Real space study of local bonding for zinc structure based on temperature-dependent x-ray pair distribution function analysis

In materials science, the revealing of the atomic structure using single crystal and powder x-ray crystallographic techniques is one of the important steps toward understanding the properties of materials. These days, the majority of interesting materials exhibit some level of local atomic distortion. Materials with distorted atomic structures are becoming scientifically and technologically important. Knowledge of the three-dimensional real space atomic structure of locally distorted materials is a prerequisite to understanding and engineering their properties.

Zinc metal crystallizes in a hexagonal close packing hcp structure and deviates from the ideal hcp by a significantly increased axial ratio c/a. This deviation from the ideal hcp was linked with unusual properties of zinc metal compared with other hcp metals as zinc has a lower melting point with respect to other 3d transition metals. Different experimental and theoretical studies have been performed to investigate the origin of the unusual structural behavior of zinc metal as non-ideal hcp, the “zinc anomaly.”^1–5 The unusual behavior in the pressure dependence of lattice constants was first observed by Lynch and Drickamer; they found that the c axis compressibility exhibits a distinct irregularity at intermediate density.⁶ This result was reproduced in experiments performed by Takemura et al., where they demonstrated that the findings depend on the experimental conditions.⁷ 

In the literature, many studies were carried out to explore the real structure of elemental zinc by employing different techniques, such as x-ray diffraction (XRD) experiments. Conventional XRD methods explore the long range-order for a given periodic structure with allowed Bragg reflections. In addition, short-range order information and local structural features are not captured by Bragg peak intensity, but they mainly affect the diffuse scattering intensity. Hence, to have information about the local structure and local bonding, a total scattering technique needs to be employed, such as the atomic pair distribution function (PDF) technique. The atomic PDF analysis is found to be a powerful technique to study the crystal structure of distorted materials at different length scales.^8,9

In this work, we have applied a three dimensional real space technique—the synchrotron-based x-ray total scattering atomic pair distribution function method—to study the local structure and local atomic bonding in a non-ideal hcp structure using temperature-dependent x-ray PDF analysis.

The PDF experiment was performed at the 11-ID-C beamline of the Advanced Photon Source (APS) at Argonne National Laboratory, Lemont, IL (USA). The zinc powder sample (325 mesh, 99.9%) was packed in Kapton capillary with 1.0 mm diameter and measured in transmission geometry in the temperature range of 100–300 K. Data were collected using the rapid-acquisition pair distribution function (RA-PDF)¹⁰ experiment setup with an x-ray energy of 114.8 keV (λ = 0.1080 Å). The two-dimensional (2D) raw data were integrated and converted to intensity vs 2θ, where 2θ is the angle between the incident and scattered x-rays, using the software Fit2D.¹¹ 

The program PDFgetX3¹² was used to correct the converted intensity I(2θ) using standard methods^8,9 to obtain the total scattering structure function S(Q), shown in Fig. 1, where the value of the momentum transfer (Q) is given by $Q=4πλsin⁡θ$⁠. Then the S(Q) data were processed to obtain the experimental reduced total scattering structure function, F(Q), and the experimental PDF, G(r), shown in Fig. 2.

In the Fourier transform step, to obtain the PDF G(r) from S(Q), the data are truncated at a finite maximum value of the momentum transfer, and such value is called Q_max. Here, Q_max = 35.0 Å⁻¹ was found to be optimal. Structural information was extracted from the PDF data using a full-profile real-space local-structure refinement method,¹³ analogous to Rietveld refinement.¹⁴ Here, we used the program PDFgui¹⁵ to fit the experimental PDF. Starting from a given structural model associated with a given set of parameters that can be refined, PDFgui searches for the best structure that is consistent with the experimental PDF data. The obtained residual function (R_w) is used to quantify the agreement level between the calculated PDF from a given model and the collected experimental data.^8,9

The structural parameters of the used model were unit cell parameters and anisotropic atomic displacement parameters (ADPs) (U₁₁ = U₂₂ ≠ U₃₃). The non-structural parameters that were refined are the correction for the finite instrumental resolution, σ_Q, low-r correlated motion peak sharpening factor (δ), and scale factor.^16,17

The XRD data for zinc metal are collected for a sufficient range of momentum transfer (Q) in the temperature range of 100–300 K. The experimental reduced structure functions, F(Q), for selected temperatures (300, 200, and 100 K) are shown in Fig. 2(a). The corresponding PDFs, G(r), are shown in Fig. 2(b), where high quality PDFs are obtained, with a sufficient Q_max value of 35.0 Å⁻¹.

