A model for the compositions of non-stoichiometric intermediate phases formed by diffusion reactions and its application to Nb₃Sn superconductors

Abstract

In this work we explore the compositions of non-stoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb₃Sn superconductors. In the first part, the governing equations for the bulk diffusion and inter-phase interface reactions during the growth of a compound are derived, numerical solutions to which give both the composition profile and growth rate of the compound layer. The analytic solutions are obtained with certain approximations made. In the second part, we explain an effect that the composition characteristics of compounds can be quite different depending on whether it is the bulk diffusion or grain boundary diffusion that dominates in the compounds and that “frozen” bulk diffusion leads to unique composition characteristics that the bulk composition of a compound layer remains unchanged after its initial formation instead of varying with the diffusion reaction system; here the model is modified for the case of grain boundary diffusion. Finally, we apply this model to the Nb₃Sn superconductors and propose approaches to control their compositions.

Introduction

Intermediate phases with finite composition ranges represent a large class of materials and their compositions may influence their performance in application, as demonstrated in a variety of materials, such as electrical conductivity of oxides (e.g., TiO_2−y¹), electromagnetic properties of superconductors (e.g., Nb₃Sn and YBa₂Cu₃O_7−y²) and mechanical properties of some intermetallics (e.g., Ni–Al_0.4–0.55³), etc. For instance, the superconducting Nb₃Sn phase, which finds significant applications in the construction of 12–20 T magnets^4,5,6, has a composition range of ~17–26 Sn at.%^7,8 and its superconducting transition temperature and magnetic field decrease dramatically as Sn content drops^4,8,9. The Nb₃Sn phase, which is generally fabricated from Cu-Sn and Nb precursors through reactive diffusion processes, is always found to be Sn-poor (e.g., 21–24 at.%^4,9,10), making composition control one of the primary concerns in Nb₃Sn development since 1970s¹⁰. Although a large number of previous experiments (e.g.,^4,9,10,11,12) have uncovered some factors that influence the Sn content, it is still a puzzle what fundamentally determines the Nb₃Sn composition. This work aims to fill that gap. Here it is worth mentioning that the composition interval of a compound layer does not necessarily coincide with its equilibrium phase field ranges – the former can be narrower (e.g., the Nb₃Sn example above) if the inter-phase interface reaction rates are slow relative to the diffusion rate across the compound, which results in discontinuities in chemical potentials at the interfaces.

There have been numerous studies regarding diffusion reaction processes, most of which have focused on layer growth kinetics (e.g.,^{13,14,15,16,17}), compound formation and instability (e.g.,^15,16,17), phase diagram determination (e.g.,¹⁸) and interdiffusion coefficient measurement (e.g.,¹⁹), while a systematic model exploring how to control compound composition is still lacking. We find it indeed possible to modify the model developed by Gosele and Tu¹⁴ for deriving the layer growth kinetics of compounds to calculate their compositions; however, certain assumptions (e.g., steady-state diffusion and first-order interface reaction rates) that the model was based on may limit the accuracy of the composition results. In this work, we aim to develop a rigorous, systematic mathematical framework for the compositions of intermediate phases.

The model

Let us consider that a non-stoichiometric A_nB compound is formed in a system of M–B/A, where M is a third element that does not dissolve in A_nB lattice²⁰. The use of the third element M is to decrease the chemical potential of B, so that unwanted high-B A–B compounds (e.g., NbSn₂ and Nb₆Sn₅ in the Nb-Sn system⁷) that would form in the B/A binary system can be avoided. With the M–B, A_nB and A-rich phases denoted as α, β and γ, respectively, a schematic of the α/β/γ system for a planar geometry is shown in Fig. 1. Let us denote the α/β and β/γ inter-phase interfaces as I and II, respectively and the mole fractions, chemical potentials, activities and diffusion fluxes of B in the β phase at interfaces I and II as X_I^β, μ_I^β, a_I^β, J_I^β and X_II^β, μ_II^β, a_II^β, J_II^β, respectively. The maximum and minimum mole fractions of B in β phase (i.e., A_nB compound) from the phase diagram are set as X_I^β,eq and X_II^β,eq, respectively. Let us also denote the μ_Bs and a_Bs of α and γ as μ_B^α, a_B^α and μ_B^γ, a_B^γ, respectively. Let us assume that the solubility of B in γ phase is negligible⁷. An isothermal cross section of such an M–A–B phase diagram at a certain temperature is shown in Fig. 2. This is the case we see for the Nb₃Sn example above (for which A stands for Nb, B for Sn and M for Cu), but the model below can be modified for other cases. Similar to the Cu-Nb-Sn system, let us assume B is the primary diffusing species in the β phase²¹ and that the α phase can act as an intensive sink for B vacancies in order for it to be an efficient source of B atoms for β layer growth and that the diffusivity of B in α is high so that the α phase remains homogeneous during the growth of β layer¹¹.

