R: Combined Electrostatic and Magnetostatic bend (Generalized Wien filter)

Parameter Description Argument Order Argument Name Units Default Value
Length 1 L cm
Magnetic Field 2 B kG
Magnetic Field Gradient 3 Gb kG/cm
Electric Field 4 E kV/cm
Electric Field Gradient 5 Ge kV/cm **2

R denotes a bend element where bending is provided by a combination of electrostatic and magnetic forces. Bending occurs in the horizontal plane and results from an electric field of magnitude $E_0$ in the $x$ direction and a magnetic field of magnitude $B_0$ in the $-y$ direction. A transverse field gradient may be present in either or both static fields. The first three arguments in the element specification are the same as those used for a conventional (strictly magnetic) combined function magnet (see: B-combined function magnet). Two additional arguments follow: the electric field on the central orbit, E [kV/cm] and its gradient, Ge [kV/cm2]. For cylindrical electrodes, the electric field near the central orbit is given by the following equation $$ \begin{eqnarray} \phi(x,y) & = & R_0 E_0 \ln(1+x/R_0) \\ E_x(x,y) & = & - E_0 (1-x/R_0) + ... ,\\ E_y(x,y) & = & + y/R_0 + ... , \end{eqnarray} $$ where $R_0$ is the bending radius set by the combined effect of the electric and magnetic fields. In this case one has $G_e = -E_0/R_0$. Magnetic edge focusing is modeled with a separate element. (see: G-magnet edge). For spherical electrostatic deflector plates, one has $G_e = - 2 E_0 /R_0$. For toroidal plates, $G_e = - (1+ k) E_0/R_0 $ with $k$ the ratio of the electrode curvature in the bending plane to that in the transverse plane. If the electrodes plates are plane and parallel, $R_0 = \infty$, the magnetic and electric bending forces cancel each other out and one has a classic Wien filter.

Examples:
  
R   L[cm]= 139.62634   B[kG]=0  	Gb[kG/cm]=0 	E[kV/cm]= 0.9999869 	Ge[kV/cm**2]=0
... 
ra L[cm]= 8.7266461 B[kG]=0 Gb[kG/cm]=0 E[kV/cm]= 0.9999869 Ge[kV/cm**2]=0

Transfer Matrix

$$ R = \left( \begin{array}{cccccc} \cos k_xL & \frac{1}{k_x} \sin k_xL & 0 & 0 & 0 & \frac{\kappa_p}{k_x^2} (1-\cos k_xL) \\ -k_x\sin k_xL & \cos k_xL & 0 & 0 & 0 & \frac{\kappa_p}{k_x} \sin k_xL \\ 0 & 0 & \cos k_yL & \frac{1}{k_y}(\sin k_yL) & 0 & 0 \\ 0 & 0 & -k_x\sin k_yL & \cos k_yL & 0 & 0 \\ -\frac{\kappa_p}{k_x}\sin k_xL & -\frac{\kappa_p}{k_x}(1- \cos k_xL) & 0 & 0 & 1 & \frac{L}{\gamma^2} - \frac{L\kappa^2_p}{k_x^2} \frac{(1 -\sin k_xL)}{Lk_x} \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right) $$ where $$ \kappa_p = 1 + \left( \frac{eE_0R_0}{m_0c^2\beta^2\gamma^3}\right)^2 $$ $$ \begin{eqnarray} k_x^2 & = & \frac{1}{R_0^2} + \left( \frac{eE_0}{m_0c^2\beta^2\gamma^2} \right)^2 + \frac{eG}{cp_0} \\ k_y^2 & = & -\frac{eG}{cp_0} \end{eqnarray} $$ $$ \frac{1}{R_0} = \frac{eE_0}{\beta c p} + \frac{eB_0}{p} = \frac{E_0}{\beta c B\rho} + \frac{B_0}{B\rho} = \frac{1}{\rho_B} + \frac{1}{\rho_E} $$ and $$ G = G_B + \frac{1}{\beta c} G_E $$