RELATIVISTIC KKR CALCULATIONS ON HEAVY METALS (LEAD)

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RELATIVISTIC KKR CALCULATIONS ON HEAVY METALS (LEAD)

C. Sommers, G. Juras, B. Segall

To cite this version:

C. Sommers, G. Juras, B. Segall. RELATIVISTIC KKR CALCULATIONS ON HEAVY METALS (LEAD). Journal de Physique Colloques, 1972, 33 (C3), pp.C3-39-C3-48. �10.1051/jphyscol:1972307�.

�jpa-00215041�

JOURNAL DE

PHYSIQUE Colloque C3, supplbment au no 5-6, Tome 33, Mai-Juin 1972, page C3-39

RELATIVISTIC KKR CALCULATIONS ON HEAVY METALS (LEAD)

C. B. SOMMERS

CECAM BBtiment 506, 91-FacultC des Sciences d'Orsay G. JURAS

Battelle Memorial Institute, Colombus Ohio, USA B, SEGALL

Case Western Reserve University, Cleveland, Ohio, USA

Resume.

- La structure de bande, la surface de Fermi et la densitk d'ktats du plomb ont kt6 calcul6es en utilisant une forme relativiste de la mkthode proposee

par

Koringa-Kohn-Rostoker, incluant un spineur

a

quatre composantes et l'hamiltonien de Dirac avec un potentiel central. Le calcul a kt6 entrepris avec un potentiel cristallin rksultant d'un calcul atomique

H.

F. D.

S.

qui inclut les effets du noyau atomique de dimensions finies. Les dkrivks logarithmiques au rayon

cr

muffin tin

⁾⁾

sont calcultks en int6grant la fonction d'onde

a

lYintMeur et a l'extkrieur de la rkgion interstitielle

a

partir de l'origine, assurant ainsi une solution convergente l'kquation de Dirac comme dans le cas atomique. Deux structures ont 6tk calculkes, l'une pour un potentiel dY6change de Kohn-Sham avec un coefficient kgal

a 213,

l'autre pour un coefficient kgal a

^1,

qui est en dksaccord avec les r6sultats expkrimentaux. Celle correspondant au coefficient Bgal

a 213

est en accord avec les mesures de Hass-Van-Halfen faites par Anderson et Gold, et les ktudes spectros- copiques de Liljenvall.

Abstract.

- The Electronic band structure, Fermi surface, and density of states for lead has been calculated using a relativistic form of the Koringa-Kohn-Rostoker method, including a four- component spinor wave function and the full Dirac central-field Hamiltonian. The calculation was performed with a crystal potential derived from a

H. F. D. S.

atomic calculation, which included the effects of a finite atomic nucleus. The logarithmic derivatives at the muffi tin radius were found by integrating the wave equations inwards from the intersticial region and outward from the origin, thus assuring a convergent solution to the Dirac equation as in the atomic case. Two band struc- tures, one for a Kohn-Sham like exchange potential with coefficient

213,

and the other with coeffi- cient

1,

were performed. The results for the coefficient

1,

gave poor agreement with experiment and were therefore disregarded. On the other band the results for

²¹³

were in good agreement with the de Haas van Alfen experiments of Anderson and Gold, and the optical transitions of Liljenvall.

The relativistic energy band structure of lead has been calculated using the Green's function method, modified [I] to treat the heavy metals (such as lead atomic number 82). This modification consisted essen- tially of replacing the Schroedinger Hamiltonian inside the muffin tin spheres by the Dirac Central field Hamil- tonian, thus necessitating the use of 4-component spinors as trial wave functions. This modification presents no additional calculational difficulties except that the irreducible representations for the double groups must be employed to reduce the size of the secular matrix.

This paper is divided into two parts. The first part

concerns itself with a brief description of the RKKR or Green's function formalism, but which includes a new method for integrating the radical wave functions in order to obtain the logarythmic derivatives a t the muffin tin radius. The second part concerns itself with a discussion and comparison of the theoretical results with available experimental data (de Haas van Alfen, Kohn Anomaly, and optical transitions in the 0-6 eV range).

