TJ-II:Heavy Ion Beam Probe: Difference between revisions

We'll denote as <math>A</math>, <math>B</math> and <math>C</math> the injection, ionization and detection points respectively. The Hamiltonian of a charge in an electromagnetic field is

  \mathcal{H}(\mathbf{p},\mathbf{x},t) = \frac{1}{2m}(\mathbf{p}-q\mathbf{A})^2 + q\phi  

  = \frac{1}{2}m\mathbf{v}^2 + q\phi = E_k + E_p

From the Hamiltonian equations we obtain the ion trajectory <math>\mathbf{x}(t)</math>. The ion's energy variation from <math>A</math> to <math>B</math> is given as

\Delta\mathcal{H}(A\rightarrow B) = \int_{t_A}^{t_B} \frac{d\mathcal{H}}{dt} dt =

\int_{t_A}^{t_B} \frac{\partial\mathcal{H}}{\partial t} dt~

and using the above expression for the Hamiltonian we obtain

\frac{\partial\mathcal{H}}{\partial t} = q\frac{\partial\phi}{\partial t} -

q\mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial t}~,

and thus

\Delta\mathcal{H}(A\rightarrow B) = q\int_{t_A}^{t_B} \frac{\partial\phi}{\partial t}dt -

q\int_{t_A}^{t_B} \mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial t} dt~.

We use the notation

\Delta\mathcal{H}(A\rightarrow B) = q\Omega_{AB}~.

In the range of ion energies and plasma temperatures commonly found in magnetic fusion experiments the main ionizing collision is electron impact <math>I^+ + e^- \to I^{2+} + 2e^-</math>. Because of the large mass ratio between the heavy ions and light electrons (<math>\sim 10^5</math>) we can regard the collision as an ionizing process with no change in the ion momentum and kinetic energy.  

The extra positive charge after the ionization gives the ion an energy increment

\Delta\mathcal{H}_{ioniz.} = (q'-q)\phi(B,t_B)~,

and therefore the total energy change between <math>A</math> and <math>B</math> is

\Delta\mathcal{H}(A\to B)\equiv \mathcal{H}_B-\mathcal{H}_A = q\Omega_{AB} + (q'-q)\phi(B)~.

Similarly, the energy change between <math>B</math> and <math>C</math> is

\Delta\mathcal{H}(B\to C)\equiv \mathcal{H}_C-\mathcal{H}_B = q'\Omega_{BC}~.

Summing the above energy variations gives

\phi(B) = \frac{\mathcal{H}_C-\mathcal{H}_A-(q\Omega_{AB}+q'\Omega_{BC})}{q'-q}~.

In most situations, the energy difference (or the electric potential energy inside the plasma) is much larger than the energy increment caused by the fluctuating electromagnetic fields <math>q\Omega</math> which tends to cancel out along the ion's trajectory. The electric potentials of the injection and detection points are both close to the vacuum vessel potentials so that <math>\mathcal{H}_C-\mathcal{H}_A\approx E_{k,C} - E_{k,A}</math>. Therefore this simplifies to

\phi(B) = \frac{E_{k,C} - E_{k,A}}{q'-q}~.

===Operational expression for the electric potential <math>\phi</math>===