Home work Tuesday June 22

1. Home work Tuesday June 22 • In the TOTEM experiment at the CERN LHC, the total pp cross section will be measured at 7 TeV per beam; to carry out this experiment, we have to detect the particles scattered in the forward direction with very small scattering angle (14 mrad). This implies that the divergence is very small, while the beam size at the IP must be very big (b*~1000 m); the detector has to be located far away from the IP as indicated below. There is a BPM on either side of the IP and at the ‘Roman-pot’ detector; the strengths of the quadrupoles next to the IP and next to the Roman pot can be varied; there is also a corrector magnet on either side of the IP. Devise a way of measuring the R12 matrix element from the IP to the detector (this element is required to infer the scattering angle of detected particles). hint: not all the available elements are needed to measure R12

2. To optimize luminosity at a circular collider, like RHIC or the LHC, it is desirable to smoothly reduce the beta function at the collision point without perturbing machine operation. This can be achieved with a beta-star tuning knob, which changes b* but keeps other parameters constant. Consider a simplified case in one plane only with two symmetrically placed quadrupoles on either side of the IP and a nominally symmetric optics. Construct a b* tuning knob by which, in linear approximation, the four quadrupole strengths are varied in a constant relative ratio, without changing a* and so that b* is changed by a specified amount Db*, while the b and a functions outside the interaction region (‘1’) remain constant hint 1: this is fulfilled if for constant initial b and a, a*=0 and Da*=0 at the IP and the quadrupoles on either side of the IP are excited symmetrically – hence it is sufficient to study the Twiss response matrix on one side of the IP. hint 2: derive the coefficients a,b,c,d in Db*=a Dk1+b Dk2 Da*=c Dk1+d Dk2 and then solve for (Dk1,Dk2) that fulfill the requirement • A ‘primitive’ way of measuring the beam size of a stored proton beam is by slowly moving a scraper into the beam. Assume that the transverse beam profile is Gaussian and derive an expression describing the fraction of surviving beam as a function of beam closed orbit x0, the scraper position xs and the horizontal beam size sx.

3. Computer lab, Tuesday 22 June 2004 BCM consider a simple storage ring consisting of a defocusing quadrupole and a focusing quadrupole and two bends (or drifts) use L=2 m, K=0.5 m-1 hints: 2cos(2pQ)=R11+R22 R12=bsin(2pQ) compute tune and b at focusing quadrupole compute tune shift as a function of quadrupole strength change DK over a range -0.1…+0.1 compute b as and plot vs. DK compute b as and plot vs DK