https://doi.org/10.22463/2011642X.3308

Recibido: march 15, 2022 - Aprobado: august 26,2022

**How to cite:**

C. A. Aguirre-Tellez, Q. Martins & J. J. Barba-Ortega, “Brief introduction to Raman study of charges particles”, *Rev. Ingenio*, vol. 20(1), pp. 1-7, 2023

An analytical development is proposed to determine the solution of the motion equations of a charged particle under the influence of an electric field. In the proposal, the advantages and properties of Laplace's transformation are used to map a system of N second-order non-homogeneous differential equations into a system composed of N linear equations. From the most general solution for the dynamics of the system, some particular cases were studied to recover, in a simple way, the results present in the literature. To motivate the study, Ehrenfest's theorem is used and we discuss how the classical results can be interpreted in their quantum version.

**Keywords:**Classic System, Electromagnetic field, Quantum system,Raman spectroscopy.

Se propone un desarrollo analítico para determinar la solución de las ecuaciones de movimiento de una partícula cargada bajo la influencia de un campo eléctrico. En la propuesta se utilizan las ventajas y propiedades de la transformación de Laplace, para mapear un sistema de N ecuaciones diferenciales no homogéneas de segundo orden en un sistema compuesto por N ecuaciones lineales. A partir de la solución más general para la dinámica del sistema, se estudian algunos casos particulares para recuperar, de manera sencilla, los resultados presentes en la literatura. Para motivar el estudio, se utiliza el teorema de Ehrenfest y se discute como los resultados clásicos pueden ser interpretados en su versión cuántica

**Palabras clave:** Sistemas clásicos, Espectroscopía Raman, Sistemas cuánticos,Campos electromagnéticos.

A decade ago since the centenary Nobel Prize in Physics for the discovery of the Raman effect, awarded to Sir Chandrasekhara Venkata Raman [1]. It is still little known and perhaps for this reason little disseminated in academia. This work comes as an effort to introduce the subject to teachers and students of of the various areas of exact sciences and enable the theoretical discussion of this important physical phenomenon.

The Raman effect can be considered a topic of contemporary physics of relative complexity in its mathematical formulation since for its correct understanding it is necessary to know specific concepts of quantum physics. On the other hand, a classical approach can be adopted to introduce and understand the phenomenon. Such an approach starts from the understanding of oscillatory phenomena, which appear throughout academic life and become familiar in the most diverse applications. So, for example, second-year undergraduate students, in general, are already having contact with equations of motion, mechanical and electromagnetic waves, differential equations, differentiation, and anti-differentiation and in future periods they are becoming familiar with concepts of quantum physics. Unfortunately, the experimental requirement of the phenomenon can be a limiting factor when exposing the subject; since, from an experimental point of view, laboratories with a high financial investment are required to deal with the matter.

Maybe that's why the topic is not approached very often or with due depth in the physics disciplines of physics, chemistry, or engineering courses of the vast majority of universities. We aim to show that there is a relatively straightforward way to introduce the Raman effect problem and obtain its main features, without falling back on experimental requirements or the deep formulations of advanced mathematics. From this perspective, an important starting point is based on knowing the generalization of the harmonic oscillator, thus making it possible to understand and describe the phenomenon classically, in a qualitative way, allowing the student to have a deeper contact with the oscillatory nature of the phenomenon. We hope that this text will serve as a basis for the teacher who wants to approach the topic of Raman spectroscopy in a certain depth, mathematically deducing important properties of the phenomenon, thus giving a reasonable theoretical basis to the student who wants to continue with research in the area. The only ones necessary prerequisites to understanding this development are that the student has already taken a course of waves, basic electromagnetism, and a foundation on differential equations.

In the following sections of our work, we have, in the second section, a brief presentation on what spectroscopy is. In Section 2, we will be concerned to present a brief history of the development of the Raman effect since 1928 and its implications in the scientific environment. In section 3 we will analyze the description of the classical formulation to explain the phenomenon. Finally, in the section 4, we show how the classical treatment can provide us with important information about the dynamics of a beam of charged particles.