Figure 1

Schematics of (a) the α/β/γ diffusion reaction system in the planar geometry and (b) X_B profiles of the system.

Figure 2

Schematic of an isothermal cross section of the M–A–B ternary phase diagram.

The shaded region shows the equilibria among M-X₁ B, A-X_I^β,eq B and γ phases and the dashed line shows the equilibrium between α and β phases.

Here we assume that diffusion occurs by vacancy mechanism and the total atomic flux is balanced by the vacancy flux. As discussed in the papers by Svoboda and Fischer et al.^22,23,24, the presence of various types of sinks or sources for vacancies may lead to quite different diffusional and conservation laws and equations. For this model, we assume that B vacancies are generated by the reaction at interface II (as will be discussed in detail later) and then diffuse across the β layer to interface I, where they are annihilated by B atoms from α phase (the B source). For the simplicity of the model, we assume that there are no sinks or sources for vacancies in the bulk or grain boundaries of β phase, while the only sink in the system for B vacancies is the α phase. The following model can be modified for cases with other types of sinks or sources for vacancies using the models by Svoboda and Fischer et al.^22,23,24.

In this work let us assume the diffusivity of B in β phase, D and the molar volume of β phase, V_m^β, do not vary with X_B, in which case the continuity equation in the A_nB layer is given by:

According to mass conservation, in a unit time the amount of B transferring across the interface I should equal to that diffusing into the β layer from the interface I and the amount arriving at the interface II should equal to that transferring across it, i.e., dn/dt|_I  = J_I^β∙A_I and dn/dt|_II = J_II^β∙A_II, where A_I and A_II are the areas of the interfaces I and II, respectively. The molar transport rate dn/dt across an interface equals to r∙A_int∙exp(−Q/RT)∙[1−exp(−Δμ/RT)], where r is the transfer rate constant for this interface with the unit of mol/(m²∙s), A_int is the interface area, Q is the energy barrier, R is the gas constant, T is the temperature in K and Δμ is the driving force for atom transfer. For the interface I, Δμ|_I = μ_B^α−μ_I^β. For the interface II, Δμ|_II = μ_II^β–μ_B^γ and μ_B^γ = μ_B(A−X_II^β,eq B). With J_B = −(D/V_m)∙(∂X_B/∂x), we have:

Eqs. (2) and (3) are the boundary conditions for Eq. (1). Note that X_B in α phase, X_B^α, drops with annealing time as B in α is used for β layer growth, so μ_B^α drops with t:

where n_M and n_B0 are the moles of M and B in the M-B precursor. For those systems without the third element, μ_B^α is constant and Eq. (4) is not needed. In addition, since the B atoms diffusing to the interface II are used to form new β layers, we have:

Eqs. (1)–(5) are the governing equations for the system set up above, solutions to which give both the X_B(x, t) and the l(t) of a growing A_nB layer. It should be noted that for the systems with large volume expansion associated with transformation from γ to β, stress effects need to be considered²⁵.

To simplify Eqs. (2) and (3), we notice that 1−exp[−(μ_B^α−μ_I^β)/RT] = 1−a_I^β/a_B^α, since μ_B^α−μ_I^β = RTln(a_B^α/a_I^β); similarly, 1−exp[−(μ_II^β−μ_B^γ)/RT] = 1−a_B^γ/a_II^β. Let us also denote D/[V_m∙r_I∙exp(−Q_I/RT)] as φ_I and D/[V_m∙r_II∙exp(−Q_II/RT)] as φ_II: clearly φ_I and φ_II represent the ratios of diffusion rate over interface reaction rates and have a unit of meter. Then Eqs. (2) and (3) can be respectively written as:

