Nanoscale Mapping of Heterogeneous Strain and Defects in Individual Magnetic Nanocrystals

Abstract

We map the three-dimensional strain heterogeneity within a single core-shell Ni nanoparticle using Bragg coherent diffractive imaging. We report the direct observation of both uniform displacements and strain within the crystalline core Ni region. We identify non-uniform displacements and dislocation morphologies across the core–shell interface, and within the outer shell at the nanoscale. By tracking individual dislocation lines in the outer shell region, and comparing the relative orientation between the Burgers vector and dislocation lines, we identify full and partial dislocations. The full dislocations are consistent with elasticity theory in the vicinity of a dislocation while the partial dislocations deviate from this theory. We utilize atomistic computations and Landau–Lifshitz–Gilbert simulation and density functional theory to confirm the equilibrium shape of the particle and the nature of the (111) displacement field obtained from Bragg coherent diffraction imaging (BCDI) experiments. This displacement field distribution within the core-region of the Ni nanoparticle provides a uniform distribution of magnetization in the core region. We observe that the absence of dislocations within the core-regions correlates with a uniform distribution of magnetization projections. Our findings suggest that the imaging of defects using BCDI could be of significant importance for giant magnetoresistance devices, like hard disk-drive read heads, where the presence of dislocations can affect magnetic domain wall pinning and coercivity.

1. Introduction

Due to their diverse range of applications—from data storage, energy harvesting and conversion to nano-mechanical devices—nanostructures are of enormous interest to academia and industry [1,2,3,4,5,6,7,8]. At the nanoscale, confinement effects, size variations, strain, defects and shape morphology have been recently shown to play a crucial role in mechanical and magnetic device applications [9,10,11,12,13]. For metallic and magnetic materials such as nickel nanoparticles, extended line defects such as edge and screw dislocations provide an important and universal catalyst-free growth mechanism of quasi one-dimensional nanostructures [14]. In FePt nanostructures, thermally activated spin current assisted de-pinning of magnetic domain walls has been controlled with the aid of defects [15].

The synthesis of nanoscale pure metallic materials is difficult as it is often accompanied by a spontaneous oxidation of the surface. This tendency for surface oxidation however results in the formation of core–shell heterogeneous nanoparticle structures with interesting proximity effects resulting from the structural modification and competition of different magnetic orderings at the interface. As a result of the inherent difficulties encountered at the nanoscale, experimental characterization of these materials and identification of strain and dislocations across buried interfaces in nanostructures is possible but often problematic.

In materials physics, a dislocation is considered as a linear crystallographic defect or irregularity within a crystal structure which contains an abrupt change in the arrangement of atoms from their ideal equilibrium positions (strain). The movement of dislocations allows atoms to slide each other at low stress levels. In polycrystalline materials, dislocation glide can be hindered by grain boundaries, precipitates and other dislocations. These dislocations are currently modeled with continuum mechanics methods, which do not explicitly account for the discrete nature of the material and are even not applicable in the core region of the dislocation (traditionally regions of highest distortions) that leads to singularities in the amplitudes or divergences in the phases of a scattered complex x-ray wave field. Moreover, this core region has abrupt and large displacement gradients in the nanostructures. Therefore, experimental probes capable of mapping the strain and stress field in the vicinity of dislocations are required in order to study such defects.

Understanding and studying defects and dislocations across buried interfaces or in individual nanostructures is challenging. Traditional characterization techniques such as laboratory X-ray diffraction (XRD) [16,17], electron microscopy: (SEM [17], EF-TEM [18], HR-TEM [16,17,19]) of Ni/NiO with the core/shell morphology have retrieved information about structure of the core and shell incorporating twinned nature [18] and the texturing and dislocation activity of Ni nanoparticles (NPs) [20].

Techniques such as X-ray topography [21] and electron microscopy [22] are able to probe the properties and dynamics of dislocations. Recently, combined studies involving high-resolution transmission electron microscopy (HR-TEM) and theoretical calculations showed that the core of a Ni/NiO nanoparticle can inherently possess a regular multi-twinned icosahedral structure that is composed with single-crystal tetrahedra with (111) faces (depending on the cooling rate in the synthesis process). As a result, the shell with crystal NiO islands with direct or twinned stacking [18] and dislocations are generated. However, to further understand the underlying structure–property relationships in such complicated nanostructures, the need for a volumetric and non-destructive probe capable of isolating an individual nanocrystal is warranted. This probe should be able to isolate and distinguish between defect types, and map displacement and strain field in the vicinity of such defects with nanoscale precision.

