On the automorphism group of a binary self-dual [120, 60, 24] code

We prove that an automorphism of order 3 of a putative binary self-dual $[120, 60, 24]$ [ 120 , 60 , 24 ] code $C$ C has no fixed points. Moreover, the order of the automorphism group of $C$ C divides $2^a\cdot 3 \cdot 5\cdot 7\cdot 19\cdot 23\cdot 29$ 2 a · 3 · 5 · 7 · 19 · 23 · 29 with $a\in \mathbb N _0$ a ∈ N 0 . Automorphisms of odd composite order $r$ r may occur only for $r=15, 57$ r = 15 , 57 or $r=115$ r = 115 with corresponding cycle structures $3 \cdot 5$ 3 · 5 - $(0,0,8;0), 3\cdot 19$ ( 0 , 0 , 8 ; 0 ) , 3 · 19 - $(2,0,2;0)$ ( 2 , 0 , 2 ; 0 ) or $5 \cdot 23$ 5 · 23 - $(1,0,1;0)$ ( 1 , 0 , 1 ; 0 ) respectively. In case that all involutions act fixed point freely we have $|\mathrm{Aut}(C)| \le 920$ | Aut ( C ) | ≤ 920 , and $\mathrm{Aut}(C)$ Aut ( C ) is solvable if it contains an element of prime order $p \ge 7$ p ≥ 7 . Moreover, the alternating group $\mathrm{A}_5$ A 5 is the only non-abelian composition factor which may occur in $\mathrm{Aut}(C)$ Aut ( C ) .     

Experimental Measurements of Thermophysical Properties of \mathrm{Al}_{2}\mathrm{O}_{3}– and \mathrm{TiO}_{2}–Ethylene Glycol Nanofluids

This paper presents measurements of the thermal conductivity and the dynamic viscosity of $\mathrm{Al}_{2}\mathrm{O}_{3}$ Al 2 O 3 –ethylene glycol and $\mathrm{TiO}_{2}$ TiO 2 –ethylene glycol (1 % to 3 % particle volume fraction) nanofluids carried out in the temperature range from $0\,^{\circ }$ 0 ° C to $50\,^{\circ }$ 50 ° C. The thermal-conductivity measurements were performed by using a transient hot-disk TPS 2500S apparatus instrumented with a 7577 probe (2.001 mm in radius) having a maximum uncertainty $(k=2)$ ( k = 2 ) lower than 5.0 % of the reading. The dynamic-viscosity measurements and the rheological analysis were carried out by a rotating disk type rheometer Haake Mars II instrumented with a single-cone probe (60 mm in diameter and $1^{\circ }$ 1 ° ) having a maximum uncertainty $(k=2)$ ( k = 2 ) lower than 5.0 % of the reading. The thermal-conductivity measurements of the tested nanofluids show a great sensitivity to particle volume fraction and a lower sensitivity to temperature: $\mathrm{TiO}_{2}$ TiO 2 –ethylene glycol and $\mathrm{Al}_{2}\mathrm{O}_{3}$ Al 2 O 3 –ethylene glycol nanofluids show a thermal-conductivity enhancement (with respect to pure ethylene glycol) from 1 % to 19.5 % and from 9 % to 29 %, respectively. $\mathrm{TiO}_{2}$ TiO 2 –ethylene glycol and $\mathrm{Al}_{2}\mathrm{O}_{3}$ Al 2 O 3 –ethylene glycol nanofluids exhibit Newtonian behavior in all the investigated temperature and particle volume fraction ranges. The relative viscosity shows a great sensitivity to the particle volume fraction and weak or no sensitivity to temperature: $\mathrm{TiO}_{2}$ TiO 2 –ethylene glycol and $\mathrm{Al}_{2}\mathrm{O}_{3}$ Al 2 O 3 –ethylene glycol nanofluids show a dynamic viscosity increase with respect to ethylene glycol from (4 to 5) % to 30 % and from 14 % to 50 %, respectively. Present experimental measurements were compared both with available measurements carried out by different researchers and computational models for thermophysical properties of nanofluids.     

