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Investigated of the optical properties for SiO2 by using Lorentz model

  • Widad Hamza Tarkhan EMAIL logo and Sundus Y. Hasan
Published/Copyright: March 11, 2024
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Open Engineering
From the journal Open Engineering Volume 14 Issue 1

Abstract

In this work, the optical characteristics of SiO2 (refractive index [ n ], dielectric constant [ ε r ], reflectivity [R], transmissivity [T], absorptivity [A], and absorption coefficient [α]) were studied using the Lorentz model. A comparison was made with the practical results of previous studies and showed a good agreement.

keywords: SiO2 ; Lorenz model; dielectric constant; refractive index; extinction coefficient

1 Introduction

Lorentz model may be used to describe the dispersion of many substances and usually indicates sturdy dispersion across the resonant frequency. It is more often apposite for substances that have bound electrons, with the possibility of having many oscillators in a given structure [1]. Lorentz tried to describe the interaction between atoms and electric fields in classical expressions, which have been proposed that the electron (a particle with a few small mass) is bound to the nucleus of the atom (with a far larger mass) through a force that behaves according to Hooke’s law, i.e., a spring-like force. An implemented electric field might then interact with the charge of the electron, causing “stretching” or “compression” of the spring, which could set the electron into oscillating motion. That is the so-called Lorentz oscillator model [2].

Silicon dioxide, SiO2, additionally called silica, is a chemical compound that is an oxide of silicon. Silicon dioxide consists of one silicon atom bonded to two oxygen atoms. The silicon–oxygen bonds create a tetrahedral shape, with each silicon atom on the middle of a tetrahedron fashioned by four oxygen atoms. This association is the basis for the different crystalline and amorphous types of SiO2 [3].

SiO2 has occupied a distinguished role each in scientific researches and industrial applications due to its ease of preparation and huge range of makes use of together with anti-resistance, corrosion resistance, hardness, dielectric, and optical transparency [4]. The synthetic silica (colloidal silica, silica gels, pyrogenic silica, and precipitated silica) is produced mainly in amorphous powder forms as compared to natural mineral silica (quartz, tridymite, cristobalite), which might be in crystalline forms [5]. SiO2 is a low refractive index material, which is widely used in optical coatings that perform from the ultraviolet to close to near-infrared radiation areas. Due to its high stability, high density, refractive index adjustability, low particulate infection, and amorphous structure, SiO2 has been the point of interest of many researchers in recent years. SiO2 films have typical applications such as high-reflection coatings, antireflection coatings, beam-splitters, band-skip filters, and polarizers [6].

2 Theory

Since the electron is substantially less massive than the nucleus of an atom, the problem can be solved as if the electron-spring system is associated with an infinite mass, which does not move, permitting to use the mass of the electron, m e = 9.11 × 10−31 kg. The assumption that the binding force behaves like a spring is a justified approximation for any sort of binding, given that the displacement is small enough (meaning that only the constant and linear terms in the Taylor expansion are relevant) [2].

In Figure 1, E⃑ net represents the vector sum of the incident electric field from an external source and the electric fields due to surrounding dipoles near the electron [7], i.e.,

(1) E → net = E → incedent ω ave + E → surrounding dipoles .

Figure 1 Lorentz forced and damped oscillation model for dielectric media [7].
Figure 1

Lorentz forced and damped oscillation model for dielectric media [7].

If an electric field is incident on an area of matter, it generates electric dipoles out of neighboring atoms. These dipoles produce additional electric fields that affect the area in return. This phenomenon is also implied in the polarization equation [8]:

(2) P = ε ° χ e E net ,

where χ e is the electric susceptibility of the material and ε ° is the permittivity of free space.

From the Lorentz oscillation model, it is given as:

(3) → F net = m e d 2 r d t 2 = → F E + → F damping + F spring ,

where m e d 2 r d t 2 , → F E , → F damping , and F spring are the acceleration, electric, damping, and spring force, respectively,

which leads to:

(4) − e E net + m e γ i ω r − k spring r = − ω 2 m e r ,

where γ is the damping frequency, and γ = 1 τ , τ is the relaxation time.

Solving equation (4) for r → ,

(5) → r = e m e E net ω 2 − k spring m e + i ω γ .

Applying the definition of (electric) resonance frequency of the system [9]:

(6) ω ° = k spring m e .

Then, the polarization per unit volume is written as [10]:

(7) → P = − Ne r → ,

where N is the number of dipoles per unit volume.

Inserting equations (5) and (6) into equation (7) gives

(8) P → = N e 2 m e E net ( ω ° 2 − ω 2 ) − i ω γ .

