Subatomic Physics

14 Accelerators
where h-bar, or Dirac’s �, is
λ = λ
2π
�
= , (2.2)
p
� = h
2π =6.5821 × 10−22 MeV sec. (2.3)
As is known from optics, in order to see structural details of linear dimensions d, a
wavelength comparable to, or smaller than, d must be used:
The momentum required then is
λ ≤ d. (2.4)
p ≥ �
. (2.5)
d
To see small dimensions, high momenta and thus high energies are needed. As
an example, we consider d = 1 fm and protons as a probe. We shall see that a
nonrelativistic approximation is permitted here; the minimum kinetic energy of the
protons then becomes, with Eq. (2.5),
Ekin = p2
2mp
= �2
. (2.6)
2mpd2 It is straightforward to insert the constants � and mp (see PDG.) However, we
shall use this example to compute Ekin in a more roundabout but also more convenient
way: Express as many quantities as possible as dimensionless ratios. Ekin
has the dimension of an energy, as does mpc2 = 938 MeV. The kinetic energy is
consequently rewritten as a ratio:
Ekin 1
=
mpc2 2d2 � �2 �
.
mpc
The quantity in parentheses is just the Compton wavelength of the proton
λp = �
mpc
so that the kinetic energy is given by
�c 197.3 MeV fm
= = = 0.210 fm (2.7)
mpc2 938 MeV
Ekin 1
=
mpc2 2
� �2 λp
=0.02. (2.8)
d
The combination �c will be found very useful throughout the text. The kinetic
energy required to see linear dimensions of the order of 1 fm is about 20 MeV.
Since this kinetic energy is much smaller than the rest energy of the nucleon, the
nonrelativistic approximation is justified. Nature does not provide us with intense