Indirect measurement of $$\mu (I)$$ relation from finite granular avalanche down an inclined narrow reservoir of smooth bed

This work examines the algebraic \(\mu -I\) relation proposed for steady uniform dry granular flows via unsteady granular avalanche experiments of finite nearly identical dry glass spheres down an inclined narrow reservoir of smooth bed. Lateral high-speed digital imaging permits particle tracking velocimetry with which we can evaluate bulk local instantaneous volume fraction and velocity components to conduct a quasi-two-dimensional control volume analysis of streamwise momentum assuming an internal shear stress based on the \(\mu -I\) rheology, a hydrostatic normal stress and a Coulomb yielding condition at lateral walls. Hence, the desired \(\mu \) is a function of flow dynamics and a wall friction coefficient \(\mu _w\). Complementary sliding table experiments were conducted to estimate an upper bound of \(\mu _w=0.17\) which was used with a chosen nonzero lower bound \(\mu _w=0.05\) to extract possible range of \(\mu \) at a local instantaneous inertial number I. The so-obtained local instantaneous \(\mu -I\) data conform to the non-linear monotonically increasing trend proposed for steady inertial flows above a crossover value \(I_c\approx 0.03\). Below \(I_c\), a peculiar segment of decaying \(\mu \) with I was revealed agreeing to the rheology tests in quasi-static regime.     

Vortex Shedding from an Object Moving in Superfluid $$^4\hbox {He}$$ at mK Temperatures and in a Bose–Einstein Condensate

Vortex shedding from a microsphere oscillating in superfluid \(^4\hbox {He}\) at mK temperatures is compared with that from a laser beam moving in a Bose–Einstein condensate as observed by other authors. In particular, in either case a linear dependence of the shedding frequency \(f_v\) on \(\varDelta v = v - v_c\) is observed, where v is the velocity amplitude of the sphere or the constant velocity of the laser beam above a critical velocity \(v_c\) for the onset of turbulent flow: \(f_v = a \,\varDelta v\), where the coefficient a is proportional to the oscillation frequency \( \omega \) above some characteristic frequency \(\omega _k\) and assumes a finite value for steady motion \(\omega \rightarrow 0\).     

Structural,dielectric and piezoelectric study of Ca-, Zr-modified $$\hbox {BaTiO}_{3}$$ lead-free ceramics

We prepared a lead-free ceramic (\(\hbox {Ba}_{0.85}\hbox {Ca}_{0.15})(\hbox {Ti}_{1-x}\hbox {Zr}_{x})\hbox {O}_{3}\) (BCTZ) using the conventional mixed oxide technique. The samples were prepared by an ordinary mixing and sintering technique. In this study we investigated how small amounts of \(\hbox {Zr}^{4+}\) can affect the crystal structure and microstructure as well as dielectric and piezoelectric properties of \(\hbox {BaTiO}_{3}\). X-ray diffraction analysis results indicate that no secondary phase is formed in any of the BCTZ powders for \(0 \le x \le 0.1\), suggesting that \(\hbox {Zr}^{4+}\) diffuses into \(\hbox {BaTiO}_{3}\) lattices to form a solid solution. Scanning electron microscopy micrographs revealed that the average grain size gradually increased with \(\hbox {Zr}^{4+}\) content from 9.5 \(\upmu \!\hbox {m}\) for \(x = 0.02\) to 13.5 \(\upmu \!\hbox {m}\) for \(x = 0.1\); Curie temperature decreased due to the small tetragonality caused by \(\hbox {Zr}^{4+}\) addition. Owing to the polymorphic phase transition from orthorhombic to tetragonal phase around room temperature, it was found that the composition \(x = 0.09\) showed improved electrical properties and reached preferred values of \(d_{33} = 148\) pC \(\hbox {N}^{-1}\) and \(K_{\mathrm{p}} = 27\%\).     