Raman spectroscopy is a technique associated with the electromagnetic scattering of atoms in a sample, whether liquid, gaseous or solid [1-5]. The phenomenon as it is known today was initially proposed in the early 1920s, being improved with Smekal's theoretical implications in 1923 [2]. Experimental verifications suggested that certain materials when irradiated with light were capable of scattering radiation in a diffuse way and such scattered radiation presented a classic change of the radiation incident on the sample (Fig.1) In 1928 the publications

Even after the visibility provided by a Nobel Prize, Raman spectroscopy presented major limitations in the following years, precisely because it was a technique that required obtaining a good and intense source of monochromatic light (In the Raman effect, the incident radiation is of monochromatic light, that is, light with a single wavelength.), something difficult to obtain before the advent of the laser in the 1960s. With the discovery of the laser and its proper use, Raman spectroscopy began to offer spectra of solid samples. with high quality and resolution, making it applicable in the most diverse areas and scientific fields [6].

Being a very powerful technique in the investigation of molecular counterparts, it has become one of the main non-destructive techniques in the physical-chemical analysis of organic and inorganic compounds, standing out in the investigation of biological systems. This highlight is due to the advantages that include the requirement of small samples, the possibility of analyzing gases and solids, and being easily applied to the analysis of aqueous solutions [7]. Among the various applications of Raman spectroscopy, we have: biomedical applications [8-13], pharmaceuticals [14,15]. Archaeometry [16,17], nanotechnology [18,19], ceramic materials [20], and polymers [21], extraterrestrial materials [22, 23], forensic applications [24,25]. Currently, more than consolidated in the scientific environment, Raman spectroscopy is well-founded and for a deeper study, some authors can be consulted [26-30].

The observed scattering is nothing more than the scattering of light by the material. The Raman effect is only characterized for scatterings where there are changes in the frequency of the scattered radiation concerning the incident radiation, otherwise, the phenomenon is known as Rayleigh scattering. Rayleigh scattering is elastic scattering, while the Raman effect is defined by inelastic scattering (figure 1).

In general, when dealing with light spectroscopy, we are dealing with some experimental technique that uses a light source to obtain information on a particular material to be studied. Such techniques allow obtaining information about molecular structure, energy levels, atomic bonds, etc. The type of spectroscopy can be differentiated according to the range of electromagnetic radiation used in a given procedure. Thus, knowing the nature of the electromagnetic spectrum is of great importance when specifying contents in spectroscopy. In the figure 2, we have a representation for the visible spectrum in the ultraviolet region 400 nm to the infrared region 700 nm [31].

In cases such as Infrared and Raman spectroscopy, monochromatic radiation is used in the visible. According to the spectral region, the transitions observed depend on the type of levels: electronic, vibrational, or rotational.

In this section, we will describe the classical treatment of the main characteristic associated with the dynamics of a molecule subject to the incidence of monochromatic E electric fields. In this context, we deal directly with a forced oscillatory phenomenon, which can be started with the objections of the classical harmonic oscillator. For this to happen naturally and correctly, we must understand the concept of polarization

When a beam of light falls on a certain material, which contains a set of atoms linked to each other by forces of an electrical nature, the light particles (photons) interact with it, colliding with the molecules of the sample. From this interaction, a response to the given stimulus is produced, that is, the incident light will be part absorbed, partly transmitted and a small fraction of the light will be scattered in all directions [6]. Regarding scattered light, if it has the same wavelength as the incident light, there is no exchange of energy between the photon and the molecule, and consequently, it will have the same frequency *υ*(0) of incidence. This is elastic scattering or Rayleigh scattering (A tribute to Lord Rayleigh who studied this phenomenon and showed that he is responsible for the bluish color of the sky).

There is also the case where scattered light υ does not have the same frequency as incident light *υ*(0), i.e., some photons can excite one mode of vibration of the molecule (or several modes), losing or gaining energy in the process. This scattering is known as inelastic scattering, where the molecule starts to vibrate due to photon-matter interaction, changing the wavelength (and frequency) of the scattered light wave concerning the incident one. This simple modification of your energy is the foundation for the so-called Raman effect. In terms of energy, the described process can be presented as follows:

Where *ω=2πυ*. Thus, the frequency of scattered light can be lower or higher than the frequency of incident light, that is, when υ, is less than *υ*(0), the scattered radiation is called Stokes radiation (equation 1), otherwise *υ≫υ _{0}*, the radiation is called anti-Stokes (equation 2):

It starts from the concept of the incident wave, characterized by a frequency υ and an oscillatory electric field E, when the electric field interacts with the electron cloud of atoms in the molecule, it induces a dipole moment (figure 3) gyven by: gy

Understand how the polarizability is a constant associated with the measurement of molecular bond deformation under incident electric field action (equation 3).

