260
An Introduction to Beam Physics
ϕ + πL/ (β 0 λ) increases. From the fact that
ϕ +
πL
β 0 λ
−
ϕ −
πL
β 0 λ
= ϕ +
πL
β 0 λ
−
πL
β 0 λ
− ϕ
= 2ϕ < 0,
we reach the same conclusion that cos (ϕ + πL/ (β 0 λ)) > cos (ϕ − πL/ (β 0 λ)) .
Although we have β 1 < β 2 , yet β 2 /β 1 is usually much smaller than cos(ϕ +
πL/(β 0 λ))/ cos(ϕ − πL/(β 0 λ)). As a result, the net effect is defocusing. The
remedy in the early dates was metal foils or grids placed on the entrance of
the drift tubes (exit of the accelerating gap) to remove the defocusing force
as shown in Fig. 1.13. Nowadays, quadrupole magnets are placed inside the
drift tubes to provide transverse focusing.
At higher energy, the particles become relativistic and the contribution of
the magnetic field has to be taken into account. Yet the matter is simplified
somewhat by the fact that the difference between β 1 and β 2 can be neglected.
Assuming also that r remains a constant in the cavity, we obtain
Δp r = −
qr
2β 0 c
E 0
cos
πL
β 0 λ
− ϕ
− cos
πL
β 0 λ
+ ϕ
+
qωr
2c 2 E 0
L/2
−L/2
dz sin
2πz
β 0 λ
+ ϕ
= −
qr
β 0 c
E 0 sin
πL
β 0 λ
sin ϕ +
qβ 0 r
c
E 0 sin
πL
β 0 λ
sin ϕ
= −
qrE 0
β 0 γ 2
0 c
sin
πL
β 0 λ
sin ϕ
= −
πqE 0 T Lr
β 2
0 γ 2
0 cλ
sin ϕ.
Again, for the stable phase of −π/2 < ϕ < 0, the net effect is defocusing. It
is worth noting that, for ultra-relativistic particles, this effect goes away. As
β 0 → 1, the defocusing from the electric field is canceled by the focusing from
the magnetic field.