Luminosity of an Ion Collider

Abstract

A formula is obtained for the luminosity of a collider at the collision of two beams that differ, generally speaking, by their parameters (asymmetric colliders). The formula is valid for counterpropagating and merging beams with coincident longitudinal axes. Three special cases of the formula are considered: collision of two identical axially symmetric bunched beams, collision of a bunch with a coasting beam, and collision of two coasting beams. Collision of intersecting beams is briefly considered, and the method for the luminosity calculation is formulated. The synchronization problem is considered for collisions of asymmetric beams. A method is presented for optimizing parameters of a cyclic collider by minimizing betatron frequency shifts caused by the effect of the space charge of the beams. Numerical examples of luminosity calculations for several types of asymmetric colliders are given.

APPENDIX: THIN LENS APPROXIMATION

The frequency shift of betatron oscillations ξ₁₂ due to the beam–beam effect can be found by multiplying the matrix of the transformation for the particle revolution in the collider ring (so-called Twiss matrix)

$$\begin{gathered} {{M}_{{{\text{Ring}}}}} = \left( {\begin{array}{*{20}{c}} {\cos {{\varphi }_{0}} + \alpha \sin {{\varphi }_{0}}}&{\beta \sin {{\varphi }_{0}}} \\ { - \gamma \sin {{\varphi }_{0}}}&{\cos {{\varphi }_{0}} - \alpha \sin {{\varphi }_{0}}} \end{array}} \right), \\ {{\varphi }_{0}} = 2\pi Q, \\ \end{gathered} $$

by the thin lens matrix

$${{M}_{f}} = \left( {\begin{array}{*{20}{c}} 1&0 \\ { - \frac{1}{f}}&1 \end{array}} \right),$$

where the focal length f of the thin lens must be related to the phase shift of the betatron oscillations.

Considering the perturbation introduced by the thin lens, we find the matrix of the transformation M* by multiplying the matrices M_Ring and M_f

$$M* = {{M}_{{{\text{Ring}}}}}{{M}_{f}} = \left( {\begin{array}{*{20}{c}} {\cos {{\varphi }_{0}} + {{\alpha }_{0}}\sin {{\varphi }_{0}} - \frac{{{{\beta }_{0}}}}{f}\sin {{\varphi }_{0}}}&{{{\beta }_{0}}\sin {{\varphi }_{0}}} \\ { - {{\gamma }_{0}}\sin {{\varphi }_{0}} - \frac{1}{f}\cos {{\varphi }_{0}} + \frac{{{{\alpha }_{0}}}}{f}\sin {{\varphi }_{0}}}&{\cos {{\varphi }_{0}} - {{\alpha }_{0}}\sin {{\varphi }_{0}}} \end{array}} \right).$$

((А.1))

Representing the terms of the matrix M* as

$$\begin{gathered} \varphi * = {{\varphi }_{0}} + \Delta \varphi ,\,\,\,\,~\Delta \varphi \ll {{\varphi }_{0}};\,\,\,\,~\beta = {{\beta }_{0}} + \Delta \beta ; \\ ~\alpha = {{\alpha }_{0}} + \Delta \alpha ,\,\,\,\,~\gamma = {{\gamma }_{0}} + \Delta \gamma , \\ \end{gathered} $$

where Δφ, Δα, Δβ, Δγ are the perturbations introduced by the thin lens, we write the matrix М* as

$$\begin{gathered} M* \equiv {{M}_{{\Delta }}} \\ = \left( {\begin{array}{*{20}{c}} {\cos \varphi {\text{*}} + \alpha \Delta \sin \varphi {\text{*}}}&{\beta \sin \varphi {\text{*}}} \\ { - \gamma \sin \varphi {\text{*}}}&{\cos \varphi {\text{*}} - \alpha \sin \varphi {\text{*}}} \end{array}} \right). \\ \end{gathered} $$

((А.2))

Equating the corresponding terms of matrices (А.1) and (А.2), we obtain in the linear Δ-term approximation

$$\begin{gathered} m_{{11}}^{*} = m_{{11}}^{\Delta } \to - \frac{{{{\beta }_{0}}}}{f}\sin {{\varphi }_{0}} \\ = - \Delta \varphi \left( {\sin {{\varphi }_{0}} - {{\alpha }_{0}}\cos {{\varphi }_{0}}} \right) + \Delta \alpha \sin {{\varphi }_{0}}, \\ \end{gathered} $$

$$m_{{12}}^{*} = m_{{12}}^{\Delta } \to 0 = {{\beta }_{0}}\cos {{\varphi }_{0}}\Delta \varphi + \Delta \beta \sin {{\varphi }_{0}},$$

$$\begin{gathered} m_{{21}}^{*} = m_{{21}}^{\Delta } \to - \frac{{\cos {{\varphi }_{0}}}}{f} + \frac{{{{\alpha }_{0}}\sin {{\varphi }_{0}}}}{f} \\ = {{\gamma }_{0}}\Delta \varphi \cos {{\varphi }_{0}} - \Delta \gamma \sin {{\varphi }_{0}}, \\ \end{gathered} $$

$$\begin{gathered} m_{{22}}^{*} = m_{{22}}^{\Delta } \to 0 \\ = - \Delta \varphi \left( {\sin {{\varphi }_{0}} + {{\alpha }_{0}}\cos {{\varphi }_{0}}} \right) - \Delta \alpha \sin {{\varphi }_{0}}. \\ \end{gathered} $$

Thus, we obtained four equations in the unknowns Δφ, Δα, Δβ, and Δγ

$$\Delta \varphi \left( {\sin {{\varphi }_{0}} - {{\alpha }_{0}}\cos {{\varphi }_{0}}} \right) - \Delta \alpha \sin {{\varphi }_{0}} = \frac{{{{\beta }_{0}}}}{f}\sin {{\varphi }_{0}},$$

$$\Delta \varphi {{\beta }_{0}}\cos {{\varphi }_{0}} + \Delta \beta \sin {{\varphi }_{0}} = 0,$$

$$ - \Delta \varphi {{\gamma }_{0}}\cos {{\varphi }_{0}} - \Delta \gamma \sin {{\varphi }_{0}} = - \frac{1}{f}\left( {\cos {{\varphi }_{0}} - {{\alpha }_{0}}\sin {{\varphi }_{0}}} \right),$$

$$\Delta \varphi \left( {\sin {{\varphi }_{0}} + {{\alpha }_{0}}\cos {{\varphi }_{0}}} \right) + \Delta \alpha \sin {{\varphi }_{0}} = 0.$$

Solving this system of linear equations by the known determinant calculation method, we find the determinant of the system ${\text{De}}{{{\text{t}}}_{\Delta }} = - 2{{\sin }^{4}}{{\varphi }_{0}}$ and the determinant with the replacement of the first column in Det_Δ by the coefficients of the right-hand side of the system ${\text{De}}{{{\text{t}}}_{\varphi }} = ~ - \frac{{{{\beta }_{0}}}}{f}{{\sin }^{4}}{{\varphi }_{{0~}}}$. Their ratio yields the desired phase shift φ in the presence of the thin lens

$$\Delta \varphi = \frac{{{\text{De}}{{{\text{t}}}_{\varphi }}}}{{{\text{De}}{{{\text{t}}}_{\Delta }}}} = \frac{{{{\beta }_{0}}}}{{2f}}.$$

This expression for Δφ exactly coincides with (2.15), since $~{{\beta }_{0}} \equiv B_{{x1}}^{*}$ is the betatron function at the location of the “perturbing” thin lens (in its absence), and the focal length is f ≡ f_BB.