Laves phases of uranium and 3d transition metals

Journal of the Less-Common ~etals,50 (1976) 57 - 71 0 EIsevier Sequoia S.A., Lausanne - Printed in the Netherlands

LAVES

PHASES

OF URANIUM

E. C. BEAHM, C. A. CULPEPPER Oak Ridge National Laboratory,

AND 3d TRANSITION

57

METALS*

and 0. B. CAVIN Oak Ridge, Tenn. 37830

(U.S.A.)

(Received March 15, 1976)

Summary Laves phases of uranium and 3d transition metals have been investigated theoretically and experimentally. Pauling’s description of metal bonds was used to calculate bond numbers and d orbital participation in bonding. Limits to compound formation were found to be related to limits in d orbital hybridization. Simple equations are presented to calculate interatomic distances and composition limits of these Laves phases. Lattice constants for Laves phases in the U-Cr-Fe, U-Cr-Co, U-Cr-Ni, and U-Cr-Fe-Ni systems are also presented and discussed.

Introduction The present investigation was prompted by a desire to understand better the formation of Laves phases of uranium with the 3d transition metals, especially chromium, iron, and nickel. Laves phases are intermetallic compounds of the AR2 type. These compounds crystallize in one of three closely-related structures: cubic Cl5 (MgCua-structure type), hexagonal Cl4 (MgZn~-st~cture type), or hexagonal C36 (MgNi~-s~uct~e type). We have used Pauling’s de~ription of metal bonds to examine Laves phases. Results of previous investigators have been examined in terms of single-bond radii and bond number. Chromium does not form binary compounds with uranium [ 11, but the present work demonstrates that chromium, with one or more of the elements manganese, iron, cobalt, and nickel, may combine with uranium to form Laves phase compounds. Lattice constants of Laves phases in the U-Cr-Fe, U-Cr-Co, U-Cr-Fe-Ni systems are presented and discussed in terms of Pauling’s description of metal bonds.

*Research sponsored by the Energy Research and ~eveIopment under contract with Union Carbide Corporation.

Admin~tration

58 ORBITALS 4s 3.d

NICKEL ‘.

-1

M

1.

F]

m

COBALT

Fig. 1. Transition metal orbital diagrams. l denotes a valence electron; 0 denotes au unfilled orbital used as a “metallic” orbital; t, 4 denotes non-bonding electrons; - denotes an empty orbital.

Discussion and calculations using literature data Background Pauling [2] used empirical methods based on bonding in non-metals and metals to derive eqn. (l), which expresses the relationship between interatomic distance and bond number as d = 2R( 1) - 0.600 a log N,

(1)

where d is the interatomic distance (W), R(1) the single-bond radius (i.e., the radius that results when a single bond is formed) (A), and N the bond number. Pauhng [3] also combined trends in interatomic distances of elemental 3d transition metals with orbital hybridization to produce an empirical equation. This equation gives the single bond radius as a function of the hybrid character of the orbit& and the atomic number of the element: R( 1) = 1.825 - 0.0432 -( 1.600 - O.lOOz)6,

(2)

where z is the number of electrons outside the noble gas shell, and 6 the amount of d character of the bond orbitals. Short-period transition metals have nine orbitals for possible use in bonding or occupancy by unshared electrons (five d orbitals, one s orbital, and three p orbit&). A characteristic of Pauling’s metal bonding theory [4] is the requirement that there be an orbital (actually 0.72 orbital) called the metallic orbital. This orbital permits unsynchronized resonance of valence bonds. For reference to Laves phases, it is important to note that each elemental component in a metallic compound need not have a metallic orbital. It is sufficient for one component in a binary system to have metallic