Here, we exploit the sensitivity of Bragg coherent X-ray diffraction imaging (BCDI) to map the three-dimensional (3D) displacement field, displacement gradients (strain) and edge dislocations in the core, shell and the core/shell interface of an individual Ni nanoparticle (NP). BCDI techniques are emerging at third and fourth-generation light sources [23] as a state of the art non-destructive tool for 3D imaging of strain inhomogeneity [24,25,26,27,28,29] and dynamics [30,31] within and in the vicinity of a nanocrystal. Recent advances in instrumentation [32,33] and phasing algorithms [34,35,36,37] used in BCDI have seen a wide range of application in imaging ferroelectric polarization [38], ferroelastic domains [39], magnetostrictive strain [40], defects [41,42] and other functional material properties [43,44,45,46,47,48] in individual nanostructures. For a review of Bragg coherent diffractive imaging of strain and a general overview of coherent diffractive imaging please see Refs. [49,50] respectively. For more studies of dislocations using BCDI in a wide variety of material systems, see Refs. [51,52,53,54,55,56].

The ability to probe the long-range strain state using BCDI enables atomic sensitivity to defects/dislocations within the specimen under study. The nickel NPs used in this experiment were grown by thermal chemical vapor deposition [57,58]. BCDI is a powerful characterization technique that allows for the imaging of internal strains, commonly related to spatial gradients of displacement fields, $\mathbf{u}\left(\mathbf{r}\right)$ within small crystals [27,59]. BCDI allows us to image not only the overall shape of the nanostructure in 3D, but also the projection of the crystal lattice displacement field onto the onto the momentum transfer $\mathbf{Q}$ vector of the measured Bragg spot. Upon an inversion (traditionally using phase retrieval algorithms) [34,35,36,37] of the Bragg coherent x-ray diffraction pattern, the resulting real-space image of the object will be complex valued $\tilde{\rho }\left(\mathbf{r}\right)={\rho }_{{\mathbf{G}}_{hkl}}\left(\mathbf{r}\right)exp\left[-i{\mathbf{G}}_{hkl}·\mathbf{u}\left(\mathbf{r}\right)\right]$, where the amplitude ${\rho }_{{\mathbf{G}}_{hkl}}$ represents the density of the object and the phase, ${\mathbf{G}}_{hkl}·\mathbf{u}\left(\mathbf{r}\right)$ is the projection of the atomic displacement vector field $\mathbf{u}\left(\mathbf{r}\right)$ relative to atomic positions in an ideal perfectly periodic lattice onto the reciprocal lattice vector ${\mathbf{G}}_{hkl}$ for the measured Bragg reflection ($hkl$).

2. Discussion and Conclusions

In a crystal, initial slip takes place on a particular slip plane in a particular slip direction. Slip traditionally occurs when the resolved shear stress becomes equal to the critical shear stress, ${\sigma }_f$, a parameter that depends upon the mesostructure and the mechanical properties of the sample. BCDI is used to characterize the mesostructure of our Ni sample from coherent X-ray diffraction (CXD) measurements. Initially, we utilized scanning electron microscopy, SEM, shown in Figure 1a to identify peculiar and different types of Ni nanocrystals such as nanowires [24,40] and nanoparticles [60]. The Ni nanocrystals were grown using Thermal Chemical Deposition (CVD) [57,60] on a Si/SiO2 substrate. For the CXD experiments, we used a confocal microscope to isolate a truncated octahedral shaped nanoparticle (identified by the black arrow in Figure 1a).

Figure 1 shows the experimental setup used to collect CXD patterns from the truncated octahedral shaped Ni nanoparticle. The asymmetrical nature of the CXD pattern namely: a lack of inversion symmetry and a broadened peak is indicative of strain inhomogeneity within the nanoparticle. This usually manifests itself as a non-zero imaginary component of the effective Bragg electron density. Iterative phase retrieval algorithms [34] were utilized to invert the CXD data to obtain the resulting complex Bragg electronic density images. The atomic displacement field scales linearly with the reconstructed phase of the complex Bragg electron density while the isosurface of the amplitude can be used to display the nanoparticle shape.

