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On the bonding types of the NbRhM phases
PHYSICA[
Physica B 179 (1992) 349-354 North-Holland
On the bonding types of the NbRh M phases K. Schubert* Max-Planck-lnstitut fiir Metallforschung, lnstitut fiir Werkstoffwissenschaft, Seestrasse 92, W-7000 Stuttgart 10, Germany Received 20 August 1991
Several new crystal chemical rules suggest an improved interpretation of the intermediate phases in the mixture NbRhM by the electron correlations model: (1) Confluence of certain lower correlations to a united ground correlation g. (2) Collectivity of the correlation of a peripheral e shell with correlations of lower bands. (3) Spin compensation in both lower and higher correlations. These new rules permit an interpretation of structural phenomena in NbRh M such as icosahedral coordination in homeotypes of Cr3Si, stacking sequence in homeotypes of Cu, and deformation of higher symmetric structures into lower symmetric homeotypes. A decisive factor for the stability in NbRh M and in other transition metal-transition metal alloys appears to be the insertion of electrons according to the rule of Hund into interstices of the spatial correlation of the peripheral electrons. Although Hund insertion is only indirectly visible in diffraction experiments it is a feature of distinct influence on the bonding type of metallic phases. The insight mediated by the interpretation of the crystal structures may be useful in the search for phases with required properties.
I. I n t r o d u c t i o n
The mixture NbRh M contains a number of remarkable intermediate structures, solved by Ritter et al. [1], suggesting an attempt of bonding type analysis by the electron correlations model [2]. A first trial has been made [3] assum58 0.9 ing the electron count Nb ' Rh M , i.e. Nb contributes its 4d 5 electrons as valence electrons with correlation b, while the Nb4sp and Rh4d electrons form the core electron correlation c. Several reasons have been enumerated for this (nonEkmanian) type of count [3, 4], but an alternative proposal assuming a d electron correlation e filled by the peripheral d electrons of both components should also be investigated because it has been used to formulate the valuable rules of Raub and Matthias (see ref. [3]). This alternative [3] count N b "5 18R h ~9 18 yields bonding types obeying several new crystals chemical rules. (1) The correlation confluence, for instance 3d + 4sp in Nb (see ref. [2], p. 11, 33), re* Deceased 19 March 1992. 0921-4526/92/$05.00
suiting in the formation of a ground correlation g comprising the lowest electrons being of influence on bonding. (2) The collectivity of correlations ([2], p. 13) describing the fact that for instance the e lattice is contained in the g lattice and g has sites enough for the e and g electrons. (3) The spin-compensation ([2], p. 9) leading in conjunction with collectivity to an unexpected bonding type discussed in the following. The merits of the earlier [3] and of the present bonding type proposals must be compared. Certainly both proposals have some similarities since they try to interpret one and the same empirical data. Therefore, the reading of ref. [3] may be useful when the present arguments are somewhat brief to avoid unnecessary repetition. The bonding type is a part of the crystal structure not immediately visible to diffraction experiments. Nevertheless, because of its close connection with the two-electron density matrix (see ref. [2]), it is a phenomenon worth to be investigated. The correlations model is related to earlier valence models and may be considered as
© 1992 - Elsevier Science Publishers B.V. All rights reserved
K. Schubert I Bonding types oJ" the NbRh~ phases
350
a synthesis of various earlier models since most valence rules have to do with spatial correlations of electrons. In its present state the model is not a model for energy calculation. Since the energy differences of neighbouring phases are very small, the expensive energy calculation is not at present an appropriate method for finding crystal chemical rules: only few of these rules [2] have been found in this way. The following analysis wishes to show that bonding types are a useful and realistic means for the interpretation of alloy phases.
2. Analysis The marginal phases of the mixture NbRh M are considered first since the electron distances d , dg in intermediate phases as a function of the mole fraction N ~ interpolate the marginal values linearly (fig. 1), according to the rule of distances (ref. [2], p. 19). This rule is of use in the search for the bonding types of the intermediate phases. For Nb was found [2]: Nb(W, SR1.56)a10,36 = 3.30 A = e F ;1.5 = gB ;3, N ' s " = 13.5,54. Here W is the prototype of the structure type of
I ~
"~
N
O0
~ ~ n'rr
z
e~
.z~
.z~
Ln
LQ o o r" r " ry, p r
" ~ (.D ~--~ r" r ~
lag ag~ ig 8 .~
Z
_Q
I 1
.
Z -
Z
Z
Z
.c
-.<' /
~5
Ir----¢--, _ ._____.._,
0.6
Nb
0,2
0,4
0.6
mole froction
0,8 Nt~ h
Fig. 1. Electron distances in the mixture NbRh~,.