59

orbitals available. Valence bonds of the other component may then use the metallic orbitals of atoms surrounding it [ 5 ] . The term “valence” is used here to mean the number of electrons per atom actually taking part in bonding. Pauling [S] uses properties of elemental metals to assign valences. In the 3d transition metal series, the properties suggest that the maximum valence of 6 is attained for chromium and this valence persists up to and including nickel. Using Pauling’s assigned valence of 6, the minimum d orbital participation in bonding is 2 of 6 bonding orbitals, or 6 = 1/3. Similarly, the maximum is 3 of 7 orbitals, or 5 = 3f7. This is shown di~mmatic~ly in Fig. 1 for nickel, cobalt, and iron. The d orbital p~ticipation in bonding can range between l/3 and 3/7. We have used Pauling’s relationship, shown in eqn. (2), to determine limits to single-bond radii based on limits of d orbital participation in bonding, 6. These limits are shown in Table 1. Applications to Cl 5 Laues phases In this Section, we apply eqns. (1) and (2) given by Pauhng to Cl5 Laves phases. It is assumed that the elements in a Laves phase compound hybridize, within limits, to an extent necessary to form that compound. These limits are shown in Table 1 using a valence of 6 for the 3d transition metals chromium to nickel; they arise from limits to d orbital participation in bonding. According to this assumption, in any given Laves phase compound the components will have hybridi~tion and thus single-bond radii which are characteristic of that compound. These single-bond radii may or may not be the same as the elemental single-bond radii, depending on the hybridization necessary for compound formation. In cubic Cl5 Laves phases, as well as in hexagonal Cl4 structures where the axial ratio is close to the ideal value of 1.633, the relationship between valence and numbers of nearest neighbors is given by: 12NAB + 4NAA = V,, GN‘4n + 6Naa = v,,

(3) (4)

where NAB is the bond number of A-B bonds, NAA the bond number of A-A bonds, Nan the bond number of B-B bonds, V, the valence of A-type atoms, and V, the valence of B-type atoms. For Laves phases involving uranium and 3d transition metals, uranium atoms occupy the A positions and the transition metal atoms occupy the B positions. Applying eqn. (1) to Laves phases yields: d AA

=

2R,(1)

- 0.600 A log NAA

(5)

d BB

=

2&(l)

- 0.600 A log Nan

(6)

d AB

=RA(l)+&(l)-

0.600A

lOgNAB,

(7)

where dA,+ iS the distance between two A-type atoms, dBB the distance between two B-type atoms, dAB the distance between an A-type atom and a Btype atom, R,(l) the single-bond radius of A-type atoms, and Ra( 1) the single-bond radius of B-type atoms.

60 TABLE

1

Limits of single-bond radii RB( 1) B

Maximum (corresponds to 6 = 3/7) (A)

Minimum (corresponds to 6 = l/3) (A)

cr

1.138 1.138 1.138 1.138 1.138

1.234 1.224 1.214 1.205 1.195

Mn Fe co Ni

In order to examine Laves phase structures in terms of bond numbers, d orbital participation in bonding, and single-bond radii, it is necessary to have bond numbers and single-bond radii which are consistent with eqns. (3) - (7) inclusive. Be&fit values have been obtained with the aid of a computer. Valence A, valence B, and the literature value of the lattice constant were used as input data to the computer. A Pauling valence of 6 was used for uranium as well as the 3d transition metals from chromium to nickel inclusive. In an iterative procedure, the computer varied the value of Nss from 0.00 to 1.00 in increments of 0.01 to calculate best-fit values for NAA, Nss, NAB, R,(l), R,(l), and 6,. A flow diagram illustrating the computer calculation is shown in Fig. 2. It is unfortunate that we do not have a relationship between singlebond radii and hybridization for uranium analogous to eqn. (2) for the 3d transition metals. Thus, the limits to formation of Laves phases based on limits of hybridization must be discussed using only the single-bond radii of the 3d transition metals. Calculations for Cl 5 Laves phases Uranium forms cubic Cl5 Laves phase compounds with manganese, with iron, and with cobalt [6, 71. In every computer calculation of Cl5 Laves phases of uranium with Mn, Fe, or Co, the best-fit bond numbers, calculated by the methods described previously, were NAA = l/2, NBB = 2/3, and NAB = l/3. This interesting result agrees with Pauling’s hypothesis [ 51 that bond numbers in ratios of small whole numbers are favored. Table 2 shows the single-bond radii and d orbital participation in bonding calculated by using the bond numbers just described. Two lattice constants are shown for each of the three compounds UMnz, UFes, and UCo,; these are the high and the low values given in the literature. In every case, the single-bond radii of each transition element are at or within the limits of hybridization shown in Table 1. Petzow et al. [8] examined the pseudobinary systems UFes-UCo,, UCos-UMns, UMns-UFez, and the pseudoternary UFez-UCoz-UMns. Computer calculations again give the bond numbers NAA = l/2, Nsa = 2/3, NAB = l/3 throughout all of these systems. The d orbital character and

