Monte Carlo Simulation for a New Proton Polarimeter

Transcription

1 Monte Carlo Simulation for a New Proton Polarimeter A thesis submitted in partial fulfillment of the requirement for the degree of Bahelor of Siene with Honors in Physis from the College of William and Mary in Virginia, by Bradley M. DeBlois Aepted for (Honors) Advisor: Dr. C.F. Perdrisat Dr. Jeff Nelson Dr. Robert Lewis Dr. John Delos Williamsburg, Virginia May 2006

2 Abstrat An experiment in HallC of Jefferson lab will determine the ratio of the elasti eletromagneti form fators G Ep /G Mp of the proton by measuring the reoil proton polarization omponents. This will require haraterizing the polarization of the reoil proton in the ep elasti sattering. The polarization of the proton is obtained by re-sattering the proton in a seondary target, the analyzer. A Monte Carlo simulation was written to produe distributions of the sattering angles, θ and φ, to be used to test the analysis ode and study systemati asymmetry. The azimuthal angle distribution reveals the amount and diretion of the proton polarization. A tehnique to remove the main soure of asymmetry was tested. The bakground physis, properties of the simulation, resulting distributions, and methods for orreting systemati asymmetries are disussed. i

3 Aknowledgements Many thanks to... My advisor, Professor Perdrisat, for the ountless hours he spent with me. I was luky to have an advisor willing to make time for me and explain things three times when I needed it. Professor Brash of Christopher Newport University for answering s on weekends and at night. His help on this projet was essential. The C-GEP group at Jefferson Lab for allowing me to take a small part in their ongoing experiment. ii

4 Contents 1 Introdution 1 2 Form Fators 1 3 Polarization 3 4 Polarization Transfer Method 5 5 The New Foal Plane Polarimeter Basi Operation of a Polarimeter The New Foal Plane Polarimeter Drift Chambers Basi Operation Wire Chambers for the New Polarimeter The Simulation Coding the Polarimeter Geometry Drift Distane Sattering Angles Analyzing the Data Single vs Multiple trak Events The Azimuthal Angle and Asymmetry The Azimuthal Angle and the Cone Test Conlusion 29 A Relevant Code 32 A.1 Soure Code for the Geometry A.2 Soure Code for Calulations iii

5 List of Figures 1 Classial ross setion Spin-orbit Asymmetry Coneptual Polarimeter The New Polarimeter Drift Cell Drift Geometry Drift Distane for one wire plane Cartesian Angles Theta distribution Phi distribution Theta Distribution for Single Traks Multipliity of Theta of Varying Multipliity Theta Distribution for Single Traks Theta Distribution for Multiple Traks Spin Preession Phi Distribution Cone Test Logi Cone Test Geometry Phi Distributions-Cone Test Theta Distributions: Full and Rejeted Comparing Cone Tests Elliptial Cone Test Geometry List of Tables 1 Polarimeter Dimensions iv

6 2 Front Analyzer Rear Analyzer Cone Test Results, the statistial unertainty on all numbers is about v

7 1 Introdution The study of Nulear Physis is dediated to understanding the behavior of strongly interating matter in terms of its basi onstituents. A key omponent toward reahing this goal is the haraterization of the struture of the nuleon. Important aspets of the struture of the nuleon are the form fators of the proton and neutron, G Ep, G Mp, G En and G Mn sine these form fators are diretly related to the harge and urrent distributions inside the nuleon. Measurements of the form fators of nuleons and the nulei were begun in the 1950 s by R. Hofstadter [1]. The planned experiment measures the ratio of the elasti eletromagneti form fators, G Ep /G Mp. An experiment suh as this relies heavily on omputer simulation. To measure the ratio of the form fators, a great amount of data is olleted and must be analyzed. An effetive way of building and testing an analysis program is to simulate data that an be tested against the analysis program. 2 Form Fators If the goal of a sattering experiment is to learn about the distribution of matter inside the nuleus, it is neessary to use a partile whih will enter the nuleus. Eletrons with wavelengths on the same order of magnitude as the nuleons, or smaller, are needed to probe the nuleus. This wavelength is governed by the de Broglie wavelength of the eletron λ = h p (1) Therefore, these eletrons will have small wavelengths and hene large momentum of the order 2 GeV for a desired wavelength of 0.1 fm. These fast eletrons will interat with the nulear matter through the eletromagneti field and yield information on the distribution of harge in the nuleus. Information about the distribution of matter inside the nuleus is typially orga- 1

8 Figure 1: Classial ross setion nized into a quantity known as a differential sattering ross setion, written σ(θ, φ). Let (θ, φ) be the polar and azimuthal angles that speify the final diretion of motion and dω be an infinitesimal solid angle around the ray at (θ, φ). Then the ross setion is the number of partiles sattered per seond into dω divided by the inident flux [2]. In other words, the differential ross setion is the area through whih a partile an be sattered and is determined by the struture of the sattering element. Analyti forms of the ross setion for partiles sattered by a point harge an be obtained. For high energy eletrons sattering from a point, the ross setion is known as the Mott ross setion and has the form [3] σ M (θ) = ( ) dσ = dω Mott α 2 4E 2 beam sin4 θ 2 E e E beam os 2 θ 2 However, sattering from the nuleus or a nuleon will not look like sattering from a point harge. Thus deviations from the ross setion for a point harge reveal information regarding the nulear struture, speifially the spaial distribution of harge. The equation above refers to the Mott sattering, σ M. The atual ross (2) 2