Goodness-of-fit tests in semiparametric transformation models

Consider a semiparametric transformation model of the form \(\varLambda _{\theta }(Y)\) \(= m(X) + \varepsilon \), where \(Y\) is a univariate dependent variable, \(X\) is a \(d\)-dimensional covariate, and \(\varepsilon \) is independent of \(X\) and has mean zero. We assume that \(\{ \varLambda _{\theta } : \theta \in \varTheta \}\) is a parametric family of strictly increasing functions, while \(m\) is an unknown regression function. The goal of the paper is to develop tests for the null hypothesis that \(m(\cdot )\) belongs to a certain parametric family of regression functions. We propose a Kolmogorov–Smirnov and a Cramér–von Mises type test statistic, which measure the distance between the distribution of \(\varepsilon \) estimated under the null hypothesis and the distribution of \(\varepsilon \) without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of \(\theta \) and a local polynomial estimator of \(m(\cdot )\). The limiting distributions of these two test statistics are established under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedures, and we apply our tests to data on the scattering of sunlight in the atmosphere.     

Nonlinear Ultrasound Propagation in Solid $$^4\hbox {He}$$ Compared with Shear Modulus Experiments

Ultrasound attenuation (\(\alpha \)) and velocity (V) at 9.6 MHz are measured in polycrystalline hcp \(^4\hbox {He}\). The ultrasound signal above 200 mK is linear and understood in terms of resonant vibration of dislocation segments pinned between network nodes with an average pinning length of 3.7 \(\mu \hbox {m}\), much shorter than 59 \(\mu \hbox {m}\) estimated from a shear modulus measurement. Dramatic changes in \(\alpha \) and V are observed below 200 mK. The changes are strongly dependent on temperature and are nonlinear and hysteretic. These effects result from pinning of dislocations by \(^3\hbox {He}\) impurities (nominal concentration of 0.3 ppm). The dislocation damping constant due to thermal phonons, the binding energy between dislocation and \(^3\hbox {He}\), and the average network pinning length obtained from the ultrasound data are compared with those from the shear modulus experiments.     

New aspects for friction coefficients of finite granular avalanche down a flat narrow reservoir

This work examines the bulk internal friction coefficient, \(\mu \), and effective wall friction coefficient, \(\mu _w\), for finite number of nearly identical dry glass spheres in avalanche down a narrow inclined reservoir of smooth frictional bed using a validated discrete element scheme. Instantaneous deviatoric strain rate tensor \(\dot{\gamma }^d_{ij}\) and stress tensor \(\tau _{ij}\) are computed locally to evaluate a three-dimensional constitutive model developed based on the rheology of steady homogeneous surface flows. On one side, the algebraic \(\mu -I\) relation conforms to conventional relation for glass beads, \(\mu =0.34+0.31/(1+0.15/I)\) (Jop et al. in J. Fluid Mech. 541:167–192, 2005, Midi in Eur. Phys. J. E 14:341–365, 2004, Jop in Comptes Rendus Phys. 16:62–72, 2015), when the inertial number \(I>I_{c}=2\times 10^{-2}\). The assumption of collinear \(\tau _{ij}\) and \(\dot{\gamma }^d_{ij}\), however, does not hold and such misalignment agrees to the findings in non-uniform inhomogeneous flows (Cortet et al. in Europhys. Lett. 88(1):14001, 2009). Below \(I_c\), we observe a decaying \(\mu -I\) as found in slowly deforming rheology tests and a simplified model is developed in view of shear-induced dilatation upon yielding. Non-constant effective wall friction coefficient is measured to grow in time and with I towards the sphere-wall sliding friction coefficient in the contact model while preserving the depth-weakening feature as in confined steady surface flows (Richard et al. in Phys. Rev. Lett. 101:248002, 2008, Brodu et al. in Phys. Rev. E 87:022202, 2013). The fact that rotation at one sphere center can divert surface relative velocity across the contact area to render lower sliding friction is considered to develop a model describing how \(\mu _w\) drops with the ratio between rotation-induced velocity and sliding velocity, \(\varOmega \). The simulation data compares fairly well to the predicted monotonic decay of \(\mu _w\) with \(\varOmega \).     

Study of $$\upbeta $$-phase development in spin-coated PVDF thick films

A study was conducted to ascertain the effect of variation in spin speed and baking temperature on \(\upbeta \)-phase content in the spin-coated poly(vinylidene fluoride) (PVDF) thick films (\({\sim }4{-}25\,\upmu \hbox {m}\)). Development of \(\upbeta \)-phase is dependent on film stretching and crystallization temperature. Therefore, to study the development of \(\upbeta \)-phase in films, stretching is achieved by spinning and crystallization temperature is adjusted by means of baking. PVDF films are characterized using Fourier transform infrared spectroscopy, X-ray diffraction, differential scanning calorimetry, and scanning electron microscopy. It is observed that crystallization temperature lower than \(60^{\circ }\hbox {C}\) and increase in spin speed increases the \(\upbeta \)-phase content in PVDF films. Crystallization temperature above \(60^{\circ }\hbox {C}\) reduces \(\upbeta \)-phase content and increases \(\upalpha \)-phase content. It was also observed that viscosity of the PVDF solution affects the \(\upbeta \)-phase development in films at a particular spin speed.     