61

(OS N,,

N,,

VB-eqn.(4)

“A -

+eqn

5 t,

mcrement

A NBB = O.Oi )

A 1 A31 1 Nap

dAA--

dAe---)eqri.(7)

_

look for

I

;t#,

- NA,“2)

minimum

t “best

fit”

RA(O,

RrJO,

NBB,

NAB, NAA

c eqn. (2) 1 88

Fig. 2. Flow diagram of computer

calculation.

single-bond radii vary, but in all cases the shale-bond radii of the transition metals are at or within the limits given in Table 1. To evaluate further this description of Laves phases, we calculated bond numbers for Laves phases involving only transition metals. These results are summarized in Table 3. The single-bond radii and values of S for the 3d transition elements in all of the compounds shown in Table 3 were within the limits given in Table 1. Reference 11 contains specific R,(l) and 6 a values for the transition metal Laves phases shown in Table 3, as well as those for some hexagonal Laves phases involving transition metals. Calculations for known Laues phases of UNiz and the pseudobinary of UNi,--UF2

UNiz has several unique features compared with the other Laves phase compounds of uranium and 3d transition metals. Although UNiz is not

62 TABLE

2

Cubic Cl5 Laves phases Compound

Lattice constant (A)

mn2

7.175 7.146

UFe2 UC02

TABLE

UMn2, UFeg, UCo2 Reference

Uranium R (1) cr;‘,

Transition metal RB( 1) (A)

Transition metal 6

8 9

1.463 1.457

1.216 1.210

0.343 0.348

7.065 7.041

7 10

1.439 1.434

1.196 1.192

0.356 0.361

7.005 6.991

7 8

1.426 1.423

1.185 1.183

0.361 0.364

3

Bond numbers

for transition

metal Laves phases

TX&, TiCo2 ZrCr,, Y&Fe,, ZrCo2, ZrNiz

Valence

A =4

Valence

%= 6

HfCrz, HfCo2, HfNi2

NAA = l/4, NAB = l/4. Nnn = 3/4

NbCr,,

NbCoz

Valence

A = 5

TaCr,,

TaCo2, TaFeNi

Valence

%= 6

I

N,,

= 0.368, N,

= 0.294, N,

= 0.706

isomorphous with UMn,, UFe,, and UCo2, it forms a hexagonal C14-type Laves phase compound. The axial ratio for UNia, c/a = 1.662 [6], differs significantly from the ideal value; there are two A-A interatomic distances, three A-B interatomic distances, and two B-B interatomic distances. In the cubic Cl5 Laves phase compounds of uranium and 3d transition metals, the interatomic distances decrease across the period, UMn, > UFe, > UCo,. The compound UNi2 is unique since the inte~tomic distances are larger than in UCo2 [6]. In the UN&-UFe, pseudobinary, Brook et al. [7] report a minimum in the cubic Cl5 lattice constant at 32 - 33 at.% Fe. This behavior of UNi2 and UNi2-UFe, wilI be discussed below. Pauling [12] has pointed out that in some crystals metal atoms are not bonded together even though the interatomic distance corresponds to a significant bond. In his examination of the Cl5 Laves phase compound LaNi2, he found it necessary to assume that the La-La contacts did not represent bonds (NAA = 0). To discuss the unusual interatomic distances in UNi, and in the UNia-UFe, pseudobinary, we must also assume that NAA = 0 in UNi,. From eqns. (3) and (4), when NAA = 0, N AB=

VA 123

(8)

63

and N BB

2vf3 =

v,

12

.

