ENERGETICS AND DYNAMICS OF CAGED Zn 4 IN i-sczn

Transcription

1 ENERGETICS AND DYNAMICS OF CAGED Zn 4 IN i-sczn Marek Mihalkovič and Christopher L. Henley [Support: U.S. Dept. of Energy] ICQ11 Poster P02-3, June 14, 2010, Sapporo Japan 1

2 Introduction Recall: cluster in CaCd-type structure. (a). Zn 4 tetrahedron (focus of this poster) (b). 20 Zn dodecahedron (very deformable!) (c). 30Zn + 12 Sc (2nd shell) At higher T, the tetrahedron can reorient; what is its dynamics, and what determines the optimum orientation(s)? 2

3 Outline: Formulate eff. potential for orientations of Zn 4 tetrahedron Find optimal orientations Theory: model the dynamics (as seen in neutron diffraction) Based on molecular dynamics (MD) simulations using pair potentials. 3

4 Framework for tetrahedron eff. Ham. Degrees of freedom are labeled by rigid-body rotation matrix (orientation) Ω i. (Actually: tetrahedra deform, but we assume this just follows the orientation.) One-body terms: an icosahedrally symmetric part a term with cubic symmetry of the 1/1 approximant. Pair terms: v ij (Ω i, Ω j ) orientations of tetrahedra in neighboring clusters. Determine structural orderings (seen in CaCd case) Probably mediated elastically. 4

5 Our single Zn4 construct This sample: all but one Zn 4 one (large) Sc atom. (Out of 16 clusters in our supercell.) Important: constrained lattice const. same as in all-zn 4 sample. Purpose: separate single-body and pair interactions. potential is acting. The 1/1 approximant with Zn 4 Sc is stable against competing phases over a wide range of Sc fraction substituted. (Explains compositional variability in expts?) Behavior in single-zn 4 and all-zn 4 samples similar we infer pair interactions negligible. (From now on, only consider single-body terms.) 5

6 Pair potentials used Our MD simulation based on these. (Ab-initio-based: computationally demanding, in progress [Euchner, Stuttgart]) Salient features of pair potentials: φ ZnSc (R) strongly attractive well at R Sc 3.0Å; φ ZnZn (R) optimal distance R Zn 2.8Å. 6

7 Sc-Sc Zn-Sc Zn-Zn E[eV] 7

8 Zn potential Handy: Zn potential Φ(r) (analog of electrical potential...) Φ Zn (r) = i φ Zn,i (r r i ), i runs over neighbors; φ Zn,i (R) are the pair potentials (Note: Al potential was used previously for analogous (fluctuating) atoms in d-alcuco [Gu, Mihalkovic, and Henley 2006]). Result: A5 = lowest energy (best distance for Zn-Sc potential to 5-fold Sc; also good Zn-Zn distance to 3-fold Zn in cage). 8

9 Each curve runs from one symmetry direction to another: Am = an m-fold symmetry axis) A2c = cubic (100) axis (6 directions), A2 = other 2-fold (24 directions) A3c = cubic (111) (8 directions) A3c = other 3-fold (12 directions) A2->A3c A2c->A2 A2c->A3 A2c->A5 A3->A2 A5 -> A2 A5-A3c A5->A3 ENERGY [ev] A3c almost as low E (Zn along 111 far away), but A3 is highest E (other 3 fold Zn are close, sterically bad) 9

10 Optimal orientations Φ Zn (r) want to place all four Zn close to a 5-efold axis, but possible for at most three:. Angle between tetrahedral directions is 109, angles between 5-efold directions are 63 and (best) 117. We found two minimum orientations (evidence below). (1) optimal orientation: 2mm symmetry around (say) (001) axis. Zn(1,2) close to (±τ, 0, 1) [117 apart] whereas Zn(3,4) go with (0, ±1, τ) directions [63 apart] this pulls the latter pair closer by 0.05Å. (2) secondary minimum: place Zn(1,2,3 ) near to 5-fold axes ( τ, 0, 1) and cyclic permutations, which are 117 apart. Put Zn(4 ) near (1, 1, 1) This has no symmetry (all four Zn atoms are inequivalent, point symmetry 1 ). 10

11 Evidence 1: movies 3 frames (352, 630, 705) from simulation of Zn 4 in 20-Zn cage. First and last are 2mm states; middle is a secondary min. state Conclude: Transition path is via secondary minimum. 11

12 Evidence 2: probability density "all Zn4" x z x y z y "single Zn4" x z x y z y T=210K, 2x2x2 supercell Probability density for Zn atoms, projected in the (001) direction. 12

13 Vibrational density of states (VDOS) Dyn.; structure factor S(q, ω) from neutron scatt VDOS G(ω) lim q 0 ω 2 S(q, ω) q partial Zn4, T=100K partial Zn4, T=210K VDOS total, 200 ps VDOS E [mev] Red/blue = partial Zn 4 contributions at T = 100K/210K. Black curve = whole spectrum. (called VDOS since it agrees with phonon spectrum if phonons are the only slow dynamics. 13

14 Previous plot matches calculated harmonic phonons except at low ω end) Very low ω (T = 100K) showing (dominant) Zn 4 contributions dynamics of occasional reorientations. In both plots the axis is hω in mev 14

15 Theory: reorientation dynamics and VDOS Can be shown G v (t)= velocity autocorrelation. G(ω) = Fourier transform(g v (t)) Our situation: a constant state for a long time, then randomly has abrupt transitions to another discrete state. Described by discrete master equation with transition rates Γ αβ from state α β. In our case, where position x i (t) is confined, it can be shown that G v (t) = d2 x i (0) x i (t) dt 2. Thus, don t need to know how large the velocities are during the short and rare transitions! 15

16 For the simplest case of a double well we obtain G(ω) = 2 π x 2 0 τ [ 1 1 ] 1 + ω 2 τ 2. This fits our data well for the 1/1 approximant (red line in low-frequency plot of VDOS). Note it does not fit simulation results for the 2/1 approximant. Interpretation: the local environment is less symmetric than cubic various transition barriers spectrum of exponential relaxation rates (not just the one term in fitting form.) 16

17 Discussion 1. Comparison to CaCd 1/1 approximant Cd 4 cluster diffraction study by C. P. Gomez et al, PRB Our similar simulations for CaCd show that (in contrast to ScZn) the optimal configuration is the asymmetric one, while the 2mm confiuration is secondary. 2. Compare central cluster in i-alpdmn i-alpdmn and i-alcufe built from pseudo-mackay icosahedron clusters with an irregular Mn Al 7 core. The orientational dynamics of this core calls for analysis like Zn 4, but much more complicated. 17