Period for 200kV Helium 1 plus in DA

 The difference between calculated one and experimental one

calculation

For this calculation, at first glance, it is as simple as C0/v. Yes, you can say that only if:

• correct circumference of the ring
• correct rest mass of Helium
• properly choose the right constant for light speed, proton mass, neutron mass, electron charge
• the kinetic energy for the beam is precisely as 200 kv

After I choose the  following parameters(as used in harada's table calculation)

• C0 = 37.71 m
• const double lightSpeed = 2.99792458e8 m/s
• const double electronCharge = 1.60217646e-19 C
• const double protonMass = 1.672621e-27 kg
• const double NeutronMass = 1.674927e-27 kg

for 200kv Helium 1+, circumference of the ring taken as 37.71m, 

lightSpeed = 2.99792458*10^8;

electronCharge=1.60217646*10^-19;

protonMass=1.672621*10^-27;

NeutronMass=1.674927*10^-27;

HeliumRest = 2*(NeutronMass+protonMass)*lightSpeed^2/electronCharge

ringLength=37.71;

extV=2*10^5;

gamma=extV/HeliumRest+1

beta=Sqrt[1-1/gamma^2]

T=ringLength/(beta*lightSpeed)

Output:

 beta = 0.0103197
T=12.189 us

if calculated with (as in Harada's "TsGEN3_20111025"):

•  m0 = A * 1.6726231e-27 
• A = 4
• gamma = 1.0 + kinetic_energy * e / ( m0 * c * c )
• beta = sqrt( 1 - 1/(gamma*gamma) )

the output is the same.

energy error

if we assume some energy differece at the injection, for example, 201kV, then the result is
T=12.1586 us

method error and ring length error

if we use atomic weight of helium as 4.002602 and atomic mass unit corresponding to energy in eV as 931.494061×10^6, the result is
T=12.1446 us
further more, if the change the ring length to be 37.75(4 cm longer)
T=12.1575 us

experiment

However, after doing FFT analysis to the bunch signal from the experimental data, the period is(2011116,1206,1222),:

T_experiment=12.1579 us

the difference is about: ( T - T_experiment )/T_experiment = 0.26%. This is a difference of 31 ns. Provided that the rise time of our Vbb is around ~80, or ~100 ns, in the experiment with Vbb this might be a problem.(in fact it may be good for commissioning, because the bunch tends to  "arrive later" when the B field is rising without acceleration to the beam.

others

I notice that for the 20111116 data, we did the experiment for different bunch length and different beam current. But the FFT analysis suggests that the period is changing as time goes by. This may suggested that the voltage is not so consistent with the B-field.

A better way is to do FFT to the part of the data ( for example, every 5 turns) until the time end(for example, 4ms as we usually use), then we can get a T changing along the time.

Strictly speaking, we cannot keep the extraction energy precisely every time we do the beam commissioning. However, as we can see by different date, the period given by the experiment is somehow repeat itself. With this we can safely say they're unnoticeable difference on the injection energy.

Note that if the injection energy is different from the what it needs to be with the B field, the period will change as time goes on.

conclusion at this momentum

as discussed above, use the atomic weight and a length of 37.75m is closer to the experiment result, if the result is reliable.