With a Pauling assigned valence of 6 for both uranium and nickel, these equations give bond numbers of NAB = l/2 and NBB = l/2. Equations (8) and (9) apply both to cubic Cl5 Laves phases and to hexagonal Cl4 Laves phases with an axial ratio close to the ideal value. In UN& the axial ratio deviates from the ideal; hence, we use the mean of the two transition metal distances, daB, to discuss single-bond radii. In UN&-UX, (X = Mn, Fe, Co) p~udobin~ies, the effective transition metal bond number for the calculation of interatomic distances will be a combination of the individual bond numbers: NBB

= T XiNBBi

(10)

9

where Naa is the transition metal bond number for use in eqn. (6), i is the transition metal in the component Laves phase (i.e., Fe in UFe,, Ni in UNi,), Nar+ are the transition metal bond numbers for the component Laves phases, and xi is the fraction of B sites occupied by atoms of type i. From the definition of Xi we may express the UN&-UFe, pseudobinary as ) It is thus clear that ~:Xi = 1, and Xi = 3f2 X (at.% i/100). U(Fe,.v, Ni,,ni 2. The tmnsition metal bond number for UFe, was found previously to be N aB,re = 2/3. From eqn. (9) for UNi2, Naa,Ni = l/2; therefore, eqn. (10) becomes: NBB = Combining

XFe

X 213 + XNi X 112.

(10’)

eqns. (6) and (10) we may write:

d BB = 2Ra(l)

For UNi2-UFes d BB = 2&(l)

- 0.6 log C XiNae,i.

(6’)

i

this becomes: - 0.6

log

[XFe

x

213 + (1-

XFe)

x

1f2].

Using data given by Brook et al. [7] for the UNia-UFe2 pseudobin~, singlebond radii for Ni and Fe calculated using eqn. (6) are at or within the limits given in Table 1. These single-bond radii will be examined in detail in a later Section of this Report. We still have questions about the formation of Laves phases of uranium and the transition metals. Nickel, cobalt, iron, and manganese form Laves phase compounds with uranium; chromium, to which Pauling again assigns the valence of 6, does not form a binary compound with uranium. However, chromium may have the ability to form Laves phase compounds of the type U(Cr,Mi_, ), M = Ni, Co, Fe, Mn. Formation of Laves phases of this type would, according to this description, require that the transition metal single-bond radii are at or within the limits given in Table 1.

64

Experimental Materials and preparation Alloys of uranium and chromium with iron, cobalt, and/or nickel

were prepared by arc melting mixtures of the respective elements and annealing them in flowing argon for 66 h at 800 “C. All materials were obtained from the target preparation group of the Solid State Division of Oak Ridge National Laboratory, and had a purity of >99,9%. After all elements had been weighed to within 0.0001 g, the metals were charged in a water-cooled copper insert and pumped to 2 X 10e6 Ton in a non-consumable arc furnace. The system was then backfilled with argon to Z/3 atm. Before melting the alloys, a zirconium button was melted to remove any reactive gas in the argon atmosphere. This was followed by arc melting the metals five times, with each button being turned over between each melt to ensure that the samples were homogeneous. The weight loss for each alloy was less than 5 mg out of 6 g. The alloys were then annealed in flowing argon for 66 h at 800 “C. The compounds thus formed were brittle in nature and were ground into powder <50 pm for examination by X-ray diffraction in a Debye-Seherrer camera. The powder ~ffraction patterns were taken with CuKa radiation. The results of the X-ray diffraction analysis are presented in Table 4. Indeed, compounds of the type U(Cr,,, Fel_x,cr)tl do exist and have the Cl 5 Laves-phase structure. The lattice constant increased with additional chromium up to a value of 7.1687 f 0.0005 A. The alloy with the greatest chromium content (55.4 at.% Cr) had chromium or iron lines present in the X-ray diffraction film in addition to those caused by the Laves-phase compound. Single-bond radii, RB( l), and d orbital participation in bonding, 6, for the transition elements in these alloys were calculated by the method shown in Fig. 1, and are shown in Table 4. The bond numbers are the same as UMna, UFe,, UCo,:N,, = l/2, NAB = l/3, Nan = 2f3. The single-bond radii increase with increasing chromium content up to a maximum of 1.214 a. This corresponds to the minimum d orbital participation in bonding of 0.333 for iron. Thus, the single-bond radii are at, or within, the limits given in Table 1, and the m~imum composition limit for chromic content corresponds to the maximum single-bond radius of iron in the ternary Laves-phase compound. U-G-CO Table 4 lists the lattice constants for U(Cr, ,cr CO~_~,& compounds. The lattice constant increases with chromium addition. The alloy containing 40 at.% Cr had chromium or cobalt X-ray diffraction lines in addition to those caused by the Laves-phase compound. The calculated bond numbers are the same as in the U&In,, UFe,, UCo, compounds. The transition metal single-bond radius calculated from the maximum lattice constant corresponds well to the maximum single-bond radius of cobalt with valence 6 1.205 a as shown in Table 1.

