Neutron Precision Polarimetry
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Neutron Precision Polarimetry Roel L. Flores
Abstract In quantum mechanics, discrete symmetries such as charge conjugation-C, spatial parity- P, and time reversal-T, offer a unique complement to the continuous symmetries and associated conservation laws of Noether’s theorem. However, unlike Noether’s theorem, violations of these symmetries have been observed. Of the four fundamental forces, only the weak force exhibits C, P or T(CP)-violation. In many weak interactions, parity-violation can be directly observed if polarized neutrons are one of interacting particles, such as the n + p -> d + γ reaction or free
- neutron decay (n -> p + e + ve). In experiments involving neutron decay, such as the abBA experiment, the precision of measurements of asymmetries caused by P-violation directly rely on precise neutron polarization measurements. So in this experiment, we attempt to get the most precise measurements of neutron polarization to date, for use in future experiments like abBA, experiments that focus on neutron decay correlations. These experiments will allow us to get a better understanding of the weak force and nucleon structure.
Introduction The concept of intrinsic angular momentum, or spin, first proposed in 1925, has been a fundamental concept in quantum mechanics. One of its most important implications is that it classifies all particles into either fermions or bosons. Bosons have integral spin, follow Bose- Einstein statistical mechanics, and do not obey the Pauli-Exclusion principle. The opposite of bosons are fermions, particles with odd half-integral spins. These particles follow Fermi-Driac statistics and most importantly do obey the Pauli-Exclusion principle: no two fermions can exist in the same quantum state. Neutrons are fermions, and this important in understanding how they behave in this experiment. In order to polarize neutrons, we must pass them through a spin filter: a region of interaction in which neutrons of the undesired spin are absorbed in nuclear interactions. This would not be possible to do if neutrons were bosons. The spin quantum number always has a degeneracy of 2 for fermions; there are two different spin sub-states that each has the same magnitude but different directions. For neutrons, electrons and most other particles, s = ± ½, and it is common to describe the positive state as “up” or “right-handed” and the negative state as “down” or “left-handed”. This comes from the idea that the particles are actually spinning, which is untrue, but convenient because a particle’s magnetic moment is associated with its spin, in the same manner as ordinary angular momentum. In this paper, I will use the terms up and down to refer to either neutrons or nuclei with spin ½ and – ½ , respectively. As aforementioned, the goal of our experiment is to precisely measure neutron polarization. So a definition of our use of polarization would be convenient: polarization is a measure of the number of neutrons (or nuclei) that are spin up versus spin down. For example, if P=1 then 100% of the neutrons are spin up, if P=0, then 50% are up and 50% down, and if P=-1 then 100% are down. This will be explicitly shown mathematically in the following section. Essentially, we want to use spin filters to produce a positively-polarized neutron beam, and measure how effective our method is. However, this has more implications than are initially apparent. Neutron polarization can be a key tool in understanding underlying aspects of the weak nuclear force: fundamental symmetry breaking. Noether’s theorem, a revolutionary principle of classical physics, states that every conservation principle corresponds to a continuous symmetry in nature. For example, systems that are invariant under transformations in space obey the conservation of linear momentum, and the fact that we can arbitrarily choose t = 0 in our experiments leads to the consequence of conservation of energy. However, Noether’s theorem doesn’t hold in discrete symmetries due to quantum transitions. Symmetries such are C (charge conjugation), P (spatial parity) and T (time reversal) can be violated on the quantum level. However, of the four fundamental forces, all haven been observed to follow these symmetries except the weak nuclear force. Parity-violation is of particular interest to our group, and polarized neutrons can be used in experiments to observe P-violation. An example of this is the n + p -> d + γ interaction, where n is neutron, p is proton, d is deuteron and γ is a high-energy photon. This reaction is parity violating because the direction the gamma rays are emitted is a consequence of the spin of the incident neutron. If parity held, we would expect to see a an equal distribution of gamma rays emitted, but it has been postulated that if the neutron is spin up, then more gamma rays will be emitted down than up. This is very strange indeed; it is analogous to performing an experiment and then seeing that it looks different in the mirror. Interactions shouldn’t depend on right-handed (spin up) or left-handed (spin-up) coordinate systems, but indeed some do. The “mirror” laboratory looks different from the original at the quantum level, a strange quantum mechanical notion that is difficult to grasp. However, we are currently trying to better understand discrete symmetries of quantum mechanics, and also trying to understand why only the weak force exhibits the observed violations. In addition to P-violation, C-violation and T-violation (or CP-violation) have also been observed in weak interactions. In experiments such as abBA, precision measurement of neutrons is needed to get a precise measurement of the asymmetries caused by P-violation. It is because of this that we desire to obtain the most precise measurement of neutron polarization yet, to better increase the results of future experiments studying parity violation and its relation with the weak nuclear force.
