al7) by =a ,) ( )= E(,,).(A13) 4 A A /" \H y "^ / y,,, )  /Y a4 Y2,1,, ( ) d121 Y J (z ) e (Al8) .e= o mn=A This last property (Al8) makes the APW (Al1) continuous at the sphere radius R. The contribution to the overlap integral (~/k'from the volume outside the spheres is just o t _i___) (Al9) and that from inside the spheres is < l^ 4>!n s 2 ., .l/ (t),j (Al10) = o m=i. o using (Al6). This can be simplified by applying (Al7) to derive the relation Similarly, for the energy matrix (Al13) because of the relation (Al3). In addition, there is a surface contribution due to the discontinuity of the derivative of the APW I> at r = R , S S d~ S S where 6[f] is the discontinuity in the gradient on the sphere S. This reduces to: 7t = & =o i . In summary, the requisite matrix elements between two single APW's are /where> where Kk.Ibjk.'>~ (r / / /> J2. /Ch(/ '/P) TC jr/ 47r (2 1 hf k ) e (k' ) J ( (At t19) (Al19) p 5 a P () +EL D, () (E) + ( E ?Ljd 01 . (Al15) (Al16) (Al17) (Al18) (Al20) O,(E) S (E) (E) 0,E( ^^) (Al21) (A122) <,k y / A r tii 4 l.lt k,100u '> 00~~/d ',J 4 Z < /;C> ' 4* k)^ E)k^ The secular equation is obtained by forming < kI/ E I kZ for all and k having the same reduced vector k, and finding the zeroes of the resulting determinant: M E ( + D (E) = 0 (Al23) where k! tI > = < k o( > . The symmetrized combinations of APW functions corresponding to the pth representation of (j, the group of the reduced wave vector ko, are defined by: c(A124) where rP, (&) is the matrix element of the group element corres ponding to the Ith row and the Jth column. The matrix element between 2 symmetrized APW's corresponding to the same reduced vector, of any operator 7 which commutes with all the group elements can be shown, setting k = k0 + g, = ko + g, to reduce to = PI Jp / l1 SSPP S < (A25) Kp Lt L (4) where Go is the order of (), np is the dimension of the representation. Thus for a particular value of P and I, the secular equation will have elements of the following form: ( Pr I RE I k,' P 'j  < Zj ^ P r j +< l( A1 2 6 ) 1 j iE We note that this expression is independent of the row subscript I, which implies an np fold degeneracy for each energy eigenvalues of the secular equation: IIB EA\ + D, (E) where P t = P< S I = o, '^ P J 7>5 PC" P T> IjZ J (A27) APPENDIX II RECIPE FOR THE CONSTRUCTION OF A LINEARLY INDEPENDENT BASIS SET The problem is to choose from among the functions Qjiko + a linearly independent set of functions for a particular row I of the representation p of F(.), the group of the wave vector ko, W i { I 30 (A21) k + g> is any Bloch function, such as an APW, corresponding to the reciprocal vector ko + g. The projection operators ^r as defined in equation (2.29) satisfy the orthogonality relation TL MN P LM N (A22) .TL Q^ = P, LM N (A22) Consider the function, +itb ) +;^,> 1^o+ ? Since gi + g is also a reciprocal lattice vector, then 0 &tk+ is a member of the basis set. Now,.+ o3 X . L pf,, ^'^^ ^ Iw ^ ^ > so that the function OBr L I L + > is linearly dependent on the set of np functions ( >IL IItj> L = 1, 2 ...np). Therefore, the first ingredient of the recipe is; RULE 1: If the function Jko+ g is chosen to be in the basis set, then no functions of the form Otik + g are to be chosen. This takes care of the case of unequal vectors. Now consider the functions with fixed g. Let = = o + g The number of linearly independent functions OPI (J = 1, 2 ...n ) is equal to the number of nonzero eigenvalues of the matrix (A24) = I j, > (independent of I) Suppose I is invariant under a subgroup P of 9t.) ( # is the stabilizing group of ), i.e., S j [ / t e % ) ( (A25) If a e then the use of (A23) gives r PP( I L IL L LV so that the corresponding matrix satisfies the relation, A= rOx)A = ( a(^) A (A2 6) P  77 where and are elements of is a representa tion for YZ, and therefore is also a representation for the sub group which will in general be reducible. Therefore, there exists a transformation 5 which will reduce rp to block form. (D9 r 0 Let A\ = A\$ be the transformed matrix, which will have the same eigenvalues as A From (A26), it follows that: A"A ^ A\ k )A< N,(A27) Consider the decomposition of A\ into the same block form as A A,, 1I A\12 A A\t A\ A\3, A33,. Application of (A27) gives the equations A\t C R cf)A\ and A\ 1 Af I' ". By Schur's lemma, since (A28) holds for all elements and belonging to then, either A\ .