Rh
Nb; SR1.56 means Structure Reports Vol. 1, p. 56; a is the crystal cell matrix of Nb; 10,36 are the numbers of 4d and 3d + 4sp electrons per a: 3.30,& is the numerical value of a in the brief notation of ref. [2]; e v is the cell matrix of the e correlation and is of the cubic face centered Bravais type F; for the type nomenclature, see ref. [2]; ';1.5' is the brief notation of the commensurability matrix between a and e: the semicolon distinguishes the brief notation of the matrix from a number; g~ is the ground correlation cell matrix of the cubic body centered Bravais type; ';3' is the commensurability matrix g ~a" N s' are the numbers of sites per crystal cell in both correlations. It is seen that e is collective to g. Neither e nor g are fully occupied. The bonding type may be named FB2 [2]. gB exhibits spin compensation while e v does not show it according to collectivity and does not need it according to Hunds rule [2]. An interatomic spin compensation might cause some broadening of the e v points of the averaged correlation [2]. Contrary to Nb the component Rh must contain H u n d insertion [2] in e. Therefore the FB2 bonding type is not admissible for Rh, and in ref. [2] F'B2 was proposed, i.e. F displays Hund insertion ('). Unfortunately collectivity between e v and gB requires the unfavourable filling of tetrahedral instead of octahedral interstices of F. Apparently nature prefers for Rh another bonding type. In a bonding type ~'t~2 the compression (~) of F and B along [00 1] may be so strong that B is transformed to U, a tetragonally compressed B type showing in (1 1 0)u and (1 1 0)u hexagonal planes [2]. When for instance consecutive gLJ layers parallel to (1 1 0)u are occupied by alternating spins and eF is collective to gu then both e and g contain spin compensation and the bonding type [vU2 appears appropriate for Rh: Rh(Cu,SR1.69,24.223)a20 + 16,72 = 3.80 = e~2 ;2.5 = gu 4 ;5, N's"= 40,160. Note that 20 +spin electrons must be assumed in the a cell, and 16 - s p i n electrons. The spin compensation in g causes a little deformation of g leading to twinning of g in a, i.e. to a domain structure as it is known in the room temperature phase of iron, Fe.r. We now consider the intermediate phases in
351
K. Schubert / Bonding types of the NbRh M phases
the order of increasing mole fraction N ~ . Nb3Rh and Nb2Rh display an icosahedral coordination of Nb to Rh. For low concentration of Hund insertion these defects form isolated arrays (globules around Rh) [5], favouring icosahedral coordination. Metrical comparison with Nb yields the bonding type equation and the numbers of electron sites per a cell: Nb3Rh(Cr3Si,SR29.125)a40 + 8,144 = 5.12 = e F, ;2.5 = gB ;5, N/s~ = 62.5,250. The 22.5 e vacancies must be associated with the 8 H u n d inserted electrons, in order to reduce the potential energy of the above-mentioned tetrahedral interstices. The array of the icosahedral centers in Nb3Rh is described in a plane parallel to ( 0 0 1 ) . through one Rh, by the Schlfifli symbol 4 4 [6, 5]. The U.h 1 = / 3 U type of Nb2Rh may be attained from Nb3Rh essentially [5] by the transformation 44----~4343z. It leads to an icosahedra fusion [4] being an expression for the growth of the above-mentioned globules of Hund insertions because of increased N ~ . The bonding type follows by metrical comparison with Nb3Rh and observation of ~/~t¢~, "'S \" " Rh]" Nb~Rh(U.h i ,SR29.125)a150 + 40,540 = 9.81 ;5.07 --- evA/24.3 ;2.5 = gB~/97 ;5, N/s a = 243,970. There are 53 + 40 e-vacancies per a so that 93/40 vacancies in e v are available for one Hund insertion. The phase NbRh.h [1] has an unknown structure and the neighbouring phase Nb0.asRh0.52 is no longer a homeotype of Cr3Si, it belongs to the close packings. Metrical comparison yields the bonding type: Nb0.48Rh0.52(CuAu,SR29.125[1])a20 + 8,72 = 4.02;3.81 A = e¢2 ;2.25 = gu4 ;4.5, N/s" = 36,144. Apparently the e correlation is highly occupied, the distribution of reversed spins is now layerlike instead of globular because of the strong increase of e electron concentration. The number of g sites per atom is t~]/at = 36 while it was ''Sg N/at = 3 2 . 3 in Nb2Rh, because the increased Sg share of - s p i n along the Rh layers strains the structure. It is quite surprising that the bonding transforms already here from FB2 to ~'U2 of Rh.
The phase Nb0.45Rh0.55 is isotypic [1] to Ta0.4Rh0.6(O6.6 ), a quasi-hexagonal close packing with 6 layers parallel to a~, a 2 p e r ]a31 length, stacked in the sequence + + + - - The ordering of the components causes the orthorhombic deformation; for simplicity the bonding type is noted only for the hexagonal cell H2.83;13.59,~ and only for g, see fig. 2 for e. Nb0.45Rh0.55 (Ta0.aRh0.6.r,O6.6,SR29.125)a60 + 26,216 = 2.83;4.77;13.59 ~ = g u ~ / 1 2 ; 2 0 / 2 . While Nb0.48Rh0.52 had N/s] = 3 6 g sites per atom, Nb0.45Rh0.55 has 40. Hund insertion in e causes the stacking sequence. Because six electrons are contained in an insertion layer, 5 layers must be in a cell since the insertions stick together similarly as in the globules and they will cause the observed sequence. According to the rules of stacking [2] the series 6 n / 5 = 0, 1.2, 2.4, 3.6, 4.8, 6.0, for n = 0 , 1 . . . . . 5 must be calculated. From these values a shift value, say 0.02, should be subtraced to discard the underdetermined values near 0 and 6. Only the values next to integers (i.e. atoms) are considered as essential for the sign of the electro dipole in the a 3 direction at an atom. Assuming that the dipole vector points from the negative to the positive charge, the sign sequence ( + ) - - + + + is obtained. The change of dipole sign in the sequence causes a change of stacking sign [2]. A plausibility consideration says that a minus charge below an atom of the stacked atomic layer sees two times three plus-charges below itself in the case of no sign change, but 3 + 1 minus-charges in the case of sign change. The next phase is a high temperature phase, less ordered than Nb0.nsRh0.55 and therefore nearer to hexagonality. Once more gull is used as simple indicator for the ~'U2 bonding type.