Theoretical Basis The primary idea in this experiment is to use He-3 spin filters to polarize the incoming neutron beam. As aforementioned, neutrons are fermions and can be in the spin up state or spin down state. Similarly, the helium-3 nucleus is also a fermion because the two protons have their spins anti-parallel, leaving the other neutron either spin up or spin down, and this is what the net spin of the nucleus is. The neutron-helium cross section is very large because the helium-3 wants to capture the slow neutron and become helium-4. However, due to the Paul-Exclusion principle, the He can only capture a neutron if its spin is anti-parallel. Thus, if we want to polarize the beam to have spin up neutrons, we first polarize rubidium vapor to be spin up (using circularly polarized laser light), and it transfers its spin to the helium nuclei which then capture spin down neutrons. The polarizer and analyzer both use this technique, except the analyzer has negative polarization (spin down nuclei). With this basic principle known, and a use of basic nuclear physics concepts, we can systematically find a way to measure the effect of this polarizer, in terms of known values and things that are measurable. Consider the following notation, we define n as the flux density of neutrons (or number density as we will see for helium) and P for the polarization. We then write these mathematically: n = n+ + n- n+ - n- P = n+ + n- And we can thus see that nP = n+ - n- And using the definition of n, we can solve for the plus and minus terms by a setting up a system for both, and so we thus find that n(1 P) nұ = ұ 2 Let us now consider the basic formula for absorption of the neutrons, ignoring polarization and the asymmetry of cross sections. Consider the He-3 atoms have a cross section of σ, and the number density of the atoms, N/V, is represented as n3. This is chosen because the number density of the helium will satisfy the same equation above, which was for neutron number. Also consider the polarizer is thin, with thickness dl and area A. Then we calculate the number of absorbed neutrons dn by the following principle: (interacting neutrons/incident neutrons) = (aggregate cross section/area): dn n As dl - = 3 = n s dl n A 3 The negative is needed because in a beam passing through a finite polarizer the number of neutrons must decrease. For a thick polarizer, we solve the already nicely separated equation by integration:
ln(n) = -n3sl + c
-n3sl Ю n1 = n0e
Where n1 is defined as the number of neutrons present after the polarizer and n0 as the initial amount. Next, let us consider that the cross section is a function of neutron velocity, because faster neutrons will have a lower probability of being absorbed than neutrons. This is because cross section is dependent on the amount of time a neutron is near a nucleus, because cross section is essentially a probability of neutron capture. So, we should try to cross section in terms of data that is already known or easy to collect such as wavelength, which we can find through time of flight measurements. So then 1 s : t : v And also from the DeBroglie hypothesis,
m h = p 1 : l Ю s = cl v So the cross section is linear with respect to the wavelength of the neutrons. Also, if we consider that this holds for all values of energy, and using a temperature defined by E=kT, we can solve for c in terms of the cross section and wavelength at room temperature, 300K (the thermal cross section). So then we define c as s c = th lth and we can now write the cross section now in terms of c, and rewrite the equation we found earlier: s s = th l lth
-n3llsth /lth n1 = n0e It is apparent that the only unknown is the wavelength of the neutron, so this is much more satisfactory. We can also find the values of the other terms and combine them to form a single constant. Let’s first consider the number density of the He-3 atoms, using the ideal gas law: PV = NkT N P = n = V 3 kT The gas cell was filled at STP, so we use 1 atm or 101.325 kPa for pressure, 300K for temperature, and k is Boltzmann’s constant, which gives us P n = = 2.45e25 atoms/m3 3 kT a reasonable figure. Next we need to solve for the thermal wavelength. The Boltzmann statistics expectation value for energy is used along with the standard formula for kinetic energy. p2 E = kT = 2m Relativistic effects are very negligible because we are using slow neutrons. We next combine the above with the DeBroglie relation: h h l = = g mv p h Ю l = 2mkT Where h is Planck’s constant and m is the neutron mass. Substituting in numerical values, we find that the thermal wavelength is 1.78e-10 m or 1.78 A (angstroms). It has also been established that the unpolarized neutron thermal cross section for He-3 is 5333 barns, or 5.333e- 25 m2. The thickness of the cell is also given to be 4 cm. So with all of these numbers, we can now solve for the constant, which will be designated as chi:
s th -1 c = n3l = .293A lth
