Beam Generation

BDSIM uses the CLHEP random number generator and associated classes to generate pseudo-random particle distributions for tracking.

Gauss Twiss Covariance Matrix

The beam in an accelerator can be described as a 6D distribution of particles. The six dimensions are three canonical conjugate coordinate pairs in three planes, and can be represented by the vector - \(\boldsymbol{x} = (x,\,x^{\prime},\,y,y^{\prime},\,t,\,\delta)\). The means of that distribution are the beam centroids. The 6x6 matrix of second order moments of the distribution is referred to as the sigma matrix and can be represented compactly as \(\Sigma_{ij}=\langle\boldsymbol{x_{i}}\,\boldsymbol{x_{j}}\rangle\) with the angle brackets denoting a central moment. More explicitly for a particle beam, this is:

\[\begin{split}\mathbf{\Sigma}= \begin{pmatrix} \langle x^{2}\rangle & \langle x\,x^{\prime}\rangle & \langle x\,y\rangle & \langle x\,y^{\prime}\rangle & \langle x\,t\rangle & \langle x\,\delta\rangle \\ \langle x^{\prime}x\rangle & \langle x^{\prime 2}\rangle & \langle x^{\prime}\,y\rangle & \langle x^{\prime}\,y^{\prime}\rangle & \langle x^{\prime}\,t\rangle & \langle x^{\prime}\,\delta\rangle \\ \langle y\,x\rangle & \langle y\,x^{\prime}\rangle & \langle y^{2}\rangle & \langle y\,y^{\prime}\rangle & \langle y\,t\rangle & \langle y\,\delta\rangle \\ \langle y^{\prime}x\rangle & \langle y^{\prime}x^{\prime}\rangle & \langle y^{\prime}y\rangle & \langle y^{\prime 2}\rangle & \langle y^{\prime}t\rangle & \langle y^{\prime}\delta\rangle \\ \langle t\,x\rangle & \langle t\,x^{\prime}\rangle & \langle t\,y\rangle & \langle t\,y^{\prime}\rangle & \langle t^{2}\rangle & \langle t\,\delta\rangle \\ \langle \delta\,x\rangle & \langle \delta\,x^{\prime}\rangle & \langle \delta\,y\rangle & \langle \delta\,y^{\prime}\rangle & \langle \delta\,t\rangle & \langle \delta^{2}\rangle \\ \end{pmatrix}\end{split}\]

The beam is typically parameterised in an accelerator using the Twiss functions. \(\beta,\alpha,\gamma\) and dispersions \(\eta_x,\eta_x^{\prime},\eta_y,\eta_y^{\prime}\). These are used in combination with beam parameters \(\epsilon_x,\epsilon_y,\sigma_{\delta},\sigma_{t}\). The combination of machine and beam parameters gives the physical observables such as beam size.

Please note that \(\sigma_{\delta}\) is used here for clarity, but its definition is equivalent to \(\sigma_{E}\) defined in Beam Parameters.

For linear optics, and in the absence of dispersion or energy spread, the three 2x2 submatrices along the diagonal of the sigma matrix fully represent each plane and can be treated independently of each other. For example, the horizontal plane is parameterised as:

\[\begin{split}\begin{pmatrix} \langle x^{2}\rangle & \langle x\,x^{\prime}\rangle \\ \langle x\,x^{\prime}\rangle & \langle x^{\prime 2}\rangle \end{pmatrix} = \epsilon_{x} \begin{pmatrix} \beta_{x} & -\alpha_{x} \\ -\alpha_{x} & \gamma_{x} \end{pmatrix}.\end{split}\]

When dispersion and energy spread are present, numerous couplings are introduced, which appear as off-diagonal elements in the sigma matrix. In this case, the energy dependence of the coordinates results in both longitudinal-transverse and transverse-transverse couplings.

To be a valid covariance matrix, the sigma (capital sigma) matrix, must be positive definite. This ensures a positive variance in each dimension.

To derive each term of the covariance matrix, we consider the effect of dispersion on single coordinates. For example, for an individual particle, the horizontal position is affected as follows:

\[x_{i} = x_{\beta i} + \eta_{s}\delta_{i},\]

where \(x_{i}\) is the observed coordinate, \(x_{\beta}\) is the betatron component and \(\eta_{x}\delta_{i}\) reflects the change due to the particle’s relative energy offset in a dispersive region.

Because \(x_{\beta}\) and \(\delta_{i}\) are random variables and are independent by construction, the second order moment in \(x_{\beta}\) can be factorised using the properties of the variance

\[\langle x^{2}\rangle = \mathrm{var}[x_{\beta}+\eta_{x}\delta] = \langle x_{\beta}^{2}\rangle + \eta_{x}^{2}\langle\delta_{i}^{2}\rangle = \epsilon_{x}\beta_{x} + \eta_{x}^{2}\sigma_{\delta}^{2},\]

where the last form is obtained by applying the Courant-Snyder equations and rewriting the variance in terms of standard deviation.

The covariance of the horizontal position and the energy offset can also be obtained as:

\[\langle x\,\delta\rangle = \textrm{cov}[x_{\beta}+\eta_{x}\delta, \delta] = \langle x_{\beta}\,\delta\rangle + \eta_{x}\langle \delta^{2}\rangle = \eta_{x}\langle\delta^{2}\rangle,\]

as \(\langle x_{\beta}\,\delta\rangle = 0\) by construction as explained above.

Finally, correlations that involve two different variables affected by dispersion have the same form. For example:

\[\begin{split}\langle x\,y\rangle &= \mathrm{cov}[x_{\beta}+\eta_{x}\delta,\,x^{\prime}_{\beta}+\eta_{x^{\prime}}\delta] \\ &= \langle x_{\beta}\,x^{\prime}_{\beta}\rangle + \eta_{x}\langle x^{\prime}_{\beta}\,\delta\rangle + \eta_{x^{\prime}}\langle x_{\beta}\,\delta\rangle + \eta_{x}\,\eta_{x^{\prime}}\langle\delta^{2}\rangle \\ &= \langle x_{\beta}\,x^{\prime}_{\beta}\rangle + \eta_{x}\eta_{x^{\prime}}\langle\delta^{2}\rangle \\ &=-\epsilon_{x}\alpha_{x} + \eta_{x}\eta_{x^{\prime}}\sigma_{\delta}^{2}\end{split}\]

All other couplings are done by analogy.

Some useful relations:

\[\begin{split}&\langle XY\rangle\equiv\mathrm{cov}[X,Y], \\ &\langle X^2\rangle\equiv\mathrm{cov}[X,X]\equiv\mathrm{var}[X]\equiv\sigma_{X}^{2}, \\ &\mathrm{var}[X+Y]=\mathrm{var}[X]+\mathrm{var}[Y]\,\,\mathrm{if\,\,X,Y \, are\,independent}, \\ &\mathrm{cov}[aX+bY,cW+dV]=ac\mathrm{cov}[X,W]+ad\mathrm{cov}[X,V]+bc\mathrm{cov}[Y,W]+bd\mathrm{cov}[Y,V].\end{split}\]