The basic symmetrical secular determinant resulting from the solution of a set of linear homogeneous equations as derived from the RKKR method are shown to be [ l ]

:

where

(i,i') ( v , v r )

f i , i ' ) , ,

aa'v',

^,

=

C azg

^{BKu,K u} i T , K u

Ka,K'a' UU'

and

~ ~ ~= b

C

, c(Z ~

4

,j,

u -

^s,

s)

A I ' I ' ) u - s j l l , u 9 - s c(Z'

4

j', ^U'

- S,

S) s =

++

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972307

C3-40 C. B. SOMMERS, G. JURAS, B. SEGALL

and

E - V

d.(fK)

=

(-- ^{+ 1}

^q,

^-

^{( K}

^{+ 11%}

c2 r

with quantum numbers

This determinant is a function of both E, the energy of the band and k the wavevector. The change in sign of this determinant for various values of E, as k is held fixed, gives the required E(k) for the energy bands. One of the salient features of the RKKR method is that this secular determinant factors into two distinct terms. The first term Skk. is a function only of the energy E and geometry of the crystal and thus can be precalculated for a given crystal structure such as FCC, BCC or HCP. In fact this term has been facto- rized in such a way that one can accuratly interpolate the Skk.(Ei) for an arbitrary

E,,

over an energy mesh of 0.1

^E

(where

E =

(a12 n)2 E). The second term of eq. (1) which appears only on the diagonal of the determinant is a function of the crystal potential V(r) and the energy E, but is independent of the crystal structure. The functions gk and f, are the radial solu- tions to the Dirac equation evaluated at r

=

r,, the muffin tin radius. That is, given the crystal poten- tial V(r) and some energy E, one solves for the appropriate gk(r) and f,(r).

The rows of this determinant are labeled by the relativistic quantum numbers K and K' which as can be seen in eq. (1) give J and L the total and orbital angular momenta. This structure of the secular deter- minent leads to a rapid convergence in the wave function and this can be shown to be the case by study- ing the properties of the diagonal term. Using simple partial wave scattering theory this term becomes

The cot 6,(E), is the phase shift of the scattered wave of angular momentum K and becomes infinite for some K and E. If one multiplies each row and column of the secular determinant by the tan 6,(E), it is easy to see that as the tan 6,(E) goes to zero (when cot 6,(K) goes to infinity) one is left with only the diagonal terms. (This vanishing of tan 6,(E) assures the continuity of the logarithmic derivative CfK/gK with J, at r,.) Thus the value of the secular determinant will not change after a certain value of

K,

and in practical calculations this is found to be K

=

2, or L

=

2, J

⁼

512, and one needs only two or three terms for the trial wave function.

Once the structure coefficients s$? have been symmetrized by group theory, i. e. factorized into sums of harmonics which transform according to the irre-

ducible representations of a particular k point in reciprocal space, the secular determinant becomes even more tractable. This factorization also insures the proper identification of the energy bands which as is shown later have been mislabeled in previous band calculations of lead [2], [3].

Normally when one integrates the Schroedinger or Dirac equation in solid state calculations, no account is taken of the fact that numerically two solution^

exist

;

one which is regular at the origin but has the wrong asymptotic behaviour at large r, and the other which has the correct behaviour as r

-+

co but blows up at r

-+ 0.

One can easily see this by rewriting the coupled differential eq. (Ib). Upon substituting t

=

log r we get 151

where and

The eigenvalues of the matrix A(t) satisfy the cha- racteristic equation

:

such that the solutions of A(t) are oscillatory for

A2

<

0

and exponential for

A2

> 0. In the case where one is solving these equations for the atom, one normally integrates outward from the origin to some t < r, where

A2

< 0.

Then the outer region log r, <

t

<

tm

is treated as a boundary value problem with P

E

rf prescribed at each end of the inteval (t,, t,).

The result of a step by a step integration gives two values of P and Q

=

erg such that the difference bet- ween the outer and inner integrations is zero. In the atom the eigenvalue E is then determined by the above condition, with P

+ 0

at infinity. In the solid one replaces this condition with the value of the radial wave functions at the muffin tin radius (and inters- ticial region).

That is one writes the potential (see fig. 1 ) as

:

V crystal

=

V(r) - Vo r r rm

= 0

1 ^r>r,,,

^{( 5 )}

where V, is the average potential in the interstitial region.