9 setion will be the Mott ross setion multiplied by a form fator σ(θ) = σ M (θ)[f (q)] 2. (3) The form fator F(q) is related to the harge density distribution ρ(r) by a Fourier transform, F (q) = 4π q ρ(r)sin(qr)rdr (4) This form fator is the quantity that represents the deviations from the point harge sattering and thus the quantity that reveals features of the nulear struture. For elasti sattering from the spin 1/2 proton, two form fators are required: F 1, the Dira form fator, and F 2, the Pauli form fator [3]. The first, F 1, desribes the non-point like nature of the eletri and magneti urrent distribution within the nuleus. The seond, F 2, desribes the non-point like nature of the distribution of the anomalous part of the magneti moment. The proton does not have the magneti moment of a Dira partile, but is 2.79 times larger. The reason for this anomalous part is found in its ompliated internal struture. These two form fators are related to the form fators of interest in this experiment by: G E = F 1 τκf 2 (5) G M = F 1 + κf 2 (6) where τ = Q 2 /4M 2 p, κ is the anomalous part of the magneti moment (in nuleon magnetons e h 2m p ) and M is the mass of the proton [3]. This experiment measures the ratio of the elasti eletromagneti form fators, G Ep /G Mp. 3 Polarization A beam of nuleons is polarized if the spins are not randomly oriented but have a preferred orientation. Let n + represent the number of partiles with a spin omponent 3

10 parallel to the preferred diretion and n represent the number of partiles with an antiparallel omponent, then the polarization is defined as P = n + n n + + n (7) If the beam is not polarized then n + n = 0 = P, and the sattering of the unpolarized beam will always be symmetri about the diretion of inidene [4]. However, if a fore ats preferentially on partiles with a given spin orientation then a beam will beome polarized. A fore that depends on spin will reate a left-right asymmetry. To illustrate how asymmetry an our, assume spin up protons are more likely to satter to the left than to the right. Then when sattering N protons, N + spin up protons will go to the left and N will go to the right, with N + > N and N = N + + N. The asymmetry in the sattering is defined as: A = N + N N + + N (8) Now, of the N inident protons, N + = N 2 (1+δ) will satter to the left and N = N 2 (1 δ) to the right so that in the absene of a spin dependene, N + = N = N/2; hene δ haraterizes the spin dependene of the sattering probability. For an inoming spin up proton A = N + N N + + N = 1 + δ (1 δ) 1 + δ + 1 δ And if the inoming proton is spin down the asymmetry beomes A = N + N = 1 δ 1 δ N + + N 1 δ δ = +δ (9) = δ (10) Therefore, measuring the asymmetry diretly determines the spin dependene of the sattering probability and produes opposite values for spin up and spin down protons. The main ause of this spin dependene is the oupling of the spin and angular momentum, s and L. 4

11 Figure 2: Spin-orbit Asymmetry Figure 2 shows two nuleons with a given spin orientation, up or down. It also shows the diretion of the angular momentum. As the nuleons approah the sattering enter, they are sattered if the spin and angular momentum are in the same diretion and not sattered if they are antiparallel. This phenomenon is given by the simplified yet illuminating equation P (θ) = L s 2 L (11) where P is the sattering probability, L is the angular momentum of the projetile relative to the target enter L = b v where b is the impat parameter, and s is the spin of the projetile. This gives P = 1 for L parallel to s and P = 0 for L antiparallel to s. 4 Polarization Transfer Method This experiment makes use of polarization and asymmetry to measure the form fator ratio, G Ep /G Mp. The G Ep /G Mp form fator ratio an be obtained from the reoil proton polarization transfer oeffiients of the ep e p reation. In this reation, the polarization of the eletrons is transferred to the reoil proton with two non zero omponents, P t perpendiular to, and P l parallel to the proton momentum in the sattering plane. The polarizations are: I 0 P n = 0 (12) 5

12 Together these equations give: I 0 P t = 2 τ(1 + τ)g Ep G M p tan θ e 2 I 0 P l = 1 (E beam + E e ) τ(1 + τ)g 2 M ptan 2 θ e M p 2 (13) (14) G Ep G Mp = P t E beam + E e tan θ e P l 2M p 2 (15) In this way, the form fator ratio an be alulated from a single measurement of the two reoil polarization omponents P t and P l, perpendiular and parallel to the proton momentum at the hydrogen target [5]. 5 The New Foal Plane Polarimeter 5.1 Basi Operation of a Polarimeter A polarimeter is an apparatus used to reonstrut the traks of sattered protons whih an then be used to alulate the polarization of the protons. A polarimeter has two main omponents, the analyzers and the detetors. Typially there are two detetors, usually wire hambers whih will be desribed in some detail in the next setion, followed by an analyzer blok, followed by two more detetors. The initial set of detetors defines the path of the inident proton. The proton is then sattered at some angle by the analyzer blok. The seond set of detetors defines the path of the sattered proton [6]. The inident and sattered paths reveal the sattering angles from whih polarization an be alulated. Refer to Figure The New Foal Plane Polarimeter In order to ollet as muh useful data as possible, the new polarimeter will use two sets of the analyzer/detetor ombination. This experiment will measure the polarization transferred to the reoil proton with a polarized eletron beam sattered 6

13 Figure 3: Coneptual Polarimeter by an unpolarized hydrogen target. The new foal plane polarimeter ollets data on the reoil protons. A magneti field direts the reoil proton around a 25 degree bend and through two foal plane drift hambers, sandwihing the foal plane where protons with the same momentum are foused to a single point. Downstream from the foal plane the protons enter the polarimeter and data is olleted on tens of millions of events. The inident trak of the proton is defined by the foal plane detetors. The proton is then sattered by the first analyzer. The outgoing partile is observed by two detetors whih define the sattered trak. In addition, these hambers also define the inident trak for the seond analyzer. The proton is then sattered by a seond analyzer and a seond set of detetors defines the final sattered trak. The detetors are wire hambers built speifially for this experiment and will be disussed later. The analyzers are bloks made of CH 2. The benefit of having onseutive analyzer/detetor ombinations is that more protons will be sattered and thus more data an be olleted. Due to the high energy of the inoming protons, about forty perent of them will not enounter any nulear sattering when traveling through the analyzer. By having two analyzers, the protons have two opportunities to satter. 7

