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The mathematical description of the velocity vectors and transverses magnetic fields now are more rigorous. For completeness, the geometrical convention of sign of curvature and its expressions for the horizontal and vertical planes were added.
Due to the negative charge of the electron and the geometrical convention of the sign of curvature, the negative sign should be carried along the expressions which use the optical magnetic functions of curvature and focusing. This were correct now.
Mainly, signs of gradient errors were corrected. Other minor corrections of symbols and characters were done.
juva143
requested review from
Gabrielrezende-asc,
JeffersonBarrosVieira,
LariComuni,
VitorSouzaLNLS,
fernandohds564,
otavio-silveira,
tainara-mareco,
vellosok75 and
xresende
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| \begin{align*} | ||
| d\theta &= -\frac{d\ell}{\varrho} = + \frac{ecB}{E}d\ell = K_1 y ds. | ||
| d\theta &= -\frac{d\ell}{\varrho} = - \frac{ecB_x}{E}d\ell = \frac{1}{1+\delta} \, K_1 \, y \, dl \simeq K_1 \, y \, ds. |
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I am not sure here... whats is the convention for this theta ?
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The step by step derivation of the curvatures and angle displacements are made in the Appendix section 6.1. After apply the lorentz equation for the electromagnetic force over the particle, the electron, the horizontal angle is
| Let's look first at the effect of the field errors. Suppose we begin by considering the effect of a field error which exists only in a small azimuthal interval $\Delta s$, which we may as well place at $s = 0$. In passing through $\Delta s$ the displacement $x$ is unchanged, but the slope $x'$ change by the amount | ||
| \begin{align*} | ||
| \Delta x' = - \dfrac{ec\Delta B}{E_0}\Delta s, | ||
| \Delta x' = \dfrac{ec\Delta B}{E_0}\Delta s, |
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assuming e is the elementary charge (positive), I think the sign is wrong here!
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The angle varies with the displacement as
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I see. the convention is that e is the eletcron charge (enegative), right ?
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I like to use e for the elementary charge. should we use it in Sands?
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I also like to use e as the elementary charge, is the natural. In the text it is the elementary charge.
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the table of symbols defines e as the electron charge, juventino. I am confused now.
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Oh, I was not aware of the table. I, and I think everybody when reading the text, assumed e as the elementary charge. The changes I implemented are all thinking this way. Should we change the table to inform e as the elementary charge?
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I think it is better not o incorporate a negative sign on the symbol e. It is easy for the reader if all signs are explict. Maybe changing the table is a better approach...
| and, after a complete revolution, look at Fig. \ref{fig:fig22}, the new angle is | ||
| \begin{align} | ||
| x_c'(L) - \Delta G \Delta s = x_c'(0). | ||
| x_c'(L) = x_c'(0) - \Delta x' = x_c'(0) + \Delta G \Delta s. |
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should it not be x_c'(L) = x_c'(0) + \Delta x' ?
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I also thought the expression would be like that, but when comparing the resultant closed orbit for it and the expression provided by Lee's Accelerator Physics book and the one derived using the transfer matrix formalism it's clear the signal of the kick should be the opposite.
I interpret this in the following way: with Fig. 2.15 in mind, inside one revolution the angle changes from
Do you agree and see a more clear way to express this in the text?