Analytic and Numerical Solutions

Let us first consider two extreme cases. First, for the case that the interface reaction rates are much higher than the diffusion rate across the β layer (i.e., diffusion-rate limited), φ_I and φ_II are near zero; according to Eqs. (2)–(3), μ_Bs are continuous at both interfaces, so X_II^β = X_II^β,eq. Suppose μ_B^α and the position of interface I, x_I, are both constant with time, then X_I^β is also constant and the solutions to Eqs. (1) and (5) are respectively X_B(x, t) = X_I^β−(X_I^β−X_II^β,eq)∙erf{(x−x_I)/[2√(Dt)]}/erf(k/2) and l = k√(Dt) for the β layer, where k can be numerically solved from k∙exp(k²/4)∙erf(k/2) = 2/√π∙(X_I^β−X_II^β,eq)/X_II^β,eq. For instance, for X_I^β = 0.26 and X_II^β,eq = 0.17, k =0.953. On the other hand, if the interface reaction rates are much lower than the diffusion rate across β (e.g., as the β layer is thin), φ_I and φ_II are large; according to Eqs. (2) and (3), X_B and J_B are nearly constant in the entire β layer. Thus, (1−a_B/a_B^α)/φ_I = (1−a_B^γ/a_B)/φ_II, from which a_B can be calculated. Integration of Eq. (5) gives: l ∝ t and the pre-factor depends on the interface reaction rates.

For a general case between these two extremes, the equations can only be solved with the μ(X) relations of α and β provided. Next, let us consider a compound with a narrow composition range, so that as a Taylor series expansion is performed around X_II^β,eq for its a(X_B) curve, high-rank terms can be neglected because |X−X_II^β,eq| ≤ (X_I^β,eq−X_II^β,eq) is small; that is, a_X ≈ a_B^γ + κ(X−X_II^β,eq), where κ is the linear coefficient of the a(X) curve. Given the complex boundary conditions for Eq. (1), to obtain the analytic solutions we introduce a second approximation if the β composition range is narrow: the X(x) profile of the β layer is linear so that at a certain time J is constant with x, such that −(∂X_B/∂x)|_I ≈ −(∂X_B/∂x)|_II ≈(X_I^β−X_II^β)/l. With these two approximations, we can solve Eqs. (6) and (7) and obtain that:

where η = φ_IIa_B^α/(φ_Ia_B^α+κl). Then a_I^β can be calculated from a_II^β and X_I^β and X_II^β can be calculated from a_I^β and a_II^β using X = X_II^β,eq +(a_X − a_B^γ)/κ.

To verify the results, the equations are solved for a hypothetical system analytically and numerically, with and without the assumption that X(x) is linear, respectively. The obtained composition profiles are shown in Fig. 3(a). For simplicity, μ_B^α of the system is set as μ_B(A−X_I^β,eq B) and is constant (for Nb₃Sn systems, this means that Nb₆Sn₅ serves as Sn source) and the other parameters are specified in the figure. The difference between the analytic and numerical solutions is <0.1%, showing that the approximation of linear X(x) is good if the composition range is small (2 at.% in this case). The l(t) result (where t is the annealing time after the incubation period) from the numerical calculations is shown in Fig. 3(b). While the analytic l(t) solution is complicated, some l(t) relations with simple forms can be used as approximations. The most widely used l(t) relation for the case of constant μ_B^α is l = bt^m, in which m =1 for reaction-rate limited and m =0.5 for diffusion-rate limited; however, a defect with this relation is that as l increases from zero, it may shift from reaction-rate limited to diffusion-rate limited, so m may vary with t. Here a new relation l = q[√(t+τ)−√τ] (where q is a growth constant and τ is a characteristic time) is proposed. Such a relation is consistent with l²/v₁+l/v₂= t (where v₁ and v₂ are constants related to diffusion rate and interface reaction rates, respectively) proposed by previous studies^14,15. This relation overcomes the above problem because as t ≪τ, l  =[q/(2τ)]∙t and as t≫ τ, l = q√t. As can be seen from Fig. 3(b), a better fit to the numerical l(t) curve in the whole range is achieved by l = q(√(t+τ)−√τ).

Figure 3

(a) The calculated X_B(x) profiles of the hypothetical system for the analytic and numerical solutions, with and without the assumption that X_B(x) is linear, respectively. (b) The l(t) results from the numerical calculations, with the fits of l = q[√(t+τ)−√τ] and l = bt^m.