4

55.4 44.5 33.3 22.2 11.3

11.3 22.2 33.3 44.5 55.4

-

-

E.%)

:t.%)

23.4 33.3 40.0

composition

Alloy

Uranium-chromium:

TABLE

Cl5

43.3 33.3 26.7

-

:.x,

33.3 at.% U, plus:

iron, cobalt constant

7.0904 * 0.0004 7.1116 f 0.0003 7.1445 f 0.0003 7.1687 f 0.0005 (7.1667 * 0.0018 Cr, Fe lines a.ko present) 7.0837 +_0.0003 7.1183 + 0.0003 (7.1237 + 0.0019 Cr, Co lines also present)

(A)

Lattice

Laves phases

1.199 1.206 1.206

1.200 1.204 1.210 1.214 1.214

transition metal

R( 1) of

0.368 0.362 0.362

0.367 0.363 0.357 0.353 0.353

6 Cr

-

-

0.351 0.346 0.339 0.333 0.333

6 Fe

0.341 0.332 0.332

-

6CO

pt.%)

61.7

56.7

51.7

51.7 43.7 33.3 30.7 22.0 11.0 31.7 21.7 11.7

:t.w,

5.0

10.0

15.0

15.0 23.0 33.3 36.0 44.7 55.7 28.0 36.0 44.0

Alloy composition

a = 4.9902 f 0.0007 c = 8.2670 f d.0008

-

7.0 9.0 11.0

7.1047 7.1153 7.1425 7.1441 7.1531 7.1519 7.1157 7.1450 7.1476

+ f + + + f * f f

0.0018 0.0002 0.0002 0.0004 0.0006 0.0025 0.0008 0.0006 0.0010

Cubic Cl5

a = 4.9856 f 0.0013 c = 8.2569 f 0.0011

-

-

a = 4.9741 2 0.0007 c = 8.2511 f 0.0007

Hexagonal Cl4

(A)

-

:.w,

Lattice constant(s)

Laves phases

33.3 at.% U, plus:

Uranium-chromium-iron-nickel

TABLE 5

Faint Fe, Ni lines also present. Fe, Ni, Cr lines also present.

Very faint Ni lines also present. Cr, Ni lines also present. Cr, Ni lines also present.

Comments

67

U-Cr-Ni and U-Cr-Fe-Ni Hexagonal Cl4 and cubic Cl5 Laves phases were detected in the U(Cr,,c, Nil-x,cr)a system. Lattice constants for these compounds are given in Table 5. Brook et al. [7] studied the pseudobinary systems UNi,-UXa, where X = Mn, Fe, or Co. Three Laves-phase structures were detected across each of these binary sections: UNia (C14) -+ (C36) -+ (C15) UX2. However, the range of the C36 structure narrowed greatly from cobalt to manganese. In the UNie-UCo, section, the- C36 structure was reported to be stable from 11.1 to 16.0 at.% Co; for the UNi,-UMna section, the range was only 15.8 16.2 at.% Mn, If this narrowing trend continued into the chromium-containing alloys, we would not expect to find the C36 structure present in the U(Cr,,crNil._x,cr)2 system. The U(Cr,,,Nil_,,c,)s alloy containing 15 at.% Cr had X-ray diffraction lines for the Cl4 structure and very faint lines for the Cl5 structure. As with the UNia-UFe, pseudobinary, we use eqn. (6’) for calculating transition metal single-bond radii in U(Crx,crNil-x,cr)a compounds. The transition metal single-bond radius for the compound coi&Gning 33.3 at.% Cr was found to be 1.192 a. The sample containing 36 at.% Cr had very faint nickel lines present in the X-ray pattern in addition to the Laves-phase compound. The transition metal single-bond radius for this compound (assuming no free nickel) was 1.194 a. Thus, the m~imum limiting value of the singlebond radius is between 1.192 pi and 1.194 8, which is very close to the maximum value for the single-bond radius of nickel, 1.195 A, as given in Table 1. Table 5 shows the lattice constants obtained for some cubic Laves phases of uranium with chromium, iron, and nickel. For calculating transition metal single-bond radii in these compounds containing three transition metals, eqn. (6’) becomes dBB = WI,(I)-