For this PDF study, a hexagonal structural model (space group P6₃/mmc) was considered. During the PDF refinement and while maintaining the imposed symmetry, the lattice constants and anisotropic thermal factors were all refined. The crystal structure of zinc metal was studied in the temperature range of 100–300 K using the total scattering PDF analysis. The results of the full-profile fitting to the PDF data are shown in Fig. 3, and the obtained structural parameters are listed in Table I. Our results show that the used hexagonal model fits the room temperature PDF data very well with an excellent agreement factor (R_w) value of 0.04 for a maximum refinement range of 35.0 Å. The obtained PDF results show an anisotropy in the thermal motion of zinc at room temperature. The values of the atomic displacement parameter (ADP) along the c-direction (U₃₃) are found to be almost three times larger than the values obtained in the ab plane (see Table I), which confirms the result obtained by Masadeh et al.¹⁸ This result also agrees with the ratio of the mean square amplitudes along the principal axes measured by Merisalo and Larsen using elastic thermal-neutron scattering data at 295 K. They obtained a ratio of 2.55 for harmonic model 1, 2.25 for anharmonic model 1, and 2.30 for anharmonic model 2 (see Table 3 in Merisalo and Larsen’s work).¹⁹ 

In Fig. 3, we point out the goodness of the fit of the short range vs long range by calculating the R_w value for the converged PDF refinement using two different length scales (2–6 Å) and (6–35 Å). As can be seen from the difference curve, for the low-r region (below 6.0 Å), there are some local structural features that cannot be captured by the crystallographic model. On the other hand, regarding the goodness of the fit R_w values, the short-r range refinement should typically provide the same R_w values as the long-r range. Interestingly, our results show that in the short range region, the calculated R_w value is found to be larger than the one obtained for the long range for all temperatures (Fig. 3). This can be seen from the low-r region of the difference curve in Fig. 3. Hence, the well-known crystallographic model (P6₃/mmc) does not reflect the local structural features for zinc metal.

To explore the local bonding at a short length scale, we fitted the first two Zn–Zn bonds—the in plane bond (2.6636 Å) and the out of plane bond (2.9120 Å)¹ in the radial distribution function (RDF), R(r)—using two Gaussian peaks for all the dataset in the temperature range of 100–300 K, as shown in Fig. 4.

Our local structural results revealed a piece of evidence for the unusual behavior of temperature dependence of the bond lengths, as can be seen from Fig. 5. We found that the first in-plane bond [short bond (SB) = 2.6636 Å], which represents the a-axis, varies linearly within the temperature range of 300–160 K and stays constant below 160 K, which indicate that at lower temperatures (below 160 K), the Zn–Zn distances are no longer thermally contractible in the ab plane [Fig. 5(b)]. This observation is related to the fact that, with the inclusion of the 3d¹⁰ shell in the treatment of the electron correlation, the bonding between the zinc atoms in the hexagonal plane is especially enforced.²⁰ This PDF result agrees with crystallographic analysis, where a similar behavior was observed.¹ 