14 Figure 4: The New Polarimeter 6 Drift Chambers 6.1 Basi Operation Drift hambers are the primary traking detetors used in high energy sattering experiments. Drift hambers are wire hambers divided up into drift ells. A drift ell has a sense wire in the enter (anode, at ground potential), and field wires left and right at negative potential, as well as athode planes above and below and also at negative potential, as illustrated in Figure 5. An ionizable gaseous mixture fills the drift ell. As a partile traverses the drift ell it ionizes atoms of the gas and the resulting eletrons drift toward the sense wire. In some ases, the eletrons have enough energy to ionize gas moleules as well. As a result, an avalanhe of ionization eletrons is produed. The partile also hits a sintillation ounter whih overs the entire area of the hamber. When the eletron hits the sintillator it starts a timer, the timer is 8

15 Figure 5: Drift Cell stopped when the drifting eletrons reah the sense wire [7]. In this manner the drift time is obtained. Figure 5 shows the drift ell. The dotted irle represents points that will have equal drift times. Assuming we know the drift veloity and the arrival time of the partile, the distane from the origin of the eletrons and the sense wire, the drift distane, is x = t0 where t 1 t 0 defines the drift time and u is the drift veloity. t 1 udt, (16) 6.2 Wire Chambers for the New Polarimeter In this partiular GEP-III Hall C experiment, a set of large wire hambers was built and delivered from Russia. The new foal plane polarimeter will make use of four wire hambers. Eah wire hamber has three separate wire planes designed in the manner explained above. However, eah wire plane has a different orientation. One plane is at +45 degrees, the enter plane is at zero degrees, and the third wire plane is at -45 degrees relative to the dispersive diretion of the spetrometer, the (x) diretion. Using three wire planes in different orientations provides three drift times and thus three drift distanes per hamber. This allows for a reonstrution of the path of the 9

16 partile that traveled through the hamber. The sense wires are spaed 2.0 m apart with a field wire in between. The wire planes are 1.6 m apart with a athode layer in between. The wires of the athode layer are 0.3 m apart. Thus the drift ell is 2.0 m by 1.6 m. The sense wires are 30 µm gold plated Tungsten and the field wires are 100 µm Beryllium and Bronze alloy. 7 The Simulation When performing an experiment that will ollet an immense amount of data, it is ruial to have a mehanism for prediting the results of the experiment so that one an gain experiene with the experimental data before atually running the experiment. Computer simulations are the preferred method for simulating experimental events. Computer methods an produe simulated data, that with areful onsideration of physial properties, an look idential to the experimental data. Thus the simulated data an be used as if it were real data with the benefit of knowing all of the properties of the data ahead of time. Using the physis simulation pakage GEANT, written speifially for high energy sattering experiments, real results of the ollision of the eletrons at the target an be simulated. This sends protons down the path and into the polarimeter. If GEANT is used to ode the properties of the polarimeter then drift times an be alulated. These drift times an then be passed to the analysis pakage, whih will reeive drift times from the atual experiment. The goal of the analysis pakage is to reonstrut the traks of the protons based on the drift times. After the analysis pakage does this for the simulated drift times, the reonstruted traks an be ompared to the known simulated traks and thus the orretness of the analysis pakage an be determined. A similar simulation was written for a prior Jefferson Lab experiment in Hall A. The ode for that experiment needed to be updated to meet the needs of the new 10

17 polarimeter. 7.1 Coding the Polarimeter Geometry The first step in updating the old simulation was to model the new polarimeter. The GEANT simulation pakage is written for high energy experiments and makes defining a pakage of detetors fairly easy. Defining the polarimeter inludes reating all elements in Figure 4. Two idential foal plane wire hambers are needed, two sintillators, two analyzing bloks, four wire hambers, the air spae in between eah element and a room to house the entire apparatus. GEANT allowed for easy reation of these elements. Table 1 shows the dimensions of eah omponent. Eah omponent was positioned Component x (m) y (m) z (m) foal plane wire hambers sintillators analyzers fpp wire hambers Table 1: Polarimeter Dimensions at the orret z-position with z=0 at the foal plane. Air volumes with z-dimensions that exatly fill the spae between the detetor elements were also added. The wire hambers are defined as boxes of air sine air behaves similarly to atual gas of the hambers, Ar-ethane. Developing the geometry of the wire hambers will be disussed when alulating drift distanes is disussed. The analyzers are defined as boxes of CH 2. Events are simulated at the target through GEANT kinematis simulation ode and the reoil protons appear at the polarimeter at z=0 or the foal plane. The protons travel through the polarimeter and satter at the analyzers. The sattering ours as a result of the interation between the proton and the nulei of the arbon 11

18 and hydrogen in the CH 2 analyzers and the amount and types of sattering will be disussed in depth later. As the proton moves through the polarimeter, the simulation reords the x,y,z position of the partile every time it enters or exits one of the omponents. Aessing this information enables one to build the traks of the protons, alulate drift distanes, and alulate sattering angles. Calulating and organizing this information produes several important distributions that will ultimately be used to alulate polarization. Examining these important distributions is ritial to ensure the auray of the simulation. The alulation and results of these distributions will be disussed at length in the following setions. 7.2 Drift Distane Reall that the drift time is the time it takes the eletron avalanhe reated by an inoming proton to reah the sense wire and the drift distane is the distane traveled. The drift time is an extremely important quantity sine it will be the data olleted in the atual experiment. The proess for produing simulated drift times is to use geometry to alulate drift distanes whih an then be onverted to drift times. Calulating the drift distane begins with defining the inident trak and alulating where it hits eah wire plane. From the ode, the x, y, and z oordinates are reorded eah time the trak enters or exits an element of the polarimeter. Define the entry and exit points of the hamber as P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) respetively. Think of these two points as defining a line in three dimensions with anonial equations x x 1 x 2 x 1 = y y 1 y 2 y 1 = z z 1 z 2 z 1. (17) The z position of a given wire plane is known so solving for x and y is simple x plane = (z plane z 1 )(x 2 x 1 ) z 2 z 1 + x 1 (18) 12