Results and Discussion

Before discussing the application of this model to a specific material system, it must be pointed out that all of the analysis and calculations above are for the case that B diffuses through β bulk. In such a case, for an α/β/γ system, as μ_B^α drops with the growth of β layer, X_B(x) of the entire β layer should decrease with μ_B^α, because μ_B^α ≥ μ_I^β ≥ μ_II^β ≥ μ_B^γ. Finally, one of two cases will occur: either μ_B^α drops to μ_B^γ (if A is in excess) so the system ends up with the equilibrium among γ, A–X_II^β,eq B and M–X₁ B (as shown by the shaded region in the isothermal M–A–B phase diagram in Fig. 2), or γ is consumed up and β gets homogenized with time and finally μ_B(β)= μ_B(α) (as shown by the dashed line in Fig. 2). In either case, β layer eventually reaches homogeneity.

However, we find that the composition could be different for a compound in which the bulk diffusion is low while grain boundary diffusion dominates. One such example is Nb₃Sn, the composition of which displays some extraordinary features. As an illustration, the X_Sns of a Cu-Sn/Nb₃Sn/Nb diffusion reaction couple after various annealing times are shown in Fig. 4. Clearly, as the X_Sn (and μ_Sn) of Cu-Sn drop with time, the X_Sns of Nb₃Sn do not drop accordingly; instead, they more or less remain constant with time. In addition, from 320 hours to 600 hours, although Nb has been fully consumed, the X_Sn of Nb₃Sn does not homogenize (i.e., the X_Sn gradient does not decrease) with time. In many other studies on Cu-Sn/Nb systems with Nb in excess (e.g.,^4,9,10), even after extended annealing times after the Nb₃Sn layers have finished growing (which indicates that the Sn sources have been depleted, i.e., μ_Sns have dropped to μ_B^γ), X_Sns of Nb₃Sn remain high above X_II^β,eq, without dropping with annealing time.

Figure 4

The measured X_Sns of a Cu-Sn/Nb₃Sn/Nb system after various annealing times at 650 °C.

The standard deviation in the Sn content measurements is around ±0.5 at.%.

The reason for these peculiarities is that grain boundary diffusion in Nb₃Sn dominates due to extremely low bulk diffusivity (e.g., lower than 10⁻²³ m²/s at 650 °C)^21,26,27 and small Nb₃Sn grain size (~100 nm). In this case, our model and equilibrium-state analysis apply only to the diffusion zones (i.e., the grain boundaries and the inter-phase interfaces) instead of the bulk. To clarify this point more clearly, a schematic of the diffusion reaction process is shown in Fig. 5. At time t₁, at the β/γ interface, high-B A_nB (L₂ layer) reacts with γ (L₃ layer) to form some new A_nB cells, leaving B vacancies (noted as V_Bs) in L₂ layer (time t₂). If bulk diffusivity is high, V_Bs simply diffuse through bulk (e.g., from L₂ to L₁, as shown by grey dotted arrows) to the B source. If bulk diffusion is frozen, the V_Bs diffuse first along β/γ inter-phase interface (as shown by green solid arrows) and then along β grain boundaries to the B source. This process continues until this L₃ layer entirely becomes A_nB (time t₃), so the reaction frontier moves ahead to L₃/L₄, while the L₂/L₃ interface now becomes an inter-plane inside A_nB lattice. If bulk diffusion is frozen, the V_Bs in the L₂ layer that have not diffused to B source will be frozen in this layer forever and will perhaps transform to A-on-B anti-site defects later (e.g., for Nb₃Sn, Nb-on-Sn anti-sites are more stable than Sn vacancies²⁸). Since these point defects determine the A_nB composition, the X_B in this L₂ layer cannot change anymore regardless of μ_B variations in grain boundaries. That is to say, X_B of any point is just the X_II^β of the moment when the reaction frontier sweeps across this point, i.e., the X_B(x) of the β layer is simply an accumulation of X_II^βs with l increase. Returning to Fig. 3(a), the dashed lines display the evolution of X_B(x) with l increase for bulk diffusion, while that for grain boundary diffusion is shown by the solid lines. Since the energy dispersive spectroscopy (EDS) attached to scanning electron microscopes (SEM) that is used to measure the compositions typically has a spatial resolution of 0.5–2 μm and thus mainly reflects the bulk composition, the composition characteristics of Nb₃Sn layers as described above can be explained. It should be noted that knowledge of the difference between bulk diffusion and grain boundary diffusion is important in controlling the final composition of a compound. For instance, if bulk diffusivity is high, one method to form high-B A_nB is increasing the starting B/A ratio so that after long annealing time for homogenization subsequent to the full consumption of A, μ_B(α)= μ_B(A−X_I^β,eq B). However, our experiments demonstrate that for compounds with low bulk diffusivity (e.g., Nb₃Sn), such an approach does not work; instead, controlling the X_II^βs while the compounds are growing is the only way. For those compounds with low but non-negligible bulk diffusivities, their compositions would be between these two extremes.

