Fig.1
Scaling Model for the Short-Range Order
Structure in Amorphous Systems.
Isakov S.L.
Abstract.
Starting from precise experimental data and simple empirical relations an approximate formula for the calculation partial coordination numbers is deduced and a kit of non-dimensional functions for computing partial structure factors of multicomponent amorphous system based on supposed atomic radii and component concentrations is proposed. Comparison of modelling functions with that from known experiments shows that the model describes qualitative and quantitative particularities of the most different amorphous systems well enough. Within the model framework an entropy of amorphous alloys is deducated.
Fig.1
1. Introduction.
For a detailed description of short-range order in amorphous metallic alloys, one needs to evaluate all partial atomic pair correlation functions. At present the most accurate information can be obtained from neutron diffraction experiments on isotopic substituted examples [1-3]. These experiments, however, expensive and applicable to the limited range of the systems.
Meantime, having sufficiently exact data on partial distributions in the amorphous systems Ni65B35 [1] and Ni62Nb38 [2], I built approximate model, well reproducing the functions of other systems: Zr-Be [3], LaNi5 [4], Ni-Ti [5], Ni-Dy [6], Ni-Y [7]. (Data [5-7] were used on the qualitative level through author’s detailed graphs). Moreover, the model is rendered suitable for the ternary system Ti-Zr-Si [8], as well as for the semiconductor system Zn-Sb [9].
It appears to be possible after mathematical examination of regularities of partial distribution functions for different, but similar systems. So, it was marked that if these functions to scale on subsystem atom size (ri+rj), first spheres having approximately same shape differ from each other by the height, defining differences in coordination numbers, while second spheres almost equal meaning one. Besides, one can find a difference between structure factors for homogeneous and heterogeneous distribution: 
(fig.5), while 
(fig.7). Alike observation was made by Siestma and Thijsse [10] but not scaling, only “shifting”, without making such conclusion as I do.
2. Basic definitions.
Total structure factor ST(Q) of multicomponent system obtained from neutron diffraction experiments may be expressed through partial functions Sij(Q), introduced by Ashcroft and Langreth [11]:
(1)
where ci,cj - concentrations of components,
bi,bj - coherent scattering amplitudes of components,
And the functions correspond to radial distribution functions gij(r)
(2)
where dij - the Kronecker symbol
n - atomic density.
For binary systems only Bhatia-Thornton formalism are introduced[12]:
(3)
where 
SNN - density-density function
SNC - density-concentration function
SCC - concentration-concentration function.
Relationship between these formalisms is following:
(4)
Partial coordination numbers zij in the first sphere may be calculated by the formula:
(5)
where r1 and r2 - are bounds of the first peak in the correspondent radial distribution function,
and r1 really may be placed to zero.
3. Coordination numbers.
Pay attention to such particularity of the structure factors as a long wave limit. In the majority of experiments Sij(0)=0, and according to work [12] SNN(0)=kBT(dn/dP)T=c - isotermic compressibility, which is for real metals <<1. Based on formulas (4) we shall consider that
(6)
According to formula (2)
Function h(r)=g(r)-1 at large r oscillates around zero with extinction, therefore having in view of mentioning above scale resemblance, we can expect that some constant x exists uniform for all systems, so that 
lies in minimum between first and second coordination spheres and
(7)
So we have
Comparing with formulas (5) and (6) we have
(8)
where
To evaluate this value we can calculate a coordination number for the closely packed crystalline lattice by this formula. It is 12. Having substituted this value to the formula (8) we can count x, it turns out to be close to 1.3. This value seems to be highly suiting for amorphous alloys, and justifies our suggestion making at derivation of the formula (7), because the value of 1 correspond to maximum in the first coordination sphere, and 1.7 - to second sphere(see also Magic Numbers). So the value of 1.3 really is half way between them. This is the place where one usually select upper limit of integrating R2 in the formula (5), i.e. in the minimum between these peaks.
4. Structure factor model.
Let us suppose, that any radial distribution function can be obtained from the universal function amount, scaling on the atomic sizes and being multiplied by factors, defining their main differences:
(9)
where bij and gij - the factors depending on atomic parameters, and
the functions of non-dimensional argument g1(x), g2(x) and g3(x) are the same for all systems.
We can co-ordinate (9) with the formulas for coordination numbers (5) and (8) by selecting 
(10)
If we define inverse functions of non-dimensional argument q
(11)
we derive a basic formula for structure factors:
(12)
where
Thus, one can construct a partial structure factor for any amorphous system by the sum of scaled predefined functions S1, S2, S3 with coefficients depending on chemical composition. For the complete definition we must obtain some new relations.
In comparison with formulas (4) we introduce new functions of nondimensional argument Snn(q), Snc(q), Scc(q), but consisting Sij(Q) will be applied with its own Q-scale, namely Sij(q/sij), and use (12). The three equation will agree with each other when
(13)
RJ
This is surprise that the scale of the partial distribution function is not linear sum of radiuses of atoms si+sj. What, however, does not influence upon preceding reasoning.
Define that the scale of structure factors are
where si is an atomic radius of the component of the alloy, and introduce average value
And we shall consider that for a closely packed structure
Introduce dilatation factor
(14)
Then partial coordination factors may be calculated as
(15)
and value vij in (12) will be
For binary system, thereby, it possible to obtain the following functions
(16)
Consequently, it is possible to define other kit of non-dimensional functions: Snn(q), Scc(q) and Snc(q)=NS1(q), and express any structure factor through the new kit of functions
(17)
This formula, unlike the formalism BT (3), can be easily generalised to multicomponent systems and it is constructive. It’s mean that if we know atomic radii and concentrations of the components then we can calculate partial structure factors of the system.