One now integrates inwards twice, assuming first a

solution rJ,(dEr) at RM and again assuming a solu-

tion r ~ , ( d ~ r ) at r,,,. These solutions are respectively

regular at the origin and regular at infinity. The solu-

RELATIVISTIC KKR CALCULATIONS O N HEAVY METALS (LEAD)

K

=

- 1, Mufin tin index JS

=

209

JJ? ^{COT dK(E)}

- - .7725069D-00

-

.7435597D-00 - .7137746D-00 - .6830990D-00 - .6514760D-00

...

(203) - 0.7725704D-00

(204)

- 0.7436148D-00 (204)

- 0.7138299D-00 (204) - 0.6831544D-00

(205) - 0.6515210D-00

E (RYD)

-

-

0.1324606D-01 - 0.1275306D-01 - 0.1234006D-01 - 0.1188706D-01

- 0.1143406D-01

...

- 0.1324606D-01

- 0.1279306D-01 - 0.1234006D-01 - 0.1 188706D-01 - 0.1 143406D-01 Results above the dotted line are for one integration from r

=

0 to r

=

JS.

Results below the dotted line are for two integrations with the classical turning point appearing in paren- thesis (ie (203)) above each entry in the cot S,(E) column.

The muffin tin index of JS

=

209 corresponds to an r

=

0.1194088D-01 on a logarithmic scale where each integer division corresponds to an r

=

.05. P(JS) and Q(JS) are the coupled solutions of the Dirac equa- tion at r

=

r,,.

The resulting eigenvalues at r6+ for the two cases are

:

...

- 1.115811 Ryds. - one integration

...

- 1.1 15843 Ryds. - two integrations.

some rJ where

A2

< 0, and matching with the two previous solutions of the inward integration, giving

:

rPr>,

=

ACE) rjK(JBr) + B(E) rNK(JEr) (7) with a similar equation for Q.

The results of this modification for the phaseshifts of lead are shown in table I. For energies less than

FIG. 1.

tion one seeks is a linear combination with coeffi- cients A(E) and B(E)

;

with

PL(~J)

=

PR,(~J) + B(E) PR,(~J) (6) QL(~J)

=

A(E) QR,(~J) + Q R ~ ( ~ J )

determined by integrating outward from the origin to

FIG. 2.

C3-42 C. B. SOMMERS, G. JURAS, B. SEGALL

- 1,O Ryd the phaseshifts are incorrect in the fourth place and this error increases rapidly with larger nega- tive energies. Although the results for lead are small, for heavier elements and especially the superheavy elements where the band energies are found to lie

much below - 2 Ryds this effect becomes rather pronounced and must be accounted for. However the main advantage of this procedure is that we now have the radiaI part of the wave function for r > r,.

In the calculation of lead the effects of a finite nucleus

been included, mainly because they were included in

RELATIVISTIC KKR CALCULATIONS OF HEAVY METALS (LEAD) C3-43

the atomic charge densities which were used to cons- We have found a discrepency with the bands of truct the crystal potential. This effect is not expected Anderson and Loucks in the labeling (Fig. 7 and 8) to have seriously changed the energy bands.

Lastly, the crystal potential was constructed using a Kohn-Sham type exchange term with coefficients of 1 and 213. The results for 1 gave poor agreement with experiment and will not be included in the next section (*).

In figure 2 the Brillouin zone for an FCC structure

^as

-

is shown with an emphasis on the points and direc- tions of high symmetry. This zone was divided into four planes and an irreducible wedge being one forty- eighth the volume of the zone was chosen to calculate the energy bands. With a mesh size AX

= 118

the dis- tance F A X a total of 89 points of varying degrees of

^1.0

symmetry is needed to map the entire wedge. These

89 points are equivalent to actually calculating

^am-

2 048 points in the zone had symmetry not been used. ,,,,

In figures 3-6 are seen the energy bands of lead with

^b)

05

- the Kohn-Sham exchange potential calculated by the

RKKR method. In figures 7 and 8 are seen the energy

bands of Anderson and Gold (AG) and Loucks (L)

^ax-

respectively.

(*) A partial list of the energy eigenvalues is shown in table V, L w r

and a complete list may be obtained upon request from C. Som-

mers. FIG. 7.

4

C. B. SOMMERS, G . JURAS, B. SEGALL

of the energy bands in the A direction, especially at X.