Figure 5

A schematic of the diffusion reaction process for grain boundary diffusion.

Then what determines the bulk composition as grain boundary diffusion dominates? From Fig. 5, it can be clearly seen that there is a competition deciding the V_B fraction in the frontier A_nB layer: at t₂ the reaction across the β/γ interface produces V_Bs in L₂ layer, while the diffusion of B along β grain boundaries and α/β interface fills these V_Bs. Thus, if the diffusion rate is slow relative to the reaction rate at interface II (i.e., φ_II is low), a high fraction of V_Bs would be left behind as the interface II moves ahead, causing low B content; if, on the other hand, the diffusion rate is high relative to the reaction rate at interface II, the A_nB layer has enough time to get homogenized with the B source, causing low X_B gradient. In this case, the μ_B of B source and the reaction rate at interface I together set a upper limit for μ_B of β.

Next, we will modify the above model for the case of grain boundary diffusion for quantitative analysis. As pointed out earlier, the chemical potentials of grain boundaries can change with μ_B^α and l, while those of the bulk cannot. In such a case, μ_I^β and μ_II^β (suppose the diffusivities along the inter-phase interfaces are large) can still be calculated using our model, provided that the μ(X) relation and D of the β grain boundary (instead of the bulk) are used in all of the equations and that φ_I and φ_II are multiplied by a factor of ∑A_GB/A_int (where ∑A_GB is the sum of the cross section areas of the grain boundaries projected to the inter-phase interfaces), because B diffuses only along β grain boundaries while reactions occur at the entire interfaces. Approximately, ∑A_GB/A_int ≈ [1−d²/(d + w)²] ≈ 2w/d (where w is the β grain boundary width and d is the grain size). Apparently, grain growth with annealing time reduces the diffusion rate. According to Eq. (8), a_II^β is determined by η and a_B^α and increases with them, as shown by Fig. 6. Since η = φ_IIa_B^α/(φ_Ia_B^α + κl)=1/[φ_I/φ_II + κl/(φ_IIa_B^α)], clearly η decreases as φ_I/φ_II and l increase and the influence of l (which reflects the X_II^β-x gradient) is mitigated as φ_IIa_B^α increases. Thus, to improve X_II^β of A_nB at l =0, one should increase μ_B^α and the reaction rate at interface I and reduce the reaction rate at interface II; meanwhile, to reduce X_II^β(x) gradient, one should increase φ_II (which means improving the diffusion rate or reducing the reaction rate at interface II) and a_B^α. Apparently, these quantitative conclusions are consistent with the above qualitative analysis.

Figure 6

The variation of a_II^β with η and a_B^α, according to Eq. (8).

