The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

optical properties of pure poly(vinyl Alcohol) films and poly(vinyl Alcohol) doped with methyl red were study, different percentage prepared with constant thickness using casting technique. Absorption, Transmission spectra have been recorded in order to study the optical parameters such as absorption coefficient, energy gap, refractive index, Extinction coefficient and dispersion parameters were measured in the wavelength range (200-800)nm. This study reveals that the optical properties of PVA affect by increasing the impurity concentration.

a-Ge: As thin films have prepared by thermal evaporation teclmique, then they were annealing at various temperatures within the

range (373-473)  K.   The  result of  X-ray di ffraction spectrum  was showing  that  all  the  specimens  remained  in  amorphous structure before and after annealing  process. This paper studied the effect of annealing  temperature as  a  function of  wavelength on  the optical energy gap and optical constants for the a-Ge:As thin  films . Results have showed that there was an increasing in the optical energy gap

{Egopt) values with the in ,;rcasing of the annealing temperatures within

Lithium doped Nickel-Zinc ferrite material with chemical formula Ni0.9−2x Zn0.1LixFe2+xO4, where x is the ratio of lithium ions Li+ (x = 0, 0.01, 0.02, 0.03 and 0.04) prepared by using sol-gel auto combustion technique. X-ray diffraction results showed that the material have pure cubic spinal structure with space group Fd-3m. The experimental values of the lattice constant (aexp) were decreased from 8.39 to 8.35 nm with doped Li ions. It was found that the decreasing of the crystallite size with addition of lithium ions concentration. The radius of tetrahedral (rtet) and octahedral (roct) site were computed from cation distribution. SEM images have been taken to show the morphology of compound. The dielectric parameters [dissipation fa

   In this research we studied  the structural and optical properties of (CdTe) thin films which have been prepared  by thermal evaporation deposition method on the glass substrate at R.T with thickness (450  25) nm., as  a function of doping  ratio with copper element  in (1,3,5) % rate .The structure  measurement  by X-ray diffraction (XRD) analyses shows that the single phase of (CdTe) with polycrystalline structure with a preferred orientation [111].   The optical  measurement shows that the (CdTe) films have a direct energy gap, and they decrease with the increase of doping ratio reaching to 5% . The optical constants are investigated and calculated, such as absorpti

In this paper the effect of thermal annealing on the structural and optical properties of Antimony Selenide (Sb2Se3) is investigated. Sb2Se3 powder is evaporated on clean amorphous glass substrates at room temperature under high vacuum pressure (4.5×10-6 mbar) to form thin films. The structural investigation was done with the aid of X-ray diffraction (XRD) and atomic force microscopy (AFM). The amorphous to polycrystalline transformation of these thin films was shown by X-ray diffraction analysis after thermal annealing. These films' morphology is explained. (UV-Vis ) spectra in ranges from 300 to 1100 nm was used to examine the optical properties of the films .The absorption coefficient and optical energy gap of the investigated films are

The semiconductor ZnO is one of II – VI compound group, it is prepare as thin films by using chemical spray pyrolysis technique; the films are deposited onto glass substrate at 450 °C by using aqueous zinc chloride as a spray solution of molar concentration 0.1 M/L. Sample of the prepared film is irradiating by Gamma ray using CS 137, other sample is annealed at 550°C. The structure of the irradiated and annealed films are analyzed with X-ray diffraction, the results show that the films are polycrystalline in nature with preferred (002) orientation. The general morphology of ZnO films are imaged by using the Atomic Force Microscope (AFM), it constructed from nanostructure with dimensions in order of 77 nm.
The optical properties o

In this research study the effect of irradiation by (CW) CO2 laser on some optical properties of (Cds) doping by Ni thin films of (1)µm thickness has been prepared by heat evaporation method. (X-Ray) diffraction technique showed the prepared films before and after irradiation are ploy crystalline hexagonal structure, optical properties were include recording of absorbance spectra for prepared films in the range of (400-1000) nm wave lengths, the absorption coefficient and the energy gap were calculated before and after irradiation, finally the irradiation affected (CdS) thin films by changing its color from the Transparent yellow to dark rough yellow and decrease the value absorption coefficient also increase the value of energy gap.