= or =F (the totally symmetric representation of ). Thus, the number of nonzero eigenvalues of A\ is equal to the frequency of in ( .This can be expressed in mathematical form to give RULE 2: The number of projections of the function = ko + that should be included in the basis set for the representation 1p is: ^ C)' = J? ^ W (A29) where is the stabilizing subgroup for @ with order G O and p (a) are the characters of the pth representation of (koI At first glance, this theorem would seem to be of little use, in choosing the basis set, since it gives only the number of projections, and not the actual columns of the representation matrix which must be used. However, due to the inherent symmetries of the functions Ik + g~, ap(l)very often vanishes. To show the usefulness of (A29), consider the following three examples: (1) rp = P the fully symmetric representation of OL.), then Xp(ait) = 1 ;1 and therefore a1()= 1 for all the functions, . From which it follows that, every function Iko + must be chosen once and only once in the basis set for r . (2) Suppose 3 has no intrinsic symmetry, i.e., P consists only of the identity element ', then: ap^ == p= .f Therefore, a nonsymmetric 4+g must be chosen np times for rp . (3) Suppose r has the full symmetry of 7(t ), i.e., = () then; ap ( # / Therefore, a fully symmetric ko+g must be chosen once for the representation PI but not for any other representation. This is the case if g = o, and ko is a point inside the Brillouin zone. In summary, we have the following recipe for constructing the basis set. For a given reduced wave vector ko, this is to be repeated for each irreducible representation, Pp of lk.) . 1. Form a list of vectors ko+ g.by running through all the values of the reciprocal lattice vectors g, up to the value at which the series is to be truncated. 2. Starting at the lowest magnitude, choose a vector k +g , cross off the list all those vectors k+g which are related to it by a relation ko+g = & (k+1), JEC(JO and continue in this fashion. 3. Find the stabilizing group of each vector I left on the list, and from the character table of 4(k ), find the frequency number a ( ) / ^ ( ) . 4. If ap ( )= 0 cross 0 off the list. If a ( ) O 0 and np > 1, then the actual projections must be formed to determine the linearly independent set. 5. As a final check, the total number of projections of all the representations must be equal to the original number of vectors on the list. In table A21 is found a list of the ap (b ) for the points of high symmetry in the Brillouin zone for the face centered cubic structure. The notation used for the representations is that of Bouckaert, Smoluchowski and Wigner, and are found along the top of each table. The general form of the V 's found in the vector list is found along the leftmost column. The small letters along the top are the type of atomic orbitals compatible with each representa tion, e.g., in the first table, it can be seen that a dorbital is compatible at ko = (0,0,0) with the two representations P and r2. Table A21. Frequency numbers for the fcc structure Gammna 1 2 12 25' 15' 2' 1' 12' 15 25 (0,0,0) s d d f pf f 000 1 0 0 0 0 0 0 0 0 0 aO0 1 0 1 0 0 0 0 0 1 0 aa0 1 0 1 1 0 0 0 0 1 1 aaa 1 0 0 1 0 1 0 0 1 0 abO 1 1 2 1 1 0 0 0 2 2 aab 1 0 1 2 1 1 0 1 2 1 abc 1 1 2 3 3 1 1 2 3 3 X 1 2 3 4 1' 2' 3' 4' 5 5' (0,2,0)*(r/a) sd d d f f pf d pf2 OaO 1 0 0 0 0 0 0 1 0 0 a0a 1 0 1 0 0 0 0 0 0 1 abO 1 1 0 0 0 0 1 1 1 1 aOb 1 1 1 1 0 0 0 0 0 2 aba 1 0 1 0 0 1 0 1 1 1 abc 1 1 1 1 1 1 1 1 1 2 Table A21 (continued) L (1, 1,)*(7/a) 1 2 1' sd