Since there is a variation of α concerning the distance **r** between atomic nuclei, we can suggest that a generalized coordinate *q* is the expansion of *α* in the form:

Where *α*_{0} is the polarizability of the bond in the distance between the equilibrium nuclei *q _{eq}* and the separation for any instant given by

Here *q _{0}* is the maximum internuclear separation relative to the equilibrium position. Thus, gathering equations 4 to equation 6, making the possible distributions we will have:

The equation 7, can be substituted into the equation 3, so we have

We have that the first term of the equation 8, represents the Rayleigh scattering, elapsed from the frequency *υ _{0}*.

The second and third terms correspond to Raman scattering, being respectively the anti-Stokes and Stokes for frequencies *υ+υ _{0}* and

Our quantum system has a classical logo as a reference for the harmonic oscillatory system that was discussed earlier in the section 3. What we will do here is nothing more and than an approximation seeking to describe the dynamics of a bundle of particles whose dynamics thermic of each particle that constitutes the system given by the Schrodinger equation in the form:

Classically, the absorption of radiation by a system is due to the periodic variation of its electric dipole moment. This absorbed frequency is identical to the dipole oscillation frequency. We see that polarizability originates from the induced dipole moment. In a quantum system, transitions between two states are characterized by the wave functions *ψ _{m}* and

The equation 10 shows us that *α _{ij}*, a Raman tensor, which is a matrix 3 ×3 correlating the electric vector

It is important to note that the polarizability tensor argument could arise naturally in a classic case of electrical systems involving molecular groups as developed in the section 3. In this case, as the solutions of the quantum model are closely linked to the effects of the Raman tensor, we chose to show it in this section. An example of this is the association between the Raman tensor and the functions *ψ _{m}* and

The components *α _{ij}* form a symmetrical tensor in the Raman effect [33], so:

In Stokes and anti-Stokes scattering, the vibrational states m and n are different and the first integral of the second member is always equal to zero, due to the orthogonality between *ψ _{m}* and

The selection rule *ΔV*=±1 can be understood by checking the solution of the equation:

Given the necessary physical conditions of *ψ(q)* whose development suggests a solution of the type:

The solution 14 tested on equation 13 give us:

which is called the Hermite equation. Your solutions must meet the *ψ(q)* requirements. After the development of equation (15) we arrive at:

The equation 16 gives the eigenvalues E of the Quantum Harmonic Oscillator. In this case, if the energy has this value from equation 14 it satisfies the equation 15. In the equation 16 we have the energy of the so-called ground state *ψ _{0}*. Energies for excited states

It is important to note how naturally the Hermite polynomials. The * H _{υ}* appears, which is an added function to obtain solutions of second-order ordinary differential equations. The various energy levels are then defined by the quantum number

In Stokes Raman scattering, the molecule in the ground state collides with the photon of energy *ℏυ _{0}* passing to an intermediate state. It then decays to an excited state of energy

Finally, as example in another physics areas, non-linear ordinary and partial differential equations can be obtained for to explain quantum model, by example, expressed by the Ginzburg- Time-dependent Landau, obtained by computational simulation in two and three dimensions [34-37].

From the Laplace transform we study the more general case of the dynamics of a particle subject to interaction with static electric and magnetic fields. Using the definition and properties of the Laplace transform, we show how to map the problem of solving a system of ordinary differential equations to a problem of finding the solution of a system of first-degree linear equations. With this, we find the most general possible solution of the equation of motion that governs the dynamics of the system in terms of the components of the electromagnetic fields from the general solution, we show how to obtain the solution of three particular cases whose importance and applicability exceed the didactic point of view. Finally, we motivated our study by doing the quantum treatment of the studied system, where we consider a beam of charged particles without spin. In this context, we show that the parameterized curve of the expected value (on average) of the position vector of a beam particle as a function of time corresponds to that curve associated with its classical analog.

Q. S. Martins thanks to Brazilian agencies FAPERO/CAPES, n◦ 008/2018 by financial support.

[1] C. V. Raman, K. S. Krishnan, “A new class of spectra due to secondary radiation”, *Indian Journal of Physics*, vol. 2, pp. 399-419, July 2012.

[2] A. Smekal, “Zur Quantentheorie der ispersion”, vol. 11, pp. 873-875, October 1923.

[3] C. V. Raman, K.S. Krishnan, “A new Radiation”, *Indian Journal of Physics*, vol 2, pp. 387-398. June, 1928.

