Neutron and X-ray diffraction studies of the structure of non-crystalline materials

Journal of Non-Crystalline Solids 76 (1985) 29-42 North-Holland, Amsterdam

29

N E U T R O N AND X-RAY DIFFRACTION S T U D I E S OF T H E S T R U C T U R E O F NON-CRYSTALLINE MATERIALS C.N.J. W A G N E R Department of Materials Science and Engineering, Universi(v of California, Los Angeles, California 90024, USA Neutron and X-ray diffraction experiments have provided useful information about the topological and chemical short-range order in non-crystalline materials. The availability of new sources and detectors for X-rays and neutrons has greatly improved the statistical accuracy of the scattered intensity and extended its range in momentum (Q) space, yielding high-resolution atomic distribution functions. The methods of isotopic and isomorphous substitution have been used to determine the partial atomic structure factors and their corresponding atomic pair distribution functions in binary metallic systems, and to evaluate the nearest-neighbor interactions in more complicated inorganic glasses. Recent results of structural investigations on Ni-based amorphous alloys and on halide glasses are discussed.

1. Introduction The structure of multicomponent, non-crystalline materials is still not yet well understood. The problem lies in the diffuse nature of the scattering pattern which permits us to determine only a weighted average of the partial atomic distribution functions [1]. The weight factors in this average depend on the scattering amplitudes of the individual components in the non-crystalline materials. They can be changed by using different radiation probes (e.g. X-rays and neutrons), and by isomorphous and isotopic substitution of at least one of the alloying elements. In this paper, a critical review will be presented of the recent efforts made to evaluate the partial structure factors in binary metallic glasses, and to elucidate the more complicated structure of multicomponent glasses. 2. Theoretical background The scattering of neutrons and X-rays by non-crystalline materials can best be described by the Van Hove scattering function [2]: the scattering cross section per unit solid angle of scattering [2 at the scattering angle 20 and per unit frequency w or energy E = h~ is proportional to the incoherent and coherent scattering functions Si( Q, ~) and St( Q, w), respectively, for a system of N identical atoms [3], i.e.

de°

d~2d~o

N(k,/ko){[(bZ)-(b)Zlsi(Q,o~)+(b)zS~(Q,w)}

0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(1)

30

C.N.J. Wagner / Neutron and X-ray diffraction studies

where k 0 and k~ are the incident and scattered wave vectors, respectively, Q = k 0 - k~ is the diffraction vector, b is the coherent scattering length of the atom which is related to the coherent scattering cross section oc = 4~rb 2, and a 1 = 4~r[(b 2) - ( 2 ) 2] is the incoherent scattering cross section. The function Si(Q, w) is the Fourier transform of the correlation function Gs(r, t) representing the probability of finding the atom at the position r at time t which was at r = 0 at t = 0, i.e.,

Si(Q, ~0) = (2~r)

'ffGs(r, t)exp[i(O-r-

tot)ldrdt.

(2)

Similarly, the function So(Q, ~o) is the Fourier transform of the correlation function G(r, t) which describes the probability of finding an atom at the position r at time t when there is an atom at r = 0 at t = 0, i.e.

Sc(Q, ~0)= ( 2 ~ r ) - '

fro(r, t) e x p [ i ( q ,

r - ~0t)ldrdt.

(3)

2.1. Static approximation of the scattering 2.1.1. Monatomic systems If the energy E 0 of the incident neutrons or X-rays is large compared to the energy exchange h~o between the radiation probe and the system, i.e., E 0 > hco, it can be assumed that the interaction is elastic, i.e., I k l ] = I k01. This assumption is well satisfied in X-ray scattering, but corrections must usually be applied to the neutron scattering data for the deviation from the static approximation, the so-called Placzek correction [4,5]. By integrating d2o/(d~2d6o) over all energies, we obtain doc/dI2 = IN(Q) =

N(b)2fSc(Q, ~0)d~o = N(b)2S(Q),

(4)

where S(Q) is the (static) structure factor of the non-crystalline material, i.e.,

s ( o ) = f a(r, 0) e x p ( i Q , r)dr.

(5)

The correlation function G(r, 0) can be replaced by the n u m b e r density function p(r) and the Dirac delta function 8(r), i.e.

G( r, O) = 8( r) + p( r ).