Figure 2a,b show the side and top view of the 3D isosurface of the reconstructed amplitude of the complex density depicting the shape morphology of a truncated octahedra confirmed by SEM. The support used for the reconstruction is depicted as transparent box. The reconstructed NP has a diameter of 240 nm, consistent with SEM measurements. The real-space phase ${\mathbf{G}}_{hkl}·\mathbf{u}\left(\mathbf{r}\right)$, used to color the density isosurface shown in Figure 2a,b, indicates the presence of some residual strains. The magnitude of the phase within the nanoparticle varies between $\pm \pi$, corresponding to the displacement of $\pm 0.1\mathrm{nm}$ which is approximately a third of the lattice spacing of Ni [61].

A slice cut through the reconstructed 3D displacement field of the NP shows a “core and shell” region (see Figure 2c). The core region is about 120 nm wide and has no visible dislocation features. The core region has a predominant compressive displacement field. We observe an average of compressive displacement of 0.01 nm in the core along the (111) direction. This value represents less than a 10% deviation from equilibrium Ni lattice implying the core is pure single crystal. We observe dislocations (quantifiable as $\pi$ phase jumps) in the shell region and at the boundary between the core and shell. According to the theory of dislocations [62], the propagations of stresses, strains and displacement fields near a dislocation source can be used to identify the dislocation type. To determine the stresses and strains associated with a given dislocation, an equation that describes how the dislocation displaces the lattice is needed. In Figure 2d–f, strong strain inhomogeneity is found within the core of the NP in the x- and y-direction while the z-direction has a predominant compressive strain distribution. These features are attributes of edge dislocations. Generally, edge dislocations create more complicated lattice distortions than screw dislocations. However, a more rigorous study of the individual dislocation types is required which is not within the scope of this work. Earlier studies [41,53] have demonstrated how CXD from samples with dislocations can be quantitatively analyzed and used to distinguish between dislocation types. Using analytical methods similar to [41,53], we identify edge dislocations along the core–shell interface and a mixture of partial and full dislocation networks within the shell boundary.

Since dislocations are responsible for slip in crystalline solids [62,63]. The nature of these dislocations are provided by the action of Frank–Reed sources [62,64] on the slip plane. The number of defect pinning points associated with the Frank–Reed source depends on the critical resolved stress. The closer the defect pinning points/sites, the higher the critical shear stress is expected. Below, we have estimated and tabulated critical shear stress (${\sigma }_f$) values for dislocations (or pinning sites) observed in Figure 2c.

We utilize the theory of dislocations [62,64] to compare an individual dislocation obtained from BCDI (see Figure 3a) with theory (see Figure 3b), with the overall region of the reconstructed core–shell structure shown ( see Figure 3c).The observed symmetrical (and inverse radial distant from dislocation line) decay of the displacement field is consistent with the theory of elasticity. Ideally, a Frank–Reed source will cause dislocation multiplication and slip when the stress field is greater than ${\sigma }_f=\frac{2G\tilde{b}}{D}$, where D is the distance between the pinning points of the source, G is the shear modulus of the sample and $\tilde{b}$ is the Burgers vector of the dislocation.

Typically, in bulk [62] structures, $D={10}^{4}b$. This provides critical stress values of the order $2\times {10}^{-4}G$. The calculated critical stress values for the observed dislocations in our Ni nanoparticle are summarized in

Table 1. The calculated values for each dislocation.

Dislocation	$\tilde{\mathit{b}}\mathbf{=}\mathit{n}\mathit{b}$	Dislocation	Source Size	Critical Shear	Stacking Fault
	($\mathit{b}\mathbf{=}$ 0.22 nm)	Type		Stress	Energy
(i)	28b	full	43 nm	${\sigma }_f$ = 12.32 GPa	-
(ii)	28b	full	43 nm	${\sigma }_f$ = 12.32 GPa	-
(iii)	27b	full	42 nm	${\sigma }_f$ = 12.16 GPa	-
(iv)	19b	partial	30 nm	${\sigma }_p$ = 11.98 GPa	1421 mJ/m${}^2$

