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## Q. 14.11

How much energy would a fixed-target proton accelerator require to match the energy available in the LHC for a $p+\bar{p}$ reaction?

Strategy The energy available in the LHC is the sum of the colliding beams, or 14 TeV. We use Equation (14.10) to determine the kinetic energy K needed for a fixed-target experiment.

$E_{ cm }=\sqrt{\left(m_{1} c^{2}+m_{2} c^{2}\right)^{2}+2 m_{2} c^{2} K}$ (14.10)

## Verified Solution

If we insert 14 TeV (14000 GeV ) into Equation (14.10), we can solve for K, the kinetic energy.

$14000 GeV =\sqrt{[2(0.938 GeV )]^{2}+2(0.938 GeV ) K}$

The second term on the right-hand side will have to be much larger than the first term, so we can neglect the first term. We then solve for K.

$K=\frac{(14000 GeV )^{2}}{2(0.938 GeV )}=1.0 \times 10^{8} GeV =1.0 \times 10^{17} eV$

Such a fixed-target accelerator cannot currently be constructed.