### Hints

A1  ?? From the Poisson equation, obtain an expression which establishes a relationship between the coefficients $\alpha$, $\beta$ and $\gamma$.

A2  ?? Calculate the maximum frequency $\omega_\text{fr,max}$ for which the wavelength $\lambda$ is $n=100$ times greater than $r_0$.

A3  ?? Using the equation $\DeclareMathOperator{\Grad}{grad} \vec{E} = - \Grad \Phi$, find the $x$-component of the electric field $\vec{E}$, in which the motion of the ion occurs.

A4  ?? Write an expression for $\xi$, $a_x$ and $q_x$ using time $t$ and parameters of the system: voltage $U$, radio frequency $\omega_\text{rf}$, the amplitude $V$ of the AC voltage, mass $m$ of the ion, it's charge $Ze$, typical size of the setup $r_0$ and coefficients $\alpha$ and $\tilde{\alpha}$.

H1 Write down the second Newton's law for the ion.
A5  ?? Qualitatively plot the subset of $a_x$ and $q_x$ parameter values that result in the stable motion along the $x$-axis. Consider the range $a_x \in [-3;\: 7]$ and $q_x \in [0; \:5]$.

B1  ?? Qualitatively plot the subset of $U$ and $V$ parameter values that result in stable ion motion in the 3D RF trap. Calculate the values of the characteristic points of the stability region for $\omega_\text{rf} = 3\cdot 10^7~\text{s}^{-1}$, $r_0=1~\text{mm}$ and the ions of rubidium $\rm ^{87}Rb^+$. Consider that the generator in the lab allows you to get this range of voltage values $|U| \in [0; \: 200]~\text{V}$ and $|V| \in [0; \: 400]~\text{V}$.

H1 Motion along each axis should be stable.
H2 The motion along the $z$-axis with parameters $a_z$, $q_z$ is stable if the motion along the $x$-axis with parameters $a_x=-a_z/2$, $q_x=-q_z/2$ is stable.
B2  ?? Qualitatively plot the subset of $U$ and $V$ parameter values that result in different stability for $\rm ^{87}Rb^+$ and $\rm ^{86}Rb^+$ ions. Consider the case where the $\rm ^{87}Rb^+$ ions fly through the trap and the $\rm ^{86}Rb^+$ ions are ejected from the trap. Calculate the values of the characteristic points of the stability region for $\omega_\text{rf} = 3\cdot 10^7~\text{s}^{-1}$ and $r_0=1~\text{mm}$.