[4] C. V. Raman, K. S. Krishnan, “A new type of secondary radiation,” Nature. Vol. 121, pp. 501-502, March 1928.

[5]A. F. Silva, “Caracterizacao espectroscópica de base de Schiff C18H17N3O investigada por cálculos DFT e espectroscopia vibracional.” Dissertacaoo de Mestrado. Cuiabá, Brasil, 2013.

[6] J. C. Ramos, “Espectroscopia Raman y sus aplicaciones”. *Optica Pura y aplicada*, vol. 46, no. 1, pp. 83-95, July 2013.

[7] James J. Bohning, “The Raman effect” American Chemical Society, Jadavpur, Calcutta, December 1998.

[8] P. Casper, G. Lucassen, R Wolthuis, H. Bruining, G. Puppels, “In vitro and in vivo Raman spectroscopy of human skin”. *Biospectroscopy*, vol. 4, no. 31, January 1998. Doi: https://doi.org/10.1002/(SICI)1520-6343(1998)4:5+

[9] A. Villanueva-Luna, J. Castro-Ramos, S. Vazquez-Montiel, A. Flores Gil, J. A. Delgado-Atencio. ”Comparison of different kinds of skin using Raman spectroscopy”. *Optical Diagnostics and Sensing X: Toward Point-of-Care Diagnostics*, vol. 7572, no. 75720M, Febraury 2010. Doi: https://doi.org/10.1117/12.842843

[10] K. Golcuka, G. Mandair, A. F. Callender. N. Sahar. D. Kohn, M. Morris, “Is photobleaching necessary for Raman imaging of bone tissue using a green laser” *Biochimica et Biophysica Acta (BBA) – Biomembranes*, vol 1758, no. 7, pp. 868-873, July 2006. Doi: https://doi.org/10.1016/j.bbamem.2006.02.022

[11] M. D. Morris W. F. Finney, R. M. Rajachar, D. H. Kohn, “Bone tissue ultrastructural response to elastic deformation probed by Raman spectroscopy”. *Faraday Discussions*, vol. 126, pp. 159-168, November 2003. Doi: https://doi.org/ 10.1039/b304905a

[12] N. Ismail, “Raman spectroscopy with an integrated arrayed-waveguide grating”. *Optical Letters*, vol. 36, no. 23, pp. 4629-4631, April 2011. Doi: https://doi.org/10.1364/OL.36.004629

[13] P. G. Spizziri, N. Cochrane, S. Prawer E. Reynold, “A comparative study of carbonate determination in human teeth using Raman spectroscopy”, *Caries Res.*, vol 46, pp. 353-360, July 2012. Doi: https://doi.org/ 10.1159/000337398

[14] P. Matousek, M. D. Morris (Edts), “Emerging Raman applications and techniques in biomedical and pharmaceutical fields”. P. Matousek, M. D. Morris Edts, Springer, New York, 2010.

[15] T. De Beer, A. Burggraeve, M. Fonteyne, L. Saerens, J.P. Remon, C. Vervaet, “Near infrared and Raman spectroscopy for the in-process monitoring of pharmaceutical production processes”, *International Journal of Pharmaceutics*, vol. 417, no.1, pp. 32-47, December 2010. Doi: https://doi.org/10.1016/j.ijpharm.2010.12.012

[16] R. P. Freitas, “Aplicacoes de técnicas nucleares e espectroscopia molecular em arqueometria”. Tese Doutorado, Engenharia Nuclear, UFRJ, Rio de Janeiro, Brasil, 2014.

[17] C. Calza, “Desenvolvimento de sistema portátil de fluorescencia de raios X com aplicacoes na arqueometria”. Tese Doutorado, Engenharia Nuclear, UFRJ, Rio de Janeiro, Brasil, 2007.

[18] M. Amer, “Raman spectroscopy, fullerenes and nanotechnology”. Royal Society of Chemistry, Cambridge, 2010.

[19] V. N. Popov, P. Lambin, “Carbon nanotubes: from basic research to nanotechnology”, Springer, Dordrecht 2006.

[20] J. H. Robin, M. L. C. Clark, Catarina Lagarana, “Ramam Miscroscopy: The identification of lapis cazuli on medieval pottery from the south Italy”, *Spectrchimica Acta Part. A*, vol. 53, pp. 597, July 1997. Doi: https://doi.org/10.1016/S1386-1425(96)01768-4

[21] R. P. Millen, D. L. A. De Faria, L. A. Marcia Temperini, “Modelos para dispersao Raman em polímeros conjugados”, *Quimica Nova*, vol. 28, no. 2, pp. 289, September 2005.