5. Results.
Functions Snn(q), Snc(q) and Scc(q) were obtained from own experimental results on amorphous systems Ni-Nb [2] and Ni-B [1] using equations (17). Charts of the functions are observed in fig.1, and their numerical value are given in the table 1. Corresponding radial distribution functions are shown in fig.2.
Fig.2
Analysis of the results shows the following:
The functions obtained from different sets of experimental data is in qualitative coincidence with each other. The differences may arise from an inaccuracy of the experimental data [2] and mathematical condition number of the linear equation systems (17).
The function Snn(q) resembles the structure factor of one-component system. Among the different experimental data with sufficient accuracy may be emphasises an amorphous Ni [13]. Provided in the article a graph for the structure factor very similar to the model (fig. 3). A fine differences with other metals can be partly explained by variation of the value N, or shows insufficiency of pure geometric models.
Fig.3
The function Snc(q) is qualitative coincide with function dS(Q)/dn, determined for liquid germanium [14] and probably it is the physical sense of the function. Thus the formula (17) is likely to be a linear approximation for the structured factor in terms of powers of density.
And finally the function Scc(q) represents full diffraction from a binary amorphous system with the zero matrix of coherent scattering and equal radiuses of components. Unfortunately such experiments are unknown for me. There is only some unpublished result for a crystalline alloy Au-Ti, on which experiment was pursued on the pulse neutron source of the LINAC “Fakel” at Kurchatov Atomic Energy Institute. The data given in fig.4 allows us to hope to obtain function Scc(q) in the frame of the model by direct experiment.
Fig.4
A number of following figures (5-10) shows applicability of the model for different systems. For the best fittings there were used small variations of atomic radii and concentrations in the range of experimental inaccuracy as well as value N. For example for semiconductors it follows to choose N=5.
Fig.5
Fig.6
Fig.7
Fig.8
Fig.9
Fig.10
In the table 2 it is shown a comparison of coordination numbers, calculated under the formula (15) with experimental data.
Table 2.
|
|
Z11 | Z12 | Z21 | Z22 | ref | Comment |
| Ni65B35 | 9.4 | 4.9 | 9.0 | 0.9 | [1] | |
| 9.9 | 4.0 | 7.4 | 1.1 | calc | ||
| Ni81B19 | 10.8 | 2.2 | 9.3 | 0.0 | [17] | |
| 11.0 | 1.9 | 8.2 | 0.02 | calc | ||
| Ni62Nb38 | 6.1 | 5.7 | 9.3 | 5.5 | [2] | |
| 5.8 | 5.2 | 8.5 | 5.2 | calc | ||
| Zr70Be30 | 10.9 | 3.2 | 7.3 | 2.2 | [3] | 1) |
| 10.3 | 3.3 | 7.6 | 0.7 | calc | ||
| Zr60Be40 | 10.0 | 4.5 | 6.8 | 2.7 | [3] | |
| 9.6 | 4.7 | 7.1 | 1.4 | calc | ||
| Zr50Be50 | 8.9 | 6.2 | 6.2 | 3.9 | [3] | |
| 8.7 | 6.5 | 6.5 | 2.3 | calc | ||
| Ni83La17 | 7.7 | 4.1 | - | - | [4] | |
| 6.7 | 3.4 | 17.0 | 4.3 | calc | ||
| Ni40Ti60 | 2.3 | 7.9 | 5.3 | 8.1 | [5] | |
| 2.9 | 7.5 | 5.0 | 8.1 | calc | ||
| Ni31Dy69 | 3.0 | 10.8 | 4.9 | 12.4 | [6] | 2) |
| 0.9 | 7.6 | 3.4 | 10.2 | calc |
| Z11 | Z12 | Z21 | Z22 | |
| Authors data | 5.3 | 6.2 | 7.6 | 6.4 |
| Calc | ||||
| Ni66Ta45 | 4.8 | 6.0 | 7.3 | 6.2 |
| Ni62Ta38 | 5.7 | 5.2 | 8.5 | 5.3 |
| Ni67Ta33 | 6.4 | 4.6 | 9.4 | 4.65 |
6. Entropy.
One another application for the models is to estimate a contribution to binary amorphous system entropy caused by the interaction between components of the alloy.
Conclusion of the expression for entropy made under the strategy, described in the book by Radu Balescu [16]:
(18)
where s(0) - entropy without interactions
kB - Boltsman constant
b=1/kBT
Vij(r) - interatomic potential, within the framework of the model we will assume that
where U(x) - some nondimensional function,
and parameters eij and sij enclosure all chemical nature of the elements,
l - parameter, defining fade-in atomic interactions, i.e.
For example to use formula (17) for one-component system it follows to put l =0.
Will now define the pair correlations functions in the frame of the scaling model
Having substituted in the formula (18) one can easy obtain for the binary system simple expression:
(19)
The formula is presented seems to be reasonable, because the entropy has a maximum at the equality of concentrations, as well as entropy increases when reducing a cross interaction.
7. Conclusion.
Formula for calculation of partial coordination numbers is deduced and a kit of three function is constructed that permit anyone to compute short-range order of amorphous alloys based on its chemical composition.
In the model described above the ultimate formulae are deduced as a result of uncomplicated transformation based on empirical admissions, obviously or implicitly used in the work. These admissions are justified only postfactum under the results given by the model.
One of the interesting application of the model is a determination of chemical composition of complex alloy by its full structure factor. For this it is needed to fit model structure factors by variation of concentrations and radiuses a component, and then fix their chemical nature by the final value. Such experience was really successful.
My acknowledgments to Kurchatov Atomic Energy Institute for the possibility of getting the experimental results used in the article and to Dr.Ishmaev S.N. for support and critics.
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