Measurement of the Nonlinearity of Heat-Flux Sensors Employing a $$\hbox {CO}_2$$ laser

Heat-flux sensors are widely used in industry to test building products and designs for resistance to bushfire, to test the flammability of textiles and in numerous applications such as concentrated solar collectors. In Australia, such detectors are currently calibrated by the National Measurement Institute Australia (NMIA) at low flux levels of 20 W \(\cdot \) m\(^{-2}\). Estimates of the uncertainty arising from nonlinearity at industrial levels (e.g. 50 kW \(\cdot \) m\(^{-2}\) for bushfire testing) rely on literature information. NMIA has developed a facility to characterize the linearity response of these heat-flux sensors up to 110 kW \(\cdot \) m\(^{-2}\) using a low-power \(\hbox {CO}_2\) laser and a chopped quartz tungsten–halogen lamp. The facility was validated by comparison with the conventional flux-addition method, and used to characterize several Schmidt–Boelter-type sensors. A significant nonlinear response was found, ranging from (\(3.2 \pm 0.9\))% at 40 kW \(\cdot \) m\(^{-2}\) to more than 8 % at 100 kW \(\cdot \) m\(^{-2}\). Additional measurements confirm that this is not attributable to convection effects, but due to the temperature dependence of the sensor’s responsivity.     

Thermo-transport properties of Zn-substituted layered Li-nickel oxide, $$\hbox {LiNiO}_{2}$$

The layered Li-TM-\(\hbox {O}_{2}\) materials have been investigated extensively due to their application as cathodes in Li batteries. The electrical properties of these oxides can be tuned or controlled either by non-stoichiometry or substitution. Hence the thermo-transport properties of Zn-substituted \(\hbox {LiNi}_{1-x}\hbox {Zn}_{x}\hbox {O}_{2}\) for \(0 \le x \le 0.16\) have been investigated in the temperature range of 300–900 K for potential application as a high-temperature thermoelectric material. For \(x < 0.08\), the compounds were of single phase belonging to the space group R-3mH while for \(x > 0.08\) an additional minority phase, ZnO forms together with the main layered phase. All the compounds exhibit a semiconducting behaviour with electrical resistivity, varying in the range of  \(\sim 10^{-4}\) to \(10^{-2}\,\,\Omega \hbox {m}\) between 300 and 900 K. The electrical resistivity is found to increase with increasing Zn-substitution predominantly due to a decrease in the charge carrier hole mobility. The activation energy remains constant, \(\sim \)10  meV, with Zn-substitution. The Seebeck coefficient of the compounds is found to decrease with increasing temperature and increase with increasing Zn-substitution. The Seebeck coefficient decreases from \(\sim \)95 to \(35\ \upmu \hbox {V K}^{-1}\) and the corresponding power factor is \(\sim \)12\(\ \upmu \hbox {W m}^{-1}\ {\hbox {K}}^{-2}\) for the \(x = 0.16\) compound.     

Effect of Pressure on Deep-Ocean Thermometers

A laboratory experiment was devised and performed to investigate the pressure dependence of Sea-Bird Electronics SBE35 and SBE3 deep-ocean thermometers. The thermometers were mounted in a massive brass comparator together with a calibrated standard platinum resistance thermometer. The measurements were performed in a pressure chamber in the pressure range 0.1 MPa to 60 MPa. The results showed that both the investigated SBE35 and SBE3 thermometers are pressure dependent, with a pressure sensitivity of +41 \(\upmu \)K\(\cdot \)MPa\(^{-1}\) and \(-77\) \(\upmu \)K\(\cdot \)MPa\(^{-1}\), respectively. Nevertheless, the results obtained in only one individual device per model (one SBE35 and one SBE3) cannot be generalized and further investigations of a larger number of devices per model are needed.     