(6)

By adding and subtracting the average atomic density P0 to G(r, 0) and neglecting the forward scattering poV(Q) = pofv(r) e x p ( i Q , r)dr, where v(r) is a slowly decreasing function of r and v ( 0 ) = 1, we obtain for an isotropic medium

S(Q) = 1 + f 4~rr2 [ p ( r ) - Po] [(sin q . r ) / ( Q , r ) l d r .

(7)

C.N.J. Wagner / Neutron and X-ray diffraction studies

31

2.1.2. Multicomponent systems It has been repeatedly shown that the coherent intensity per atom Ia(Q) = IN(Q)/N in a multicomponent system, consisting of n elements 1, 2, 3. . . . . i, can be expressed as the weighted sum of the partial atomic pair structure factors Lj(Q) [1], i.e.,

ia(Q ) = ( [ f ( Q ) ] 2 ) + E ~_,Gf~(Q)cJj(Q)[Iij(O) - 1], i

(8)

j

where f,(Q) is the atomic scattering amplitude of element i [ f ~ ( Q ) - b, for neutrons], ([f(Q)]2) = Eei[f~(Q)]2, and ci is the atomic fraction of element i. The partial structure factor Iij(Q ) is the Fourier transform of the partial number density function O,j(r) which describes the number of j-type atoms per unit volume at the distance r from an /-type atom, i.e.,

I ,j ( Q) - l = f 4Trr{[o,j(r)/cs] -Oo }(sin Q'r)dr.

(9)

Since the scattering amplitude f(Q) is a function of Q for X-rays, it has been customary to introduce the total structure factor (or interference function), defined in such a way that it will modulate about a constant, usually chosen to be unity. It is readily seen that the following definitions will satisfy this requirement:

I(O) = { Ia(Q) - [ ( f 2 ( O ) ) _ ( f ( O ) ) 2 ] }/(f(O))Z

(10)

S(Q) = Ia(Q)/(fZ(Q)).

(11)

and Eqs. (10) and (11) represent two alternative definitions of a total structure factor, each of which has certain advantages. Thus:

I(Q) - 1 = Y'~ Y'~ W~j(Q)[ Io(Q ) - 1], S(Q)- 1

=EEw~/(Q)[(/(Q))2/(f2(Q))][I,s(Q)-I],

(12) (13)

where

W~j( Q ) = cj~( Q )cJj( Q ) / ( f ( Q ) ) 2. (14) The Fourier transforms of Q[Iij(Q)-I], Q[I(Q)-1], and Q[S(Q)-I] yield the total and partial reduced pair correlation functions G,/(r), G i (r), and Gs(r), respectively, i.e.

G,:(rl=(2/~r) f Q [ I , j ( Q ) - l](sin Q'r)dQ=4rrr([o,j(r)/cj] -Oo},

(15) Gi(r )

=

(2/Tr)fQ[I(Q)

- al (sin Q. r)dQ = Y'~ • W,j(0)G,j (r),

06)

Gs(r ) = ( 2 / = ) f o [ s ( Q ) - 1] (sin Q. r)dQ

= E E W,:(O)[(f)2/(f2)] G,s(r).

(17)

32

C.N.J. Wagner / Neutron and X-ray diffraction studies

It is readily seen that the differential pair correlation function D(r) can be expressed as

D( r ) = (f(o))zG,( r ) = ( f 2(O))Gs( r)

The total pair-correlation function T(r) is given by: r ( r ) = O ( r ) + 47rr0o(f(0)) 2 = ~ ~cj~(O)cJ~(O)4~rro~j(r)/q.

(19)

2.2. Evaluation of the partial structure factors. The structure of binary systems is characterized by three atomic distribution functions oH(r), pz2(r), and 012(r) which are the Fourier transforms of the corresponding partial structure factors II1(Q ), I22(Q), and 112(Q). In order to determine these partial structure factors, at least three independent scattering experiments must be carried out. Since the total structure factor I(Q) or S(Q) is the weighted sum of the partial structure factors Iij(Q ), we must change the scattering amplitude f,(Q) in the weight factor W,j(Q) (eq. (14)) in each experiment. This can be accomplished by: (1) Isomorphous substitution: One or both elements i in the amorphous system are partially or totally replaced by physically and chemically similar elements. (2) Isotopic substitution: In neutron scattering experiments, the scattering amplitude of some elements can be changed by choosing different isotopes. (3) Anomalous dispersion of the scattering amplitude: In highly absorbing elements, the scattering amplitude is changed due to resonance effects (e.g. Cd at the neutron resonance of 0.178 eV, or elements with K or L absorption edges at energies larger than 5 keV for X-rays). When three or more independent scattering experiments are available, we can solve the set of linear equations in Itj(Q), which can be written in matrix form as [ J ( Q ) ] = [W(Q)] [ P ( Q ) ] ,