In general at points of high symmetry (such as X) the bands must become flat as they approach and cross the axis. For the upper two bands in the x direction of both previous band calculations, labeled and X: respectively, this is not the case. In our calculation we have found these bands to be of a different symme- try, namely belonging to the xi and X; irreducible representations. That these are the proper represen- tations is further verified by the A , and A , bands

which are compatible (by group theory) with these points, approach the axis with zero slope, and give the same eigenvalue at the points and X; respec- tively. However the overall global band structure is in close agreement with (AG) and (L) showing a band gap between the first and the second bands throughout the Brillouin zone. The Fermi level is at - 0.32 Ryd with the lowest band T : at - 1.1 15 84Ryd.

The de Haas van Alfen data of Anderson and Gold

has been studied extensively in their paper and almost

RELATIVISTIC KKR CALCULATIONS OF HEAVY METALS (LEAD)

all of the orbits near the Fermi surface have been

acurately identified. In figures 9 and 10 we have supe- rimposed our orbits (dashed lines) on those of (AG).

Our results show slight budges and identations as

compared with (AG), but the total volume of elec-

trons and holes remains the same. These results are

numerically presented in [4] table 11, along with those

of Loucks and Stedman, the latter results being obtain-

C. B. SOMMERS, G. JURAS, B. SEGALL

Dimensions ( r Ax)

U'AL) ( r w (xsu) (TCksx) (TCksx) (xzw) 2 - hh

Optical transitions Group theory

in direction (080) D4h.

not allowed not allowed allowed in A direction (OaO) 0 < a < 8 (C4V).

in W direction (480) (D2d).

allowed allowed

allowed allowed

S.

et al.

-

1.41 ,934 1.218 .516 1.40

1.534 1.054

Loucks

-

1.306 .941 1.124

1.361 .568

.937

1.178 1.706

RKKR - 1.35

.987 1.187 .448 .522 1.561 .720 1.280 .382 1.713 1.796 1.032 1.238 1.156 .280 1.268 9 1.839

ed from an analysis of the Kohn anomalies. As Sted- man et al. have pointed out, the Anderson and Gold results show a discrepancy in the kk and pp dimen- sions, since by a simple geometric argument one can obtain these dimensions

in

terms of

hh.

Our results do check with the geometric formulae.

On an overall scale, our results fluctuate within 10 percent of the other calculations and a more detailed study of these experiments will be undertaken to resolve the difficulties. We also pian to increase the number of mesh points so as to have a finer band calculation (to within .OO 1 Ryd numerically).

In figure 9 we present the optical absorption data of H. G. Liljenvall et al. In general, the structure of the absorption curve gives one information as to the onset of direct optical transitions. These transitions are directly proportional to the joint density of states (8) of the filled and empty bands below and above the Fermi level. At the time of this paper we had not yet completed the calculation of the joint density of states and thus can only speculate on the nature of these transitions (from viewing the bands as Liljenvall had done using Loucks calculation).

Since we are considering direct dipole transitions we can eliminate a certain number of transitions which are not allowed by group theory. In table I1 we present the results of eq. 8 which states that a dipole transition between two states i and k is possible if the sum of the product of the characters of the represen- tation of the states i and k with the representation of the dipole or gradient operator

j

is not zero. In table IV we present our results of this analysis. In general the magnitude of the gaps which correspond to certain optical transitions are in close agreement with experi- ment. The small gap .06 eV is predicted to be in the WZX direction. However, the peaks at larger energies correspond to transitions which could arise from several bands and thus are not localizable without the density of states information.

One concludes this study optimistically noting that in general the Fermi surface and optical transitions have been resonably identified. It remains for a closer study of the density of states to pinpoint accurately the bands giving rise to these dipole transitions

;

this study is presently under way.

Acknowledgments. - The author would like do express his appreciation to B. Segall and G. Juras for

consultations, and to J. P. Desclaux for providing a program to calculate the atomic charge density.

RELATIVISTIC KKR CALCULATIONS OF HEAVY METALS (LEAD)

Structure in eZ/A - Shoulder at (.8 eV) - ^{.06 Ry}

(Peak in g2/A)

Absorption band (1.5

^-,

3.5 eV)

(.

