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.

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

Tin oxide films (SnO2) of thickness (1 ?m) are prepared on glass substrate by post oxidation of metal films technique. Films were irradiated with Nd:YAG double frequency laser of wavelength (532 nm) pulses of three energies (100, 500, 1000) mJ. The optical absorption, transmission, reflectance, refractive index and optical conductivity of these films are investigated in the UV-Vis region (200-900) nm. It was found that the average transmittance of the films is around (80%) at wavelength (550 nm) and showed high transmission (? 90 %) in the visible and near infrared region. The absorption edge shifts towards higher energies, which is due to the Moss-Burstien effect and it lies at (4 eV). The optical band gap increased with increasing of ene

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.

The effect of heat treatment using different annealing temperatures on optical properties of bulk heterojunction blend (BHJ) Alq3: C60 thin films which are fabricated by the spin coating technique were investigated in this study. The films have been coated on a glass substrate with speed of 2000 rpm for one min and treated with different annealing temperature (373, 423 and 473) K under vacuum. The optical properties and the chemical bonds structure of blends as-deposited and heat treated have been studied by UV-Vis spectroscopic and Fourier Transform-Infra Red (FTIR) measurements respectively. The results of UV visible show that the optical energy gap decreasing with increasing the annealing temperature for the ratio (100:1) while decrea

Thin films of the blended solution of (NiPc/C60) on glass substrates were prepared by spin-coated method for three different ratios (100/1, 100/10 and 100/100). The effects of annealing temperature and C60 concentration on the optical properties of the samples were studied using the UV-Vis absorption spectroscopy and FTIR spectra. The optical absorption spectrum consists of two main bands, Q and B band, with maxima at about (602-632) nm and (700-730) nm for Q1 and Q2 respectively, and (340-375) nm for B band. The optical energy gap were determined from optical absorption spectra, The variation of optical energy gap with annealing temperature was nonsystematic and this may be due to the improvement in crystal structure for thin films. Whi

Cadmium sulfide (CdS) thin films with n-type semiconductor characteristics were prepared by flash evaporating method on glass substrates. Some films were annealed at 250 ^oC for 1hr in air. The thicknesses of the films was estimated to be 0.5µ by the spectrometer measurement. Structural, morphological, electrical, optical and photoconductivity properties of CdS films have been investigated by X-ray diffraction, AFM, the Hall effect, optical transmittance spectra and photoconductivity analysis, respectively. X-ray diffraction (XRD) pattern shows that CdS films are in the stable hexagonal crystalline structure. Using Debye Scherrerś formula, the average grain size for the samples was found to be 26 nm. The transmittance of the