19 Figure 6: Drift Geometry y plane = (z plane z 1 )(y 2 y 1 ) z 2 z 1 + y 1 (19) where x plane and y plane are the (x, y) oordinates of the inident trak at the given wire plane. After alulating where the inident trak hit the wire plane, the next step is to figure out whih wire is losest to the trak sine that is the wire whih will register a hit. This is done through areful examination of the geometry of the wire planes. Eah wire of a wire plane is given a number and plaed at an appropriate position. An algorithm was written by Dr. Edward Brash, of Christopher Newport University, to determine whih wire number is losest to the alulated (x, y) point of the inident trak at the wire plane. Next, the drift distane, the shortest distane between the trak and the wire, is alulated. This is another three dimensional geometry problem involving two skew lines, the trak and the wire. The desired quantity is the losest distane between the two lines. Dr. Brash also wrote an algorithm for alulating this distane, the ode an be found in the appendix. In this manner drift distanes were alulated and their distributions observed. Figure 7 shows a sample distribution of drift distanes for one of the wire planes 13

20 Figure 7: Drift Distane for one wire plane in the first hamber. Drift distanes lose to 0 m are more likely and note the drop off as distane inreases. The largest possible drift distane given the dimensions of the drift ell is approximately 1.3 m whih the distribution also shows. 7.3 Sattering Angles The relevant variables will be analyzed by examining the spherial sattering angles θ and φ after sattering in eah of the analyzer bloks. These angles are the polar and azimuthal angles between the inident trak and the sattered trak. The method for obtaining these angles is as follows: alulate the artesian angles for the inident trak. Build a rotation matrix from those angles. Rotate the sattered trak by the rotation matrix. Now the sattered trak is in the same oordinate system as the inident trak and the spherial angles are the polar and azimuthal angles [8]. Consider the inident trak f: θ f is the angle between the projetion of the trak onto the xz plane and the z-axis, φ f is the angle between the projetion of the trak onto the yz plane and the z-axis, and ψ f is the angle between the projetion of the trak onto the yz plane and the trak. These three angles represent the artesian angles. 14

21 Figure 8: Cartesian Angles Using small angle approximations θ f = x/z (20) φ f = y/z (21) where x, y, and z are given by the diretion vetor of the inident trak. Also tan ψ f = tan θ f os φ f (22) Now the rotation should be broken into two steps, a rotation of the yz plane around the x-axis by an angle θ f and the seond a rotation by an angle ψ f. This rotation sets the inident trak along the z diretion. The new projetion of the sattered trak r is now [8]: r x r y r z = os ψ f 0 sin ψ f sin ψ f 0 os ψ f os φ f sin φ f 0 sin φ f os φ f r x r y r z (23) We an now alulate the spherial angles θ and φ 15

22 r θ = tan 1 x 2 + r y 2 (24) r z ) ( r φ = tan 1 x (25) r y The values of θ and φ for eah sattered event at eah analyzer is olleted. They are plaed into a histogram so that the θ and φ distribution for eah analyzer an be examined. Figures 9 and 10 are examples of the distributions. Figure 9: Theta distribution Figure 10: Phi distribution There is a lot to be learned about the simulation from the θ and φ distributions it produes. In general, θ determines the momentum transfer. The azimuthal angle, φ 16

23 is a partiularly important angle sine its distribution will determine the polarization. It is also a very important quantity that helps to refine and orret the simulation. 8 Analyzing the Data Building an appropriate simulation requires an iterative proess. A basi simulation is written, run, and the results are analyzed. Based on the initial results, the simulation is refined and made more omprehensive. For this partiular experiment the θ and φ distributions were repeatedly analyzed and the simulation was orreted aordingly. 8.1 Single vs Multiple trak Events As harged and heavy partiles like protons travel through matter, they undergo multiple Coulomb sattering in the eletri fields of the atoms/moleules, and strong nulear interation with the nulei/nuleons of these atoms/moleules (as well as muh weaker interation with the harges of the nulei/nuleons). Every proton traversing the analyzer undergoes multiple Coulomb sattering; if nothing else happens, they appear in the Coulomb peak at zero degrees, with a width inversely proportional to their momentum, and proportional to the square root of the thikness of material. If they do undergo nulear sattering, then they will be sattered as some finite angle, and Coulomb sattering will only slightly smear their θ and φ distribution. Initial θ distributions showed that only about thirty perent of the events were experiening nulear sattering and produed a single trak. This number was muh less than the expeted value of about sixty perent. Notie in Figure 11 that the Coulomb peak ontains a high perentage of the events and not enough sattered traks appear in the tail. In fat, there are only events sattered by a few degrees. The expeted sixty perent sattering is alulated from the probability of survival, e x/λ where λ is the mean free path in the sattering blok and x = ρd where ρ is 17