(20)

where [J(Q)] is the column matrix or vector of the total structure factors [W(Q)] is the matrix of the weighting factors Wii(Q), and [P(Q)] is the vector of the minor partial structure factors P~j(Q) = Iij(Q ) - 1. Applying the least-squares method, we find the solution

J(Q) = I ( Q ) - 1 ,

[P(Q)]=[[W(Q)]T[w(Q)]]

'[W(Q)]T[J(Q)I,

(21)

where [W(Q)] T and [W(Q)] -1 are the transpose and the inverse of the matrix [ W(Q )], respectively.

C.N.J. Wagner / Neutron and X-ray diffraction studies

33

Unfortunately, Eq. (20) is often ill-conditioned because the determinant of the weight-factor matrix [W(Q)] is small. A quantitative measure of the conditioning of a set of linear equations is given by the figure of merit T, also called the Turing number, defined as [6] T = IIW(Q)II IIW-~(Q)II---IIW(Q)IIEIIW ' ( Q ) l l z ,

(22)

where I[W(Q)II is the norm of the matrix [w(Q)], and IIW(Q)II E is the Euclidean norm, i.e., II W(Q)II E = ( E E l ( ~ i ) 2 I) 1/2In addition, the total structure factors J(Q) include experimental errors AJ(Q). The error Ap(Q) in the partial structure factors P(Q) is related to the error AJ(Q) and Turing's number T, i.e., II Ap(Q) II/11P(Q)

[I <~T 11AJ(Q)

II/11J(Q) II.

(23)

As a consequence, eq. (21) usually does not produce reasonable solutions for the individual partial structure factors P,j(Q) = / , j ( Q ) - 1 = J,j(Q). For these reasons, additional constraints have been used based on the sum rule which follows from eq. (15) when r goes to zero, and on the fact that eq. (8) represents a positive-definite quadratic form in f, [7]. If we consider experimental errors /~J(Q) in the total minor structure factors J(Q), then we have to solve the following equation:

[ J ( Q ) ] = [ W(Q)] [ P ( p ) ] + [ , a J ( Q ) ] .

(24)

One of the problems with eq. (24) is the fact that small errors AJ(Q) might lead to large errors in the solution P(Q) if [W(Q)] is ill-conditioned. The ridge analysis [8,9] is a mathematical treatment to minimize errors in the solution of an ill-conditioned set of linear equations. The solution P'(Q) of eq. (24) is given by [P'(Q)]=[[W(Q)]x[W(Q)]

+k[I]]

I[W(Q)]X[J(Q)],

(25)

where k is an arbitrary constant, with values of k in the range of l 0 - 4 tO 1 being typical, and [I] is the identity matrix. There is an optimal value of k, chosen so that the total error in P'(Q) is minimized. The solutions P'(Q) obtained with the ridge analysis are related to the least-squares solutions P(Q), i.e. [P'(Q)] = [Zk(Q) ] [P(Q)],

(26)

where [Zk(Q) ] =

[[W(Q)]T[w(Q)] +k[I]]-l[W(Q)]v[w(o)].

(27)

It is usually found that the off-diagonal elements of [Zk(Q) ] are not negligible, and that the sum of the row elements is in general not equal to one. This means that the ridge analysis solutions for the partial structure factors [ P'(Q)] are linear combinations of the least-squares solutions [ P (Q)].