Effect of Previous Milling of Precursors on Magnetoelectric Effect in Multiferroic {\mathrm{Bi}_{5}\mathrm{Ti}_{3}\mathrm{FeO}_{15}} Ceramic

$\mathrm{Bi}_{5}\mathrm{Ti}_{3}\mathrm{FeO}_{15}$ Bi 5 Ti 3 FeO 15 magnetoelectric (ME) ceramics have been synthesized and investigated. The ME effect can be described as an induced electric polarization under an external magnetic field or an induced magnetization under an external electric field. The materials in the ME effect are called ME materials, and they are considered to be a kind of new promising materials for sensors, processors, actuators, and memory systems. Multiferroics, the materials in which both ferromagnetism and ferroelectricity can coexist, are the prospective candidates which can potentially host the gigantic ME effect. $\mathrm{Bi}_{5}\mathrm{Ti}_{3}\mathrm{FeO}_{15}$ Bi 5 Ti 3 FeO 15 , an Aurivillius compound, was synthesized by sintering a mixture of $\mathrm{Bi}_{2}\mathrm{O}_{3}, \mathrm{Fe}_{2}\mathrm{O}_{3}$ Bi 2 O 3 , Fe 2 O 3 , and $\mathrm{TiO}_{2}$ TiO 2 oxides. The precursor materials were prepared in a high-energy attritorial mill for (1, 5, and 10) h. The orthorhombic $\mathrm{Bi}_{5}\mathrm{Ti}_{3}\mathrm{FeO}_{15}$ Bi 5 Ti 3 FeO 15 ceramics were obtained by a solid-state reaction process at 1313 K. The ME voltage coefficient ( $\alpha _\mathrm{ME}$ α ME ) was measured using the dynamic lock-in method. The highest ME voltage coefficient ( $\alpha _\mathrm{ME} = 8.28\,\text{ mV }{\cdot }\text{ cm }^{-1}{\cdot }\text{ Oe }^{-1})$ α ME = 8.28 mV · cm ? 1 · Oe ? 1 ) is obtained for the sample milled for 1 h at $H_\mathrm{DC }= 4$ H DC = 4  Oe (1 Oe = 79.58  $\text{ A }{\cdot }\text{ m }^{-1})$ A · m ? 1 ) .     

Measurement and Modeling of Water-Vapor Diffusion in Elastomers with Impact in Humidity and Vacuum Measurements

The dynamics of water-vapor dissolution in Viton O-rings is measured with a gravimetric method using a precise mass comparator. A sample gasket was degassed in high vacuum for a sufficiently long period to remove more than 99 % of the dissolved water vapor. After that, it was exposed to the ambient atmosphere with a controlled temperature, and relative humidity and water-vapor uptake curves were measured gravimetrically with a precise balance. The dynamics of a water-vapor release into vacuum from another sample that was previously saturated with water vapor at room temperature was determined. The sample was placed in a vacuum outgassing rate measurement apparatus. The time dependence of the evolved water vapor was calculated by integrating the measured outgassing rate. The physical process of water absorption can be described by the diffusion equation. The geometry of the samples required solving the diffusion equation in cylindrical coordinates. This was done numerically using a finite-difference method. As a result of the modeling, room temperature values of the diffusion constant $D$ D , the solubility $s$ s , and the permeability $K = D\times s$ K = D × s of water vapor in the sample material (Viton A-401C) were obtained. For sample 1, we obtained $D = 8.0 \times 10 ^{-8}$ D = 8.0 × 10 ? 8  cm $^{2}\,\,{\cdot }\,\,$ 2 · s $^{-1}$ ? 1 and $s = 6.5 \times 10^{-7}$ s = 6.5 × 10 ? 7  g $\,{\cdot }\,$ · cm $^{-3}\,{\cdot }\,$ ? 3 · Pa $^{-1}$ ? 1 , while for sample 2, $D = 3.0 \times 10^{-7}$ D = 3.0 × 10 ? 7  cm $^{2}\,{\cdot }\,$ 2 · s $^{-1}$ ? 1 and $s = 3.5 \times 10^{-7}$ s = 3.5 × 10 ? 7  g $\,{\cdot }\,$ · cm $^{-3}\,{\cdot }\,$ ? 3 · Pa $^{-1}$ ? 1 .     

On optimal binary codes with unbalanced coordinates

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either $\lfloor M/2\rfloor $ ? M / 2 ? or $\lceil M/2 \rceil $ ? M / 2 ? (that is, exactly $M/2$ M / 2 for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining $A(17,8) = 36$ A ( 17 , 8 ) = 36 , which is shown to be unique here; $A(n,d)$ A ( n , d ) denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that $A(18,8) \le 68$ A ( 18 , 8 ) ≤ 68 .     