24 Figure 11: Theta Distribution for Single Traks the density of the analyzer and d is the thikness. For CH 2, λ = 57g/m 2 and x = = 51.7 so that e 51.7/ If forty perent are expeted to survive then a total of sixty perent should interat. It beame immediately apparent that limiting the simulation to only single trak events was not physially orret and produing inomplete results. At high energy the proton-nuleus interation is highly inelasti and the sattering results in one or several partiles being reated as in pp pnπ +, pp ppπ + π et. Also, the inident proton may satter from another nuleon before exiting the analyzer. In fat a single inident proton ould produe a hain of sattering within the analyzer. This effet is defined as sattering multipliity, suh that if the inident proton results in two partiles after sattering then it has multipliity two, three partiles after sattering results in multipliity three and so on. Figure 12 shows one manner in whih a multipliity of three ould our. In GEANT, traking the multiple sattered traks is simply a physial setting, thus multiple traks are easily observed. The diffiulty lies in seleting only the sattered partiles of interest. This was aomplished by traking only the partiles that experiened the smallest sattering angle θ. Adding multiple traks provided the data shown in Table 2 for the first hamber with 10,000 events where the first olumn 18

25 Figure 12: Multipliity of 3 multipliity # sattered % sattered higher Table 2: Front Analyzer is the multipliity, the seond olumn is the number of events that were sattered with the given multipliity and the third olumn is the perentage of the 10,000 events that were sattered with the given multipliity. Adding the perentages, 57 perent of events were sattered, very lose to the expeted value of sixty perent. For the seond hamber, only events whih experiened no nulear sattering in the first hamber are onsidered, for this partiular run 4600 events are onsidered as inident partiles to be analyzed by the seond hamber. Table 3 shows the result from these 4600 events. multipliity # sattered % sattered higher Table 3: Rear Analyzer 19

26 Adding up the third olumn shows that when inluding sattering multipliity, about 51 perent of the events experiened nulear sattering again lose to the expeted amount. Thus in eah analyzer, when multiple traks were allowed, the number of sattered events was lose to the expeted value of sixty perent. Figure 13 shows the θ distributions for the first two multipliities on the same plot as the full θ distribution. The distributions have been set so that the Y axis is a logarithmi sale so that it is easier to see the differene in the number of events in the Coulomb peak and tail. The highest plot, represented by the full line, is the full theta distribution. Notie how eah onseutive multipliity ontains fewer and fewer events. Figure 13: Theta of Varying Multipliity When multiple traks are inluded, the θ distribution looks muh nier. Compare Figures 14 and 15. Notie that there are many more events in the tail of the multitrak distribution (Figure 15) whih are the events of interest. Eah distribution is taken from events reorded from the front analyzer. Now that the orret number of analyzable events have been produed, attention is turned to the φ distribution. 20

27 Figure 14: Theta Distribution for Single Traks Figure 15: Theta Distribution for Multiple Traks 8.2 The Azimuthal Angle and Asymmetry The result of sattering in the polarimeter is an azimuthal asymmetry in the sattering distribution. Analyzing this asymmetry provides information on the polarization of the proton. The azimuthal angular distribution of the number of partiles sattered, N, is: N = N 0 (1 + A y (θ) P fpp n) (26) where N 0 is the number of partiles sattered in the absene of polarization, P fpp is the proton polarization vetor at the polarimeter, A y is the analyzing power of the analyzing reation, and ˆn is a unit vetor normal to the sattering plane, ˆn = ˆk ˆk / ˆk ˆk where ˆk and ˆk, are the unit vetors in the diretion of the inident and sattered proton [5]. Now, P fpp has three omponents, longitudinal, normal, and transverse. It is important to notie how the omponents of the polarization hange from the target to the polarimeter. The proton spin preesses as it travels from the target resulting in new polarization omponents at the polarimeter as shown in Figure

28 Figure 16: Spin Preession The transverse polarization omponent P t is parallel to the magneti field and does not undergo any preession. The longitudinal polarization omponent P l is perpendiular to the magneti field and preesses with angle χ θ. In terms of the polarization omponents at the polarimeter, P l ˆn = 0 sine the longitudinal polarization is perpendiular to the normal vetor, Pn ˆn = os(φ) and P t ˆn = sin(φ). This produes an azimuthal distribution whih, for spin-1/2 partiles is of the general form: N(θ, φ) = N 0 (θ)[1 + A y (θ)(p t sin(φ) P n os(φ))] (27) where θ and φ are the polar and azimuthal sattering angles, N(θ, φ) and N 0 (θ) are the number of partiles reorded after the analyzer for the polarized and unpolarized beam, respetively; A y (θ) is the analyzing power whih depends on the material of the analyzer, the energy of the partile, and the polar angle. It is useful to onvert the azimuthal distribution to a detetion probability distri- 22

29 bution: f ± (θ, φ) = ɛ(θ, φ) 2π (1 ± A y(p t sin(φ) P n os(φ)) (28) where the ± refers to the sign of the beam heliity and ɛ(θ, φ) is the effiieny of the polarimeter. It is lear that the distributions are sinusoidal and out of phase. Adding the distributions will show instrumental asymmetries and subtrating the distributions removes heliity independent effets. 8.3 The Azimuthal Angle and the Cone Test In an attempt to ensure that the simulation is orret, unpolarized protons are sattered in the polarimeter. Without polarization there is no preferred diretion and should therefore be no asymmetry or struture in the azimuthal, φ, distribution. However, when the simulation is run with high statistis, 20 million events, a lear struture is observed. See Figure 17. Figure 17: Phi Distribution It is important to understand this struture. Think of this distribution as a flat distribution with large dips at 0 and π and smaller dips at π/2 and 3π/2. The dips represent some sort of missing data at those points. Now onsider the bak 23