a,
9uH
aet
a
/\/\ Fig. 2. The cells a, e, g of Nb3Rhs as projected on (00 1)~,.
352
K. Schubert I Bonding types of the NbRh M phases
Nbn.40Rh0.60.h(AuCd,SR29.125,)a20 + 9.6,72 = 2.81;4.81;4.51 ,~ = gUHX/12;6.5/2. W h e n there are 3 - s p i n layers in eB then the atomic dipoles caused by e parallel to a 3 have the sequence + - + - and determine by the above m e t h o d the atomic stacking. There are 39g sites per a t o m , and 39 e sites per cell. The phase Nb3Rh~ is a deformation h o m e o t y p e ( D h t p ) of Sm (ref. [3], drawing 4) and exhibits before the stacking + + - + + ++-. N b 3 R h s ( D h t p Sm,M9.9,SR29.70)a90 + 45,324 = M90.53°4.77;2.81 ;20.25 ,~ = eB3;2;15 = gUHX/12;30/2. T h e monoclinic angle is set to 90°C for simplicity. The slight compression of gull may be caused by a spin distribution with alternating sign in neighbouring g planes parallel to (0 0 1)~, for eB because of collectivity also spin compensation results. The eB correlation causes the monoclinic s y m m e t r y , as may be seen from fig. 2. The n u m b e r of e and g sites is N/s" = 180,720, correctly 90 sites for +spin are available. The observed stacking is reproduced: 9n/15=3n/5=O, (0.6), 1.2, 1.8, (2.4), 3.0. Possibly this bonding type decreases the upper decomposition t e m p e r a t u r e as c o m p a r e d with that of the following Nb3Rh 7. For this phase the F U 2 bonding type appears to be valid once more: Nb3Rh7 ( V C o 3.r,H6.18,SR29.125,[1])a120 + 67,432 = H5.48;13.41 A = ef34;6;10 = gull X/48;20, ,,s/V/~'= 10,40, N"s~ = 240,960. T h e commensurability e ~ a is written for the o r t h o r h o m b i c base centered cell (Q) of a. The stacking sequence + + + - - is the same as that for Nb0.4sRh0.5.s(O6.6 ). If it is assumed that an insertion layer parallel to (0 0 1)~ tends to be filled, i.e. takes 12 electrons into the hexagonal mesh, then about 5 of the ten Hund insertion layers parallel to (0 0 1)~ are occupied and favour the observed stacking: 6 n / 5 = 0, 1.2, 2.4, 3.6, 4.8, 6, after subtracting the shift value the sequence ( + ) - - - + + + appears. The n u m b e r of e sites per a is N/s~ = 240 so that sufficient sites for +spin e electrons are available. The e correlation must be twinned in a to yield the hexagonal symmetry.
The phase N b R h 3 is closely homeotypic to Nb3Rh 7, therefore a homeotypic bonding type should be tried. NbRh3(Cu3Au,SR29.126)a20 + 12,72 = 3.86 = %2; 2.5 = gu4; 5,J,s~S/<'= 40,160. Since Ns"e = 4 0 , there is no spin sign change necessary. The n u m b e r of g sites per atom is /at Nsg = 40. Once more the increased e concentration causes the denser g correlation. The slight deformation of gu must be attributed to the spin ordering in g. Since the close packed g layers are not parallel to the close packed a layers a stacking h o m e o t y p e of Cu is not formed. An interesting illustration of the interaction between H u n d insertion and F-vacancies in a FB2 bonding type is presented in the mixture V I r M investigated by Giessen et al. [7]: V(W, SR1.56,26.275)a10,32 = 3.02 ,~ = e v ;1.5 = gB ;3 [21, Us <'= 13.5,54. V3Ir(Cr3Si,SR23,233)a40 + 8,140 = 4.79 A = e v ;2.5 = gB ;5, N"<'s= 62.5,250. T h e r e are 15.6 F-vertices in the cell but only 8 electrons in H u n d insertion. Therefore, globules are f o r m e d and a h o m e o t y p e of Cr3Si becomes stable. VIr(Q2.2,SR30.6418])a40 + 16,152 = 5.79;6.76; 2.80/k = ev-,3;4;1.5 = gB6;8;3, N~" = 72,288. The phase is homeotypic to C u A u and a~ corresponds to 2a3(CuAu ). The compression (~) of the bonding type lies along a~. From the assumed commensurability follows that there are 72ev sites per cell. Subtracting 40 sites occupied by +spin electrons yields 32 vacancies being exactly two times the n u m b e r of H u n d insertions. This exact relation is favourable for ordering. Since one e~ layer parallel to (0 1 0)~ contains 9 sites the vacancies may be found in two times two neighbouring (0 1 0)a layers near the Ir layers parallel to (0 1 0)a and the H u n d insertions m a y be near. The layer-like distribution of vacancies causes the compression of e~ and the one +spin electron in the vacancy layer allows an ordering doubling the [0 0 1]CuAu. V0.45Ir0.55(CuAu,SR30,150)a20 + 8.8,60 = 3.89;3.65 ,~t = et2;2.25 = gu4;4.5, VIr3(Cu3Au,SR23.267)a20 + 12,82 = 3.81 it = e~2;2.5 = gu4;5, as in N b R h ~ above.
