1000 Solved Problems in Modern Physics

9.3 Solutions 533

9.108 The CMS energy is calculated from the invariance ofE^2 −p^2

E∗^2 =E^2 −p^2 =

(

Ep+Ee

) 2

−

(

pp−pe

) 2

=E^2 p−pp^2 +Ee^2 −p^2 e+ 2

(

EpEe+pppe

)

≈mp^2 +me^2 + 4 EpEe

Sincemp<<Epandme<<Ee

E∗≈

√

4 EpEe=

√

4 × 820 × 30 =314 GeV

Note that the HERA accelerator is different from other colliders in that the

energy of the colliding particles (protons and electrons) is quite asymmet-

rical. It has been possible to achieve high momentum transfer in the CMS

(20,000 GeV^2 ), necessary for the studies of proton structure.

9.109 Total CMS energy,E∗≈[2E 1 E 2 (1+cosθ)/2]^1 /^2
SubstitutingE 1 =20 GeV,E 2 =300 GeV, andθ= 100
we findE∗=109 GeV
If the same energy (E∗=109 GeV) is to be achieved in a fixed target
experiment, then the electron energy in the lab-system would be
(2EM+M^2 +m^2 )^1 /^2 =E∗
NeglectingM^2 andm^2
E=(E∗)^2 / 2 M=(109)^2 /(2× 0 .94)≈ 6 ,300 GeV