The plasma nanosphere cooling rate simulation in the presence of the coherent electromagnetic waves with Gaussian profile

Abstract

Using cold atmospheric plasma (CAP) could promise a breakthrough in the fight against cancer cells, providing an alternative to surgical and chemical therapies. Here, the reduction rate of the temperature of plasma nanosphere is simulated and studied after the injection of electromagnetic waves into it. For this purpose, the plasma nanosphere is irradiated by a laser beam with Gaussian profile and the variation rate of the temperature is investigated. Diagrams of the variation rate of the plasma temperature are presented in terms of some parameters such as the plasma frequency, the radius of the plasma nanosphere, the wavelength of the laser beam, the temperature of the electrons, the relative distance between the front mirror of laser structure and nanosphere center as well as the position of the beam waist center with respect to the center of the nanosphere. In this regard, the temperature of the plasma nanosphere can be reduced by laser irradiation, thus observing the cooling of the plasma. These calculations are performed for the plasma in the collisional approximation. The results obtained can be potentially employed for designing a new type of CAP-based cancer treatment, being useful for the destruction of cancerous tissues and the delivery of drugs to them.

Graphic abstract

Appendix A

As can be seen in formulas (7–9), the coefficients $c_{n}^{{{\text{pw}}}}$ (pw for plane wave) are isolated because they appear in the Bromwich formulation of the pure LMT. They are given by:

$$ c_{n}^{{{\text{pw}}}} = \frac{1}{ik}\left( { - i} \right)^{n} \frac{2n + 1}{{n\left( {n + 1} \right)}} $$

(A1)

It allows to disconnect the $c_{n}^{{{\text{pw}}}}$’s and the $g_{n}^{m}$’s (n from 1 to infinity, and m from − n to n). Also, the generalized Legendre functions have been used in formulas (7–9) which are introduced as:

$$ \tau_{n}^{m} \left( {\cos \theta } \right) = \frac{{\text{d}}}{{{\text{d}}\theta }}P_{n}^{m} \left( {\cos \theta } \right){ } $$

(A2)

$$ \pi_{n}^{m} \left( {\cos \theta } \right) = \frac{{P_{n}^{m} \left( {\cos \theta } \right)}}{\sin \theta } $$

(A3)

Generalized Legendre functions $\tau_{n}^{1}$ and $\pi_{n}^{1}$ identify with usual Legendre functions of the LMT, namely τ_n and π_n, respectively.

The basic solutions of the spherical Bessel equation for n given are four spherical Bessel functions that denoted by $\psi_{n}^{\left( i \right)} ,{ }i = 1,2,3,4$ as follows:

$$ \psi_{n}^{\left( 1 \right)} \left( x \right) = \sqrt {\frac{\pi }{2x}} J_{n + 1/2} \left( x \right) $$

(A4)

$$ \psi_{n}^{\left( 4 \right)} \left( x \right) = \sqrt {\frac{\pi }{2x}} H_{n + 1/2}^{\left( 2 \right)} \left( x \right) $$

(A5)

in which J and H⁽²⁾ are the ordinary Bessel functions and Hankel functions of the second kind, respectively.

Here, the size parameter α and the optical size parameter β are introduced by:

$$ \alpha = \frac{\pi d}{\lambda } $$

(A6)

$$ \beta = M\alpha $$

(A7)

in which λ is the wavelength in the surrounding medium.