The maximum current the target can take is limited by the beam heating
of the target. The magnetization of a material drops with temperature,
going to zero at the Curie's point (940C° for supermendur).
Heating by a Beam Pulse
A short beam pulse (0.5 ps at CEBAF) heats the area it overlaps with. The
heat flow is negligible
during this period. The temperature change can be calculated as: ΔT=dNe/ds·α/CP ,
where
ΔT is the temperature jump,
dNe/ds=d²N/dx/dy is the beam pulse profile in e-/cm²,
α=1.5MeV/(g/cm²) = 2.4·10-13 J/(g/cm²)
- the energy deposit by one minimum ionizing particle,
CP is the specific heat of the target material,
CP=0.4J/g/K for iron.
The effect depends neither on the target thickness nor on the target
angle with respect to the beam.
The beam profile is presented by a Gaussian, as d²N/dx/dy=Ne/(2·π·rb²)·exp(-(x²+y²)/2/rb²).
The number of electrons in the pulse is Ne = Ibeam/1.6·10-19/F, where:
Ibeam is the beam current in A,
F is the repetition rate.
At CEBAF:
rb is 30-100 µm - the intrinsic beam size, the fast raster can increase
the effective size;
F=500 MHz
Ne(max)=1.25·106 at Ibeam=100 µA
We obtain:
ΔTmax=0.013 K / pulse
for rb=30 µm and at x=y=0
This effect of one pulse is negligible.
Heating by a continuous beam
The beam gradually heats up the foil, a thermal wave is propagated across the foil
surface, until heat dissipation compensates the beam heating.
Here we assume that a thin foil is normal to Z
and the problem is 2-dimensional.
The heat equation, taking into account the heat conduction, radiation, and the
beam heating, looks like: ∂T/∂t = ∇²T·κ/(ρCP) -
2·σε((T+To)4-To4)/(ΔzρCP) +
Bfluxα/CP ,
where
κ is the thermal conductivity, κ=0.75W/cm/K for iron;
ρ is the density, ρ=7.87 (8.14) g/cm³ for iron (supermendur);
σ=5.67·10-12 W/cm²/K4 is the Stefan-Boltzmann constant;
ε is the foil emissivity. It depends on the surface structure and may range from 0.05 for
polished material to 0.8 for rough surfaces.
We assume a lower limit for emissivity of 0.1. The factor 2 in the
radiation term is used for two surfaces of the foil;
To=300 K is the outside temperature;
Δz is the foil thickness;
Bflux=d³Ne/ds/dt= is the density of the
beam flux in e-/cm²/s.
This term can take into account the
angle between the foil and the beam β:
Bflux = Ibeam/1.6·10-19/(2·π·rb²)·exp(-(x²·sin²β
+y²)/2/rb²). The full power released in the foil is ∝1./sin β;
α=1.5MeV/(g/cm²) = 2.4·10-13 J/(g/cm²)
- the energy deposit by one minimum ionizing particle;
CP is the specific heat of the target material,
CP=0.4J/g/K for iron.
We solved this equation numerically, on a grid with a variable cell size.
Heating during a short time period
Let us consider a foil at a room temperature To.
At the moment t=0 the beam starts. For a certain time period
the heat wave does not reach yet the foil boundaries. During this period
the heating
does not depend on the boundary conditions or the size of the foil
(if it is large enough).
The effect of a 50 μA beam of different spatial distributions
is shown on this plot.
Radiation starts to play a role when a large foil area is heated.
After 1 second the width of the heated spot is about 1 cm.
A fast raster can considerably reduce the heating for the first millisecond
of the beam pulse. In this case a simple 1-dimensional raster was simulated -
a circle of a 1 mm diameter.
At CEBAF, one can use a pulsed beam with a pulse length of 0.1-1ms and
a repetition rate of 30-120 Hz. This provides a low average beam current, while
having the "regular" current in the pulse (i.e. the current used in the
experiment). Let us consider using a foil perpendicular to the beam,
magnetized in a high field of 4 T (a clone of Hall C polarimeter).
The maximum temperature of the foil during the first beam pulse
is shown on this plot.
With a 1.4×1.4mm² fast raster the maximum foil temperature increases
by about
ΔT(t=1ms)max=12K;
The average beam current is <Ibeam>=1.5 μA.
The effect of this current is estimated analytically, assuming
an axial symmetry:
Flat beam density inside a spot r<RR = 0.7 mm
Room temperature at Rfoil = 1 cm
Neglecting the radiation, the final temperature at the foil center is:
With the raster, the total temperature rise of about
ΔTmax=25K;
should cause the foil depolarization of less than 0.25%.
With a 12 μm thick target at 20° to the beam,
the coincidence Moller counting rate at 0.850 GeV was about 40 kHz
at Ibeam = 0.3 μA. The LG single arm counting rates were ∼ 1 MHz,
while the aperture counters rates were ∼ 2 MHz. The projected rates will be
larger by a factor (50μA/0.3μA)·(1μm/12μm·sin(20°))=4.75
We can reduce the single rates by about a factor of 2 with a collimator,
which would reduce the coincidence rate by a factor of ∼3. Another
factor of 2.8 should be gained by upgrading the aperture counters and
a part of the electronics used.
About first 150μs of the pulse should not be used for the measurement
because of various transitional processes.
The time needed for a 1% absolute statistical error measurement will be (the
foil polarization is 0.08 and the analyzing power is 0.76): t = 1/(0.01·0.08·0.76)²/(40·10³Hz·4.75/3.*0.85ms*30Hz) = 28 min