Study of Lanthanide Complexes with BTFA in Silica Gels by Photoacoustic Spectroscopy

In this work, lanthanide \(\beta \)-diketonate complexes Ln(btfa)\({}_{3} \cdot 2\hbox {H}_{2}\)O (Ln\(^{3+}\): Eu\(^{3+}\), Sm\(^{3+ }\), and Tb\(^{3+}\); btfa: 4,4,4-trifluoro-l-phenyl-1,3-butanedione) were incorporated into silica gels by a sol–gel method. Photoacoustic (PA) spectra of these complex-doped silica samples were measured and studied. The PA intensity of the \(\beta \)-diketonate ligand is nearly the same for lanthanide complexes in wet gels. After heat treatment at 150 \(^{\circ }\)C, however, the PA intensity of the ligand increases for Eu\(^{3+}\), Sm\(^{3+}\), and Tb\(^{3+}\) complexes in silica gels, respectively. Different PA intensities of the samples are interpreted by comparison with their luminescence spectra. The luminescence result is consistent with the PA spectra. The result indicates that lanthanide \(\beta \)-diketonate complexes cannot be formed in silica gels without a suitable heat treatment. Moreover, the relaxation process model is proposed based on the PA and luminescence results.     

Mesoscale evolution of non-graphitizing pyrolytic carbon in aligned carbon nanotube carbon matrix nanocomposites

Polymer-derived pyrolytic carbons (PyCs) are highly desirable building blocks for high-strength low-density ceramic meta-materials, and reinforcement with nanofibers is of interest to address brittleness and tailor multi-functional properties. The properties of carbon nanotubes (CNTs) make them leading candidates for nanocomposite reinforcement, but how CNT confinement influences the structural evolution of the PyC matrix is unknown. Here, the influence of aligned CNT proximity interactions on nano- and mesoscale structural evolution of phenol-formaldehyde-derived PyCs is established as a function of pyrolysis temperature (\(T_{\mathrm {p}}\)) using X-ray diffraction, Raman spectroscopy, and Fourier transform infrared spectroscopy. Aligned CNT PyC matrix nanocomposites are found to evolve faster at the mesoscale by plateauing in crystallite size at \(T_{\mathrm {p}}\) \(\sim\)800 \(^{\circ }\hbox {C}\), which is more than \(200\,\,^{\circ }\hbox {C}\) below that of unconfined PyCs. Since the aligned CNTs used here exhibit \(\sim\)80 nm average separations and \(\sim\)8 nm diameters, confinement effects are surprisingly not found to influence PyC structure on the atomic-scale at \(T_{\mathrm {p}}\) \(\le \)1400 \(^{\circ }\hbox {C}\). Since CNT confinement could lead to anisotropic crystallite growth in PyCs synthesized below \(\sim\)1000 \(^{\circ }\hbox {C}\), and recent modeling indicates that more slender crystallites increase PyC hardness, these results inform fabrication of PyC-based meta-materials with unrivaled specific mechanical properties.     

A comparative study on dielectric behaviours of Au/(Zn-doped PVA)/<Emphasis Type="Italic">n</Emphasis>-4H-SiC (MPS) structures with different interlayer thicknesses using impedance spectroscopy methods

Three different thicknesses (50, 150 and 500 nm) Zn-doped polyvinyl alcohol (PVA) was deposited on n-4H-SiC wafer as interlayer by electrospinning method and so, Au/(Zn-doped PVA)/n-4H-SiC metal–polymer–semiconductor structures were fabricated. The thickness effect of Zn-doped PVA on the dielectric constant (\(\varepsilon ^{\prime }\)), dielectric loss (\(\varepsilon ^{{\prime }{\prime }}\)), loss-tangent (tan \(\delta \)), real and imaginary parts of electric modulus (\(M^{\prime }\) and \(M^{{\prime }{\prime }})\) and ac electrical conductivity \((\sigma _{\mathrm{ac}})\) of them were analysed and compared using experimental capacitance (C) and conductance (\(G/\omega \)) data in the frequency range of 1–500 kHz at room temperature. According to these results, the values of \(\varepsilon ^{\prime }\) and \(\varepsilon ^{{\prime }{\prime }}\) decrease with increasing frequency almost exponentially, \(\sigma _{\mathrm{ac}}\) increases especially, at high frequencies. The \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) values were obtained from the \(\varepsilon ^{\prime }\) and \(\varepsilon ^{{\prime }{\prime }}\) data and the \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) vs. f plots were drawn for these structures. While the values of \(\varepsilon ^{\prime }\), \(\varepsilon ^{{\prime }{\prime }}\) and tan \(\delta \) increase with increasing interlayer thickness, the values of \(M^{\prime }\) and \(M^{{\prime }{\prime }}\) decrease with increasing interlayer thickness. The double logarithmic \(\sigma _{\mathrm{ac}}\) vs. f plots for each structure have two distinct linear regimes with different slopes, which correspond to low and high frequencies, respectively, and it is prominent that there exist two different conduction mechanisms. Obtained results were found as a strong function of frequency and interlayer thickness.     