[22] P. W. M. Ehrenfreund, “Astrobiology: future perspectives”, Springer, Dordrecht, 2004.

[23] H. Edwards, “Raman spectroscopic approach to analytical astrobiology: the detection of key geological and biomolecular markers in the search for life”. *Philosophical Transactions of the Royal Society A*, vol. 368, pp. 3059, April. 2010. Doi: https://doi.org/10.1098/rsta.2010.0100

[24] E. Ali, H. Edwards, M. Hargreaves, H. Scowen, “Raman spectroscopic investigation of cocaine hydrochloride on human nail in a forensic context”, *Anal Bioanal Chemical*, vol. 390, pp. 1159-1166, June 2007. Doi: https://doi.org/10.1007/s00216-007-1776-z

[25] E. L. Izake, “Forensic and homeland security applications of modern portable Raman spectroscopy”, *Forensic Science International*, vol 202, no. 1, pp. 1-8, October 2010. Doi: https://doi.org/10.1016/j.forsciint.2010.03.020

[26] E. Smith, G. Dent, “Modern Raman spectroscopy: A Practical Approach”, John Wiley & Sons Ltd.Copyright, 2005.

[27] D. A. Long, Raman Spectroscopy, McGraw-Hill International, New York (1977).

[28] B. P. Stoicheff, “Advances in Spectroscopy, Interscience”, Publishers Ltd., New York, 1959.

[29] A. G. Rodrigues, J. C. Galzerani. “Espectroscopias de infravermelho, Raman e de fotoluminescencia: potencialidades e complementaridades”, *Revista Brasileira de ensino de Física*, vol. 34, no. 4, pp. 4309-1, 4309-9, July 2012. Doi: https://doi.org/10.1590/S1806-11172012000400009

[30] P. Larkin, Infrared and Raman spectroscopy, Elsevier, Amstardam, 2011.

[31] D. A. Skoog, F. J. Holler, T. A. Nieman, “Principios de análise instrumental”, 5 ed. Bookman editora, Brasil, 2002.

[32] O. Sala, “Uma molécula didática”, Quimica Nova, vol. 31, no. 4, pp. 914, July 2008.

[33] M. Tsuboi, J. M. Benevides, G.J. Thomas, “Raman tensors and their application in structural studies of biological systems”, *Proceedings of the Japan Academy. Series B, Physical and Biological Sciences*, vol. 85 no. 3, pp. 83-97, August 2009.Doi: https://doi.org/10.2183/pjab.85.83

[34] J. Barba-Ortega, M. Rincón-Joya, J. Faúndez-Chaura, “Curva voltaje-tiempo en un proceso aniquilación-creación de pares vórtice-antivórtice”. *Rev. Ingenio*, vol. 15, no. 1, pp. 31–37, July 2018. Doi: https://doi.org/10.22463/2011642X.3122

[35] C. Aguirre-Tellez, E. Valbuena-Niño, J. Barba-Ortega, “Estado de vórtices en un cuadrado superconductor de dos-orbitales con condiciones de contorno mixtas”. *Rev. Ingenio*, vol. 15, no 1, pp. 38–43, July 2018. Doi: https://doi.org/10.22463/2011642X.3118

[36] C. Aguirre-Tellez, Quesle Martins, J. Barba-Ortega, “Desarrollo analítico de las ecuaciones Ginzburg-Landau para películas delgadas superconductoras en presencia de corrientes”. *Revista UIS Ingeniería*, vol. 18, no 2, pp. 213–220, Febraury 2019. Doi: https://doi.org/10.18273/revuin.v18n2-2019020

[37] C. Aguirre-Tellez, J. Faundez, S. Magalhaes, J. Barba-Ortega, “Vortex Matter in a Two-Band SQUID-Shaped Superconducting film”, *Journal of Low Temperature Physics*, pp.85–96, March 2022. Doi: https://doi.org/10.1007/s10909-022-02701-3

* Doctor.Correo: cristian@fisica.ufmt.br

** Doctor.Correo: quesle@fisica.ufmt.br

*** Doctor. Correo: jjbarbao@unal.edu.co

Licencia Creative Commons Reconocimiento-NoComercial 4.0 Internacional