wakefield

K.Yokoya derived a theoretical expression of wakefield induced by the collimators, which shall be compared to a measurement. The emittance growth due to the wakefield is expressed by,  

$$\Delta \varepsilon_y = ( {\Delta y^2 \over{\beta_y}} ) \cdot... ...2}} + {L \over{a^3}}) + 0.69 {\theta \over{a \sigma_z}}\biggr]$$ (3)

where $\lambda \equiv 1.0 / \mu_o c \rho = 1/ 120 \pi \rho$ is the resistive depth of material of the collimator, and $\rho$ is the conductivity, and N, $\varepsilon_y$, $\sigma_y$, $\sigma_z$ are the intensity/bunch, (vertical) invariant emittance, vertical beam size, bunch length of the beam, respectively. r_e is the electron classical radius. The geometrical parameters of the collimator are displayed in Fig.3.

  

Figure 3: Geometry of a collimator.

There is the optimal angle ($\theta_{opt}$) minimizing the $\Delta$, that is, $(2 \lambda \sigma_z)^{1/4} / a^{1/2}$. With this angle and $\beta_y$ instead of $\varepsilon_y$ and $\sigma_y$, the $\Delta$ is,  

$$\Delta = 1.77 N r_e {\beta_y \over{\gamma}} \biggl[ \sqrt{\l... ...\over{a^3}} + \sqrt{2} {\theta_{opt} \over{a \sigma_z}} \biggr]$$ (4)

where $\gamma$ is the Lorentz factor, E_beam /m_e.

The emittance growth should be less than 1% so that the total growth due to 6 collimators is less than 6%. Assuming that the surface of the collimator is plated by copper, the aperture (a) of the collimator must be 250 (204)$\mu$m corresponding to 70.4 (56.9) $\sigma_y$ with the length of 20 (10)cm, N = 10¹⁰/bunch $\sigma_z$=80$\mu$m and $\beta_y$=400m at E_beam=750GeV, where the optimal angle ($\theta_{opt}$) is 1.05^o (1.17^o) .