. Here, we observe that the critical stress values in NPs are larger than the bulk values reported in [62]. It is thus feasible that mechanical properties and ultimately magnetic functionality of core shell systems can be altered by the presence of dislocations, the dislocation type, the behavior of the elastic displacements field near dislocations and dislocation densities. However, more rigorous study should be carried out to understand the significance of each of the parameters above and their effect on magnetic functionality such as coercivity. To understand the role played by the presence of dislocations on the equilibrium structure of the core Ni region, we utilized density functional Theory (DFT) computations within the framework of 3D periodic structure using the Vienna Ab initio Simulation Package (VASP). [65,66], and and atomistic simulations using the software package Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [67]. We determined the equilibrium volume of the Ni nanoparticle by allowing the bulk lattice to change isotropically. This led to the observations of residual stress values that are less than 0.06 GPa. Further relaxation of fcc-Ni to its equilibrium structure gives an equilibrium lattice parameter that corresponds to previous experiments and theory [68,69]. In this study, the predicted strain and energy needed for changing the magnetic moments in fcc-Ni corresponds to similar strain values and displacement fields observed near the dislocations sources in Figure 3a.

The spatial resolution of our BCDI experiments is 30 nm as shown by the phase retrieval transfer function (PRTF) in Figure 4. This is a typical value for the BCDI data acquired in 34-ID-C beamline at Advanced Photon Source (APS). The PRTF allows us to analyze correlations in reconstructed phases over a number of converged solutions. The less correlated the phases are, the lower is the value of PRTF. In the analysis, the PRTF is weighted for a given spatial frequency, with conventional threshold being set at $50\%$ of PRTF for the frequencies that are reconstructed with reproducible phases.

To understand the correlation between structure (atomic displacement field) and functional property (magnetism), we utilize Landau-Lifshitz-Gilbert (LLG) theory [70,71,72] to model the distribution of magnetization in a core-shaped Ni nanoparticle. The distribution the of magnetization and atomic displacement field along the (111) direction in a 120 nm wide Ni nanoparticle is shown in Figure 5a,b respectively. Here we observe that the distribution of the displacement field flows with the distribution of magnetization. However, in this model, we have taken only the core of the Ni nanoparticle into consideration.

From experimental BCDI results, we have observed the presence of dislocations within the shell, and along the boundary between the core and the shell. We have also observed that such dislocation sources can strongly influence the the lattice distortions and displacement field. We propose future studies in which Ni (core) and NiO (shell) nanocrystals will be grown via controlled methods [73]. The lattice mismatch and post growth annealing conditions shall be used to control misfit dislocations between the core and the shell region. Since NiO is anti-ferromagnetic and Ni is ferromagnetic, We shall investigate dislocation pinning of magnetic domains, exchanged bias coupling and the role of core size and shell thickness using theoretical models and advanced synchrotron characterization techniques.

Numerous experimental studies [74,75] using electron microscopy have demonstrated that magnetic domain walls can be pinned by dislocations.The pinning was observed to be preferentially along the length of the dislocations, rather than being penetrated by them. Most of these studies have demonstrated that domain walls are more strongly pinned at networks of dislocations than at single dislocations. In addition, the domain walls pinned by larger dislocation sources have higher microcoercivities than those pinned at shorter dislocations. This implies the ability of dislocations to not only alter the coercivity of the magnetic system but also modify functionality such as exchanged bias. To further understand the influence of dislocations, and their associated stress field on exchanged bias, Ni-NiO core shell nanocrystals could serve as an ideal template for future studies. Using controlled growth techniques, suitable nanocrystals can be isolated for both BCDI studies and device applications. Since it is anticipated that dislocation types, the dislocation network, dislocation location, nanocrystal size, shape and faceting and interfacial mismatch-strain will play an ultimate role in the functionality of core–shell Ni-NiO systems advances in BCDI is highly relevant. Our future studies will thus involve complementary growth, volumetric characterization using BCDI and theoretical validation of interactions of dislocations in n ferromagnetic (FM) core and an antiferromagnetic (AFM) shell structure such as Ni-NiO.

Our results open up new avenues for the imaging and mapping of deformation and elastic properties in buried magnetic interfaces to help unlock the potential for a synthesis-characterization feedback loop. Future works should focus on understanding how various structural defects in the nanoparticle may influence the magnetic domain structures and the heterogeneities of magnetization.