-cl Ю n1 = n0e Next we will consider that the cross section is actually dependent on whether the spins are aligned or anti-aligned. The combination of the neutrons being spin up or down as well as the helium nuclei being up or down gives us 4 combinations to assign cross section to: s ++ ,s +- ,s -+ ,s -- But we have two pairs that are non-unique in probability, so we will now use the following conventions: s ++ = s -- = s + s +- = s -+ = s - Where the plus represents spins aligned and the minus represents spins anti-aligned. Now we want to solve for the plus and minus values like we did earlier with n. So thus we need to set up two similar equations. The first one is the relation between unpolarized cross section and the above cross sections. s + +s - s = 2 The equation must be divided by two because it as an average of the spin up and spin down cross sections: a neutron will be affected by either one of the cross sections, but not both. Next we define the asymmetry factor A, which is defined similar to the polarization. s + -s - A = s + +s - And we can solve for the plus and minus factors with the same method as above. It is then found that
s ұ = s (1ұ A) So we can now setup equations that take into account the above definitions. Notice how they are not symmetric; this is because neutrons will be absorbed more in one sub-state than the other. In the following equations, a subscript of 0 means initial value and a subscript of 1 means value after going through the polarizer. A subscript of 3 refers to the number density or polarization of the helium-3. dn+ - = (n+s +dl + n-s -dl) n+ 3 3 dn- - = (n-s +dl + n+s -dl) n- 3 3 We can solve these equations again like earlier to find
+ + - - + + -(n3 s l+n3 s l) n1 = n0 e
- + + - - - -(n3 s l+n3 s l ) n1 = n0 e For now, we will focus on the plus term. We can now insert the following formulas from earlier: n (1 P ) nұ = 3 ұ 3 3 2
s ұ = s (1ұ A)
1 1 -( n (1+P )s (1+ A)l+ n (1-P )s (1- A)l) + + 2 3 3 2 3 3 n1 = n0 e
n3sthl - l ((1+P3 )(1+ A)+(1-P3 )(1-A)/2) + + lth Ю n1 = n0 e
+ + -cl (1+P3 A) n1 = n0 e Where χ was defined above and the other terms were multiplied out and reduced. Similarly, it can be shown that
- - -cl (1-P3 A) n1 = n0 e So the next step is to formulate in terms of n only, no minuses or pluses. This is again shown for the plus factor: n (1 P ) nұ = 0 ұ 0 0 2 1 n+ = n (1+ P )e-cl (1+P3 A) 1 2 0 0 n n+ = 0 (e-cl e-clP3 A + P e-cl e-clP3 A ) 1 2 0 n e-cl n+ = 0 (e-clP3 A + P e-clP3 A ) 1 2 0 And similarly for the spin down neutrons: n e-cl n- = 0 (eclP3 A - P eclP3 A ) 1 2 0 And now we can finally describe the new number of neutrons in terms of the original number and polarization:
+ - n1 = n1 + n1 n e- cl 0 й cl P3 A - cl P3 A cl P3 A - cl P3 A щ n1 = (e + e )- P0 (e - e ) 2 л ы They have been written in this order so that it is much easier to see we can rewrite them as hyperbolic functions:
-cl n1 = n0e (cosh(clP3 A) - P0 sinh(clP3 A)) We can also find the new polarization:
+ - n1P1 = n1 - n1 n e- cl 0 й cl P3 A - cl P3 A cl P3 A - cl P3 A щ n1P1 = - (e - e )+ (P0e + P0e ) 2 л ы
-cl n1P1 = n0e (-sinh(clP3 A) + P0 cosh(clP3 A)) -sinh(clP3 A) + P0 cosh(clP3 A) Ю P1 = cosh(clP3 A) - P0 sinh(clP3 A) So now we have two equations describing the number of neutrons and the new polarization of the beam after going through a general polarizer, with initial polarization. To determine A, consider that if a neutron has the same spin as the helium nucleus, it will not get absorbed into the ground state of the nucleus, due to the Pauli-Exclusion principle. In other words the σ+ cross section should be nearly zero (remember that for the cross sections, + means aligned, - means anti-aligned). It has been experimentally confirmed that for helium-3, A=-1 so our above equations become:
-cl n1 = n0e (cosh(clP3 ) + P0 sinh(clP3 ))
sinh(clP3 ) + P0 cosh(clP3 ) P1 = cosh(clP3 ) + P0 sinh(clP3 ) because cosh is an even function while sinh is odd. Our goal is to measure the neutron polarization after the polarizer using two different methods, thus we must briefly describe the setup Thus we need two polarizers, with three beam monitors, and we also make use of a spin flipper. The neutron beam passes first through the first beam monitor, then the polarizer, then the second beam monitor, then through a spin flipper, then through another polarizer that we designate as an analyzer and finally through the last beam monitor. The spin flipper does exactly what it says: it flips spin up neutrons into spin down and vice versa. The neutron beam also passes through detectors at certain points: before the polarizer, after the polarizer, and after the spin flipper and analyzer. The experimental setup will be described in more detail in the next section.
So, let us first look at the polarizer. Let P3->PP, because will want to distinguish the polarizer from the analyzer in this section. Also, consider that before the first polarizer, there is an equal number of neutrons spin up and spin down so that P0 = 0. So then we have:
-cl n1 = n0e cosh(clPP )
P1 = tanh(clPP ) And now it will be useful to define transmission, because it is what we can directly measure with the detectors: n1 -cl T = = e cosh(clPP ) n0
And we also consider the special case when the polarizer is off, or PP = 0.