30 of the seond detetor. Projet the inident proton to the hamber. Now, without polarization, sattering is equally likely on a irle around the inident projetion, meaning any azimuthal angle is equally likely. The irle radius is defined by the polar angle, θ. Consider what happens when the inident projetion is near one edge of the detetor, say the right side whih orresponds to 0 radians. Eah φ is equally likely but only the ones to the left side of the inident projetion will be seen by the detetor. There are an equal number sattered to the right of the inident point but these will not be reorded by the detetor. See Figure 18. These unreorded points are the missing data that reate the dips in the distribution. Figure 18: Cone Test Logi To orret for this effet, there must be a test on the data that only aepts events for whih a full φ distribution is possible. A one test is defined with this sort of test. The one has its origin at the point of interation and opens to a irle at the bak of the seond detetor defined by θ. Using the geometry of this one, events for whih a full phi distribution is not possible are eliminated. This effetively shaves off the peaks of the original φ distribution. The proess begins by loating the projetion of the inident proton on the bak of the hamber. The x and y position of this projetion is defined by the following 24

31 Figure 19: Cone Test Geometry equations: x proj = (z z 1)(x 2 x 1 ) z 2 z 1 + x 1 (29) y proj = (z z 1)(y 2 y 1 ) z 2 z 1 + y 1 (30) where (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) are points on the inident trak and z is the z position of the bak of the detetor. The height of the one is just the distane between the point of interation and the projetion of the inident trak onto the bak of the detetor. h = (x proj x) 2 + (y proj y) 2 + (z plane z) 2 (31) where (x,y,z) is the point of interation. 25

32 The radius of the one is easily omputed r = h tan(θ) (32) The irle at the bak of the detetor defined by this one must be totally ontained within the detetor. To hek this, add the radius to x and y oordinates of the projeted point of the inident partile in the plus and minus diretion. If this number is outside the dimension of the detetor then the event is not aepted, otherwise it is aepted. Adding the one test redued the struture onsiderably as shown in Figure 20. The first plot is the final distribution after the one test was applied, the seond Figure 20: Phi Distributions-Cone Test plot is the distribution of the φ values of the rejeted events, and the final plot is the original φ distribution without a one test. After the one test the φ distribution has a muh smaller struture than it did before the one test. To see whih events are being rejeted refer to Figure 21. The full line represents all θ values and is the usual logarithmi distribution of θ. The dotted line shows those events whih were rejeted by the one test. As expeted, most of the events rejeted had fairly high θ values whih orrespond to larger ones. As this paper was being written, a Fourier analysis of the data was performed and a struture still observed. In addition, Professor Brash of Christopher Newport 26

33 Figure 21: Theta Distributions: Full and Rejeted University wrote a seond, and improved one test that again dereased the struture of the φ distribution. Figure 22 shows both one tests: the left olumn represents the improved one test and the right olumn represents the one test desribed above. The first row is the final distribution after the one test was applied, the seond plot is the distribution of the φ values of the rejeted events, and the final plot is the original φ distribution without a one test. The Fourier parameters are shown in the upper right hand orner. Table 4 summarizes the results. No Cone Test Cirular Cone Test Elliptial Cone Test os(2φ) sin(2φ) os(4φ) sin(4φ) RMS Deviation Improvement Table 4: Cone Test Results, the statistial unertainty on all numbers is about

34 Figure 22: Comparing Cone Tests The numbers in Table 4 are the oeffiients of funtion N(φ) N 0 = 1 + Aos(2φ) + Bsin(2φ) + Cos(4φ) + Dsin(4φ) (33) that fits the φ distribution, and the RMS Deviation is RMS = A 2 + B 2 + C 2 + D 2. (34) As Table 4 shows, the elliptial one test dereased the struture. The improvement was made by dropping the earlier assumption that all traks were perpendiular to the detetor. For the earlier one test, the azimuthal distribution is obtained by essentially utting the one with the detetor plane. When the traks are perpendiular to the detetor plane the result is a irle and hene the so alled irular one test. However, this is a bad assumption. Traks are usually not perpendiular and utting the one with the detetor plane reates an ellipse and hene the so alled elliptial one test. In the irular one test, the radius was the variable of interest to ensure that the azimuthal distribution was fully ontained. In the elliptial one test this is no longer the ase. As Figure 23 shows, it is atually x 1 and x 2 that must be taken into 28

35 Figure 23: Elliptial Cone Test Geometry aount. When inluding this geometrial phenomenon, the azimuthal distribution gets better, the magnitudes of the distribution fit are all smaller than 10 3 whih is aeptably low. 9 Conlusion A foal plane polarimeter has been onstruted for use in an experiment to obtain the ratio of the elasti eletromagneti form fators G Ep /G Mp by measuring the reoil proton polarization omponents. While the polarimeter is still in a testing phase, a omputer simulation was written and desribed above. It produes data and predits the results of the future true data olletion. Analyzing the polar sattering angle distribution revealed that multiple trak events must be taken into aount. initial distribution of the azimuthal angle revealed a systemati asymmetry. The The main soure of this asymmetry was removed by a tehnique alled the one test, whih rejets events for whih a full azimuthal distribution is not measurable. There is still a great deal of information to be learned by studying the results 29

36 of the simulation. A main area that has not been studied has to do with the drift times. The eletri field inside the drift hamber is not uniform and therefore the drift veloity is not uniform within a drift ell. The effets of this variable eletri field must be inluded in the ode and analyzed. Also, the drift time and drift distane are related mathematially by a fairly ompliated relationship that must be aurately established. 30

37 Referenes [1] R. Hofstadter, Rev. Mod. Phys. 28, 214 (1956) [2] M.A. Preston, Physis of the Nuleus. Addison-Wesley Publishing, Reading, Massahusetts, [3] C.F. Perdrisat, V. Punjabi, and M. Vanderhaeghen, Nuleon Eletromagneti form Fators, in preparation (2006). [4] L.R.B. Elton, Introdutory Nulear Theory. Seond ed. W.B. Saunders Company, Philadelphia [5] V. Punjabi, C.F. Perdrisat and Hall A Collaboration, Proton elasti form fator ratios to Q 2 = 3.5GeV 2 by polarization transfer, Phys. Rev. C (2005) [6] C.F. Perdrisat, Polarimetry in the Few GeV Region, Ot 26, 1994, unpublished. [7] W.R. Leo, Tehniques for Nulear and Partile Physis Experiments. Seond Revised ed. Springer-Verlag, [8] O. Gayou, Proton Form Fators, Measurement of the Proton Form Fators Ratio by Reoil Polarimetry, Ph.D. Thesis, College of William and Mary, unpublished. (2002) [9] Stewart, James, Multivariable Calulus. 4th ed. Paifi Grove: Gary W. Ostedt,

