... | @@ -203,9 +203,9 @@ to the entrance of that element. |
... | @@ -203,9 +203,9 @@ to the entrance of that element. |
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Denoting with latexmath:[i] a beam line element, one can compute
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Denoting with latexmath:[i] a beam line element, one can compute
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latexmath:[\mathbf{v}_i] and latexmath:[\mathcal{W}_i] by the
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latexmath:[\mathbf{v}_i] and latexmath:[\mathcal{W}_i] by the
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recurrence relations
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recurrence relations
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[latexmath]
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[latexmath]
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++++
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++++
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\label{eq:surv}
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\mathbf{v}_i = \mathcal{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
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\mathbf{v}_i = \mathcal{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
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\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,
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\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,
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++++
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++++
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... | @@ -896,12 +896,15 @@ b_z (z) = B_z (-z) \qquad B_x (z) = - B_x (-z) \qquad B_s (z) = - B_s (-z) |
... | @@ -896,12 +896,15 @@ b_z (z) = B_z (-z) \qquad B_x (z) = - B_x (-z) \qquad B_s (z) = - B_s (-z) |
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++++
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++++
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The most general form of the expansion is
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The most general form of the expansion is
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.Equation 1
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[#eq-01]
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[latexmath]
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[latexmath]
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++++
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++++
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\begin{aligned}
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\begin{aligned}
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B_z & = \sum_{i,k=0}^{\infty} b_{i,k} x^i z^{2k} \label{eq:01} \\
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B_z & = \sum_{i,k=0}^{\infty} b_{i,k} x^i z^{2k} \\
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B_x & = z \sum_{i,k=0}^{\infty} a_{i,k} x^i z^{2k} \label{eq:02}\\
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B_x & = z \sum_{i,k=0}^{\infty} a_{i,k} x^i z^{2k} \\
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B_s & = z \sum_{i,k=0}^{\infty} c_{i,k} x^i z^{2k} \label{eq:03}
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B_s & = z \sum_{i,k=0}^{\infty} c_{i,k} x^i z^{2k}
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\end{aligned}
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\end{aligned}
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++++
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++++
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... | @@ -924,8 +927,8 @@ yield |
... | @@ -924,8 +927,8 @@ yield |
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\end{aligned}
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\end{aligned}
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++++
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++++
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The substitution of ([eq:01]), ([eq:02]) and
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The substitution of (<<eq-01, 1>>)
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([eq:03]) into Maxwell’s equations allows for the derivation of
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into Maxwell’s equations allows for the derivation of
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recursion relations. ([eq:23]) gives
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recursion relations. ([eq:23]) gives
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[latexmath]
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[latexmath]
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... | @@ -1007,7 +1010,7 @@ _(variable radius of curvature)_ |
... | @@ -1007,7 +1010,7 @@ _(variable radius of curvature)_ |
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The difference between this case and the above is that
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The difference between this case and the above is that
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latexmath:[\rho] is now a function of latexmath:[s],
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latexmath:[\rho] is now a function of latexmath:[s],
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latexmath:[\rho(s)]. We can obtain the same result starting with the
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latexmath:[\rho(s)]. We can obtain the same result starting with the
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same functional forms for the field ([eq:01]), ([eq:02]), ([eq:03]). The
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same functional forms for the field (<<eq-01, 1>>). The
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result of the previous section also holds in this case since no
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result of the previous section also holds in this case since no
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derivative latexmath:[\frac{\partial}{\partial s}] is applied to the
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derivative latexmath:[\frac{\partial}{\partial s}] is applied to the
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scale factor latexmath:[h_s]. If the radius of curvature is set to be
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scale factor latexmath:[h_s]. If the radius of curvature is set to be
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... | | ... | |