3. Materials and Methods

3.1. Sample Growth

Ni nanostructures were grown on an amorphous SiO${}_2$ templated Si (100) substrates via a one-step catalyst-free thermal chemical vapor deposition method. A powdered NiCl${}_2$. 6H${}_2$O precursor was first dried at 200 ${}^{\circ }$C in a nickel boat in a quartz tube along with 5 Si (100) substrates lined up next to the boat. The temperature of the system was then raised to 650 ${}^{\circ }$C, and the growth was allowed to occur for 30 min. Argon flow was maintained throughout the process. Afterwards, the entire system was cooled to room temperature while maintaining the Ar flow in the closed system. Scanning electron microscopy (JEOL JSM-6332- Field Emission Scanning Electron Microscope) revealed well defined nickel nano-objects.

3.2. CXD Experiments and BCDI Data Reconstructions

Coherent X-rays of $E=8.919$ keV with 1 eV bandwidth, resulting in a longitudinal coherence length of about 0.7 $\mathsf{\mu }$m from the 34-ID-C beamline of the Advanced Photon Source (APS) were focused onto the sample using Kirkpatrick–Baez (KB) mirrors that achieved a focus around about 2 $\mathsf{\mu }$m × 2 $\mathsf{\mu }$m. The diffraction patterns in the vicinity of the (111) Bragg reflections were measured using a direct-detection charge-coupled device (CCD) (Roper/PI direct detection CCD) with 20.5 $\mathsf{\mu }$m pixels located on the detector arm, 1.05 m away from the sample. 3D CXD pattern was then acquired as a $\mathsf{\theta }$-scan, rotating in 81 steps of 0.006 degrees about a vertical axis. The CXD patterns were inverted to images with a 3D Fourier transform and coordinate transformation, following support-based phasing using a version of Fienup’s hybrid input-output algorithm. The iterative reconstructions were performed with 30 iteration of error reduction (ER), 300 iterations of hybrid input-output (HIO) followed by 30 iterations of ER with a random amplitude and a flat phase for the initial starting guesses.

To ensure and confirm the reproducibility and uniqueness of the reconstructed complex Bragg electronic density, we performed a series of phase retrieval procedures with different random input phases for the measured scattered radiation. We define a measure for the error ${ϵ}^{\left(i\right)}$ in the BCDI reconstruction in the following form:

$$\begin{array}{c}\hfill {ϵ}^{\left(i\right)}=\frac{{\sum }_{\mathbf{q}}{\left(\left|F_{\left(sim\right)}^{\left(i\right)}\left(\mathbf{q}\right)\right|-I_{exp}\left(\mathbf{q}\right)\right)}^2}{{∥I_{exp}∥}_2^2},\end{array}$$

where $F_{\left(sim\right)}^{\left(i\right)}\left(\mathbf{q}\right)$ is the magnitude of the reconstructed amplitude and $I_{exp}\left(\mathbf{q}\right)$ is the experimental intensity in the reciprocal space. An estimate of the resolution of our BCDI reconstruction is obtained using the phase retrieval transfer function (PRTF) [76,77]. The spatial resolution of the reconstructed results is estimated to be around 30 nm as given by the PRTF shown in Figure 4.

3.3. Classical Potential Simulations

In order to confirm the equilibrium shape of the nanoparticle, we utilize atomistic MEAM [78] and EAM [79] calculations. All computations were performed with the software package Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [67]. We constructed a 9 shell Mackay cluster [80,81,82] containing 2057 atoms. The cluster was placed in a periodic box and surrounded by vacuum that was sufficiently large to eliminate interactions between modeled cluster and its periodic images. Subsequently the ground state structure of the cluster was determined by relaxing the cluster using a conjugate gradient algorithm as implemented in LAMMPS. density-functional-theory (DFT) computations within the framework of 3D periodic structure computations using VASP [65,66]. Interactions between nuclei and electrons were described within the all-electron like the projector augmented-wave (PAW)-formalism [68,83]. The valence electron configuration was $\left[\mathrm{Ar}\right]4s^{2}3d^8$$\left({\mathrm{R}}_{\mathrm{core}}=2.300{\mathrm{a}}_{\mathrm{B}};1{\mathrm{a}}_{\mathrm{B}}=0.529\times {10}^{-10}\stackrel{\circ }{\mathrm{A}}\right)$. Electronic exchange and correlation effects were treated within the Generalized-Gradient-Approximation (GGA) as parameterized by Perdew–Burke–Ernzerhof. All computations were performed for the conventional unit cell fcc-Ni containing four Ni atoms using a plane-wave energy-cutoff of 500 eV and a $\Gamma$-centered 16 × 16 × 16 k-point gird. The Fermi-level was slightly smeared using a Fermi–Dirac distribution function with a width of 0.026 eV. These computational settings lead to total energies that are converged to better than 1 meV/atom. The equilibrium volume was determined by allowing the lattice to change isotropically. Residual stresses are less than 0.06 GPa. Structural relaxation of fcc-Ni to its equilibrium structure gives an equilibrium lattice parameter that corresponds to previous experiments and theory [68,69]. The predicted strain and energy needed for changing the magnetic moments in fcc-Ni corresponds to those obtained experimentally.