f aaa 1 0 0 1 0 0 aab 1 0 0 1 1 1 abc 1 1 1 1 2 2 W (1,2,0)*(Tr/a) 1 1' sdf d 2' 3 pdf2 abO 1 0 0 1 1 a0b 1 0 0 1 1 abc 1 1 1 1 2 1 spdf2 3 pd2f2 Sigma (x,x,0) 1 spd2f2 aaa 1 0 0 aba 1 0 1 abc 1 1 2 aa0 1 0 0 0 aab 1 0 1 0 abO 1 0 0 1 abc 1 1 1 1 2' pf2 Lambda (x,x,x) 3 pdf2 4 pdf2 Table A21 (continued) Delta (O,x,0) 2' df 1 spdf 1' 5 pdf2 OaO 1 0 0 0 0 aba 1 0 1 0 1 Oab 1 1 0 0 1 abO 1 1 0 0 1 abc 1 1 1 1 2 z (x, 2T/a, 0) 1 spd2f2 3 pdl2 4 pdf2 abO 1 0 1 0 a0b 1 0 0 1 abc 1 1 1 1 S 1 2 3 4 (x,21T/a,x) sd2f2 pdf p2df2 df2 a0a 1 0 0 0 aba 1 0 1 0 a0b 1 0 0 1 abc 1 1 1 1 APPENDIX III APW CHARGE DENSITY The electronic wave function corresponding to the ith row of the pth irreducible representation of the group of the reduced wavevector k is FL I P a .o N Pr > (A3 0 k., y,5 where Ik PIJ> are symmetrized APW's as defined in equation (Al24), pIT and Cie are the elements of the eigenvector as obtained from the secular equation (Al26). The sum is carried over those combinations of lattice vector f and column index J which give a linearly indepen dent set of symmetrized APW's, as described in Appendix II. N is the normalization integral, which will be derived in equation (A3U). To see the explicit spatial dependence of the wave function inside the spheres we can express (A31) in the form in Y E) (A32) where a 03g^j J(A3 3) The coefficients aA are defined in equation (Al2). The spherically averaged charge density inside the sphere is S,. (A34) a0E ^ ^. JS.SO i^L~~ where we have used the orthogonality of the spherical harmonics (Al6). addition n theorem (A ), this can be reduced to A. rRF ( : 8+ + ltl ) Ml I + 5 1 / lAA ,' (A35) By using the defiorthogonity properties of the min equatrix el2) ments the additexpression inside the curly brackets be reduced to yrk ) &.(? 'J 1t, CR, EC X 4U 2. r z + I p'ty I P) j pL I*7:S'I apelL (A36) By using the orthogonality properties of the matrix elements, the expression inside the curly brackets becomes CT. 9^ 3'.J Cj, )A')J "qZ P+( A z S< t I I* ^t PJ>r 7 (A37) where the matrix is exactly the same quantity that was calculated for the secular equation (Al27). Therefore, we have as a final expression for the inside charge r density (spherically averaged) The outside wave function has the form oi (A39) The average charge density outside the sphere involves the same calculation as in the derivation of (Al9), and turns out to be o ? c (A310) where the (, matrix was also calculated for the secular equation. The normalization integral can now be obtained by the equation 0 kZ c fj<^ o JtjIPFJ/> (A3l1) 0 where I(E)= t  o The equations (A38), (A39) and (A311) can now be used to find the "muffintin" part of the charge density corresponding to an APW wave function. Once the secular equation (Al27) has been diagonal prJ ized to find the C. we have only to sum up over quantities that are already available. The determination of these quantities for a suitable number of occupied levels gives a total charge density from which a new oneelectron potential can be determined. In Section 3.2, we refer to the amount of slike, plike and dlike charge inside the sphere. These terms refer to the quantities, SJ" c I l IP TJIT> ..(E) (A312) for .= 0, 1 and 2 respectively. The total charge outside the spheres is just J T JJJ/ APPENDIX IV DERIVATION OF THE COULOMB POTENTIAL FROM A GIVEN CHARGE DENSITY The generation of an averaged potential (muffintin type) from an averaged charge density is a simple electrostatics problem 18 The equation to be solved is VV 8 IJ (A41) which is Poisson's equation in atomic units, where p =f, (constant) outside the spheres and p = y (r) (spherically symmetric) inside each sphere, subject to the condition 4rJ r^() di ^ P'(A42) 0 R is the sphere radius and J2 = 11 4 r J is that part of the unit J cell outside the sphere. There is also a point charge Z at each lattice site. (An electronic charge is taken as positive). Since (A41) is a linear equation, we can express the solution as a sum