K. Schubert / Bonding types of the NbRh Mphases
3. Discussion
Several assumptions of the first interpretation [3] of NbRh M phases are corroborated. (1) Participation of at least two correlations in the bonding type. (2) Harmony of the commensurabilities b-Ca, -i e a, etc., i.e. these matrices contain many integral elements. (3) Stacking sequence of homeotypes of closed packings may be assessed from the bonding type by an analysis of the electro-dipole distribution along the stacking normal. Contrary to these conserved assumptions the electron count of ref. [3] had to be changed because in the bonding type analysis of other mixtures between transition elements it had become apparent that an e correlation containing the peripheral d electrons of both components leads to simpler bonding types. Accordingly, various questions on the structures in the alloys NbRh M find a simple answer by the electron correlations model. (1) Why are structures with icosahedral coordination formed? Because of globular array of Hund insertion into an FB2 bonding. (2) Why is the U.h~ type stable at higher d electron concentrations than the Cr3Si type? Because the icosahedral fusion allows greater insertion globules. (3) Why has Nb0.48Rb0.52(CuAu) a larger ~/a, ,,Sg value than Nb3Rh and Nb2Rh? The globules of Nb3Rh and Nb2Rh have grown together to infinitely extended layers along the Rh layer of the CuAu type straining the structure. (4) Why are in Nb0.45Rh0.55 the Rh layers pleated? In Nb0.nsRho.52 the maximum of Hund insertion in a plane layer is reached. This strong structural transformation entails a change of bonding type commensurability. The new bonding type contains an e correlation which causes electro-dipoles and these cause the pleating. (5) Why has Nb0.nsRh0.s5 the stacking sequence + + + - - - ? This observation obeys the rule [2] that a change in the sign of the electro-dipole vector of an atom in
(6)
(7)
(8)
(9)
(10)
353
the direction of the stacking normal causes a change in the stacking sequence. This is an independent indication for the influence of the spatial correlation of the electrons. The band model arguments for the stacking of Sm.r [10] have not yet been applied to Nb0.nsRho.55, but the present argument for Nbo.4sRho.ss, has been applied to Sm.r [2] because it is simpler. Why has Nb3Rhs(Dht p Sm.r) a monoclinic cell? The orthorhombic e correlation breaks the rhombohedral symmetry. Also this phenomenon was not explained so far by the band model. Why change Nbo.4~Rho.~z(CuAu), Nbo.45 Rho.55(TaaRh~.r), Nb0.40Rh0.60.h(AuCd), Nb3Rhs(Sm), Nb3Rh7(VCo3.r ) and NbRh3(Cu3Au ) from the bonding type FB2 of the previous phases to I~U2? The increased mole fraction N ~ decreases the atomic volume and increases the e electron concentration and thus favours a closer packed g correlation. Why is VIr(Q2.2) a D homeotype of CuAu? One Hund insertion at Ir requires exactly two vacancies in e v. These vacancies are ordered and break the tetragonal symmetry of the CuAu type. Why does the mixture NbRh M exhibit only the two bonding types FB2 and I~U2? The bonding type is a consequence of strong interactions of electrons of various atoms and shells. If the interaction results in a low internal energy, it may be conserved in neighbouring phases. Why does the electron correlations model provide easier bonding types than the band model? The influence of electron correlation on structure is much stronger than the distribution of momenta of electrons and the account of the band model for correlation is not satisfactory.
References
[1] D.L. Riner, B.C. Giessen and N.J. Grant, Trans. AIME 230 (1964) 1250.
354
K. Schubert / Bonding types o f the NbRh M phases
[2] K. Schubert, Bonding types of two-component phases 1 (Max-Planck-lnst. f. Metallforschung, Inst. f. Werkstoffwissenschaft, Stuttgart, 1990). [3] K. Schubert, Z. Metallk. 76 (1985) 326. [4] K. Schubert, Kristallstrukturen und Zweikomponentige Phasen, (Springer, Berlin, 1964). [5] K. Schubert, Z. Metallk. 82 (1991) 582. [6] F.C. Frank and J.S. Kasper, Acta Cryst. 11 (1958) 184; 12 (1959) 483.
[71 B.C. Giessen, P.N. Dangel and N.J. Grant, J. LessCommon Met. 13 (1967) 62. [8] B.C. Giessen and N.J. Grant, Acta Cryst. 18 (1967) 10811.
[91 K. Schubert, J. Solid State Chem. 53 (1984) 246. [l()] J.C. Duthie and D.G. Pettifor, Phys. Rev. Lett. 38 (1987) 564.