38 A Relevant Code A.1 Soure Code for the Geometry This is part of the routine that defines all of the omponents of the polarimeter. data hh1_size/75.00,60.00,8.5345/ data aira_size/75.00,60.00,32.131/ data hh2_size/75.00,60.00,8.5345/ data airb_size/60.96,40.64,12.387/ data si1_size/60.96,40.64,0.50/ data air_size/60.96,40.64,9.297/ data si2_size/60.96,40.64,0.50/ data aird_size/71.12,53.975,14.114/ data anl1_size/71.12,53.975,27.94/ data aire_size/83.0,67.0,2.542/ data fh1_size/83.0,67.0,5.60/ data airf_size/83.0,67.0,5.00/ data fh2_size/83.0,67.0,5.60/ data airg_size/71.12,53.975,2.277/ data anl2_size/71.12,53.975,27.94/ data airh_size/83.0,67.0,2.282/ data fh3_size/83.0,67.0,5.60/ data airi_size/83.0,67.0,5.00/ data fh4_size/83.0,67.0,5.60/ zhh1= zaira=zhh1+hh1_size(3)+aira_size(3) zhh2=zaira+aira_size(3)+hh2_size(3) zairb=zhh2+hh2_size(3)+airb_size(3) zsi1=zairb+airb_size(3)+si1_size(3) zair=zsi1+si1_size(3)+air_size(3) zsi2=zair+air_size(3)+si2_size(3) zaird=zsi2+si2_size(3)+aird_size(3) zanl1=zaird+aird_size(3)+anl1_size(3) zaire=zanl1+anl1_size(3)+aire_size(3) zfh1=zaire+aire_size(3)+fh1_size(3) zairf=zfh1+fh1_size(3)+airf_size(3) zfh2=zairf+airf_size(3)+fh2_size(3) zairg=zfh2+fh2_size(3)+airg_size(3)-0.1 zanl2=zairg+airg_size(3)+anl2_size(3) zairh=zanl2+anl2_size(3)+airh_size(3) zfh3=zairh+airh_size(3)+fh3_size(3) zairi=zfh3+fh3_size(3)+airi_size(3) zfh4=zairi+airi_size(3)+fh4_size(3) the mother volume will be the HALL volume, it is filled with air write(6,*) defining volumes now write(6,*) defining HALL now all gsvolu ( HALL, BOX, nair, HALL_size, ndim, iv_hall ) if ( iv_hall.le.0 ) then write ( 6,* ) ugeom: error, iv_hall=,iv_hall write ( 6,* ) HALL geometry setup failed stop end if we do not see the HALL volume in the pitures 32

39 all gsatt ( HALL, SEEN, 0 ) define the null rotation write(6,*) defining null rot now irot = 1 all gsrotm(irot,nul_rot(1),nul_rot(2),nul_rot(3), x nul_rot(4),nul_rot(5),nul_rot(6) ) irotnull = irot define the rotation neessary to position the x-straws write(6,*) defining rotx now irotx = 2 all gsrotm(irotx,rotx(1),rotx(2),rotx(3), x rotx(4),rotx(5),rotx(6) ) define the rotation neessary to position the u-straws write(6,*) defining rotu now irotu = 3 all gsrotm(irotu,rotu(1),rotu(2),rotu(3), x rotu(4),rotu(5),rotu(6) ) define the rotation neessary to position the v-straws write(6,*) defining rotv now irotv = 4 all gsrotm(irotv,rotv(1),rotv(2),rotv(3), x rotv(4),rotv(5),rotv(6) ) write(6,*) defining hh1 volume now all gsvolu ( hh1, BOX, nt_air, x hh1_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) hh1 geometry setup failed stop end if write(6,*) defining aira volume now all gsvolu ( aira, BOX, nt_air, x aira_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) aira geometry setup failed stop end if write(6,*) defining hh2 volume now all gsvolu ( hh2, BOX, nt_air, x hh2_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) hh2 geometry setup failed stop end if 33

40 write(6,*) defining airb volume now all gsvolu ( airb, BOX, nt_air, x airb_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) airb geometry setup failed stop end if write(6,*) defining sintillator now all gsvolu ( si1, BOX, nt_si, x si1_size, ndim, iv_targ ) write(6,*)iv_targ if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ stop end if write ( 6,* ) si1 geometry setup failed write(6,*) defining air volume now all gsvolu ( air, BOX, nt_air, x air_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) air geometry setup failed stop end if write(6,*) defining the seond sintillator now all gsvolu ( si2, BOX, nt_si, x si2_size, ndim, iv_targ ) write(6,*)iv_targ if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ stop end if write ( 6,* ) si2 geometry setup failed write(6,*) defining aird volume now all gsvolu ( aird, BOX, nt_air, x aird_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) aird geometry setup failed stop end if write(6,*) DEFINING THE FIRST ANALYZER NOW! all gsvolu ( anl1, BOX, nt_analyz, x anl1_size, ndim, iv_targ ) write(6,*)nt_analyz,anl1_size,iv_targ if ( iv_targ.le.0 ) then 34