Transfer Partial Molar Isentropic Compressibilities of (l-Alanine/l-Glutamine/Glycylglycine) from Water to 0.512 {\mathrm{mol}}\!\cdot \!{\mathrm{kg}}^{-1} Aqueous {\mathrm{KNO}}_{3}/0.512\, {\mathrm{mol}}\!\cdot \! {\mathrm{kg}}^{-1} Aqueous {\mathrm{K}}_{2}{\mathrm{SO}}_{4} Solutions Between 298.15 K and 323.15 K

Speeds of sound of (l-alanine/l-glutamine/glycylglycine $\,+\, 0.512\, {\mathrm{mol}}\cdot {\mathrm{kg}}^{-1}$ + 0.512 mol · kg ? 1 aqueous ${\mathrm{KNO}}_{3}/0.512\, {\mathrm{mol}}\cdot {\mathrm{kg}}^{-1}$ KNO 3 / 0.512 mol · kg ? 1 aqueous ${\mathrm{K}}_{2}{\mathrm{SO}}_{4}$ K 2 SO 4 ) systems have been measured for several molal concentrations of amino acid/peptide at different temperatures: $T$ T = (298.15 to 323.15) K. Using the speed-of-sound and density data, the parameters, partial molar isentropic compressibilities $\phi _{\kappa }^{0}$ ? κ 0 and transfer partial molar isentropic compressibilities $\Delta _{\mathrm{tr}} \phi _{\kappa }^{0}$ Δ tr ? κ 0 , have been computed. The trends of variation of $\phi _{\kappa }^{0}$ ? κ 0 and $\Delta _{\mathrm{tr}} \phi _{\kappa }^{0}$ Δ tr ? κ 0 with changes in molal concentration of the solute and temperature have been discussed in terms of zwitterion–ion, zwitterion–water dipole, ion–water dipole, and ion–ion interactions operative in the systems.     

Optimal (v,5,2,1) optical orthogonal codes of small v

We classify up to multiplier equivalence optimal $(v, 5, 2, 1)$ ( v , 5 , 2 , 1 ) optical orthogonal codes (OOC) with $v\le 114$ v ≤ 114 . Examples of optimal $(v, 5, 2, 1)$ ( v , 5 , 2 , 1 ) OOCs are presented for all $v\le 155$ v ≤ 155 , for which an optimal OOC exists.     

Theoretical Investigations on Electrocaloric Properties of \mathrm{PbZr}_{0.95}\mathrm{Ti}_{0.05}\mathrm{O}_{3} Thin Film

Based on a phenomenological model, the electrocaloric effect (ECE) accompanied with the ferroelectric-to-paraelectric phase transition in a PbZr $_{0.95}$ 0.95 Ti $_{0.05}$ 0.05 O $_{3}$ 3 thin film was investigated. The extracted data reveal many features of the ECE such as electrocaloric entropy changes, heat capacity changes, and temperature changes as functions of temperature due to different electric fields shifts. From the behavior of the PbZr $_{0.95}$ 0.95 Ti $_{0.05}$ 0.05 O $_{3}$ 3 thin film in phase transitions, it leads to a large change of heat capacity of 105.94 J  ${\cdot }\,$ · kg ${^{-1}}\,{\cdot }\,{^{\circ }}$ ? 1 · ° C, a temperature change of 22.44 K, and a relative cooling power of 1469 J  ${\cdot }$ ·  kg $^{-1}$ ? 1 .     

Path-Integral Calculation of Cross Second Virial Coefficients for Hydrogen Isotopologues

The path-integral Monte Carlo technique and a recent high-accuracy six-dimensional potential are used to compute the cross second virial coefficients for all unlike pairs among the hydrogen isotopologues $\mathrm{H}_{2}, \mathrm{D}_{2}, \mathrm{T}_{2}$ H 2 , D 2 , T 2 , HD, HT, and DT. Values are calculated from 15 K to 2000 K for these quantities where experimental information is almost completely absent. It is found that the commonly assumed arithmetic mean of the pure-component values does not provide a good approximation for the cross coefficients below approximately 50 K, especially for pairs containing $\mathrm{H}_{2}$ H 2 .     

