Faraday Cup Temperature

Calculating the temperature of a Faraday Cup Rod

Number of particles per second hitting one rod

Assume electron beam parameters at FC location are:

Frequency:     f=300 Hz
Peak current:  I=3 Amps
Pulse width:   t= 50 ps
Beam energy:   E=45 MeV

The number of electrons per second at FC location is:

[math] N = \frac {Q} {e} = \frac {f I ∆t} {e} = \frac {(300\ Hz)(3 A) (50\ ps)} {1.6*10^{-19}\ C} = 0.28*10^{12} [/math]

Power deposited in 1 FC rod

Assume the whole power of beam is absorbed by just one Faraday Cup rod. For 45 MeV electron beam this power is:

[math] P = (0.28*10^{12}\ \frac{e}{sec}) \dot (44\ MeV) \dot  (1.6*10^{-19} \frac{J}{eV})[/math]

The energy deposited per pulse:

[math] (0.0061 \frac {MeV}{electron})(15.625*10^9 \frac {electrons}{pulse}) = 95.3125*10^6 \frac {MeV}{pulse} [/math]

The energy deposited per second:

[math] (95.3125*10^6 \frac {MeV}{pulse})(300 \frac {pulses}{second}) = 28.6*10^9 \frac {MeV}{second} [/math]

Calculating the temperature increase

The power deposited in 1/2 mil Al is:

[math] P = (28.6*10^{15} \frac {eV}{second})(1.6*10^{-19} \frac {C}{eV}) = 4.575*10^{-3} W [/math]

Stefan-Boltzmann Law (Wien Approximation) says [math] P = (0.924)(Area)(\sigma)(T^4) [/math]

Solving for Temperature and taking into account the two sides of the converter we get:

[math] T^4 = \frac {P}{(0.924)(2A)(\sigma)} [/math]

where [math] \sigma [/math] is the Stefan-Boltzmann constant, [math] \sigma = 5.67*10^{-8} \frac {W}{m^2 K^4} [/math]. Assume a beam spot diameter on the converter surface of 5mm, or an area of [math] A = 19.62 mm^2 = 19.62*10^{-6} m^2 [/math].

Plugging in the numbers we see that the temperature will increase [math] 217.2 K [/math]. Now, adding in the temperature of the converter at room temperature we get :

[math] T = 300 + 217.2 = 517.2 K[/math]

The melting temperature of Aluminum is [math] 933.5 K [/math].

Conclusion

An Aluminum converter that is 1/2 mil thick being struck by a 44 MeV electron beam with a 50 picosecond pulse width, 300 Hz rep rate, and 50 Amp peak current is found to be safe from melting.

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