41 write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) analyzer geometry setup failed stop end if write(*,*) analyzer 1 thikness =,anl1_size(3)*2.0 write(*,*) nt_si =,nt_si write(*,*) nt_h2 =,nt_h2 write(*,*) nt_analyz =,nt_analyz write(6,*) defining aire volume now all gsvolu ( aire, BOX, nt_air, x aire_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) aire geometry setup failed stop end if write(6,*) defining fh1 volume now all gsvolu ( fh1, BOX, nt_air, x fh1_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) fh1 geometry setup failed stop end if write(6,*) defining airf volume now all gsvolu ( airf, BOX, nt_air, x airf_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) airf geometry setup failed stop end if write(6,*) defining fh2 volume now all gsvolu ( fh2, BOX, nt_air, x fh2_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) fh2 geometry setup failed stop end if write(6,*) defining airg volume now all gsvolu ( airg, BOX, nt_air, x airg_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) airg geometry setup failed stop end if 35

42 write(6,*) DEFINING THE SECOND ANALYZER NOW! all gsvolu ( anl2, BOX, nt_analyz, x anl2_size, ndim, iv_targ ) write(6,*)nt_analyz,anl2_size,iv_targ if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) anl2 geometry setup failed stop end if write(*,*) analyzer 2 thikness =,anl2_size(3)*2.0 write(*,*) nt_si =,nt_si write(*,*) nt_h2 =,nt_h2 write(*,*) nt_analyz =,nt_analyz write(6,*) defining airh volume now all gsvolu ( airh, BOX, nt_air, x airh_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) airh geometry setup failed stop end if write(6,*) defining fh3 volume now all gsvolu ( fh3, BOX, nt_air, x fh3_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) fh3 geometry setup failed stop end if write(6,*) defining airi volume now all gsvolu ( airi, BOX, nt_air, x airi_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) airi geometry setup failed stop end if write(6,*) defining fh4 volume now all gsvolu ( fh4, BOX, nt_air, x fh4_size, ndim, iv_targ ) if ( iv_targ.le.0 ) then write ( 6,* ) ugeom: error, iv_targ=,iv_targ write ( 6,* ) fh4 geometry setup failed stop end if 36

43 make the target "SEEN" in the pitures and give it COLOr number 3 (probably green) and then position it inside the mother volume all gsatt ( aira, SEEN, 1 ) all gsatt ( aira, COLO, 1 ) all gspos ( aira, 1, HALL, 0.,0.,real(zaira),irot, only ) all gsatt ( airb, SEEN, 1 ) all gsatt ( airb, COLO, 1 ) all gspos ( airb, 1, HALL, 0.,0.,real(zairb),irot, only ) all gsatt ( air, SEEN, 1 ) all gsatt ( air, COLO, 1 ) all gspos ( air, 1, HALL, 0.,0.,real(zair),irot, only ) all gsatt ( aird, SEEN, 1 ) all gsatt ( aird, COLO, 1 ) all gspos ( aird, 1, HALL, -5.08,0.,real(zaird),irot, only ) all gsatt ( aire, SEEN, 1 ) all gsatt ( aire, COLO, 1 ) all gspos ( aire, 1, HALL,-5.08,0.,real(zaire),irot, only ) all gsatt ( airf, SEEN, 1 ) all gsatt ( airf, COLO, 1 ) all gspos ( airf, 1, HALL,-5.08,0.,real(zairf),irot, only ) all gsatt ( airg, SEEN, 1 ) all gsatt ( airg, COLO, 1 ) all gspos ( airg, 1, HALL,-5.08,0.,real(zairg),irot, only ) all gsatt ( airh, SEEN, 1 ) all gsatt ( airh, COLO, 1 ) all gspos ( airh, 1, HALL,-5.08,0.,real(zairh),irot, only ) all gsatt ( airi, SEEN, 1 ) all gsatt ( airi, COLO, 1 ) all gspos ( airi, 1, HALL,-5.08,0.,real(zairi),irot, only ) all gsatt ( anl1, SEEN, 1 ) all gsatt ( anl1, COLO, 4 ) all gspos ( anl1, 1, HALL, -5.08,0.,real(zanl1),irot, only ) all gsatt ( anl2, SEEN, 1 ) all gsatt ( anl2, COLO, 4 ) all gspos ( anl2, 1, HALL, -5.08,0.,real(zanl2),irot, only ) all gsatt ( hh1, SEEN, 1 ) all gsatt ( hh1, COLO, 2 ) all gspos ( hh1, 1, HALL, 0.,0.,real(zhh1),irot, only ) all gsatt ( hh2, SEEN, 1 ) all gsatt ( hh2, COLO, 2 ) all gspos ( hh2, 1, HALL, 0.,0.,real(zhh2),irot, only ) all gsatt ( fh1, SEEN, 1 ) all gsatt ( fh1, COLO, 5 ) all gspos ( fh1, 1, HALL, -5.08,0.,real(zfh1),irot, only ) all gsatt ( fh2, SEEN, 1 ) all gsatt ( fh2, COLO, 5 ) all gspos ( fh2, 1, HALL, -5.08,0.,real(zfh2),irot, only ) 37

44 all gsatt ( fh3, SEEN, 1 ) all gsatt ( fh3, COLO, 5 ) all gspos ( fh3, 1, HALL, -5.08,0.,real(zfh3),irot, only ) all gsatt ( fh4, SEEN, 1 ) all gsatt ( fh4, COLO, 5 ) all gspos ( fh4, 1, HALL, -5.08,0.,real(zfh4),irot, only ) all gsatt ( si1, SEEN, 1 ) all gsatt ( si1, COLO, 3 ) all gspos ( si1, 1, HALL, 0.,0.,real(zsi1),irot, only ) all gsatt ( si2, SEEN, 1 ) all gsatt ( si2, COLO, 3 ) all gspos ( si2, 1, HALL, 0.,0.,real(zsi2),irot, only ) write(6,*) everything is positioned 38