Dislocation structures in a { \bar{1} 104}/��11 \bar{2} 0�� low-angle tilt grain boundary of alumina (��-Al2O3)

An alumina (??-Al₂O₃) bicrystal with a ( $ \bar{1} $ 104)/[11 $ \bar{2} $ 0] 2^o low-angle tilt grain boundary was fabricated by diffusion bonding at 1500 °C in air, and the grain boundary was observed by transmission electron microscopy (TEM). High-resolution TEM observations revealed that the grain boundary consists of at least two kinds of dislocations. One is a perfect dislocation which has a Burgers vector of 1/3[ $ \bar{1} $ 2 $ \bar{1} $ 0]. The other is dissociated into two partial dislocations with a stacking fault on the (0001) plane, and each partial dislocation has a 1/6[ $ \bar{1} $ 101] edge component. It is suggested from structural considerations that the dissociated-dislocation pair originates from a b = 1/3[02 $ \bar{2} $ 1] perfect dislocation (i.e., 1/3[02 $ \bar{2} $ 1] ?? 1/6[02 $ \bar{2} $ 1] + 1/6[02 $ \bar{2} $ 1]). This dissociation produces a stacking fault in the anion sublattice. The stacking fault energy is estimated to be roughly 1.3 Jm^?2 based on the elastic theory. The authors discuss the dislocation structures and the stacking fault formed on the (0001) plane in detail.     

Improved microwave dielectric properties of La(Mg0.5Sn0.5)O3 ceramic with Ba2+ substitution

The microwave dielectric properties of La(Mg_0.5?xBa_xSn_0.5)O₃ ceramics were examined with a view to their exploitation for wireless communications. The La(Mg_0.5?xBa_xSn_0.5)O₃ ceramics were prepared by the conventional solid-state method with various sintering temperatures. The La(Mg_0.5?xBa_xSn_0.5)O₃ ceramics contained La₂Sn₂O₇. An apparent density of 6.54 g/cm³, a dielectric constant ( $ \varepsilon_{r} $ ε r ) of 20.1, a quality factor (Q  $ \times $ ×  f) of 51,600 GHz, and a temperature coefficient of resonant frequency ( $ \tau_{f} $ τ f ) of ?82 ppm/°C were obtained for La(Mg_0.43Ba_0.07Sn_0.5)O₃ ceramics that were sintered at 1,550 °C for 4 h.     

Generalized Wien’s Displacement Law in Determining the True Temperature of \text{ ZrB }_{2}–SiC-Based Ultrahigh-Temperature Ceramic: Thermodynamics of Thermal Radiation

The temperature dependence of the generalized Wien displacement law is investigated. For determining the true temperature of a $\text{ ZrB }_{2}$ ZrB 2 –SiC-based ultrahigh-temperature ceramic, the experimental values of the position of the maximum of the spectral density power are needed. Thermodynamics of the thermal radiation of $\text{ ZrB }_{2}$ ZrB 2 –SiC is constructed by using the temperature dependence of the generalized Stefan–Boltzmann law. The calculated values of the normal total emissivity for $\text{ ZrB }_{2}$ ZrB 2 –SiC at different temperatures are in good agreement with experimental data. The total radiation power emitted from a surface of $\text{ ZrB }_{2}$ ZrB 2 –SiC specimens at different temperatures is calculated. The temperature dependences of the Helmholtz free energy, entropy, heat capacity at constant volume, pressure, enthalpy, and internal energy of the thermal radiation of $\text{ ZrB }_{2}$ ZrB 2 –SiC are obtained. For determining the true temperature, experimental values of either the normal total emissivity or the normal total energy density are needed. The uncertainty in the determination of the true temperature is no greater than 1 %. A new universality class of bodies with a new relationship between the temperature $T$ T and the position of the spectral energy density maximum is established.     