C.N.J. Wagner/ Neutron and X-ray diffraction studies

34

3. Experimental techniques In any scattering experiment, the intensity IN(Q) is measured as a function of the length of the scattering vector I Q I = I k a - k 0 I- This can be accomplished by varying the scattering angle 20 when using m o n o c h r o m a t i c radiation of wavelength ?~ (variable-20 method), or by varying the wavelength ~ at a fixed scattering angle 20 (variable-~, method). In X-ray diffraction, it is possible to determine the wavelength ?~ of the radiation by measuring its energy E = hc/~ using a solid-state detector, where h is Planck's constant and c is the velocity of light. In neutron diffraction using a pulsed source, it is possible to correlate the time-of-flight t with the wavelength ?~, i.e., t = ( m / h ) ( L / ? Q , where m is the mass of the neutron and L is the total flight path of the neutrons. Thus we can write Q = ( 4 7 r / ~ ) sin 0 = [4~r/(hc)] (sin O)E = (4~rmL/h)(sin O)/t. (28)

I

2

I

i

I

I

i

I

I

I

1

i

I

1

--

R

(a)

2 ZrF4Ba F2

(b)

2 HfF4 B a F 2

1-

0A

O

0

2

4

6 Q

8

10

12

14

in A -1

Fig. 1. Total structure factors I(Q) (eq. (10)) of the halide glass 2(Zr-Hf)F4BaF2. Curves (a) and (b) were measured with Ag-K~ radiation using the variable-2# method. Curve (c) was obtained with the pulsed-neutron spallation source at Argonne National Laboratory [13].

C.N.J. Wagner / Neutron and X-ray diffraction studies

35

The variable-20 method, i.e. the conventional scanning diffractometer technique, has been a powerful method to elucidate the structure of amorphous solids. Modern position-sensitive detectors for X-rays and neutrons, combined with efficient radiation sources of relatively short wavelength (synchrotron radiation and hot-neutron reactor sources) have shortened the counting time, but the Q range is still limited to less than 20 ,~-1 when the wavelength ?~ is larger than 0.6 A. With the variable-~ method, it became possible to extend the Q range to values of 50 A-~, when using epithermal neutrons from a pulsed spallation source. Detailed descriptions of the experimental techniques for the variable ?~ and 20 methods have been given in the literature [10-12]. In principle, neutron diffraction should be more advantageous for structural studies of amorphous materials, because the coherent scattering amplitudes of neutrons do not exhibit the fall-off with increasing Q which is so troublesome with X-rays. However, corrections of the neutron data must be made for the departure from the static approximation, the so-called Placzek correction, and for multiple scattering [5]. Short-wavelength neutrons, available from pulsed neutron sources, minimize both corrections, but present-day neutron diffractometers do not permit us to measure the scattering at sufficiently small angles to cover a Q range as low as 0.3 A 1. An example of the total structure factor I(Q) is given in fig. 1 for the halide glass 2(Zr, Hf)F4BaF2, determined with X-rays using the variable-20 method and Ag-K~ radiation, and with neutrons, using the variable-~ method and the pulsed neutron source at Argonne National Laboratory [13]. There are still some uncertainties about the low- and high-Q data, because of the lack of detectors at low-20 positions, and the limited intensity a n d / o r counting time available for the experiment.

4. Experimental results and discussion

4.1. Metallic glasses In the last few years, the partial structure factors of several Ni-based amorphous alloys have been determined using the methods of isomorphous and isotopic substitution. Ni is one of the elements which possess several isotopes with positive and negative scattering lengths for neutrons. Examples of other elements are Ti, Dy and W. Because of the availability of several isotopes, it is possible to vary extensively the weight factors Wij(Q ) in eq. (12) for amorphous Ni alloys. Lamparter et al. [14,15] have determined the partial structure factors in amorphous Ni81B19 and Ni80P20 alloys. They prepared a mixture of Ni isotopes such that b = 0, i.e., a null element. The alloys containing the null nickel yield directly the partial structure factors laB(Q ) and Ipp(Q), respectively, as can be seen from eq. (8). Unfortunately, B, and to some extent P, are light elements which makes the Placzek correction rather difficult and, as a

36

C.N.J. Wagner / Neutron and X-ray diffraction studies 12

I

I

I

i

I

I

8 I ~

I

I

I

I

I

-

~

I

Ni-Ni

4

p_p

I

,~"~

o

5_--

.jf'~_._ _

12 C "~"--"

o J

0

2

NI P

8 Ni-P

0

/~k/

M /", A .-. V '--,J v

-4

4

6

8

10

12

14

16

0

I 2

1 4

I 6

I 8 r inA

I 10

I 12

Fig. 2. Partial structure factors lij(Q ) (eq. (9)) of the amorphous Nis0P20 alloy [15]. Fig. 3. Reduced, partial pair correlation functions G,j(r) (eq. (15)) of the amorphous NisoP2o alloy, evaluated from the partial structure factors presented in fig. 2 [15].

