Effect of Magnetic Field on the Spectral Characteristics of Thermal Motion of Charged Particles in an Isotropic Trap

Abstract

Results are presented from analytical and numerical studies of the effect of a dc magnetic field on the spectral characteristics of thermal motion of charged particles in an isotropic electrostatic trap. An analytic expression for the spectral density of the shifts of the center of mass of the systems under study is obtained. The analytic expression is verified by numerically simulating ensembles with different numbers of particles interacting via the Coulomb potential in a wide range of parameters.

APPENDIX

Any solution F(t) to problem (2) and (3) can be represented as the superposition

$$F(t) = {{C}_{0}} + \sum\limits_{i = 1}^4 {{{C}_{i}}\exp ({{\lambda }_{i}}t)} ,$$

((A.1))

where for the function $\left\langle {x{{{(t)}}^{2}}} \right\rangle$ and for the function $\left\langle {x({{t}^{*}})x({{t}^{*}} + t)} \right\rangle$. The coefficient С_i ($i = 1,2,3,4$) for these functions can be found from the initial conditions of the problem: , $dF\left( 0 \right){\text{/}}dt = 0$, , ${{d}^{3}}F\left( 0 \right){\text{/}}d{{t}^{3}} = 0$, where Т₀ = 0 for $\left\langle {x{{{(t)}}^{2}}} \right\rangle$, for $\left\langle {x({{t}^{*}})x({{t}^{*}} + t)} \right\rangle$, Т₂ = 2T/М for $\left\langle {x{{{(t)}}^{2}}} \right\rangle$, and Т₂ = ‒2T/М for $\left\langle {x({{t}^{*}})x({{t}^{*}} + t)} \right\rangle$. As a result, we obtain the following set of equations for the coefficients С_i:

$$\sum\limits_{i = 1}^4 {{{C}_{i}} = \pm \;\frac{{2T}}{{M\omega _{t}^{2}}}} ,$$

((A.2))

$$\sum\limits_{i = 1}^4 {{{\lambda }_{i}}{{C}_{i}} = 0} ,$$

((A.3))

$$\sum\limits_{i = 1}^4 {\lambda _{i}^{2}{{C}_{i}} \pm \frac{{2T}}{M}} = 0,$$

((A.4))

$$\sum\limits_{i = 1}^4 {\lambda _{i}^{3}{{C}_{i}} = 0} .$$

((A.5))

Hereinafter, the minus sign refers to the function $\left\langle {x{{{(t)}}^{2}}} \right\rangle$, while the plus sign refers to the function $\left\langle {x({{t}^{*}})x({{t}^{*}} + t)} \right\rangle$. Thus, for the coefficients С_i, we have

((A.6))

((A.7))

((A.8))

((A.9))

Using formulas (5a) and (5b) for the eigenfrequencies λ_i and assuming that and ${{\nu }^{2}} - {{D}_{1}}^{2}{\text{/}}2 \ll 2{{\omega }_{{1,2}}}$ (where ${{\omega }_{1}} = {{(\Psi _{1}^{2} + \Omega _{1}^{2})}^{{1/2}}}$ and ${{\omega }_{2}} = {{(\Psi _{2}^{2} + \Omega _{2}^{2})}^{{1/2}}}$), we represent formulas (6)–(9) as

((A.10))

((A.11))

((A.12))

((A.13))

Here, and ${{B}_{2}}\, = \,\sqrt 2 (\omega _{1}^{2}\, - \,\omega _{t}^{2}){\text{/}}$.

Note that the above-mentioned conditions $D_{1}^{2} \ll$ and ${{\nu }^{2}} - D_{1}^{2}{\text{/}}2 \ll 2{{\omega }_{{1,2}}}$ are satisfied when ${{({{\Psi }_{{1,2}}})}^{2}} \ll {{({{\Omega }_{{1,2}}})}^{2}}$.