-cl TU = e
The subscript U donates unpolarized transmission. Now we notice if we divide T by TU, we are just left with the cosh function, so we also define this as the ratio of transmissions: T R = = cosh(clPP ) TU So now all of our hard work working through the equations has paid off; we now have a simple cosh function to fit data to. Now, as we are interested in the polarization, we must solve for P1 in terms of this by using the fundamental identity of the hyberolic functions: cosh 2 (x) - sinh2 (x) =1 And combining this with the above equations we find 1 1 P1 = tanh(clPp ) = 1- 2 = 1- 2 cosh (clPP ) R which we will be able to find by curve fitting the data. Now we want to develop a mathematical description of the spin flipper. In our experiment, the spin flipper is useful because it allows use to make increase our sensitivity when fitting functions; we can use it to help us find the polarization in the analyzer very simply, as we shall soon see. The spin flipper is also important in many experiments that study polarized neutron decay and parity violation, because it allows the experimenters to compare sets of spin up neutrons and spin down neutrons with more precision. Precision is increased because the spin flipper activated on every other 50 ms pulse of the neutron beam, runs that have spin up neutrons are only 50 ms apart from those that have spin down neutrons due to the spin flipper This pulsed nature of the neutron beam will be discussed more in the next section. Developing a mathematical description is easily done by just thinking of the definition: a perfect spin flipper will reverse all of the incident neutrons’ spin. However, there are limitations on any instrument, so now we must define ε, the spin flipper efficiency. This value ε ranges from 0 to 1; 0 meaning the spin flipper flips 0% and 1 meaning 100% of the neutrons. So thus, a perfect spin flipper has an efficiency of 1. With this in mind, we can deduce the following equation, using 0 subscripts again as the initial neutrons before the spin flipper and subscripts with a 1 representing neutrons that have gone through the spin flipper:
+ + - n1 = (1-e )n0 + en0 At first, the equation seems quite arbitrary, but it is easily verified when we consider two things: the (1- ε) factor is necessary because we are using fractions of the total neutron flux, so the two n0 terms must always add up to the original n. The spin flipper thus does not change the total neutron flux. The SF also flips spin up neutrons to spin down neutrons as it does for down to up. We can test this equation considering an ideal spin flipper: e =1
+ - \n1 = n0 This is precisely what we wanted. The equation for the new spin down neutrons is now fairly obvious:
- - + n1 = (1-e )n0 + e n0 Now, we want to find the new polarization resulting from the neutron flux passing through the spin flipper, so we use the familiar equation:
+ - n1 - n1 P1 = + - n1 + n1
+ - - + (1-e )n0 + e n0 -[(1-e )n0 + en0 ] P1 = + - - + (1-e )n0 + en0 + (1-e )n0 + en0 And now factoring will yield the nice results we are looking for:
+ - (n0 - n0 )(1- 2e ) P1 = + - n0 + n0
P1 = P0 (1- 2e ) Let’s see if this equation makes sense. In a perfect spin flipper, ε = 1 and the polarization is perfectly reversed, just as we would expect. This ε factor will be helpful when discussing the next part, the analyzer. Although the general formulas that were derived earlier hold true for the analyzer, we must take care in taking the specific approach because the spin flipper complicates things a little. There are two important things that must be said here: as aforementioned the spin flipper is constantly turning on and off, so that every other pulsed beam that passes through it is off. It is on for 50 ms, then off for 50 ms, and this repeats all while the experiment is running. So this will be important in finding relevant equations. Also, because it is an analyzer, we are not interested in the new polarization that the analyzer produces, but rather we are interested in finding the initial polarization that comes in to it, which is the same polarization that is resultant from the polarizer. This gives us a goal in finding and manipulating the equations. As with the last section, we will use 0 subscripts for the initial values and 1 for the new values. This means the 0 subscript values represent the beam right after the spin flipper but before going the analyzer. Let us look at how the equations will differ for spin flipper on and spin flipper off:
-cl SFOFF : n1 = n0e (cosh(clPA ) + P0 sinh(clPA ))
-cl SFON : n1 = n0e (cosh(clPA ) + (1- 2e )P0 sinh(clPA ))
PA is the polarization of the analyzer. Remember that P0 is actually a function: it is equivalent to the P1 we found from the polarizer:
P0 = tanh(clPP ) So, now we will look at the transmissions:
OFF -cl T = e (cosh(clPA ) + P0 sinh(clPA ))
ON -cl T = e (cosh(clPA ) + (1- 2e )P0 sinh(clPA )) And we recall the definition of transmission ratio as: T R = TU To find the following equations:
OFF R = cosh(clPA ) + P0 sinh(clPA )
ON R = cosh(clPA ) + (1- 2e )P0 sinh(clPA ) And we can manipulate these until we find an equation that we can fit to:
OFF ON (2e -1)R + R = ((2e -1) +1)cosh(clPA ) + ((2e -1) + (1- 2e ))P0 sinh(clPA )
1 OFF ON Ю cosh(clPA ) = й(2e -1)R + R щ 2e л ы We can use this equation to find the polarization in the analyzer. And now we can solve for the neutron polarization resulting from the polarizer in the following method: ROFF - RON P sinh(clP ) = 0 A 2e ROFF - RON Ю P0 = 2e sinh(clPA ) And now recalling the hyperbolic identity again: cosh 2 (x) - sinh2 (x) =1 1 sinh(x) = 2 ұ cosh (x) -1 ROFF - RON P = tanh(clP ) = - Ю 0 P 2 2e cosh (clPA ) -1 The negative has been chosen because the polarization of the helium-3 in the analyzer is negative, so thus a negative sinh would be produced when fitting the data. Using this analyzer technique, we have two separate methods on how to measure the neutron polarization (after the polarizer). This is important because we will be able to compare the two independent methods to check for precision. We will use an approximate value for the spin flipper efficiency, but it is worth investigating how it can be directly measured. Consider a set of data in which the polarization of the helium nuclei in the analyzer has been completely reversed. Using adiabatic fast passage, or AFP, this can be achieved. Although the method is different from that of the spin flipper, the purpose of AFP is the nearly same: to reverse the polarization of the helium. This is opposed to reversing the polarization of the neutrons, which the spin flipper does. So now, we can use this AFP analyzer data and the original data to find an expression for the spin flipper efficiency. Consider again the ratios of the transmissions through the analyzer:
OFF R = cosh(clPA ) + P0 sinh(clPA )
ON R = cosh(clPA ) + (1- 2e )P0 sinh(clPA ) In reality, this AFP technique will also have an efficiency associated with it, but that will make things much more complicated, so we will assume that the AFP is perfect while the spin flipper is not. So now consider the equations for the AFP data:
PA ® -PA
OFF RAFP = cosh(cl(-PA )) + P0 sinh(cl(-PA )) OFF Ю RAFP = cosh(clPA ) - P0 sinh(clPA )
ON RAFP = cosh(cl(-PA )) + (1- 2e )P0 sinh(cl(-PA ))
ON Ю RAFP = cosh(clPA ) + (2e -1)P0 sinh(clPA ) So now we can combine these equations in the following manner to solve for the efficiency:
OFF ON R - R = 2e P0 sinh(clPA )
OFF OFF R - RAFP = 2P0 sinh(clPA )
ROFF - RON Ю e = OFF OFF R - RAFP As noted earlier, transmission ratios are the most ideal because they are things that we can easily measure and fit to. However, this does give us a set method to get a good approximation, which we can use in the future. Because this method was not planned, it will not be in the experiment and will thus not be discussed in the analysis.