0.6

IOg [(xc~

+ XF~)

X 2/3 + XNi X l/2] *

(6”)

The alloy containing 33.3 at.% U, 28 at.% Cr, 7 at.% Fe, and 31.7 at.% Ni showed only the X-ray diffraction lines of the cubic Laves-phase compound. From eqn. (6”), the transition metal single-bond radius in this compound is 1.189 W. The sample cont~ning 33.3 at.% U, 36 at.% Cr, 9 at.% Fe, and 21.7 at.% Ni had faint iron and nickel lines in the X-ray diffraction pattern in addition to those from the Laves-phase compound. The single-bond radius of the transition metals, assuming no free iron or nickel, is 1.199 a. Thus, the maximum single-bond radius is between 1.189 A and 1.199 8, which is close to the maximum value for the single-bond radius of nickel indicated in Table 1. Linearity of single-bond radii and interatomic distance In calculations of single-bond radii using eqn. (6) or eqn. (6’), we have observed a linear variation in single-bond radius with composition. Thus,

MAXIMUM

I.(95

SINGLE

BOND RADIUS

OF NICKEL

/ 1.190 -

U-Cr-Ni

I.185 -

0

.

/

/ 0

1.180 -

/

U-Fe-N1 f.475 -

0 .

I.470 -

/

0

/

. 0

A i

I

1

IO

I

I

20

30

ATOM

I 50

I

40

I 60

I

% Fe or Cr

Fig. 3. Linearity of single-bond

radii. RB( 1) us. at.% Fe or Cr.

using eqn. (6’) and the interatomic distances given by Brook et al. [7] for the UN&-UFe, pseudobinary system, a plot of RB (1) us. at.% Fe yields a straight line, as shown by the circular points in Fig. 3. This indicates a linear relationship with composition of the transition metal single-bond radii of Fe in UFe,, R,(l), and Ni in UNis, R,,(l). Using the value of Baenziger et al. [6] for the lattice constant of UFe,, 7.058 A, the single-bond radius of iron is R,,(l) = 1.195 A. For UNia, we take the mean of the two transition metal interatomic distances given by Baenziger et al. [6] to obtain dBB,Ni = 2.498 A. With a bond number of l/2, eqn. (6) gives 1.158 R as the single bond radius of Ni in UN&. The linear relation with composition of the singlebond radii is thus expressed by: RB

(l)

For the UNia-UFe, Ra(1)

(11)

= T xiRi(1)e

system this becomes:

=xFe (1.195)

+ (1 -~r,)(1.158)

A.

(11’)

The straight line in the RB (1) us. at.% Fe plot of Fig. 3 is given by eqn. (11’). Combining eqns. (6’) and (11) gives: d BB = 27 XiRi(I) - 0.6

log

F i%iNBB,i. 1

(12)

69

For UNis-UFe,

this is represented

d BB = { 2[x,,(1.195)

by

+ (1- x,,)(1.158)]

+ (1- x&/2]

- 0.6 log [xFe X 2/3

} f 0.002 A.

(12’)

Equation (12’) reproduces d BB values observed by Brook et ~2. [7] within ~0.002 A, and the calculated values reflect the minimum they observed at 32 - 33 at.% Fe. Using eqn. (6’) with our data for the U-Cr-Ni Laves phases, a plot of R,(l) us. at.% Cr yields a straight line, as shown by the triangular points in Fig. 3. Analogous to eqn. (12’), the transition metal interatomic distance in Laves phases may be expressed as: U(%,c,Nil-,,c& d BB = { 2[xc,(1.226)

+ (1- xc,)(1.158)]

+ (1- xcr) x l/2] } f 0.002 A,

- 0.6 log [xc, X 2/3 (12”)

where, as determined earlier, the single-bond radius of Ni in UNis is 1.158 A; the single-bond radius of Cr in a hypothetical “UCrs” compound is assumed to be 1.226 A. Transition metal single-bond radii in solutions of UMn,, UFe,, and UCo, may also be examined for linearity with composition. Petzow et al. [8] noted that the lattice constants of the pseudobinaries of UMn,, UFe,, and UCos deviated only slightly from a linear variation with composition. For continuous solid solutions, linear additivity of interatomic distances may be expressed as: dgg = Z XidBB,i s