3.4. Landau–Lifshitz–Gilbert (LLG) Simulation

The free energy of the nickel particle is F is described as $F=F_{exch}+F_{anis}+F_{dip}+F_{ela}$, where $F_{exch}$ is the exchange, $F_{anis}$ denotes the anisotropy energy, $F_{dip}$ is the dipole–dipole interaction energy and $F_{ela}$ is the elastic energy. The exchange energy density in micromagnetism is written as $F_{exch}=A[{(\nabla m_1)}^2+{(\nabla m_2)}^2+{(\nabla m_3)}^2]$, where A is the exchange coefficient, and $m_i$ (i = 1, 2, 3) are components of the unit magnetization vector. The anisotropy energy density of a cubic magnetic crystal is given by $F=K_1(m\frac{2}{1}m\frac{2}{2}+m\frac{2}{1}m\frac{2}{3}+m\frac{2}{2}m\frac{2}{3})+K_{2}m\frac{2}{1}m\frac{2}{2}m\frac{2}{3}$, where $K_1$ and $K_2$ are anisotropy constants. The dipole–dipole energy is given by $F_{dip}=\frac{1}{2}{\mu }_0{M}_s{\mathbf{H}}_d·\mathbf{m}$ Where ${\mu }_0$ is the permeability of the vacuum, ${\mathbf{H}}_d$ is the demagnetization field and $M_s$ is the saturation magnetization. The elastic energy is written as $F_{ela}=\frac{1}{2}{C}_{ijkl}({ϵ}_{ij}-{ϵ}_{ij}^0)({ϵ}_{kl}-{ϵ}_{kl}^0)$, where $C_{ijkl}$ is the elastic constant, ${ϵ}_{ij}$ is the elastic strain and ${ϵ}_{ij}^0$ is the spontaneous strain which is given by $\left\{\begin{array}{c}\frac{3}{2}{\lambda }_{100}\left(m_i{m}_j-\frac{1}{3}\right)(i=j)\\ \frac{3}{2}{\lambda }_{111}\left(m_i{m}_j\right)(i\ne j)\end{array}\right.$.

The temporal evolution of the magnetization is described by the Landau–Lifshitz–Gilbert (LLG) equation $(1+{\alpha }^2)\frac{\partial \mathbf{M}}{\partial t}=-{\gamma }_0\mathbf{M}\times {\mathbf{H}}_{eff}-\frac{{\gamma }_0\alpha }{M_s}\mathbf{M}\times (M\times {\mathbf{H}}_{eff})$. Here, $\mathbf{M}=M_s\mathbf{m}$, ${\gamma }_0$ is the gyromagnetic ratio, $\alpha$ is the damping constant and ${\mathbf{H}}_{eff}$ is the effective magnetic field ${\mathbf{H}}_{eff}=-\frac{1}{{\mu }_0}\frac{\partial F}{\partial \mathbf{M}}$. For the nickel particle, the parameters we use in the simulation are $K_1=-5.7$×${10}^3$ Jm${}^{-3}$, $K_2=-2.6$×${10}^3$ Jm${}^{-3}$, ${\lambda }_{100}=-4.6\times {10}^{-5}$, ${\lambda }_{111}=-2.4\times {10}^{-5}$, $M_s=4.8\times {10}^5$ A/m. The elastic constants are $C_{11}=2.5\times {10}^{11}$ N/m, $C_{11}=1.6\times {10}^{11}$ N/m and $C_{11}=1.18\times {10}^{11}$ N/m.