consequence, the corresponding partial structure factors might contain some errors. The partial structure factors I i j ( Q ) for the NisoP20 are shown in fig. 2. The Fourier transforms of Q [ I i j ( Q ) - 1], i.e., the reduced partial pair correlation functions G , j ( r ) [eq. (15)], are shown in fig. 3. It is clearly seen that metalloid-metalloid are never nearest neighbors in the glass. Recently Wright et al. [16] have studied the amorphous alloy Ni30Dy70, using both Ni and Dy isotopes. For this system they could prepare also a null dysprosium. Thus it became possible to determine directly the scattering functions due to N i - N i and D y - D y . However, the evaluation of the partial structure factors is rather involved because Dy has a relatively high absorption cross section, and it shows the magnetic fall-off of the scattering amplitude. The structure of the amorphous Ni33Y67 alloy was determined by Maret et al. [17] using the variable-20 neutron technique. Amorphous alloys were prepared with nat. Ni, 58Ni, and 6°Ni. This combination permitted them to deduce the partial structure factors, shown in fig. 4, and the corresponding partial radial distribution functions R i j ( r ) = 4~'rZpu( r ) / c j = rGij( r ) + 4~rZpo, shown in fig. 5. In this system, there are very few N i - N i nearest neighbors.

14

C.N.J. Wagner / Neutron and X-ray diffraction studies

37

+ i

~

i

-t-...

i

i

\

~"

~

II

©

%

r--

Z

>-

Z

~

m

~-

.~ r,

I

I

I

o

o

o

o

I

I

o

o

o

I

I

o

o

,~

o o

~ . °

Z 0 e-, 0

I

I

I

I

I

I

I

I

E

t~

0 0

_.=

1

.

.

.

.

(0)!!1

I

I

0

38

C.N.J. Wagner / Neutron and X-ray diffraction studies

i

i

i

i

i

i

i

i

i

,4

i

Ni35Zr65

2

2

/

, '

--

Z r - - Ni

Zr -- Ni

i

o,

: ,., ~

0 ~

v ',

~

/ /

', \

k

1', ~" ".c-~

."

"'--"

1

D 1

2-

o I

0

i

I

2

4

I

I

I

6 O

I

I

8 in

I

10

I

I

I

12

A"

Fig. 6. Partial structure factors Iij(Q ) of the amorphous (Ni-Co)35(Zr-Hf)6 ~ alloy [18].

14

0

2

4

6

8 o

r

in

A

Fig. 7. Reduced, partial pair correlation functions Gij(r ) of the amorphous (Ni-Co)35(Zr-Hf)65alloy [18].

The method of isomorphous substitution of Zr by Hf in X-ray diffraction and Ni by Co in neutron scattering has been used by Lee et al. [18] to determine the partial structure factors in the amorphous Ni35Zr65 alloy, shown in fig. 6. The partial reduced pair correlation functions Gij(r), shown in fig. 7, indicate that there also are only few N i - N i neighbors in this alloy. The structure of amorphous N i - T i alloys has been determined by Ruppersberg et al. [19] and by Fukunaga et al. [20]. The partial structure factors of the glassy Ni4oTi6o alloy, determined by Fukunaga et al. [20], using the isotopes 58Ni and 6°Ni, are presented in fig. 8. Since natural Ti has a negative scattering amplitude for neutrons, it is easy to prepare a so-called "zero alloy", whose average scattering amplitude ( b ) is zero. In this case, one measures directly the concentration-concentration structure factor Scc(Q ), whose Fourier transform yields the concentration-correlation function poe(r), which describes the chemical short-range order (CSRO). As shown by Bhatia and Thornton [21],

10

C.N.J. Wagner / Neutron and X-ray diffraction studies I

t

I

I

I

]

[

I

I

I

I

i

39

i

2

0

-2 2 |

2=o 4

O i 0

I 2

i

i 4

l

i 6

I

I 8

I

t 10

~

I 12

I

I 14

Q in A-' Fig. 8. Partial structure factors lii(Q ) of the amorphousNi40Ti60 alloy [201.

the structure factor S(Q) [eq. (11)] can be expressed as

S( Q ) - 1 = [ { f ) 2 / ( f 2 ) ] [ S N N ( Q ) - 1] + 2( A f ) { f ) S y c ( Q ) / ( f 2 ) + c,c2(Af)2( [Scc(Q)/(clc2)] - 1 } / ( f 2 ) ,