(13)

i where dBB,i = transition

metal interatomic distances for the components of the solution, in this case UMns, UFe,, and UCos. Equation (13) is a statement of what is often called “Vegard’s Law” in terms of interatomic distance. Combining eqns. (6) and (13) gives: d Bz = 2x xiRi(l) i

- 0.6 7 xi 1OgNaB.i

(14)

Equation (14) expresses “Vegard’s Law” in terms of single-bond radii and bond number of the components of the solid solution. Comparing eqn. (12) with eqn. (14) and noting that, in general, --OS6 C 3Ci1ogNsn.i + -0.6 i

log Z Xi Naz,i, i

we see that “Vegard’s Law” is obeyed only when the bond numbers are the same and when there is a linear relationship of the single-bond radii with composition. In the UNis-UFe, system, the minimum in the lattice constant may be attributed to different bond numbers in UNis and UFes. In the UMn,, UFe,, UCo, pseudobinaries and pseudoternary where the bond numbers are the same, Petzow’s observation of near linearity in the lattice constants implies a linearity in single-bond radii. Table 2 shows that variations exist in the reported lattice constants for cubic Cl5 Laves phases. However, the data of Petzow et al. [ 81 can be represented in eqn. (14) as:

70

dgg = {2[~,,(1.215)

+x,(1.195)

+ xc,(1.181)]

-log

2/3} rt 0.002 8, (14’) radius

where the single-bond radius of Fe in UFe, is 1.195 W, the single-bond of Mn in UMn, is 1.215 A, and the single-bond radius of Co in UCo, is 1.181 A. For Cl5 Laves phases in the U-Cr-Fe system, the transition metal interatomic distance can be expressed in the form of eqn. (14) as: dgg = { 2[~,,(1.226)

+ (1- xcr)( 1.195)]

- 0.6 log 2/3} f 0.003 a. (14”)

Equation (14”) reproduces our experimentally determined interatomic distances, dgg ,within f 0.003 A. The observation of a linear relationship with composition of the singlebond radii enables us to write simple equations to predict the formation of Laves phases in multicomponent systems involving uranium and 3d transition metals. Thus, using the maximum single-bond radius of nickel (1.195 A) as a limit, we expect Laves phases to form in the U-Cr-Fe-Ni system if: ~cr(1.226) Similarly,

+ Xre(l.195)

+ XNi(1.158) < 1.195 A.

Laves phases form in the U-Cr-Fe

x&1.226)

+ x,,(1.195)

(15)

system if:

<, 1.214 A.

(16)

From eqns. (12’), (12”), (14’), and (14”) we write a general equation for calculating the transition metal interatomic distance in Laves phases of uranium with 3d transition metals: d aB = 2C xiRi( 1) - 0.6 log [2/3 X iZNi C xi + It2 x

i

XNilP

(17)

where i = Cr, Mn, Fe, Co, Ni, and R,,(l) = 1.226 W, R,,(l) = 1.215 A, R,,(l) = 1.195 ,&,.Rc,(l) = 1.181 A, andR,i(l) = 1.158 8.

Conclusions Most discussions of Laves phases have included a discussion of an “ideal” radius ratio based on metallic radii [ 13,141. The “ideal” ratio of the metallic radius of the A component over that of the B component is (3/2)lj2. It is usually noted that there are wide variations in the metallic radius ratios of actual Laves phases [14,15]. The approach to Laves phases we have given offers an alternative to the usual practice of discussing these compounds in terms of variations about “ideal” radius ratios. In this description, single-bond radii and d orbital participation in bonding produce limits to compound formation. Experimental detection of Laves phases in the U-Cr-Fe, U-Cr-Co, U-Cr-Ni, and U-CrFe-Ni systems are therefore of great significance. To the authors’ knowledge, these are the first cases of ohromium taking part in a uranium-transition

71

metal compound. to the maximum of 6.

In all cases, the maximum chromium content corresponds single-bond radius of iron, cobalt, or nickel, with a valence

Acknowledgements The authors express their gratitude to Marcia L. Beahm for computer programming and Terry B. Lindemer for a critical appraisal of the work.

References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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