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TJ-II:Heavy Ion Beam Probe: Difference between revisions
→Formal expression of the electric potential \phi
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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 | 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 | ||
:<math> | |||
\mathcal{H}(\mathbf{p},\mathbf{x},t) = \frac{1}{2m}(\mathbf{p}-q\mathbf{A})^2 + q\phi | \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 | = \frac{1}{2}m\mathbf{v}^2 + q\phi = E_k + E_p | ||
</math | </math> | ||
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 | 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 | ||
:<math> | |||
\Delta\mathcal{H}(A\rightarrow B) = \int_{t_A}^{t_B} \frac{d\mathcal{H}}{dt} dt = | \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~ | \int_{t_A}^{t_B} \frac{\partial\mathcal{H}}{\partial t} dt~ | ||
</math | </math> | ||
and using the above expression for the Hamiltonian we obtain | and using the above expression for the Hamiltonian we obtain | ||
:<math> | |||
\frac{\partial\mathcal{H}}{\partial t} = q\frac{\partial\phi}{\partial t} - | \frac{\partial\mathcal{H}}{\partial t} = q\frac{\partial\phi}{\partial t} - | ||
q\mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial t}~, | q\mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial t}~, | ||
</math | </math> | ||
and thus | and thus | ||
:<math> | |||
\Delta\mathcal{H}(A\rightarrow B) = q\int_{t_A}^{t_B} \frac{\partial\phi}{\partial t}dt - | \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~. | q\int_{t_A}^{t_B} \mathbf{v}\cdot\frac{\partial\mathbf{A}}{\partial t} dt~. | ||
</math | </math> | ||
We use the notation | We use the notation | ||
:<math> | |||
\Delta\mathcal{H}(A\rightarrow B) = q\Omega_{AB}~. | \Delta\mathcal{H}(A\rightarrow B) = q\Omega_{AB}~. | ||
</math | </math> | ||
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. | 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. | ||
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The extra positive charge after the ionization gives the ion an energy increment | The extra positive charge after the ionization gives the ion an energy increment | ||
:<math> | |||
\Delta\mathcal{H}_{ioniz.} = (q'-q)\phi(B,t_B)~, | \Delta\mathcal{H}_{ioniz.} = (q'-q)\phi(B,t_B)~, | ||
</math | </math> | ||
and therefore the total energy change between <math>A</math> and <math>B</math> is | and therefore the total energy change between <math>A</math> and <math>B</math> is | ||
:<math> | |||
\Delta\mathcal{H}(A\to B)\equiv \mathcal{H}_B-\mathcal{H}_A = q\Omega_{AB} + (q'-q)\phi(B)~. | \Delta\mathcal{H}(A\to B)\equiv \mathcal{H}_B-\mathcal{H}_A = q\Omega_{AB} + (q'-q)\phi(B)~. | ||
</math | </math> | ||
Similarly, the energy change between <math>B</math> and <math>C</math> is | Similarly, the energy change between <math>B</math> and <math>C</math> is | ||
:<math> | |||
\Delta\mathcal{H}(B\to C)\equiv \mathcal{H}_C-\mathcal{H}_B = q'\Omega_{BC}~. | \Delta\mathcal{H}(B\to C)\equiv \mathcal{H}_C-\mathcal{H}_B = q'\Omega_{BC}~. | ||
</math | </math> | ||
Summing the above energy variations gives | Summing the above energy variations gives | ||
:<math> | |||
\phi(B) = \frac{\mathcal{H}_C-\mathcal{H}_A-(q\Omega_{AB}+q'\Omega_{BC})}{q'-q}~. | \phi(B) = \frac{\mathcal{H}_C-\mathcal{H}_A-(q\Omega_{AB}+q'\Omega_{BC})}{q'-q}~. | ||
</math | </math> | ||
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 | 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 | ||
:<math> | |||
\phi(B) = \frac{E_{k,C} - E_{k,A}}{q'-q}~. | \phi(B) = \frac{E_{k,C} - E_{k,A}}{q'-q}~. | ||
</math | </math> | ||
===Operational expression for the electric potential <math>\phi</math>=== | ===Operational expression for the electric potential <math>\phi</math>=== | ||