(29)

where A f = f l ( Q ). When ( f ) = 0, only the term Scc(Q ) remains. The correlation function oct is related to the partial number density functions pij(r):

Pcc(r)=cle2{[Pll(r)/cl] + [P22(r)/c2] - 2[P12/c2]} = c2Pl(r) + c ' o 2 ( r ) - O]2(r)/c2 , (30) where o,(r) = E&j(r). The number-number structure factor Syy(Q) describes the topological short-range order (TSRO), and its Fourier transform Pnn(r) is related to &j(r) and 0,(r): o . . ( r ) = c, ot, + c2022 + c,0,2 = c10, + c202.

(31)

The Fourier transform of S y c ( Q ) is a measure of the difference in the atomic distribution functions about a 1- and a 2-type atom:

O.c(r) = c,c 2[Or(r)- o2(r)].

(32)

In amorphous materials, the first peak in R~j(r)=rGs(r)+4~r2po= 4~rr20,a(r)/cj permits us to determine the partial coordination numbers ~ j :

Nij = fcjRij (r)dr = f 4 = r % j ( r ) d r .

(33)

40

C.N.J. Wagner / Neutron and X-ray diffraction studies

Table 1 Chemical short-range order parameter cq in amorphous Ni alloys 1

2

C1

NII

N12

N22

N1

N2

N,m

N,~.

a]

Ref.

B P B Zr Ti Y

Ni Ni Ni Ni Ni Ni

0.19 0.20 0.36 0.65 0.60 0.67

-

9.3 9.3 8.7 4.6 5.3

10.8 9a.4 9.2 3.1 2.3

9.3 9.3 9.8 11.6 10.2

10.g 9.4 14.1 13.8 13.3

10.5 9.4 12.6 13.0 12.1

9.6 9.3 11.3 12.4 11.4

-0.21 -0.23 -0.20 -0.05 -0.15 -0.15

14 15 24 18 20 17

1.1 9.2 8.0

Consequently, N i = Y~N,/, Nnn = ~EciN, t = ~ G N , , No,. = Nv,, - N 1 2 / c 2 and N,~ = c 2 N~ + c~ N 2. Thus, we can express the chemical short-range order parameter ~1 of the first coordination shell as: cq = 1 - N 1 2 / [ c 2 ( c 2 N , + CLN2)] = 1 - N , 2 / ( c 2 N , ~ ) = N~.c/N w.

(34)

It has been shown previously [22] that the short-range order parameter rh2 defined by Cargill and Spaepen [23] is, to a first approximation, equal but opposite in sign to the CSRO parameter c~1. Values of the partial coordination numbers N,i, N,, N,,n and Nw are given in table 1 for Ni-based amorphous alloys. Using these values, it is possible to calculate the CSRO parameter c~. It is clearly seen that cq is negative in amorphous Ni alloys, indicating a preference for unlike nearest neighbors. 4.2. Inorganic glasses

A definite unit of structure is usually observed in inorganic glasses. This unit might be an oxide or halide compound. Most of the inorganic glasses are mixtures of several of these compounds. Recent examples of neutron diffraction studies of oxide glasses are (Li-Na)22SiO 2 [25] and K2OTiO22SiO 2 [26], where the fact that 7Li and nat. Ti possess negative scattering lengths for neutrons has made it possible to resolve the ( L i - N a ) - O and T i - O distances, respectively. Crystalline, heavy-metal fluoride compounds are characterized by a high metal fluorine coordination. It was of interest to see whether heavy-metal fluoride glasses also possess such a high first-shell coordination, which is quite common in metallic glasses. The structure of 2ZrF4BaF 2 glass has been evaluated by employing the method of isomorphous substitution of Zr by Hf, and combining X-ray and neutron diffraction data [13,27]. Using the structure factors presented in fig. la, the total pair correlation functions T ( r ) (eq. (19)] were calculated, shown in fig. 9. The first peak can be readily correlated with the Z r - F nearest neighbors at r = 2.08 A, yielding a coordination number CN of approximately 7, whereas the second peak is a mixture of the Ba F (CN = 15) and F - F (CN = 5) correlations. The sharp peak at r = 4.1 A is most likely due to Z r - Z r correlations, indicating almost linear Zr F - Z r bonds.