By equating equations (2) and (8) and defining the plasma frequency [1,11],

ω p = N e 2 m e ε ° ,

(9) χ e = ω p 2 ( ω ° 2 − ω 2 ) − i ω γ .

By the relation ε r = 1 + χ e , the refractive index of the dielectric medium becomes [12]

(10) n 2 = ε r ≈ 1 + χ e ,

where n is the complex refractive index:

(11) ε r = 1 + ω p 2 ( ω ° 2 − ω 2 ) − i ω γ ,

where ε r is the relative permittivity.

3 Deriving the equation of dielectric constant ( ε r ) and refractive index ( n )

When the interaction between light and the charges of the medium occurs due to the process of energy absorption in the material, this interaction results in a polarization of the charges of that medium, and this polarization is usually described by the complex dielectric constant ( ε r ), which is defined by the following relation [13]:

(12) ε r = ( ε R + i ε I ) ,

where ε R and ε I are the real and imaginary part of relative permittivity, respectively, and the real and imaginary parts, from equation (11), are

(13) ∴ ε R = 1 + ω p 2 ( ω 0 2 − ω 2 ) ( ω 0 2 − ω 2 ) 2 + ω 2 γ 2 , ∴ ε I = ω p 2 ω γ ( ω 0 2 − ω 2 ) 2 + ω 2 γ 2 ,

The complex refractive index ( n ′ ) is defined by the following relation [12]:

(14) n ′ = ( n + ik ) ,

where n and k are the real and imaginary parts of the complex refractive index, which represent the normal refractive index and extinction coefficient, respectively.

(15) n ′ 2 = ( n + ik ) 2 = n 2 + 2 ink − k 2 .

From equations (10) and (15), the real and imaginary parts are

(16) n 2 − k 2 = 1 + ω p 2 ( ω 0 2 − ω 2 ) ( ω 0 2 − ω 2 ) 2 + ω 2 γ 2 = Re ,

(17) 2 nk = ω p 2 ω γ ( ω 0 2 − ω 2 ) 2 + ω 2 γ 2 = Im ,

(18) 2 nk = Im → n = Im 2 k .

By inserting equation (18) in equation (16),

(19) Re = n 2 − k 2 = Im 2 4 k 2 − k 2 ,

(20) 4 k 4 + 4 Re k 2 − Im 2 = 0 .

By solving equation (20), using the constitution method:

(21) k 2 = − 4 Re ± 16 Re 2 + 16 Im 2 8 ,

(22) k 2 = − Re ± Re 2 + Im 2 2 .

The real and imaginary parts of relative permittivity can be calculated from:

(23) ε R = n 2 − k 2 ,

(24) ε I = 2 nk .

4 Optical properties (reflectivity [R], transmissivity [T], absorptivity [A], and absorption coefficient [α])

Optical properties of a material is meant as a material’s response to exposure of electromagnetic radiation and, in particular, to visible light. All electromagnetic radiation traverses a vacuum at the same velocity that of light (c = 3 × 108 m/s). This velocity is related to the electric permittivity of a vacuum ε ° and the magnetic permeability of a vacuum μ ° through c = 1 μ ° ε ° [14].

When light passes from one medium into another, several things may happen. Some of the incident light radiation may be transmitted through the medium, some will be absorbed, and some will be reflected at the interface between the two media, as explained in the following sections [14].

4.1 Reflectivity (R)

Reflectivity (R) is the ratio of the reflected radiation intensity, I R , from the material to the incident radiation intensity I ° :

(25) R = I R I ° .

If the light is normal (or perpendicular) to the interface, then [15]:

(26) R = 1 − n ′ 1 + n ′ 2 .

4.2 Absorption coefficient (α)

Absorption coefficient (α) is defined as the ratio of the decrease in the incident radiation energy with respect to the unit distance toward the wave within the medium, and the absorption depends on the incident photon energy and the properties of the semiconductor [14].

The absorption coefficient depends on the extinction coefficient as [13]:

(27) α = 4 π k λ ( cm ) .

4.3 Transmissivity (T)

Transmissivity (T) is the ratio of the transmitted radiation intensity , I T , to the the incident radiation intensity , I ° [15]:

(28) T = I T I ° .

We can be found transmissivity from the following equation [14]:

(29) T = e − α t ,

where α is the absorption coefficient and t is the thickness of the film assumed to be 1.

4.4 Absorptivity (A)

Absorptivity (A) is the ratio of the absorbed radiation intensity , I A , to the incident radiation intensity , I ° , [16]:

(30) A = I A I ° .

The relationship between transmittance (T) and absorbance (A) is given by the Beer–Lambert law (Beer’s law). According to Beer’s law, the absorbance is formulated as [14]:

(31) A = 2 − log ( T ) .

