in the interest of maximizing luminosity, the average beam current should
be maximized. if one assumes an exponential beam current fall-off rate
such as:
I = I_0 * exp(-t/lambda)
then it can be shown with some basic calculus that the average beam
current is given by the following:
I_avg = - lambda * exp(-t_hvup/lambda) * (I_stop - I_0)
/ [t_hvdown + t_fill + t_dead - lambda * ln(I_stop/I_0)]
where:
I_avg = average beam current
I_stop = current you choose to stop the data-taking at
I_0 = maximum beam current
lambda = beam lifetime
t_hvup = time to ramp up HV after injection maximum
t_hvdown = time to ramp down HV before injection begins
t_fill = time for current to go from 0 to beam current maximum
t_dead = time after HV ramp down but before beam filling begins
(i welcome someone to check this. note that the "exp(-t_hvup/lambda)"
factor arises because we don't start taking data until the HV has ramped
up AFTER the injection maximum is reached.)
anyway, assuming the above equation is correct (or about so; i don't think
we're quite ready to worry about lambda changing with run-time), i have
calculated the beam current at which data taking should be stopped
(I_stop) in order to maximize the average current. note that i assumed
the following:
t_hvup + thvdown + tfill = 2 minutes
from my shift-taking experience, i know that this is a good estimate.
these stop-current values are listed below as functions of I_0 and
lambda, both of which will switch depending on the target, the beam
behavior, a sneeze in japan.
maybe, at the beginning of each shift, the shift-takers should do a few
fills, find out what the beam maximum is for the night and get an estimate
for lambda, and then set the fill current accordingly.
i have taped a copy of the results below to one of the computers in the
counting bay. if my calculations are wrong, tear it down; i don't care
(personal pride was lost a long time ago).
aaron
Imax = maximum beam fill current (in mA)
lamb = beam lifetime (in minutes)
Istp = current to refill at in order to maximize average current (in mA)
Iavg = average current if you refill at Iavg (in mA)
---------------------------------------------------------------------------
Imax || lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg
70 || 10 42 40 | 12 44 42 | 14 46 44 | 16 47 45
75 || 10 45 43 | 12 47 45 | 14 49 47 | 16 50 49
80 || 10 48 46 | 12 50 48 | 14 52 50 | 16 53 52
85 || 10 51 49 | 12 54 51 | 14 55 53 | 16 57 55
90 || 10 54 52 | 12 57 54 | 14 59 57 | 16 60 58
95 || 10 58 55 | 12 60 57 | 14 62 60 | 16 63 61
100 || 10 61 58 | 12 63 60 | 14 65 63 | 16 67 65
105 || 10 64 60 | 12 66 63 | 14 68 66 | 16 70 68
110 || 10 67 63 | 12 69 67 | 14 72 69 | 16 73 71
115 || 10 70 66 | 12 72 70 | 14 75 72 | 16 77 74
120 || 10 73 69 | 12 76 73 | 14 78 75 | 16 80 78
125 || 10 76 72 | 12 79 76 | 14 81 78 | 16 83 81
130 || 10 79 75 | 12 82 79 | 14 85 82 | 16 87 84
135 || 10 82 78 | 12 85 82 | 14 88 85 | 16 90 87
140 || 10 85 81 | 12 88 85 | 14 91 88 | 16 93 91
145 || 10 88 83 | 12 91 88 | 14 94 91 | 16 97 94
Imax || lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg
70 || 18 48 46 | 20 49 47 | 22 49 48 | 24 50 49
75 || 18 51 50 | 20 52 51 | 22 53 52 | 24 54 53
80 || 18 55 53 | 20 56 54 | 22 56 55 | 24 57 56
85 || 18 58 56 | 20 59 58 | 22 60 59 | 24 61 60
90 || 18 61 60 | 20 63 61 | 22 64 62 | 24 64 63
95 || 18 65 63 | 20 66 64 | 22 67 66 | 24 68 67
100 || 18 68 66 | 20 69 68 | 22 71 69 | 24 72 70
105 || 18 72 70 | 20 73 71 | 22 74 72 | 24 75 74
110 || 18 75 73 | 20 76 75 | 22 78 76 | 24 79 77
115 || 18 78 76 | 20 80 78 | 22 81 79 | 24 82 81
120 || 18 82 80 | 20 83 81 | 22 85 83 | 24 86 84
125 || 18 85 83 | 20 87 85 | 22 88 86 | 24 90 88
130 || 18 89 86 | 20 90 88 | 22 92 90 | 24 93 91
135 || 18 92 90 | 20 94 92 | 22 95 93 | 24 97 95
140 || 18 96 93 | 20 97 95 | 22 99 97 | 24 100 98
145 || 18 99 96 | 20 101 98 | 22 102 100 | 24 104 102
Imax || lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg | lamb Istp Iavg
70 || 26 51 50 | 28 51 50 | 30 52 51 | 32 52 51
75 || 26 54 53 | 28 55 54 | 30 56 55 | 32 56 55
80 || 26 58 57 | 28 59 58 | 30 59 58 | 32 60 59
85 || 26 62 60 | 28 62 61 | 30 63 62 | 32 64 63
90 || 26 65 64 | 28 66 65 | 30 67 66 | 32 67 66
95 || 26 69 68 | 28 70 68 | 30 70 69 | 32 71 70
100 || 26 73 71 | 28 73 72 | 30 74 73 | 32 75 74
105 || 26 76 75 | 28 77 76 | 30 78 76 | 32 78 77
110 || 26 80 78 | 28 81 79 | 30 81 80 | 32 82 81
115 || 26 83 82 | 28 84 83 | 30 85 84 | 32 86 85
120 || 26 87 85 | 28 88 86 | 30 89 87 | 32 90 88
125 || 26 91 89 | 28 92 90 | 30 93 91 | 32 93 92
130 || 26 94 92 | 28 95 94 | 30 96 95 | 32 97 96
135 || 26 98 96 | 28 99 97 | 30 100 98 | 32 101 99
140 || 26 102 100 | 28 103 101 | 30 104 102 | 32 105 103
145 || 26 105 103 | 28 106 104 | 30 107 106 | 32 108 107
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