C.N.J. Wagner / Neutron and X-rc~v diffraction studies I

[

T

-

41

-

2ZrF4.BaF 2

i

£ w,,. v

2HfFo.B. I--

0 ,/',v 0~

_ 2

4

r(A)

J 6

I 8

10

Fig. 9. Total pair correlation functions T ( r ) / < f ( O ) ) 2 of the halide glass 2Zrf4BaF 4, evaluated from the total structure factors presented in fig. 1 [13].

The author would like to thank G. Etherington and A.E. Lee for helpful discussions. This research was supported by the grant D M R 83-10025 from the National Science Foundation.

References [1] [2] [3] [4] [5]

C.N.J. Wagner, J. Non-Cryst. Solids 43 (1980) 3. L. van Hove, Phys. Rev. 95 (1954) 249, 1374. G.E. Bacon, Neutron Diffraction, 3rd ed. (Clarendon, Cambridge, 1975). G. Placzek, Phys. Rev. 86 (1952) 377. D i . Price, Analysis of time-of-flight neutron diffraction data from isotropic samples, IPNS Note 19, Argonne Nat. Lab. (1985). [6] A.K. Livesey and P.H. Gaskell, in: Proc. 4th Int. Conf. on Rapidly Quenched Metals, eds. T. Masumoto and K. Suzuki, Japan Inst. Metals, Sendal (1982), Vol. 1, p. 335.

42 [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17]

[18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

C.N.J. Wagner / Neutron and X-ray diffraction studies J.E. Enderby, D.M. North and P.A. Egelstaff, Phil. Mag. 14 (1966) 961. A. Hoerl and R. Kennard, Technometrics 12 (1970) 55, 69. D.W. Marquardt, Technometrics 12 (1970) 591. C.N.J. Wagner, J. Non-Cryst. Solids 41 (1978) 1. R.N. Sinclair, D.A.G. Johnson, J.D. Dore, H.H. Clarke and A.C. Wright, Nucl. Instr. and Meth. 117 (1974) 445. K. Suzuki, M. Misawa, K. Kai and N. Watanabe, Nucl. Instr. and Meth. 147 (1977) 519. G. Etherington, C.N.J. Wagner, R. Almeida and J. Faber, in: Syrup. on Neutron Scattering, eds. M.S. Lehmann, W. Saenger, G. Hildebrandt and H. Dachs (Reports Hahn-Meitner Institut, Berlin, HMI-B 411, 1984) p. 64. E. Nold, P. Lamparter, H. Olbrich, G. Rainer-Harbach and S. Steeb, Z. Naturforsch. 36a (1981) 1032. P. Lamparter and S. Steeb, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont, Wuerzburg 1985 (North-Holland, Amsterdam, 1985) p. 459. A.C. Wright, A.C. Hannon, R.N. Sinclair, W.L. Johnson and M. Atzman, J. Phys. F: Met. Phys. 14 (1984) L201. M. Maret, P. Chieux, P. Hicter, M. Atzman, and W.L. Honson, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont, Wuerzburg, 1985 (North-Holland, Amsterdam, 1985) p. 521. A.E. Lee, G. Etherington and C.N.J. Wagner, J. Non-Cryst. Solids 61/62 (1984) 349. H. Ruppersberg, D. Lee and C.N.J. Wagner, J. Phys. F: Met. Phys. 10 (1980) 1645. T. Fukunaga, N. Watanabe, and K. Suzuki, J. Non-Cryst. Solids 61/62 (1984) 343. A.B. Bhatia and D.E. Thornton, Phys. Rev. B2 (1970) 3004. C.N.J. Wagner, in: Proc. 5th Int. Conf. on Rapidly Quenched Metals, eds. S. Steeb and H. Warlimont, Wuerzburg, 1985 (North-Holland, Amsterdam, 1985) p. 405. G.S. Cargill and F. Spaepen, J. Non-Cryst. Solids 43 (1981) 91. N. Cowlam, W. Guoan, P.P. Gardner and H.A. Davies, J. Non-Cryst. Solids 61/62 (1984) 337. M. Misawa, D.L. Price and K. Suzuki, J. Non-Cryst. Solids 37 (1980) 85. A.C. Wright and A.J. Leadbetter, Phys. Chem. Glasses 17 (1976) 122. G. Etherington, L. Keller, A. Lee, C.N.J. Wagner and R.M. Almeida, J. Non-Cryst. Solids 69 (1984) 69.