Physica B 179 (1992) 349-354 North-Holland
On the bonding types of the NbRh M phases K. Schubert* Max-Planck-lnstitut fiir Metallforschung, lnstitut fiir Werkstoffwissenschaft, Seestrasse 92, W-7000 Stuttgart 10, Germany Received 20 August 1991
Several new crystal chemical rules suggest an improved interpretation of the intermediate phases in the mixture NbRhM by the electron correlations model: (1) Confluence of certain lower correlations to a united ground correlation g. (2) Collectivity of the correlation of a peripheral e shell with correlations of lower bands. (3) Spin compensation in both lower and higher correlations. These new rules permit an interpretation of structural phenomena in NbRh M such as icosahedral coordination in homeotypes of Cr3Si, stacking sequence in homeotypes of Cu, and deformation of higher symmetric structures into lower symmetric homeotypes. A decisive factor for the stability in NbRh M and in other transition metal-transition metal alloys appears to be the insertion of electrons according to the rule of Hund into interstices of the spatial correlation of the peripheral electrons. Although Hund insertion is only indirectly visible in diffraction experiments it is a feature of distinct influence on the bonding type of metallic phases. The insight mediated by the interpretation of the crystal structures may be useful in the search for phases with required properties.
I. I n t r o d u c t i o n
The mixture NbRh M contains a number of remarkable intermediate structures, solved by Ritter et al. [1], suggesting an attempt of bonding type analysis by the electron correlations model [2]. A first trial has been made [3] assum58 0.9 ing the electron count Nb ' Rh M , i.e. Nb contributes its 4d 5 electrons as valence electrons with correlation b, while the Nb4sp and Rh4d electrons form the core electron correlation c. Several reasons have been enumerated for this (nonEkmanian) type of count [3, 4], but an alternative proposal assuming a d electron correlation e filled by the peripheral d electrons of both components should also be investigated because it has been used to formulate the valuable rules of Raub and Matthias (see ref. [3]). This alternative [3] count N b "5 18R h ~9 18 yields bonding types obeying several new crystals chemical rules. (1) The correlation confluence, for instance 3d + 4sp in Nb (see ref. [2], p. 11, 33), re* Deceased 19 March 1992. 0921-4526/92/$05.00
suiting in the formation of a ground correlation g comprising the lowest electrons being of influence on bonding. (2) The collectivity of correlations ([2], p. 13) describing the fact that for instance the e lattice is contained in the g lattice and g has sites enough for the e and g electrons. (3) The spin-compensation ([2], p. 9) leading in conjunction with collectivity to an unexpected bonding type discussed in the following. The merits of the earlier [3] and of the present bonding type proposals must be compared. Certainly both proposals have some similarities since they try to interpret one and the same empirical data. Therefore, the reading of ref. [3] may be useful when the present arguments are somewhat brief to avoid unnecessary repetition. The bonding type is a part of the crystal structure not immediately visible to diffraction experiments. Nevertheless, because of its close connection with the two-electron density matrix (see ref. [2]), it is a phenomenon worth to be investigated. The correlations model is related to earlier valence models and may be considered as
© 1992 - Elsevier Science Publishers B.V. All rights reserved
K. Schubert I Bonding types oJ" the NbRh~ phases
350
a synthesis of various earlier models since most valence rules have to do with spatial correlations of electrons. In its present state the model is not a model for energy calculation. Since the energy differences of neighbouring phases are very small, the expensive energy calculation is not at present an appropriate method for finding crystal chemical rules: only few of these rules [2] have been found in this way. The following analysis wishes to show that bonding types are a useful and realistic means for the interpretation of alloy phases.
2. Analysis The marginal phases of the mixture NbRh M are considered first since the electron distances d , dg in intermediate phases as a function of the mole fraction N ~ interpolate the marginal values linearly (fig. 1), according to the rule of distances (ref. [2], p. 19). This rule is of use in the search for the bonding types of the intermediate phases. For Nb was found [2]: Nb(W, SR1.56)a10,36 = 3.30 A = e F ;1.5 = gB ;3, N ' s " = 13.5,54. Here W is the prototype of the structure type of
I ~
"~
N
O0
~ ~ n'rr
z
e~
.z~
.z~
Ln
LQ o o r" r " ry, p r
" ~ (.D ~--~ r" r ~
lag ag~ ig 8 .~
Z
_Q
I 1
.
Z -
Z
Z
Z
.c
-.<' /
~5
Ir----¢--, _ ._____.._,
0.6
Nb
0,2
0,4
0.6
mole froction
0,8 Nt~ h
Fig. 1. Electron distances in the mixture NbRh~,.