Experimental Setup The precision polarimetry experiment was performed at the Los Alamos Neutron Science Center or LANSCE, in Los Alamos, NM. LANSCE houses an 800 MeV proton linear accelerator, a high energy beam that is directed toward several targets. Our experiment was conducted on Flight Path 12 (FP12), in the Lujan Neutron Scattering Center. FP12 uses a tungsten spallation target (Target 1), in which the high-energy protons collide with tungsten and eject about 30 neutrons per proton. The beam is a pulsed source with a frequency of 20Hz, or in other words pulses come every 50 ms. These neutrons are slowed down by moderators immediately surrounding the target. FP12 uses a cold hydrogen moderator to slow the neutrons by eight orders of magnitude—to energies of less than 5 meV. Hydrogen is ideal due to its mass that is very close to the neutrons, and thus reduces the energy of the neutrons with elastic collisions. This is an important feature because elastic collisions are the best way to reduce the energy of the neutrons (or of any system). Hydrogen is also a good choice due very small cross- section: it does not readily absorb neutrons. A supermirror neutron guide was specifically designed for FP12, and it guides the neutrons along the flight path from the moderator to the experimental area, or cave as it is commonly referred to. The guide is also equipped with a beam chopper, a rotating plate covered with gadolinium (a strong absorber of neutrons) except in one wedge-shaped area called the window. This narrow window allows neutrons of the desired velocity range, and thus wavelength range, to pass through. This is achieved by selecting the chopper to rotate at 20 Hz, the same frequency of the neutron beam, and setting the phase to 19.2 degrees, also the same phase of the neutron beam. Thus, only a select range of neutrons are not absorbed by the gadolinium. This wavelength range will be experimentally determined in the analysis section. The beam guide ends in the cave, where all of the apparatus for the experiment is put in place. To the hold the main apparatus, a table was set up and aligned with the axis of the beam (hereafter referred to as the z-axis, as is common in beam physics). At the end of the beam guide, we setup up a lithium-6 collimator mounted on the beam to narrow the beam as desired. Also mounted on the beam guide is the first detector or beam monitor, referred to as m1, in which the beam passes through a gas chamber filled with helium-3 and in which signals are produced that represent the neutron flux density as a function of time of flight (or wavelength). About half a meter from m1, an “oven” was put in place, and also aligned along the z- axis. This oven is named so because it is an insulator that is designed to keep heat inside, as well as laser light. The oven houses the polarizer cell: the glass cell containing rubidium vapor and helium-3. The insulation of the oven is needed so the rubidium vapor can achieve the optimal density for polarization. The vapor must be heated until it reaches this density, and at this density it can be polarized. The top of the oven is surrounded by a laser light enclosure, which connects it to an optical instrument box, placed above the oven. Optical fibers connect the optical instrument box to a laser diode device that is kept outside of the cave for optimal cooling. Also inside the cell is a pickup coil used for NMR, specifically for using free induction decay (FID). This gives a method in directly determining the polarization of the helium without the neutron beam. Also surrounding the oven are coils that will be used for adiabatic fast passage (AFP). AFP is the technique used to flip the spin of the helium nuclei, as mentioned in the theoretical basis section. To achieve polarization, we first tuned the Coherent FAP laser to the D1 absorption wavelength of Rb, 795 nm, and maximum power, about 30W. This laser is then turned on and transmitted through the fibers and enters into a box containing the optics needed for circular polarization. The optics creates right-circular (left-circular for the analyzer) polarization of the laser beam, and directs it toward the cell in the oven. This part of the laser is surrounded by an enclosure, making it a Class 1 laser. The circularly polarized light shines on the rubidium-vapor atoms, which also become polarized. Then the rubidium atoms transfer spin to the helium nuclei, through quantum mechanical spin exchange. To heat the cell, a 700 W Omega air heater is used. This is connected with high temperature tubing to the cell, and exhaust tubes are also connected which take the outgoing air out of the cave. The heaters are regulated at the optimal temperature by temperature controllers with RTD thermometers in the cell. Thus, the temperature of the cell was kept at 145-155 C. This heating is necessary to achieve the optimal vapor pressure of the rubidium. High polarization cannot be achieved without this condition. Next along the z-axis comes the second monitor m2, which is mounted to the front of the radio frequency spin flipper. m2 is important because it is the first measurement of polarized neutrons. The spin flipper does exactly what it says: it uses nuclear magnetic resonance techniques to rotate a spin up neutron to spin down and vice versa. As mentioned earlier, this a crucial part of the experiment spin flippers are used in many other experiments involving neutron polarization and P-violation, so it is important to see how precise we can measure the polarization when the SF is on. In order to compare beams that are affected by the spin flipper and those that aren’t precisely, we use a signal generator that for 50 ms generates the signal necessary for the SF to operate, then for the next 50 ms generates a 0 signal. This means that every other pulse of the neutron beam will be flipped, instead of all pulses. The next part of the setup is the analyzer, an oven and laser/optical box setup that is essentially identical to the polarizer oven setup, except it has opposite polarization. Just like in optics, when a polarizer is used as an analyzer to determine the polarization of a light beam, our setup contains two polarizers, the second one referred to as the analyzer. The analyzer’s primary purpose has already been shown in the equation: in provides another method in which to analyze the beam and extract the polarization the neutrons due to