Partial permutation decoding for MacDonald codes

We show how to find s-PD-sets of the minimal size \(s+1\) for the \(\left[ \frac{q^n-q^u}{q-1},n,q^{n-1}-q^{u-1}\right] _q \) MacDonald q-ary codes \(C_{n,u}(q)\) where \(n \ge 3\) and \(1 \le u \le n-1\). The construction of [6] can be used and gives s-PD-sets for s up to the bound \(\lfloor \frac{q^{n-u}-1}{(n-u)(q-1)} \rfloor -1\), of effective use for u small; for \(u \ge \lfloor \frac{n}{2} \rfloor \) an alternative construction is given that applies up to a bound that depends on the maximum size of a set of vectors in \(V_u(\mathbb {F}_q)\) with each pair of vectors distance at least 3 apart.     

NRC Microwave Refractive Index Gas Thermometry Implementation Between 24.5 K and 84 K

The implementation of microwave refractive index gas thermometry at the National Research Council between 24.5 K and 84 K is reported. A new gas-handling system for accurate control and measurement of experimental gas pressure has been constructed, and primary thermometry measurements have been taken using a quasi-spherical copper resonator and helium gas at temperatures corresponding to three defining fixed points of the International Temperature Scale of 1990 (ITS-90). These measurements indicate differences between the thermodynamic temperature T and ITS-90 temperature \(T_{90}\) of \(\left( T - T_{90} \right) = -0.60 \pm 0.56\) mK at \(T_{90} = 24.5561\) K, \(\left( T - T_{90} \right) = -2.0 \pm 1.3\) mK at \(T_{90} = 54.3584\) K, and \(\left( T - T_{90} \right) = -4.0 \pm 2.9\) mK at \(T_{90} = 83.8058\) K. The present results at \(T_{90} = 24.5561\) K and \(T_{90} = 83.8058\) K agree with previously reported measurements from other primary thermometry techniques of acoustic gas thermometry and dielectric constant gas thermometry, and the result at \(T_{90} = 54.3584\) K provides new information in a temperature region where there is a gap in other recent data sets.     

Microwave-hydrothermal synthesis of mesoporous $$\upgamma $$-$$\hbox {Al}_{2}\hbox {O}_{3}$$Al2O3 and its impregnation with AgNPs for excellent catalytic oxidation of CO

Mesoporous \(\upgamma \)-alumina was synthesized by the microwave-hydrothermal process with a shorter duration time at 150\({^{\circ }}\)C/2 h followed by calcination at 550\({^{\circ }}\)C/1 h. Ag nanoparticles (AgNPs) were impregnated into \(\upgamma \)-alumina under a reducing atmosphere at 450\({^{\circ }}\)C. The synthesized product was characterized by X-ray diffraction (XRD), thermogravimetric (TG)/differential thermal analysis (DTA), X-ray photoelectron spectroscopy (XPS), \(\hbox {N}_{2}\) adsorption–desorption study, field-emission scanning electron microscopy (FESEM) and transmission electron microscopy (TEM). The BET surface area values of \(\upgamma \)-alumina and Ag-impregnated \(\upgamma \)-alumina were found to be 258 and 230 m\(^{2}\) g\(^{-1}\), respectively. FESEM images showed the formation of grain-like particles of 50–70 nm in size with a flake-like microstructure. The XRD, XPS and TEM studies confirmed the presence of Ag in the synthesized product. Catalytic properties of the product for CO oxidation was studied with the \(T_{50}\) (50% conversion) and \(T_{100}\) (100% conversion) values of 118 and 135\({^{\circ }}\)C, respectively; the enhanced values were compared with the literature reported values.     