Rh
Nb; SR1.56 means Structure Reports Vol. 1, p. 56; a is the crystal cell matrix of Nb; 10,36 are the numbers of 4d and 3d + 4sp electrons per a: 3.30,& is the numerical value of a in the brief notation of ref. [2]; e v is the cell matrix of the e correlation and is of the cubic face centered Bravais type F; for the type nomenclature, see ref. [2]; ';1.5' is the brief notation of the commensurability matrix between a and e: the semicolon distinguishes the brief notation of the matrix from a number; g~ is the ground correlation cell matrix of the cubic body centered Bravais type; ';3' is the commensurability matrix g ~a" N s' are the numbers of sites per crystal cell in both correlations. It is seen that e is collective to g. Neither e nor g are fully occupied. The bonding type may be named FB2 [2]. gB exhibits spin compensation while e v does not show it according to collectivity and does not need it according to Hunds rule [2]. An interatomic spin compensation might cause some broadening of the e v points of the averaged correlation [2]. Contrary to Nb the component Rh must contain H u n d insertion [2] in e. Therefore the FB2 bonding type is not admissible for Rh, and in ref. [2] F'B2 was proposed, i.e. F displays Hund insertion ('). Unfortunately collectivity between e v and gB requires the unfavourable filling of tetrahedral instead of octahedral interstices of F. Apparently nature prefers for Rh another bonding type. In a bonding type ~'t~2 the compression (~) of F and B along [00 1] may be so strong that B is transformed to U, a tetragonally compressed B type showing in (1 1 0)u and (1 1 0)u hexagonal planes [2]. When for instance consecutive gLJ layers parallel to (1 1 0)u are occupied by alternating spins and eF is collective to gu then both e and g contain spin compensation and the bonding type [vU2 appears appropriate for Rh: Rh(Cu,SR1.69,24.223)a20 + 16,72 = 3.80 = e~2 ;2.5 = gu 4 ;5, N's"= 40,160. Note that 20 +spin electrons must be assumed in the a cell, and 16 - s p i n electrons. The spin compensation in g causes a little deformation of g leading to twinning of g in a, i.e. to a domain structure as it is known in the room temperature phase of iron, Fe.r. We now consider the intermediate phases in
351
K. Schubert / Bonding types of the NbRh M phases
the order of increasing mole fraction N ~ . Nb3Rh and Nb2Rh display an icosahedral coordination of Nb to Rh. For low concentration of Hund insertion these defects form isolated arrays (globules around Rh) [5], favouring icosahedral coordination. Metrical comparison with Nb yields the bonding type equation and the numbers of electron sites per a cell: Nb3Rh(Cr3Si,SR29.125)a40 + 8,144 = 5.12 = e F, ;2.5 = gB ;5, N/s~ = 62.5,250. The 22.5 e vacancies must be associated with the 8 H u n d inserted electrons, in order to reduce the potential energy of the above-mentioned tetrahedral interstices. The array of the icosahedral centers in Nb3Rh is described in a plane parallel to ( 0 0 1 ) . through one Rh, by the Schlfifli symbol 4 4 [6, 5]. The U.h 1 = / 3 U type of Nb2Rh may be attained from Nb3Rh essentially [5] by the transformation 44----~4343z. It leads to an icosahedra fusion [4] being an expression for the growth of the above-mentioned globules of Hund insertions because of increased N ~ . The bonding type follows by metrical comparison with Nb3Rh and observation of ~/~t¢~, "'S \" " Rh]" Nb~Rh(U.h i ,SR29.125)a150 + 40,540 = 9.81 ;5.07 --- evA/24.3 ;2.5 = gB~/97 ;5, N/s a = 243,970. There are 53 + 40 e-vacancies per a so that 93/40 vacancies in e v are available for one Hund insertion. The phase NbRh.h [1] has an unknown structure and the neighbouring phase Nb0.asRh0.52 is no longer a homeotype of Cr3Si, it belongs to the close packings. Metrical comparison yields the bonding type: Nb0.48Rh0.52(CuAu,SR29.125[1])a20 + 8,72 = 4.02;3.81 A = e¢2 ;2.25 = gu4 ;4.5, N/s" = 36,144. Apparently the e correlation is highly occupied, the distribution of reversed spins is now layerlike instead of globular because of the strong increase of e electron concentration. The number of g sites per atom is t~]/at = 36 while it was ''Sg N/at = 3 2 . 3 in Nb2Rh, because the increased Sg share of - s p i n along the Rh layers strains the structure. It is quite surprising that the bonding transforms already here from FB2 to ~'U2 of Rh.
The phase Nb0.45Rh0.55 is isotypic [1] to Ta0.4Rh0.6(O6.6 ), a quasi-hexagonal close packing with 6 layers parallel to a~, a 2 p e r ]a31 length, stacked in the sequence + + + - - The ordering of the components causes the orthorhombic deformation; for simplicity the bonding type is noted only for the hexagonal cell H2.83;13.59,~ and only for g, see fig. 2 for e. Nb0.45Rh0.55 (Ta0.aRh0.6.r,O6.6,SR29.125)a60 + 26,216 = 2.83;4.77;13.59 ~ = g u ~ / 1 2 ; 2 0 / 2 . While Nb0.48Rh0.52 had N/s] = 3 6 g sites per atom, Nb0.45Rh0.55 has 40. Hund insertion in e causes the stacking sequence. Because six electrons are contained in an insertion layer, 5 layers must be in a cell since the insertions stick together similarly as in the globules and they will cause the observed sequence. According to the rules of stacking [2] the series 6 n / 5 = 0, 1.2, 2.4, 3.6, 4.8, 6.0, for n = 0 , 1 . . . . . 5 must be calculated. From these values a shift value, say 0.02, should be subtraced to discard the underdetermined values near 0 and 6. Only the values next to integers (i.e. atoms) are considered as essential for the sign of the electro dipole in the a 3 direction at an atom. Assuming that the dipole vector points from the negative to the positive charge, the sign sequence ( + ) - - + + + is obtained. The change of dipole sign in the sequence causes a change of stacking sign [2]. A plausibility consideration says that a minus charge below an atom of the stacked atomic layer sees two times three plus-charges below itself in the case of no sign change, but 3 + 1 minus-charges in the case of sign change. The next phase is a high temperature phase, less ordered than Nb0.nsRh0.55 and therefore nearer to hexagonality. Once more gull is used as simple indicator for the ~'U2 bonding type.