the polarizer. After the analyzer comes the last part of the primary setup: the third monitor m3. m3’s primary purpose is to measure the remaining neutron flux after the beam as gone through all parts, and to utilize the analyzer. The entire set up is surrounded by four large coils that were wound from copper. These coils are connected in series, and then a current of ~7 A is run through the circuit. By Ampere’s law, this creates a magnetic field, measured to be ~10 G. This magnetic field acts as a holding field for the polarized neutrons, because the magnetic moments that are correlated to spin align with this holding field. It is also an important part of how the spin flipper achieves its purpose. The final part of the setup is the one most important to the analysis: the data acquisition system or DAQ. Each of the monitors is connected to a current to voltage preamplifier, which has a gain on it according to the monitor’s position. The farther down the beamline, the higher the gain needed. This is due to the reduction of neutron flux as the beam goes through the different apparatus and thus the signal needs to be higher in the subsequent monitors. m1 had a gain of 1 MΩ, m2 of 10 MΩ and m3 of 30MΩ. Each of these preamps is connected to a separate channel on a CAEN v1724 ADC (analog to digital converter), controlled by a VME controller board. The VME is connected to the main workstation which collects and stores the data.
Experimental Analysis Now it will be useful to describe the computer methods we used in collected and analyzing data. First it must be noted that the OS used was RedHat Linux, because the terminal was used frequently. In order to collect data we used a BASH shell script called ‘takerun’, a script that interacts with the DAQ to collect binary data in real-time. Takerun and the DAQ make use of the proton pulse trigger (T0 signal), a trigger system that is in sync with the 20 Hz neutron pulses. The user can select how many pulses the DAQ records. The binary data is stored as hexadecimal in a file, which is “crunched” by another C program called ‘crunch’, into a ROOT histogram profile. This histogram profile over time-of- flight (TOF) is an average of all the pulses from that run for each neutron velocity, a separate profile for each monitor, since the DAQ has separate channels and ‘takerun’ is programmed to keep these channels separate. To actually view and manipulate these histograms, two C scripts called ‘setup.C’ and ‘getprof.C’ were written. The primary analysis can be done using ROOT along with ‘setup.C’, although another C script, ‘emanalysis.C’ was written to make the process easier. All of these programs were written by Dr. Chris Crawford or Elise Martin, with modifications and input from other members of the experiment, including myself. ROOT is a set of classes as well as a command line interpreter (CINT). Most of these classes were designed for data analysis in the sciences. The main classes we make use of are the TH1 classes, which allow us to make histograms (with error bars) that on a large scale look like continuous curves, and are also useful because it provides a simple way to make use of the derived equations shown earlier. Also particularly important is the class TF1, which allows the fit of the histograms. I wrote a C script called 'rfanalysis.C’ which utilizes the other scripts as well as ROOT classes. The following analysis was done using this script in ROOT. Now that the programs have briefly been explained, we can use them to analyze the main points of interest, using an aggregate of runs. There are three main types of runs: pedestal, unpolarized, and polarized. Pedestal runs are taken when the neutron beam is off, and so they represent background noise. Unpolarized runs are taken when the polarizers are off (lasers off) and polarized runs are taken when the polarizers are on. Most of the analysis will focus on polarized runs, but we will also need to make use of pedestals (to define the zero point) and unpolarized runs (for the transmission ratios used in the equations). A selection of runs (about 400) was averaged for this following analysis, for better accuracy. Unless otherwise noted, pedestal runs have been automatically subtracted from the main runs. First, let us concentrate on finding the polarization of the helium in the polarizer, and then from this find the polarization of the neutron beam as a result of the polarizer. Recall that the ratio will be T R = = cosh(clPP ) TU
Where T is the transmission, or m2/m1 and TU is the unpolarized transmission. Now first we want consider runs in which the polarizer is off, PP = 0 (also TU). In this case, cosh(0) = 1, so we should have a straight line. Before we look at graphs and fits, it should be noted that all fits will be made with a normalization constant, aka A*cosh(x). This is should be very close to 1 in all cases, but it is a parameter that must be accounted for. For the following fit, we will use two random unpolarized runs. So now, the fit for polarization of zero is shown below: This is how we can determine the permitted neutron wavelength range from the chopper. We can see that from the above graph, that the range 2-6 is the most ideal, however the range 1.5-7 would also be acceptable. We can thus see that the chopper opens roughly between 0 and 1.5 and roughly closes between 7 and 8.5. We will use this when fitting our other data. Now let us first find the polarization in the polarizer. We will again use the transmission ratio, and again fit it to A*cosh(.293*PP*λ), a function of wavelength with two parameters. This will use the aggregate runs mentioned earlier. As shown, the fit is from 2-7. The fit was made using the TF1 ROOT class, and the resulting solution for the parameters is: A = 1.00142 ± 3.62628e-6
PP = 0.25258 ± 8.51809e-6 It is first obvious that are statistical errors are quite small, so they will be neglected unless otherwise noted. Although we want A to be exactly one, it is very close (percent error = .142%) so we know that this is quite a good fit. Thus, we can see that the polarization of the helium-3 in the polarizer is ~25.3%. We will now compare this to the fit for polarization of the neutrons. Recall from earlier that this should be 1 1 P1 = tanh(clPp ) = 1- 2 = 1- 2 cosh (clPP ) R The above graph was R, the transmission ratio, so the below graph will now be the formula for
P1, using R, fit to A*tanh(.293* PP* λ).