110

-+

.257 Ry)

Bump in absorption band

at (2.3 eV)

=

1.69 Ryd. up to 3 eV shoulder continues

Extension by Kramers-Kronig shows 2 weak peaks at

:

(3.9 eV)

=

.286 Ry (5.3 eV)

=

.389 Ry

Optical transitions

Assignement from bands -

WZX direction, k,

=

.35 Parallel bands, gap -- .816 eV Z,

+

Z, (Z, near Fermi level)

l?

ZK direction, K,

=

425 Parallel bands, gap -- .12 Ry

Also I'W direction, near W, k,

=

-45 Parallel bands, gap -- .12 Ry

W,

-+

W, gives gap -- .17 Ry bands again parallel in r W direction,

at k, - -4 gap of .220 Ryd. (3 eV) KSX direction, k, - .775 (near K) Gap - ^{3.9 eV (K, - K, or S,}

^-,

^S,)

r AX direction A,

-,

A, or A ,

⁺

A, near K, - ^-85

⁺

^{.9 gap -- .38 Ryd.}

also

C, -+

X,, K, - ^.7

Energy bands of lead given in dimensionless units of K

⁼

2

z / A ,

epsilon in units of El(2 nlA)' and energy in rydbergs with respect to a constant potential of - 0.6904061D1-00 Type

- FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB FCC-PB

Epsilon

-

- 0.10331698D-01 - 0.17766763D-00 0.31294513D-00 0.1 1415536D-01 0.13612194D-01 - 0.10166679D-01 - 0.96698822D-00 0.15216854D-01 - 0.884001 58D-00 0.13475421D-01 - 0.768 18909D-00

0.1 1465665D-01 0.14984174D-01 0.91 5071 52D-00

-

0.62199436D-00 0.13395975D-01

-

0.45140026D-00 0.66442424D-00 0.12282890D-01 - 0.275731 80D-00 0.43 1603OOD-00 0.1 163053 1 D-01 0.1 668 1 847D-01 0.15369342D-01 0.14402793D-01 0.13811312D-01 - 0.41 168337D-00 0.14533079D-00 - 0.59755480D-00 0.43261 694D-00

Energy (Ryd.)

- 0.1 1584317D-01

- 0.77088947D-00

- 0.54864200D-00

- 0.17328256D-00

- 0.73774000D-01

-

0.11509564D-01 - 0.1 1284515D-01

- 0.10829298D-02 - 0.10908586D-01 - 0.79969836D-01 - 0.10383956D-01

-

0.17101171D-00

- 0.11623331D-01

- 0.27587889D-00 - 0.97216938D-00 - 0.83568731D-01

- 0.89489029D-00 - 0.38942205D-00 - 0.13399143D-00

-

0.8 153 1252D-00 - 0.49489002D-00 - 0.16354328D-00 0.65281 199D-01 0.58247666D-02 - 0.37959872D-01

- 0.64753949D-01 - 0.87689855D-00

- 0.62457126D-00

- 0.96109826D-00 - 0.49443070D-00

Representation

-

FCC GAM 6 + FCC X6+

FCC X6- FCC X6- FCC X7- FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA6 FCC DELTA7 FCC DELTA7 FCC DELTA7 FCC DELTA7 FCC L6+

FCC L6-

FCC* LAMBDA 6

FCC* LAMBDA 6

C3-48 C. B. SOMMERS, G . JURAS, B. SEGALL

Type KX KY

-

-

-

FCC-PB 0.2500 0.2500 FCC-PB 0.2500 0.2500 FCC-PB 0.1250 0.1250 FCC-PB 0.1250 0.1250 FCC-PB 0.7500 0.7500 FCC-PB 0.7500 0.7500 FCC-PB 0.7500 0.7500 FCC-PB 0.7500 0.7500 FCC-PB 0.6250 0.6250 FCC-PB 0.6250 0.6250 FCC-PB 0.6250 0.6250 FCC-PB 0.6250 0.6250 FCC-PB 0.5000 0.5000 FCC-PB 0.5000 0.5000 FCC-PB 0.5000 0.5000 FCC-PB 0.3750 0.3750 FCC-PB 0.3750 0.3750 FCC-PB 0.3750 0.3750 FCC-PB 0.2500 0.2500 FCC-PB 0.2500 0.2500 FCC-PB 0.2500 0.2500 FCC-PB 0.1250 0.1250 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0

FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB 1.0000 0.0 FCC-PB - 0.8750 0.1250 FCC-PB - 0.8750 0.1250 FCC-PB -0.8750 0.1250 FCC-PB -0.8750 0.1250 FCC-PB -0.7500 0.2500 FCC-PB -0.7500 0.2500 FCC-PB - 0.7500 0.2500 FCC-PB -0.7500 0.2500 FCC-PB - 0.6250 0.3750 FCC-PB -0.6250 0.3750 FCC-PB -0.6250 0.3750

TABLE V (suite) Epsilon

- - 0.83395859D-00

0.96190235D-00 - 0.98148307D-00 0.151 80224D 01 - 0.18552543D-00 0.44889543D-00 0.67200548D-00 0.13996299D 01 - 0.30514740D-00 0.50594129D-00 0.71258515D-00 0.15662778D 01

-

0.52194772D-00 0.65404775D-00 0.95595342D-00

-

0.73614315D-00 0.92727907D-00 0.12682401D 01 - 0.90038471D-00 0.12708955D 01 0.16262248D 01 - 0.10001073D 01 - 0.15528406D-00 0.47031 175D-00 0.80549300D-00 0.95645730D-00

-

0.164952 10D-00 0.38781060D-00 0.93790107D-00 0.10841 188D 01

-

0.17404983D-00 0.33219080D-00 0.10723972D 01 0.12538844D 01

-

0.15109331D-00 0.52986265D-00 0.72805898D-00 0.91 154288D-00 0.21 11 4340D-00 0.45430190D-00 0.785481 62D-00 0.11184351D 01 - 0.31367090D-00 - 0.2949281 7D-00 0.10891503D 01 0.14467757D 01 - 0.38661513D-00 0.1839791 6D-00 0.14591994D 01

Energy (Ryd.)

-

- 0.10681891D-01

- 0.25466453D-00

- 0.11350177D-01 - 0.27488000D-02 - 0.77444905D-00 - 0.48705655D-00

- 0.38598775D-00 - 0.56374051D-01 - 0.82863778D-00 - 0.46121479D-00 - 0.367605 17D-00 0.191 17385D-01 - 0.9268482733-00 - 0.3941225913-00

-

0.25735940D-00 - 0.10238788D-01

-

0.27034887D-00

-

0.11589360D-00 - 0.10982801D-01 - 0.11469069D-00 0.46273363D-01 - 0.1 1434544D-01 - 0.76074972D-00 - 0.47735496D-00

-

0.32551793D-00 - 0.25713114D-00 - 0.76512934D-00

-

0.51472797D-00 - 0.2655371 ID-00 - 0.19930052D-00 - 0.76925061D-00 - 0.53992372D-00 - 0.20461039D-00

-- 0.12239672D-00

- 0.75885131D-00 - 0.45037842D-00 - 0.36059553D-00 - 0.27747736D-00

-

0.78605399D-00 - 0.48460742D-00 - 0.33458308D-00 - 0.18375522D-00 - 0.83249892D-00

-

0.55680368D-00 - 0.19702125D-00 - 0.35017010D-01

-

0.86554264D-00

-

0.60706356D-00

-

0.29389098D-01

References

Representation WT

- -

FCC* LAMBDA 6 08 FCC* LAMBDA 6 08

FCC LAMBDA6 08

FCC LAMBDA6 08

FCC SIGMA5 04

FCC SIGMA5 04

FCC SIGMA5 04

FCC SIGMA5 04

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMAS 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC SIGMA5 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC Z 12

FCC W6 06

FCC W7 06

FCC W6 06

FCC W7 06

FCC

Q

24

FCC Q 24

FCC Q

24

FCC

Q

24

FCC Q 24

FCC

Q

24

FCC

Q

24

FCC Q 24

FCC

Q

24

FCC Q 24

FCC Q 24

[I ] SOMMER (C. B.), and AMAR (H.), Phys.

Rev.,

1969,3,188. LILJENVALL et Al. Phil. Mag., 1970,243, 121 L o u c ~ s (J. L.), Phys. Rev. Letters, 1965, 26, 14. [4] STEDMAN (R.) et al., Phys. Rev., 1967, 3, 163.

[3] ANDERSON (J. R.) and GOLD (A. V.), Phys. Rev., 1965, [5] GRANT (I. P.), Advances in Physics, 1970, 19,

5A, 139. 747.