a,
9uH
aet
a
/\/\ Fig. 2. The cells a, e, g of Nb3Rhs as projected on (00 1)~,.
352
K. Schubert I Bonding types of the NbRh M phases
Nbn.40Rh0.60.h(AuCd,SR29.125,)a20 + 9.6,72 = 2.81;4.81;4.51 ,~ = gUHX/12;6.5/2. W h e n there are 3 - s p i n layers in eB then the atomic dipoles caused by e parallel to a 3 have the sequence + - + - and determine by the above m e t h o d the atomic stacking. There are 39g sites per a t o m , and 39 e sites per cell. The phase Nb3Rh~ is a deformation h o m e o t y p e ( D h t p ) of Sm (ref. [3], drawing 4) and exhibits before the stacking + + - + + ++-. N b 3 R h s ( D h t p Sm,M9.9,SR29.70)a90 + 45,324 = M90.53°4.77;2.81 ;20.25 ,~ = eB3;2;15 = gUHX/12;30/2. T h e monoclinic angle is set to 90°C for simplicity. The slight compression of gull may be caused by a spin distribution with alternating sign in neighbouring g planes parallel to (0 0 1)~, for eB because of collectivity also spin compensation results. The eB correlation causes the monoclinic s y m m e t r y , as may be seen from fig. 2. The n u m b e r of e and g sites is N/s" = 180,720, correctly 90 sites for +spin are available. The observed stacking is reproduced: 9n/15=3n/5=O, (0.6), 1.2, 1.8, (2.4), 3.0. Possibly this bonding type decreases the upper decomposition t e m p e r a t u r e as c o m p a r e d with that of the following Nb3Rh 7. For this phase the F U 2 bonding type appears to be valid once more: Nb3Rh7 ( V C o 3.r,H6.18,SR29.125,[1])a120 + 67,432 = H5.48;13.41 A = ef34;6;10 = gull X/48;20, ,,s/V/~'= 10,40, N"s~ = 240,960. T h e commensurability e ~ a is written for the o r t h o r h o m b i c base centered cell (Q) of a. The stacking sequence + + + - - is the same as that for Nb0.4sRh0.5.s(O6.6 ). If it is assumed that an insertion layer parallel to (0 0 1)~ tends to be filled, i.e. takes 12 electrons into the hexagonal mesh, then about 5 of the ten Hund insertion layers parallel to (0 0 1)~ are occupied and favour the observed stacking: 6 n / 5 = 0, 1.2, 2.4, 3.6, 4.8, 6, after subtracting the shift value the sequence ( + ) - - - + + + appears. The n u m b e r of e sites per a is N/s~ = 240 so that sufficient sites for +spin e electrons are available. The e correlation must be twinned in a to yield the hexagonal symmetry.
The phase N b R h 3 is closely homeotypic to Nb3Rh 7, therefore a homeotypic bonding type should be tried. NbRh3(Cu3Au,SR29.126)a20 + 12,72 = 3.86 = %2; 2.5 = gu4; 5,J,s~S/<'= 40,160. Since Ns"e = 4 0 , there is no spin sign change necessary. The n u m b e r of g sites per atom is /at Nsg = 40. Once more the increased e concentration causes the denser g correlation. The slight deformation of gu must be attributed to the spin ordering in g. Since the close packed g layers are not parallel to the close packed a layers a stacking h o m e o t y p e of Cu is not formed. An interesting illustration of the interaction between H u n d insertion and F-vacancies in a FB2 bonding type is presented in the mixture V I r M investigated by Giessen et al. [7]: V(W, SR1.56,26.275)a10,32 = 3.02 ,~ = e v ;1.5 = gB ;3 [21, Us <'= 13.5,54. V3Ir(Cr3Si,SR23,233)a40 + 8,140 = 4.79 A = e v ;2.5 = gB ;5, N"<'s= 62.5,250. T h e r e are 15.6 F-vertices in the cell but only 8 electrons in H u n d insertion. Therefore, globules are f o r m e d and a h o m e o t y p e of Cr3Si becomes stable. VIr(Q2.2,SR30.6418])a40 + 16,152 = 5.79;6.76; 2.80/k = ev-,3;4;1.5 = gB6;8;3, N~" = 72,288. The phase is homeotypic to C u A u and a~ corresponds to 2a3(CuAu ). The compression (~) of the bonding type lies along a~. From the assumed commensurability follows that there are 72ev sites per cell. Subtracting 40 sites occupied by +spin electrons yields 32 vacancies being exactly two times the n u m b e r of H u n d insertions. This exact relation is favourable for ordering. Since one e~ layer parallel to (0 1 0)~ contains 9 sites the vacancies may be found in two times two neighbouring (0 1 0)a layers near the Ir layers parallel to (0 1 0)a and the H u n d insertions m a y be near. The layer-like distribution of vacancies causes the compression of e~ and the one +spin electron in the vacancy layer allows an ordering doubling the [0 0 1]CuAu. V0.45Ir0.55(CuAu,SR30,150)a20 + 8.8,60 = 3.89;3.65 ,~t = et2;2.25 = gu4;4.5, VIr3(Cu3Au,SR23.267)a20 + 12,82 = 3.81 it = e~2;2.5 = gu4;5, as in N b R h ~ above.
