FIG. 4. The probability distribution function of the free vol- ume associated with a sphere v$_{f}$ normalized by the mean free volume 〈 v$_{f}$ 〉 plotted for various φ . The smooth black curve is obtained after averaging over all the experimental data. Γ function corresponding to elastic frictionless spheres, the smooth red/grey curve, is shown for comparison, and is ob- served to deviate systematically at higher v$_{f}$ . Allowing the fitting parameters to float improves the fit, but systematic differences persist for v$_{f}$ > 〈 v$_{f}$ 〉 (blue dashed curve). 〈 v$_{f}$ 〉 at that φ along with the mean distribution obtained after averaging over all the measured φ . The errors asso- ciated with the slightly polydispersity and errors in find- ing particle centers is of order of symbol size. Further, we plot Eq. 3 using m = 15 . 5, δ = 1 . 3 reported in Ref. [6], and allowing α to float to obtain best fit. Clearly, sys- tematic deviations are observed from the frictionless case, complementing the results for Q$_{6}$ . We thank V. Kumaran, Ashish Orpe, and Michael Berhanu for many stimulating discussions, and Chris Rycroft for providing us Voro++ software library. This work was supported by the National Science Foundation under NSF Grant No CBET 0853943. In conclusion, we measured packing of granular spheres and compared experimentally obtained two-point density correlations and free volume distributions before and af- ter application of shear. We find that the radial distri- bution function is captured overall by the Percus-Yevick equation, which is important because it is fundamental to calculating the strength, heat conduction, and electro- magnetic wave scattering properties of a material. How- ever, angular correlation can be observed using the ori- entational order metric. In comparing with numerical simulations reported for frictionless sphere at compara- ble volume fractions, we find systematic differences in packing as measured by lower angular correlations, and deviation of free volume distributions from Γ distribu- tions postulated for frictionless hard spheres. In order to check if a Γ-distribution can capture the experimentally observed free volume distributions, we tested both the three fitting parameter distribution, and the two fitting parameter distribution, which corresponds to setting δ = 1 in Eq. 3. The best fit obtained with m = 12 . 3, α = 24 . 5, and δ = 0 . 73 is also shown in Fig. 4. Even in this case we obtain enhanced probabilities for v$_{f}$ greater than the mean. Therefore, our distribution differ from the experimental distributions used to give a sim- ple thermodynamic interpretation of granular packing [5]. While it is possible that such deviations arise because of the differences in preparation protocol, we note no signif- icant differences in the distributions obtained before and after application of cyclic shear in our experiments. 4 ACKNOWLEDGMENTS [1] G.D. Scott, Nature 194 , 956 (1962); G.D. Scott, and D.L. Mader, Nature 201 , 382 (1964). [2] D.M. Mueth, H.M. Jaeger, and S.R. Nagel, Phys. Rev. E 57 , 3164 (1998). [3] L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey, and D. Levine, Phys. Rev. E 65 , 031304 (2002). [4] S. Slotterback, M. Toiya, L. Goff, J.F. Douglas, and W. Losert, Phys. Rev. Lett. 101 , 258001 (2008). [5] T. Aste and T.Di Matteo, Phys. Rev. E 77 , 021309 (2008). [6] V.S. Kumar and V. Kumaran, J. Chem. Phys 123 , 114501 (2005). [7] J.-C. Tsai, G.A. Voth, and J.P. Gollub, Phys. Rev. Lett. 91 , 064301 (2003). [8] S. Siavoshi, A.V. Orpe, and A. Kudrolli, Phys. Rev. E 73 , 010301 (2006). [9] J. K. Percus and G. J. Yevick, Phys. Rev. 110 , 1 (1957). [10] A.V. Orpe and A. Kudrolli, Phys. Rev. Lett. 98 , 238001 (2007). [11] O. Pitois, P. Moucheront and X. Chateau, J. Colloid. Interface Sci. 231 , 26 (2000). [12] M. Nicolas, P. Duru, and O. Pouliquen, Europhys. Lett., 3 , 309 (2000). [13] C.H. Rycroft, G.S. Grest, J.W. Landry, and M.Z. Bazant, Phys. Rev. E 74 , 021306 (2006). [14] L. Tsang, J.A. Kong, K.H. Ding, and C.O. Ao, Scattering of Electromagnetic Waves: Numerical Simulation (John Wiley & Sons, Inc., 2001). [15] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys. Rev. B 28 , 784 (1983). [16] M.D. Rintoul and S. Torquato, J. Chem. Phys 105 , 9258 (1996).