| \begin{equation} | ||
| \bm{F} = q \; \bm{v} \times \bm{B}, | ||
| \end{equation} | ||
| In the case of the electron, $q=-e$, where $e = 1.6021766208(98)\times10^{-19}\;C$ is the elementary charge fundamental constant. If the magnetic field is colinear with the velocity, the force vanishes. Moreover, the Lorentz force in a pure magnetic field does no work on the charged particle, as can be mathematically derived, |
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I would remove this "fundamental constant" after "elementary charge"
| \frac{d}{d\ell}(\hat{v}_{\parallel}\cos{\theta} ~ \bm{\hat{z}} + \hat{v}_{\parallel}\sin{\theta} ~ \bm{\hat{x}} + \hat{v}_{\perp} ~ \bm{\hat{y}}) = \frac{1}{R_m} (\hat{v}_{\parallel}\cos{\theta} ~ \bm{\hat{z}} + \hat{v}_{\parallel}\sin{\theta} ~ \bm{\hat{x}} + \hat{v}_{\perp} ~ \bm{\hat{y}}) \times (B\bm{\hat{y}}) | ||
| \end{align*} | ||
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| In the ultrarrelativistic context considered here, the particle is moving along the longitudinal direction close to \bm{$\hat{z}}$, then $\hat{v}_{\parallel}$ not significantly changes with $\ell$ in comparison with the angle and the transverse component. The velocity in the $\bm{\hat{y}}$ direction remains unchanged whereas the two other components reduce to a single expression for the evolution of the deflected angle of the particle along the trajectory: |
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i think this construction does not give a good justification for the approximation. maybe:
In the case of storage rings, where the beam is ultrarelativistic and the deviations from the design orbit are small, we can further assume that the velocity is mainly aligned with the longitudinal direction, i. e.
Besides, I think the equation below for the angle of the particle is not particular to the case of ultra-relativistic particles, is it? Isn't it exact for any uniform field?
Maybe you could rewrite this paragraph.
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| For a numerical example, consider the 3 GeV beam at SIRIUS storage ring. Electrons in the beam have, for this nominal energy, a magnetic rigidity of approximately 10 T$\cdot$m. In order to deflect them by a complete turn, 360 degrees, the integrated field experienced must be of $2\pi R_m \approx$ 62.8 T$\cdot$m. Typical magnetic fields in dipoles are around 1 T, which means that if SIRIUS were to be build with such uniform field dipoles it would require exactly 62.8 meters of them. | ||
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| In general case of an arbitrary transverse field $\bm{B}=B_x\bm{\hat{x}}+B_y\bm{\hat{y}}$ its useful to express the movement direction of the particle close to the longitudinal $\bm{\hat{z}}$ using the small angles approximation. |
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I think it is better to remove this whole part. There is no generalization here, the previous construction is as general as this one, because the choice of the coordinate system is arbitrary, so you could take those expressions and apply any rotation to them. i think this part only complicates notation, unnecessarily.
| \end{equation} | ||
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| This last equation shows that the deflection angle per unit path length is the ratio between the magnetic field intensity and the magnetic rigidity of the particle, which depends solely on its energy. The integral form of the equation, used to compute the angular displacement accumulated along a path $\Gamma$, reads | ||
| \begin{equation} |
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I don't like this formulation, because since the field is constant, the path of integration is a circle and the integral can be solved exactly. This makes things look more complicated, unnecessarily. Maybe:
\Delta \theta = B / R_m L
and then you can readly assign \rho = B / R_m, due to the relation between the radius, the arc length and the angle of a circle.
| A deflection in angle comes with a bending radius $\rho$ of curvature, conventionaly considered positive when the rotation direction is in the clockwise sense and negative when it is counter-clockwise. Together with the convention of angles measured counter-clockwise, the geometry of the problem is syntetized by the following equation: | ||
| \begin{equation} | ||
| d\theta = -\frac{dl}{\rho} = -G dl | ||
| \end{equation} | ||
| where the curvature function $G$ is defined as | ||
| \begin{align} | ||
| G = \frac{1}{\rho} | ||
| \end{align} | ||
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| This relation now leads to an interesting expression for the magnetic rigidity | ||
| \begin{equation} | ||
| \label{eq:rigidity} | ||
| B\rho = {R_m} = p/q. |
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If you follow my suggestion above, you don't need the diferential form for the angle and don't need the justification of signs in terms of clockwise and counter-clockwise that, for me, was obscure and difficult to understand. You can readly obtain eq. 6.13, and follow with the examples.
…1 for further clarification of sign convention
Edited text to clarify assumptions and add references to appendix.
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Following the convention of positive curvature when the rotation of particles are in the clockwise direction and negative when counter-clockwise, the signs of the field coefficients who are normalized by the magnetic rigidity, namely G, K1 and S, must be correct to hold the negative sign of the electron charge. In this sense, the correction was made through the text until the section 3.1 and the appendix section 6.1.