K. Schubert / Bonding types of the NbRh Mphases
3. Discussion
Several assumptions of the first interpretation [3] of NbRh M phases are corroborated. (1) Participation of at least two correlations in the bonding type. (2) Harmony of the commensurabilities b-Ca, -i e a, etc., i.e. these matrices contain many integral elements. (3) Stacking sequence of homeotypes of closed packings may be assessed from the bonding type by an analysis of the electro-dipole distribution along the stacking normal. Contrary to these conserved assumptions the electron count of ref. [3] had to be changed because in the bonding type analysis of other mixtures between transition elements it had become apparent that an e correlation containing the peripheral d electrons of both components leads to simpler bonding types. Accordingly, various questions on the structures in the alloys NbRh M find a simple answer by the electron correlations model. (1) Why are structures with icosahedral coordination formed? Because of globular array of Hund insertion into an FB2 bonding. (2) Why is the U.h~ type stable at higher d electron concentrations than the Cr3Si type? Because the icosahedral fusion allows greater insertion globules. (3) Why has Nb0.48Rb0.52(CuAu) a larger ~/a, ,,Sg value than Nb3Rh and Nb2Rh? The globules of Nb3Rh and Nb2Rh have grown together to infinitely extended layers along the Rh layer of the CuAu type straining the structure. (4) Why are in Nb0.45Rh0.55 the Rh layers pleated? In Nb0.nsRho.52 the maximum of Hund insertion in a plane layer is reached. This strong structural transformation entails a change of bonding type commensurability. The new bonding type contains an e correlation which causes electro-dipoles and these cause the pleating. (5) Why has Nb0.nsRh0.s5 the stacking sequence + + + - - - ? This observation obeys the rule [2] that a change in the sign of the electro-dipole vector of an atom in
(6)
(7)
(8)
(9)
(10)
353
the direction of the stacking normal causes a change in the stacking sequence. This is an independent indication for the influence of the spatial correlation of the electrons. The band model arguments for the stacking of Sm.r [10] have not yet been applied to Nb0.nsRho.55, but the present argument for Nbo.4sRho.ss, has been applied to Sm.r [2] because it is simpler. Why has Nb3Rhs(Dht p Sm.r) a monoclinic cell? The orthorhombic e correlation breaks the rhombohedral symmetry. Also this phenomenon was not explained so far by the band model. Why change Nbo.4~Rho.~z(CuAu), Nbo.45 Rho.55(TaaRh~.r), Nb0.40Rh0.60.h(AuCd), Nb3Rhs(Sm), Nb3Rh7(VCo3.r ) and NbRh3(Cu3Au ) from the bonding type FB2 of the previous phases to I~U2? The increased mole fraction N ~ decreases the atomic volume and increases the e electron concentration and thus favours a closer packed g correlation. Why is VIr(Q2.2) a D homeotype of CuAu? One Hund insertion at Ir requires exactly two vacancies in e v. These vacancies are ordered and break the tetragonal symmetry of the CuAu type. Why does the mixture NbRh M exhibit only the two bonding types FB2 and I~U2? The bonding type is a consequence of strong interactions of electrons of various atoms and shells. If the interaction results in a low internal energy, it may be conserved in neighbouring phases. Why does the electron correlations model provide easier bonding types than the band model? The influence of electron correlation on structure is much stronger than the distribution of momenta of electrons and the account of the band model for correlation is not satisfactory.
References
[1] D.L. Riner, B.C. Giessen and N.J. Grant, Trans. AIME 230 (1964) 1250.
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K. Schubert / Bonding types o f the NbRh M phases
[2] K. Schubert, Bonding types of two-component phases 1 (Max-Planck-lnst. f. Metallforschung, Inst. f. Werkstoffwissenschaft, Stuttgart, 1990). [3] K. Schubert, Z. Metallk. 76 (1985) 326. [4] K. Schubert, Kristallstrukturen und Zweikomponentige Phasen, (Springer, Berlin, 1964). [5] K. Schubert, Z. Metallk. 82 (1991) 582. [6] F.C. Frank and J.S. Kasper, Acta Cryst. 11 (1958) 184; 12 (1959) 483.
[71 B.C. Giessen, P.N. Dangel and N.J. Grant, J. LessCommon Met. 13 (1967) 62. [8] B.C. Giessen and N.J. Grant, Acta Cryst. 18 (1967) 10811.
[91 K. Schubert, J. Solid State Chem. 53 (1984) 246. [l()] J.C. Duthie and D.G. Pettifor, Phys. Rev. Lett. 38 (1987) 564.