Crystal structure,thermal behaviour,vibrational spectroscopy and optical properties of new compounds K$$_{}$$Ca(HAsO$$_{4}$$4)$$_{}\cdot $$·H$$_{}$$O with kröhnkite-type chain

The new kröhnkite compound called potassium calcium-bis-hydrogen arsenate dihydrate K\(_{2}\)Ca(HAsO\(_{4})_{2}\cdot \)2H\(_{2}\)O was obtained by hydrothermal method and characterized by X-ray diffraction, infrared spectroscopy, Raman scattering, thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC) analysis and optical (photoluminescence and absorption) properties. It crystallizes in the triclinic space group P\(\bar{1}\) and unit cell parameters \(a = 5.971(3)\) Å, \(b =6.634(3)\) Å, \(c = 7.856(4)\) Å, \(\alpha =104.532(9)\) \(^{\circ }\), \(\beta = 105.464(9)\) \(^{\circ }\) and \(\gamma = 109.698(9)\) \(^{\circ }\). The structure of K\(_{2}\)Ca(HAsO\(_{4})_{2}\cdot \)2H\(_{2}\)O built up from this infinite, (Ca(HAsO\(_{4})_{2}\)(H\(_{2}\)O)\(_{2})^{2+}\), was oriented along an axis resulting from the association of CaO\(_{6}\) octahedra alternating with each two HAsO\(_{4}\) tetrahedra by sharing corners. Each potassium atom links two adjacent chains by three oxygen atoms of HAsO\(_{4}\) tetrahedra. TGA and DSC have shown the absence of phase transition. The existence of vibrational modes corresponding to the kröhnkite is identified by the IR and Raman spectroscopies in the frequency ranges of 400–4000 and 20–4000 cm\(^{-1}\), respectively. The photoluminescence measurement show one peak at 507 nm, which is attributed to band–band (free electron–hole transitions) and (bound electron–hole transitions) emissions within the AsO\(_{4}\) inorganic part.     

Bootstrap in semi-functional partial linear regression under dependence

This paper deals with the semi-functional partial linear regression model \(Y={{\varvec{X}}}^\mathrm{T}{\varvec{\beta }}+m({\varvec{\chi }})+\varepsilon \) under \(\alpha \)-mixing conditions. \({\varvec{\beta }} \in \mathbb {R}^{p}\) and \(m(\cdot )\) denote an unknown vector and an unknown smooth real-valued operator, respectively. The covariates \({{\varvec{X}}}\) and \({\varvec{\chi }}\) are valued in \(\mathbb {R}^{p}\) and some infinite-dimensional space, respectively, and the random error \(\varepsilon \) verifies \(\mathbb {E}(\varepsilon |{{\varvec{X}}},{\varvec{\chi }})=0\). Naïve and wild bootstrap procedures are proposed to approximate the distribution of kernel-based estimators of \({\varvec{\beta }}\) and \(m(\chi )\), and their asymptotic validities are obtained. A simulation study shows the behavior (on finite sample sizes) of the proposed bootstrap methodology when applied to construct confidence intervals, while an application to real data concerning electricity market illustrates its usefulness in practice.     

Study of optical properties of potassium permanganate ($$\hbox {KMnO}_{4}$$) doped poly(methylmethacrylate) (PMMA) composite films

We have developed films of pure polymethylmethacrylate (PMMA) (0.5, 1, 2 and 5%) and potassium permanganate \((\hbox {KMnO}_{4})\)-doped PMMA composite films of thickness (\(\sim 100\, \upmu \hbox {m}\)) using the solution-cast technique. To identify the possible change that happen to the PMMA films due to doping, the optical properties were investigated for different concentrations of \(\hbox {KMnO}_{4}\) by recording the absorbance (A) and transmittance (\(T\%\)) spectra of these films using UV–Vis spectrophotometer in the wavelength range of 300–1100 nm. From the data obtained from the optical parameters viz. absorption coefficient (\(\alpha \)), extinction coefficient (\(\kappa \)), finesse coefficient (F), refractive index (\(\eta \)), real and imaginary parts of dielectric constant (\(\varepsilon _{\mathrm{r}}\) and \(\varepsilon _{\mathrm{i}})\) and optical conductivity (\(\sigma \)) were calculated for the prepared films. The indirect optical band gap for the pure and the doped-PMMA films were also estimated.