Using the total scattering PDF analysis, we studied the dependency of the short and long Zn–Zn bond lengths as a function of temperature, as shown in Fig. 5. The linear dependencies of the bond length can be fitted by least-squares according to B(T) = b₀ + b₁T, as can be expressed by Eq. (1) (short bond = SB) and Eq. (2) (long bond = LB). Based on our PDF results, the bond lengths can be linearly fit from 160 to 300 K, but not below 160 K, and the local thermal expansion for the a-axis is found to be three times larger than the one obtained using the conventional crystallographic method.¹ This PDF result shows that locally, the zinc structure has more tendency for changing the length of the a-axis in response to a change in temperature. This result indicates that the average thermal vibrations in zinc metal are more pronounced locally. This PDF study demonstrates that multiple peak fitting is a useful approach for interpreting the local structural features of such a non-ideal hcp metal. It is worth clarifying that bond lengths can be fitted linearly [Eqs. (1) and (2)] from 160 to 300 K, but not below 160 K,

Furthermore, based on our presented PDF analysis, an unusual behavior was also observed for the first out-of-plane bond (long bond = 2.9120 Å) in the studied temperature range, where it shows a discontinuous increase in length below 160 K [Fig. 5(a)], suggesting a presence of atomic dislocation in the studied hcp structure. In materials science, the slip plane is a common defect that occurs in hcp metals much more limited than in bcc and fcc crystal structures. Usually, hcp crystal structures allow a slip on {0001} basal planes.

For such an hcp structure, a supercell model was introduced in the study by Wu et al.,²¹ where they investigated the generalized-stacking-fault energy and surface properties of hcp metals. They found that the adequate convergence with respect to the supercell size consists of 12 layers [see Fig. 2(a) in Ref. 21]. Based on that, we have created a supercell model with a size of (1 × 1 × 6) to explore the disagreement between the crystallographic model and the PDF data in the low r-region (2–3.5 Å) with an R_w value of 0.054, as shown in the inset of Fig. 6(a), which is larger than R_w = 0.031 for the full r-range (2–35 Å). Using the proposed supercell model, the x, y-fraction coordinates for one layer were refined. The PDF fitting shows a better agreement factor (R_w) for the low r-region with a value of 0.028 for the same fitting range (2–3.5 Å), as shown in Fig. 6(b). This PDF result supports the existence of a local slip on the {0001} plane system as a local structure model (slip model) that can reflect the real space features in the low r-region. The parameters of the obtained slip model are summarized in Table II. This slip model does not represent the feature in the high r-region due to a possible irregularity of slipping in the studied polycrystalline system. This result supports the existence of the slip mechanism in such an hcp structure that can affect the behavior of the first out-of-plane bond (long bond) and can be the possible cause of the unusual long bond behavior shown in Fig. 5(a).

In this study, we have investigated the local atomic structure of zinc metal using synchrotron-based, temperature-dependent, total scattering x-ray pair distribution function analysis. This total scattering study presents an evolution of the local bonding in the zinc structure in the temperature range of 100–300 K. The PDF analysis revealed that the a-axis varies linearly within the temperature range of 300 to 160 K and remains constant below 160 K. This result indicates that at lower temperatures, the Zn–Zn bonds are no longer thermally contractible in the ab plane. As a further significant observation, the local thermal expansion for the a-axis is found to be three times larger than the one obtained using the conventional crystallographic method.¹ This PDF study presents the structural features of elemental zinc at short length scales and points out that the average structural features are not reflected locally due to the presence of imperfections in the Zn crystal structure.

The PDF data were collected at the 11-ID-C beamline at the Advanced Photon Source (APS). Use of the APS at Argonne National Laboratory was supported by the U.S. DOE, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. A. S. Masadeh and Moneeb T. M. Shatnawi acknowledge the financial support from the Deanship of Academic Research (DAR) at the University of Jordan.

The authors have no conflicts to disclose.

Ahmad S. Masadeh: Data curation (equal); Project administration (lead); Writing – original draft (lead); Methodology (equal); Software (equal); Validation (equal). Moneeb T. M. Shatnawi: Methodology (equal); Writing – review & editing (equal); Validation (equal). Ziad Y. Abu Waar: Resources (equal); Validation (equal); Visualization (equal). Gassem M. Alzoubi: Validation (equal); Visualization (equal); Writing – review & editing (equal). Yang Ren: Data curation (equal); Methodology (equal); Validation (equal).

The data that support the findings of this study are available within the article.