Density dependent macro-micro behavior of granular materials in general triaxial loading for varying intermediate principal stress using DEM

This study presents the density dependent behavior of granular materials for varying intermediate principal stress $(\sigma _{{{2}}})$ ( σ 2 ) in general triaxial loading using the discrete element method (DEM). The variation of intermediate principal stress is represented by a non-dimensional parameter $b[={(\sigma _{{{2}}}-\sigma _{{{3}}})}/{(\sigma _{{{1}}} -\sigma _{{{3}}})}]$ b [ = ( σ 2 ? σ 3 ) / ( σ 1 ? σ 3 ) ] , where $\sigma _{{{1}}}$ σ 1 and $\sigma _{{{3}}}$ σ 3 are the major and minor principal stresses, respectively. Isotropically compressed dense and loose samples were prepared numerically using the periodic boundaries. The numerical dense and loose samples were subjected to shear deformation under strain controlled condition for different $b$ b values ranging from 0 to 1. The simulated macro results depict that the friction angle increases with $b$ b until it reaches a peak value and beyond the peak, the friction angle decreases with $b$ b regardless of the density of sample. A unique relationship between dilatancy index and equivalent deviatoric strain exists at small strain level for different $b$ b values when dense sample is considered. By contrast, the same relationship for loose sample does not show uniqueness. The relationships among the major, intermediate and minor principal strains depict non-linear behavior. The non-linearity is dominant for loose sample. The fluctuation in the evolution of strain increment vector direction is dominant in loose sample than dense sample. The evolution of different micro results is presented as well. It is noted that a unique relationship exists between the stress ratio and the fabric measure regardless of $b$ b and the density of sample when strong contacts are considered.     

Spin-Polarized Calculations of Magnetic and Thermodynamic Properties of the Full-Heusler \mathrm{{Co}}_{2}MnZ (Z = Al, Ga)

The purpose of this study is to further understanding of the structural, electronic, magnetic, and thermal properties of the full-Heusler compounds, ${\mathrm{{Co}}}_{2}$ Co 2 MnAl and ${\mathrm{{Co}}}_{2}$ Co 2 MnGa, using density functional theory. Electronic structure calculations will be performed using the full potential linear augmented plane wave. The electronic structures and magnetic properties of ${\mathrm{{Co}}}_{2}$ Co 2 MnZ (Z = Al, Ga) compounds with ${\mathrm{L}}2_{1 }$ L 2 1 structure are studied. It is shown that the calculated lattice constants and spin magnetic moments are in good agreement with experimental values using the general gradient approximation method. Thermal effects on some macroscopic properties of ${\mathrm{{Co}}}_{2}$ Co 2 MnZ (Z = Al, Ga) compounds are predicted using the quasi-harmonic Debye model, in which the lattice vibrations are taken into account. The variations of the lattice constant, volume expansion coefficient, heat capacities, and Debye temperature with pressure and temperature in the ranges of 0 GPa to 18 GPa and 0 K to700 K have been obtained.     

({p, \rho,T, x}) Properties of \text{ CO}_{2}/Propane Binary Mixtures at 280 K to 440 K and 3 MPa to 200 MPa

The (p, \(\rho \) , T, x) properties of binary mixtures of CO \(_{2}\) (volume fraction purity 0.99999) and propane (mole fraction purity 0.9999) ( \(x_{1}\) CO \(_{2}+x_{2}\) propane; \(x_{1} = 0.1744\) , 0.3863, 0.5837, and 0.7732) were measured in the compressed liquid phase using a metal-bellows variable volumometer. Measurements were conducted from 280 K to 440 K and 3 MPa to 200 MPa. The expanded uncertainties ( \(k = 2\) ) were estimated to be temperature, \(<\) 3 mK; pressure, 1.5 kPa ( \(p\le 7\)  MPa), 0.06 % (7 MPa \(< p\le 50\)  MPa), 0.1 % (50 MPa \(< p\le 150\)  MPa), 0.2 % ( \(p> 150\)  MPa); density, 0.10 %; and composition, \(4.4\times 10^{-4}\) . At \(p >100\)  MPa and 280 K or 440 K, the uncertainties in density measurements increase to 0.14 % and 0.22 %, respectively. The data were compared with available equations of state. The excess molar volumes, \(v_\mathrm{m}^\mathrm{E}\) , of the mixtures were calculated and plotted as a function of temperature and pressure.     