5 Results and discussion

In this study, the optical characteristics of the insulator silicon dioxide, SiO2, were studied using the Lorentz model and using an MATLAB code for programming the equations. Figure 2 shows the values of real and imaginary parts of the complex refractive index found from equations (18) and (22). The figure shows that the resonant frequency, ω 0 = 1.5160 × 1016. At wavelength λ ≫ λ ° ( ω ≪ ω ° ), the value of n is nearly equal to 1.4, and the value of k = 0; however, at wavelength λ ≪ λ ° ( ω ≫ ω ° ) , the value of n ≈ 1 and the value of k = 0. When λ = λ ° ( ω = ω ° ), n and k are maximum (Figure 2).

Figure 2 Theoretical values using the Lorentz model of (a) real part of complex refractive index (normal refractive index) and (b) imaginary part of complex refractive index (extinction coefficient), compared with experimental data [17].
Figure 2

Theoretical values using the Lorentz model of (a) real part of complex refractive index (normal refractive index) and (b) imaginary part of complex refractive index (extinction coefficient), compared with experimental data [17].

Figure 3 shows the values of real and imaginary parts of permittivity, εR and εI, using equations (23) and (24) (or equation (13)). It is obvious that when λ ≫ λ ° ( ω ≪ ω ° ), the value of ε R = 2 and ε I = 0 , while at wavelength λ ≪ λ ° and ω ≫ ω ° , the value of ε R ≈ 1 and the value ε I = 0 . For the case λ = λ ° , the values of ε R and ε I are maximum.

Figure 3 Theoretical values using the Lorentz model of (a) real part of dielectric constant ( ε r {\varepsilon }_{{\rm{r}}} ) and (b) imaginary part of dielectric constant, compared with experimental data for SiO2 [17].
Figure 3

Theoretical values using the Lorentz model of (a) real part of dielectric constant ( ε r ) and (b) imaginary part of dielectric constant, compared with experimental data for SiO2 [17].

Figure 4 displays the reflectivity curves using equation (26) when the angle of incidence is equal to 90°. At the wavelength λ ≫ λ ° , the value of R = 0.01, while at wavelength λ ≪ λ ° , the value of R = 0. When λ = λ ° , R is maximum.

Figure 4 Reflectivity (R) of SiO2 compared with experimental data [17].
Figure 4

Reflectivity (R) of SiO2 compared with experimental data [17].

Figure 5 shows the values of absorptive constant, α, using equation (27). The theoretical values show that the absorptive constant is maximum at resonant frequency and equal to zero when λ ≫ λ ° or λ ≪ λ ° .

Figure 5 Absorption coefficient (α) of SiO2 compared with experimental data [17].
Figure 5

Absorption coefficient (α) of SiO2 compared with experimental data [17].

Figure 6 shows the values of transmissivity using equation (29). The values explain that the minimum transmissivity is at resonance frequency and equal to 1 when λ ≫ λ ° or λ ≪ λ ° .

Figure 6 Transmittance (T) of SiO2, compared with experimental data [17].
Figure 6

Transmittance (T) of SiO2, compared with experimental data [17].

Figure 7 shows the values of absorptivity using equation (31). The values explain that the maximum absorptivity is at resonance frequency and equal to 0 when λ ≫ λ ° or λ ≪ λ ° .

Figure 7 Absorptivity (A) of SiO2, compared with experimental data [17].
Figure 7

Absorptivity (A) of SiO2, compared with experimental data [17].

6 Conclusions

  1. Lorentz model is suitable for insulator like SiO2 as shown in the comparison for the theoretical values and the empirical one.

  2. When λ = λ ° , ( ω = ω ° ), n, k, ε R , ε I , A, and R values are maximum and the and transmissivity value is minimum.

  3. At the wavelength λ ≫ λ ° , ( ω ≪ ω ° ), the value of n is nearly equal to 1.4, k = 0, ε R = 2 , ε I = 0 , R = 0.01, α = 0, A = 0, and T = 1.

  4. At the wavelength λ ≪ λ ° , ( ω ≫ ω ° ) , the value of n ≈ 1 and the value of k = 0 and the value of ε R ≈ 1 and the value ε I = 0 , λ ≪ λ ° , the value of R = 0, α = 0, A = 0, and T = 1.