The parameter values for this fit are: A = 0.85123 ± 1.66496e-4
PP = 0.30669 ± 6.96297e-5 The graph shows the neutron polarization as a function of wavelength. We thus verify that the higher the neutron wavelength, the higher the polarization. However, we notice is that A is not close to 1. The percent error on A is almost 15%, which affects the fit for the helium polarization. This polarization value is noticeably different from the value found by the first fit. However, we can resolve this in several ways. The first way is to consider the power series for tanh: x3 2x5 17x7 tanh(x) = x - + - ... 3 15 315 So for small values of x, we can approximate tanh as a linear function. For small values like this, the fit cannot be accurate because it independently fits the two variables, when really they are just one linear coefficient. Thus, we make the approximation that A*tanh(.293* PP* λ) ~
A*0.293*PP*λ, where A can be set to 1 and thus PP is the only parameter to fit to and will be equal to the polarization. Doing this fit yields:
PP = 0.24216 ± 2.46354e-6 a much better value. This fit is shown in the attached page, as all subsequent fits of a certain graph will be. It is a good measurement of the helium polarization but probably not as accurate as the above measurement using the cosh fit. So, we are more concerned about the shape of the tanh, and how reasonable its shape is. This tells us whether it is a good measurement of neutron polarization, something we could not measure in the previous plot. So, to get the best shape of the tanh, while still getting a polarization of helium value close to the above value, we can now restrict the normalization factor when fitting to the curve. Consider that if the A value measured in the first graph is 1.00142, then the A value for the polarization of neutrons should be 1/1.00142, because the first graph is a cosh and the second a tanh. Restricting this parameter also yields a better fit than the linear approximation, as well as the following results, again with the fit shown on the attached page: A = 0.9985820135 (fixed)
PP = 0.25588 ± 2.77855e-6 This method is probably the most reasonable, as it gives a helium polarization value in accordance with the first value found, and also gives a very reasonable tanh that we can interpret as neutron polarization. However there is another way to produce more precise results. If we restrict both A parameters to 1, we get a percent difference of nearly 0%, as we would expect. This method is not shown in detail because it is not reliable. This is because the A values should never be exactly one, because of systematic effects that occur due to polarization of the helium being time dependent, as well as the flux of the neutrons varying from run to run. Because we are using averaged runs that occurred over a long time span, this must be taken into account, so the A values should never be exactly 1. In doing analysis on the analyzer, it should be noted that the efficiency of the spin flipper ε was approximated as .95, a value that is fairly reliable as seen in past experiments involving this RFSF. As we fit to find the polarization in the polarizer, we now want to fit to find the polarization in the analyzer. Recall the following equation:
1 OFF ON cosh(clPA ) = й(2e -1)R + R щ 2e л ы And recall the off superscripts designates spin flipper off, and on for spin flipper on. Also the ratio transmissions are now between m3 and m2. We will now plot this histogram, using .95 for the efficiency, and then fit to a cosh in the same manner as above:
A = 0.99122 ± 3.23171e-5
PA = 0.41685 ± 1.17255e-6 We see that our A value is close to 1, with .8783% error, so this is probably a reasonable fit. From this fit we see that the analyzer polarization is about 41.7%, much higher than the polarizer polarization. This is because the laser used was better tuned to the right wavelength necessary for maximum polarization. We now want to see how the fit changes when we change the range. The graph was again fit, but this time between 2 and 6, with the following results: A = 0.99033 ± 5.13500e-6 PA = 0.41943 ± 1.17255e-6 So we see that the fit is not extremely sensitive to range. The percent difference between the two normalization values is .0893%, and the percent difference between the two polarization values is .615%. This is a good result because it shows that our original fit is likely accurate. Now to get an idea of our actual error in the analyzer polarization, as opposed to the statistical error shown, we force fit with a A parameter value of 1. This fit was done between 2 and 7 like the original, and yields the following values: A = 1.000 (fixed)
PA = 0.40017 ± 5.0775e-6 So we now compare this polarization value to the original fit value. The percent difference is 4.0848%, versus a percent error in the A values of .8783%, with percent error used since 1 is the theoretical value. This can now give us an idea of the range of helium polarization in the analyzer:
PA = 0.41685 ± 4.048%
PA = 0.41685 ± .016874 So the polarization in the analyzer is thus between about 40% and 43%. Precision measurements cannot be made as they were with polarizer polarization because a second method to fit to analyzer polarization has not been developed. This is not a major issue however, because future experiments only use one polarizer and thus only need precision in one polarizer. However, neutron polarization precision is also required. To do this we use the analyzer technique to extract neutron polarization (after the polarizer). Recall the following equation: ROFF - RON P = tanh(clP ) = - Ю 0 P 2 2e cosh (clPA ) -1 where P0 is the polarization of the neutrons after the analyzer. So we can now plot this, using the histogram used above for the cosh function: A = 0.29225 ± 3.23171e-5