Influence of temperature and frequency on electrical properties of electron beam evaporated bromoaluminum phthalocyanine, BrAlPc, thin films

The AC electrical properties of electron beam evaporated Bromoaluminum phthalocyanine (BrAlPc) thin films have been studied in the frequency range 10²–10⁵ Hz and in the temperature range of 303–413 K. The BrAlPc thin films are characterized by field emission scanning electron microscopy (FESEM). The capacitance is found to be sensitive to the frequency and increases with increasing temperature and decreases with increasing frequency. A loss minimum has been observed in the frequency dependence of the dissipation factor. Such behavior is found to be in good qualitative agreement with the model of Goswami and Goswami. The AC conductivity $\sigma \left( \omega \right)$ σ ( ω ) is found to vary as $\omega^{s}$ ω s in the studied frequency range. At frequencies 10–10² Hz, s is less than unity and decreases with increase in temperature indicating a dominant hopping process. At frequency ranges 10²–10⁴ Hz, exponent s lies very close to the unity and is independent of temperature, which shows the quantum mechanical tunneling is dominated conduction mechanism. At higher frequencies 10⁴–10⁵ Hz, s is found to be temperature independent. The temperature dependence of AC conductivity shows a linear increase with the increase in temperature. Moreover, the activation energies of device are determined as a function of frequency.     

Phase stability of \upbeta-MoSi2−x prepared by the Na flux method against thermal, oxidative, and mechanical treatments

The single $\upbeta$ -MoSi₂ phase was prepared by the Na flux method and its stability against thermal, oxidative, and mechanical treatments was investigated. The X-ray diffraction results show that the single $\upbeta$ phase is formed at 600 °C within 1 h using pre-mixed Mo and Si powders with a Si/Mo molar ratio of 2.00–2.25. By energy-dispersive X-ray spectroscopy, the produced powder is found to be Si-deficient with a Si/Mo molar ratio of 1.87–1.96. The differential thermal analysis shows that the $\upbeta$ phase transforms into the $\upalpha$ -MoSi₂ phase at 815 °C at 10 K/min with the segregation of a small amount of Mo₅Si₃. The transformation heat is ?5.5 kJ/mol and the activation energy calculated by the Kissinger method is 290 kJ/mol. Thermogravimetry reveals that the $\upbeta$ -MoSi_2?x powder oxidizes significantly at 400–600 °C via the pest oxidation mechanism while it is resistant to oxidation at 700 °C for 5 h similarly to the $\upalpha$ -MoSi₂ phase. At last, mechanical milling on the $\upbeta$ -MoSi_2?x powder with a planetary ball mill up to 216 h demonstrates that this powder is stable under a severe mechanical treatment.     

Permutation decoding of codes from generalized Paley graphs

The generalized Paley graphs $\text{ GP }(q,k)$ GP ( q , k ) are a generalization of the well-known Paley graphs. Codes derived from the row span of adjacency and incidence matrices from Paley graphs have been studied in Ghinellie and Key (Adv Math Commun 5(1):93–108, 2011) and Key and Limbupasiriporn (Congr Numer 170:143–155, 2004). We examine the binary codes associated with the incidence designs of the generalized Paley graphs obtaining the code parameters $[\frac{qs}{2}, q-1, s]$ [ q s 2 , q - 1 , s ] or $[qs, q-1,2s]$ [ q s , q - 1 , 2 s ] where $s=\frac{q-1}{k}$ s = q - 1 k . By finding explicit PD-sets we show that these codes can be used for permutation decoding.     

Strain tensor determination of compressed individual silica sand particles using high-energy synchrotron diffraction

The three-dimensional X-ray diffraction (3DXRD) nondestructive technique was used to measure lattice strains within individual sand particles subjected to compressive loading. Three experiments were conducted on similar single columns of silica sand particles with particle sizes between 0.595 and 0.841 mm. In each experiment, three sand particles were placed inside an acrylic mold with an inner diameter of 1 mm. Multiple in situ 3DXRD scans were acquired for each sand column as compressive load was increased. The volume-averaged lattice strain tensor was calculated for each sand particle. In addition, particle orientation and volumetric strain were calculated for individual sand particles. The axial normal strain $\upvarepsilon _\mathrm{zz}$ ε zz exhibited a linear response in the range of 0 to $10^{-3}$ 10 ? 3 when the applied compressive axial load (F) increased from 0 to $\sim $ ～ 30 N when one particle in the sand column fractured. Stress tensor of individual particles was calculated from the acquired lattice strain measurements and elastic constants of silica sand that were reported in the literature. To the best of our knowledge, there have been no reported experimental measurements of the lattice strain tensor measurements within individual silica sand particles. The quantitative measurements reported in this paper at the particle level are very valuable for developing, validating or calibrating micromechanics-based finite element and discrete element models to predict the constitutive behavior of granular materials. 3DXRD represents an exciting new non-destructive technique to directly measure constitutive behavior at the scale of individual particles.