  1. Conflict of interest: Authors state no conflict of interest.

  2. Data availability statement: The most datasets generated and/or analysed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2023-11-22
Revised: 2024-01-02
Accepted: 2024-01-06
Published Online: 2024-03-11

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  168. Vibration suppression of smart composite beam using model predictive controller
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  170. In-depth analysis of critical factors affecting Iraqi construction projects performance
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  172. Energy absorption and impact response of ballistic resistance laminate
  173. Effect of water-absorbent polymer balls in internal curing on punching shear behavior of bubble slabs
  174. Effect of surface roughness on interface shear strength parameters of sandy soils
  175. Evaluating the interaction for embedded H-steel section in normal concrete under monotonic and repeated loads
  176. Estimation of the settlement of pile head using ANN and multivariate linear regression based on the results of load transfer method
  177. Enhancing communication: Deep learning for Arabic sign language translation
  178. A review of recent studies of both heat pipe and evaporative cooling in passive heat recovery
  179. Effect of nano-silica on the mechanical properties of LWC
  180. An experimental study of some mechanical properties and absorption for polymer-modified cement mortar modified with superplasticizer
  181. Digital beamforming enhancement with LSTM-based deep learning for millimeter wave transmission
  182. Developing an efficient planning process for heritage buildings maintenance in Iraq
  183. Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle
  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
  185. Calculations of mass stopping power and range of protons in organic compounds (CH3OH, CH2O, and CO2) at energy range of 0.01–1,000 MeV
  186. Investigation of in vitro behavior of composite coating hydroxyapatite-nano silver on 316L stainless steel substrate by electrophoretic technic for biomedical tools
  187. A review: Enhancing tribological properties of journal bearings composite materials
  188. Improvements in the randomness and security of digital currency using the photon sponge hash function through Maiorana–McFarland S-box replacement
  189. Design a new scheme for image security using a deep learning technique of hierarchical parameters
  190. Special Issue: ICES 2023
  191. Comparative geotechnical analysis for ultimate bearing capacity of precast concrete piles using cone resistance measurements
  192. Visualizing sustainable rainwater harvesting: A case study of Karbala Province
  193. Geogrid reinforcement for improving bearing capacity and stability of square foundations
  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
  199. Analyzing the impact of transfer learning on explanation accuracy in deep learning-based ECG recognition systems
  200. Effect of scale factor on the dynamic response of frame foundations
  201. Improving multi-object detection and tracking with deep learning, DeepSORT, and frame cancellation techniques
  202. The impact of using prestressed CFRP bars on the development of flexural strength
  203. Assessment of surface hardness and impact strength of denture base resins reinforced with silver–titanium dioxide and silver–zirconium dioxide nanoparticles: In vitro study
  204. A data augmentation approach to enhance breast cancer detection using generative adversarial and artificial neural networks
  205. Modification of the 5D Lorenz chaotic map with fuzzy numbers for video encryption in cloud computing
  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
https://doi.org/10.1515/eng-2022-0577
Keywords for this article
SiO2; Lorenz model; dielectric constant; refractive index; extinction coefficient
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  176. Estimation of the settlement of pile head using ANN and multivariate linear regression based on the results of load transfer method
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  179. Effect of nano-silica on the mechanical properties of LWC
  180. An experimental study of some mechanical properties and absorption for polymer-modified cement mortar modified with superplasticizer
  181. Digital beamforming enhancement with LSTM-based deep learning for millimeter wave transmission
  182. Developing an efficient planning process for heritage buildings maintenance in Iraq
  183. Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle
  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
  185. Calculations of mass stopping power and range of protons in organic compounds (CH3OH, CH2O, and CO2) at energy range of 0.01–1,000 MeV
  186. Investigation of in vitro behavior of composite coating hydroxyapatite-nano silver on 316L stainless steel substrate by electrophoretic technic for biomedical tools
  187. A review: Enhancing tribological properties of journal bearings composite materials
  188. Improvements in the randomness and security of digital currency using the photon sponge hash function through Maiorana–McFarland S-box replacement
  189. Design a new scheme for image security using a deep learning technique of hierarchical parameters
  190. Special Issue: ICES 2023
  191. Comparative geotechnical analysis for ultimate bearing capacity of precast concrete piles using cone resistance measurements
  192. Visualizing sustainable rainwater harvesting: A case study of Karbala Province
  193. Geogrid reinforcement for improving bearing capacity and stability of square foundations
  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
  199. Analyzing the impact of transfer learning on explanation accuracy in deep learning-based ECG recognition systems
  200. Effect of scale factor on the dynamic response of frame foundations
  201. Improving multi-object detection and tracking with deep learning, DeepSORT, and frame cancellation techniques
  202. The impact of using prestressed CFRP bars on the development of flexural strength
  203. Assessment of surface hardness and impact strength of denture base resins reinforced with silver–titanium dioxide and silver–zirconium dioxide nanoparticles: In vitro study
  204. A data augmentation approach to enhance breast cancer detection using generative adversarial and artificial neural networks
  205. Modification of the 5D Lorenz chaotic map with fuzzy numbers for video encryption in cloud computing
  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
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