PP = 0.89156 ± 1.89740e-4 The first to notice is how the graph doesn’t really resemble a tanh. This is verified by the resulting parameter values, with the supposed 90% polarization in the polarizer. The A value is way below 1, so we know that this result cannot be accurate. One way to mathematically verify that this graph should not be fit to a tanh is by using a chi-square test. Chi-square tests were done on the previous 3 plots, with the values being around 400 for reduced chi-square. This is high, but only because our plots only consider small statistical errors for the error bars and do not take into account our systematic errors which are much larger. However, the chi-square test for this plot yields a reduce chi-square value of about 3000. Thus, it is 10 times bigger error in fitting than our previous plots, and we verify that this is not really a tanh. Fitting between 2 and 5 yields the best fit possible, but the results are still unpleasant. The graph is shown attached as usual, and the following are the resulting fit parameters: A = 0.34288 ± 9.53035e-5
PP = 0.69913 ± 2.83435e-4 These values are still not near the 1 and 25% values we expect. Another attempt to relieve this is to fit the graph to a linear as we did before. The results from this fit are:
PP = 0.20395 ± 8.73654e-6 So although this yields a somewhat reasonable result, the percent difference is still 20.3% between this and the approximate value of 25%. Also, the fit is as good as hoped, not nearly as good as the second fit. So thus we must concede that there is something wrong with the data or perhaps the analysis. A first guess is that the spin flipper is less efficient than .95, and this is reasonable. However, this can be ruled out by changing the value and then looking at the plot used to find the analyzer polarization. Although the details will not be shown, this plot was graphed with several different efficiency values. The results were disappointing in the sense that they did not propose a solution, but pleasant because it was found that varying this efficiency does not change the polarization fit much. As low as ε = .75, the graph was still very reasonable, and the polarization was ~35%, a reasonable value. So thus, as long as our spin flipper isn’t terrible, the above plot shouldn’t be terrible, especially since it uses the cosh plot, which we have seen is reasonable. However, this is assuming that our spin flipper does not have wavelength dependence. If it did fluctuate with wavelength, this would be very noticeable in this 4th plot, even if it was not very noticeable in the third plot. Random statistical errors are not likely to cause such a large effect, so these can be ignored. This is true for all parts of the experiment, because the preamps and ADC are very accurate and precise. So these produce small random error. Systematic errors in the setup of the spin flipper, analyzer, and m3 can also be the cause of the unreasonable plot. However, these effects should’ve also been seen in the analyzer polarization plot. So we are presented with the same dilemma as above: if the analyzer polarization plot is so reasonable, why is the analyzer technique plot not? This question has yet to been answered. Another possibility is that our pedestal subtraction is wrong. For some reason this is not very noticeable in the earlier graphs because they use much less monitor data. The last plot uses the most amount of transmission ratios, and at one point they are squared (see above equations), so this effect may be much more pronounced in this equation than in the others. The equation use to curve fit might also be wrong. The mathematics of the derivation seems sound, so perhaps there is error in the concepts used to arrive at the formula. Another possible cause of the unreasonableness is the programming of the plot and fit. Although I have checked extensively and could not find an error, there may be an error in the way something is calculated that is not apparent just by looking at the source code. This is to say that the histograms were manipulated improperly, even though they seem to not be. However, this is the most unlikely of the systematic errors. The most likely cause, or at least the one that would make the most sense considering how our 4th plot seems to have functional dependence that we did not predict.
Conclusion The polarizer and analyzer polarization seemed to be measured fairly accurately in this paper, however our main goal is precision measurements rather than accuracy measurements. Neutron polarization cannot be measured precisely until systematic errors are resolved with the analyzer technique of extracting the polarization. So this will be very important the next time the experiment is performed. The upside is that although the results for helium polarization were not as precise as we hoped, their accuracy was fairly reasonable from the information we know about the two different lasers. Also, it should be noted that this is really just the beginning stage of the analysis, plotting the simplest equations involving transmission ratios and then curve fitting to them. There are many more detailed and accurate ways to perform the overall analysis, but these take much more time, and will done over the next few months. There are some things that we can consider in this more detailed analysis that can improve our results. Working with data pulse by pulse will be better than working with runs that averaged 1000 pulses. Subtracting pedestals pulse by pulse and then averaging the pulses will likely yield more precise results than first averaging the pulses then subtracting. Also, Fourier analysis can aid in subtracting noise that the pedestals do not show. Overall, 0.1% precision be possible if we develop another method for comparison or figure out the systematic error associated with analyzer technique. However, these will only come in time, but the positive note is that the other results we have come to seem promising. Linear Fit To Neutron Polarization
Restricted Fit to Neutron Polarization Second Fit For Analyzer Polarization (Range 2-6)
Restricted Parameter Fit For Analyzer Polarization Second Fit To Analyzer Technique Plot (Range 2-5)
Linear Fit To Analyzer Technique Plot