of two potentials V = V, V, IvV 8 (A43) J) consists of a charge Z+Q at each lattice site, a spherical charge p(r) p, inside each sphere, and zero charge outside. Y2 consists of a charge Q at each lattice site and a constant charge 6 elsewhere (cf. figure A41). Q is defined so that the total charge in both problems is zero, i.e., where QU is the charge V outside each sphere. V, Figure A4l The Ewald problem We now consider the two problems separately: 1) the spherically symmetric charge in each sphere produces a zero field outside it, since the total charge inside is zero. There fore V1 inside each sphere has contributions only from the charge density inside that sphere. Choosing V1 = 0 outside the spheres, we have the solution for V1 S2 v Vo v0I = 0 + i f' l 2 4t 2j. [olJL (A45) where a(r) = 41T p( r) To( ') 4"lKg F I (A44) 2) the charge density Y 2 produces a nonspherically symmetric potential, whose derivation is less straightforward. Formally it is given by: V ()=2. + constant, (A (AC6) where t are the lattice vectors. However, the sum over t diverges, and the constant is infinite. To avoid these difficulties, we can use a mathematical trick to extract a convergent series from the two infini ties in (A46). By using the identity; Xe w A (A47) 0 we can express the sum in (A46) as V.2 O i + e constant.(A48) orr The lattice sum in (A48) is periodic in r and can therefore be expressed as Fourier series, i.e., a sum of reciprocal lattice vectors, g, which is easily shown to be: e e , e (A9) The trick now is to notice that the lattice sum on the left hand side of (A49) is rapidly convergent for large u, and the reciprocal lattice sum on the right hand side is rapidly convergent for small u. Therefore we divide the integral over u in (A47) into two parts, introducing an arbitrary constant : V,0)J 477I .e e If & .(A10 J2 d (A40l) v i.A The infinity in this expression arises from only one term, i.e., that corresponding to 3=0 in the reciprocal sum. Subtracting this term off leaves a finite potential, which is dependent on E However, the derivative of (A4l1) with respect to E is a constant independent of r; ae I ~o VTo / S7 <(A411) 4TQ/JSS3 , where we have used the relation of (A49). Thus, by adding the constant 2 IT Q//IE1 to V2, we will have a final potential independent of E ; v? (^) 7Q E' 2q e eE_ l ] Y 1 ~~~ et ^ ""p r~' ?/ Q J 3 (A412) where erfc is the complementary error function. The constant E is chosen so as to optimize the convergence of the two sums. Equation (A412)can be put in dimensionless form, by setting g = 2 K, = aR, = ax, ` o=Ea, where a is the lattice para a. meter and 2= a 3/4 for the fcc structure: a Vt 4 el X 7 K LK 71 1' *j S 4 7T7r k 4L Kjo e I [f x j i ] 4 r (A413) Empirically, it is found that if the sums in (AA12) are truncated at the 6th nearest neighbors, then foro(= 3 the error in a V2/2Q is less than 108 for all values of r. The values of this function for the feec structure are tabulated in reference 18. These values were recalculated during the course of this work and were found to be in agreement up to the fifth decimal place. V2 has a singularity at the origin due to the charge Q situated there. From the calcula tion the behavior near the origin was found to be, Atm La2 7] (A414) where b = 4.584862. The spherical average of the potential inside a sphere (A413) will be just the expression (A414) plus a term due to the constant charge density pJ, < v"n 2.q 2qb / ( /47r ). S a. 3 (A415) Since (A412) is valid as an expression for V2 for all values of 6 it holds in the limit as E 4 o V 2Q e I Z (A416) This is an exact expression which does not diverge, but is too slowly convergent for actual calculation. However, it can be easily seen from this form that the average value of V2 taken over a unit cell is zero. This allows us to calculate the average of V2 in the region outside the spheres, from the relation < 0 >d s oV,'> a*d 0 from which we can find <.( L 3* /322 J ri ( .18 2L 3a / 5a a3 