Next let us compare this model with the example of Nb₃Sn. It has been well established from experimental work that there are mainly two factors that can significantly influence the Sn content of Nb₃Sn in a Cu-Sn/Nb₃Sn/Nb diffusion reaction couple: heat treatment temperature and Cu-Sn source. The heat treatment temperature can simultaneously influence multiple factors of Eq. (8), such as a_B^α, D and reaction rates at both interfaces, etc. Thus, the explanation of the influence of temperature on Sn contents using this theory requires knowledge of the quantitative variations of these factors with temperature. For the other factor, Cu-Sn source, the diffusion reaction couples can be classified into two types based on the Cu-Sn source: the type I uses bronze (with Sn content in Cu-Sn typically below 9 at.%) as Sn source and the type II uses high-Sn Cu-Sn (e.g., Cu-25 at.% Sn). Previous measurements^4,9,10,11,29 show that both types of samples have Sn contents above 24 at.% for the Nb₃Sn layer next to the Cu-Sn source; however, they have distinct Sn content gradients as the Nb₃Sn layers grow thicker: the type I generally has Sn content gradients above 3 at.%/μm²⁹, while those of the type II are below 0.5 at.%/μm^4,9,10. Such a difference in the Sn content gradients leads to distinct grain morphologies and superconducting properties. The different X_Sn gradients in the two types of samples with different Cu-Sn sources can be easily explained by our theory above: according to Eq. (8), increased μ_B^α can decrease X_Sn gradients. It may also need further investigation regarding whether Cu-Sn source can also influence diffusion rates in Nb₃Sn layer (e.g., via thermodynamic factor), because greater D leads to greater φ_II, which helps decreasing X_Sn gradients. As to the phenomenon that different wires have similar X_Sn in the Nb₃Sn layer next to the Cu-Sn source, the relation between μ_Sn(Cu-Sn) and μ_Sn(Nb-X_Sn Sn) is required. The Cu-Sn system has been well studied and the phase diagram calculated by the CALPHAD technique using the thermodynamic parameters given by ref. 30 is well consistent with the experimentally measured diagram³¹. Thus, the parameters from ref. 30 are used to calculate μ_Sn of Cu-Sn, which is shown in Fig. 7. On the other hand, although thermodynamic data of Nb-Sn system were proposed by refs. 30 and 32, in these studies Nb₃Sn was treated as a line compound. However, some information about μ_Sn of Nb₃Sn can be inferred from its relation with μ_Sn of Cu-Sn: since Cu-7 at.% Sn leads to the formation of Nb-24 at.% Sn near the Cu-Sn source²⁹, we have μ_Sn(Cu-7 at.% Sn) ≥ μ_Sn(Nb-24 at.% Sn). Thus, the expected approximate μ_Sn(Nb-X_Sn Sn) curve in a power function is shown in Fig. 7. Furthermore, we can also infer that the Sn transfer rate at the Cu-Sn/Nb₃Sn interface must be much faster than that at the Nb₃Sn/Nb interface, so μ_Sn discontinuity across the interface I is small. These explain why low-Sn Cu-Sn can lead to the formation of high-Sn Nb₃Sn. It is worth mentioning that from Fig. 7, it is clear that the Taylor series for the true a(X) relation of Nb₃Sn have more high-rank terms than a(X)≈a_B^γ + κ(X−X_II^β,eq); however, our numerical calculations show that adding high-rank terms to the a(X) relation does not lead to different conclusions regarding the influences of a_B^α, φ_I, φ_II and l on X_II^β. Thus, the above qualitative and quantitative analysis still applies.

Figure 7

The variation of μ_Sn with X_Sn for Cu-Sn calculated based on thermodynamic data given in ref. 30 and a rough, speculative μ_Sn(X_Sn) relation for Nb₃Sn sketched according to the phase formation relation between Cu-Sn and Nb₃Sn.

In summary, a mathematical framework is developed to describe the compositions and layer growth rates of non-stoichiometric intermediate phases formed by diffusion reactions. The governing equations are derived and analytic solutions are given for compounds with narrow composition ranges under certain approximations. We also modify our model for compounds in which bulk diffusion is frozen, of which the bulk is not in equilibrium with the rest of the system. Based on this model, the factors that fundamentally determine the compositions of non-stoichiometric compounds formed by diffusion reactions are found and approaches to control the compositions are proposed.

Methods

For the Cu-Sn/Nb₃Sn/Nb diffusion reaction couples that were used for Sn content measurements (the results of which are shown in Fig. 4), the initial composition of the precursor Cu-Sn alloy was Cu-12 at.% Sn. The samples were reacted at 650 °C for 65 h, 130 h, 320 h and 600 h. Then the surfaces of the samples were polished to 0.05 μm and the compositions were measured using an EDS system attached to an SEM. An accelerating voltage of 15 kV was used for the quantitative line scans. A standard Nb-25 at.% Sn bulk sample provided by Dr. Goldacker from Karlsruhe Institute of Technology was used for calibrating the Sn content of the samples. The standard deviation in the measurements was found to be about ± 0.5 at.%.