one can take into account in the adiabatic approximation.

Another limitation is connected with the influence of value of transverse momentum arising from the electromagnetic field $E=|{\bf E}|$ of colliding (target) beam on the photon formation length. This value should be smaller than the characteristic transverse momentum transfer $m\sqrt{\zeta}$ in the photon emission process: $$\begin{aligned}
&& \frac{e E l_f}{m\sqrt{\zeta}} \sim \frac{\alpha
N_c}{(\sigma_z+\sigma_y)l \gamma_c} \frac{1}{m\sqrt{\zeta}}
\frac{4\varepsilon' \gamma_c \varepsilon_r}{\omega \zeta m^2}
\nonumber \\
&& \sim \frac{2 \alpha N_c}{(\sigma_z+\sigma_y)l}
\frac{1}{m\sqrt{\zeta}} \frac{2\varepsilon' \varepsilon_r}{\omega
\zeta m^2}=\frac{4 \alpha N_c \gamma_r \lambda_c^2
\varepsilon'}{(\sigma_z+\sigma_y)l \zeta^{3/2} \omega} \ll 1,
\label{2.2}\end{aligned}$$ here $\alpha = 1/137$, $N_c$ is the number of particles in the target beam, $\sigma_z$ and $\sigma_y$ are the vertical and horizontal transverse dimensions of target beam. Note that the ratio $\gamma/l$ is the relativistic invariant. This condition can be presented in invariant form $$\frac{2 \chi}{u \zeta^{3/2}} \ll 1,
\label{2.3}$$ where $\displaystyle{\chi=\frac{\gamma }{E_0}|{\bf
E}_{\perp}+{\bf v}\times {\bf H}|,~
u=\frac{\omega}{\varepsilon'},~ E_0=\frac{m^2}{e}=1.32 \cdot
10^{16} {\rm V/cm}}$. Since the main contribution to the spectral probability of radiation gives angles $\vartheta \sim 1/\gamma$ ($\zeta \sim 1$) this condition takes the form $\chi/u \ll 1$. For the case $\chi/u \gg 1$ the condition (\[2.3\]) can be satisfied for the large photon emission angles $\zeta \simeq
