Diffusion of Ultra-Cold Neutrons in Randomly Rough Channels

This thesis deals with ultra cold neutrons, or, more precisely, with beams of ultra-cold neutrons. ultra-cold neutrons are longwave particles produced in a reactor from which they are coming to experimental cells through narrow channels. The beams are collimated so that the distribution of longitudinal and transverse velocities is narrow. The energies of the neutrons that we consider as ultra cold are somewhere around 100neV . Neutrons with such low energies have long wavelengths; λ ∼ 100nm. Neutral particles with such large wavelengths exhibit nearly (locally) specular reflection when reflected by the solid surfaces at almost any angle of incidence. The number of ultra-cold neutrons available for experiment is extremely small. Therefore, a major experimental challenge is not to lose any particles while they travel from the reactor to the lab. Some of the main losses occur in the channel junctions when the neutrons disappear into the gaps between the overlapping channels. We explore the possibility of recovering some of these otherwise ”lost” neutrons by making the inside surfaces of the junctions rough: scattering by the surface roughness can send some of the neutrons back out of the gap. This practical goal made us to re-examine diffusion of neutrons through rough channels which is by itself an interesting problem. We assume that the correlation function of random surface roughness is either Gaussian or exponential and investigate the dependence of the mean free path on the correlation radius R of the surface inhomogeneities. My results show that in order to ensure better recovery of the ”lost” neutrons the walls of the junction should be made rough with the exponential correlation function of surface roughness with as small a correlation radius as possible. The results also show that the diffusion coefficient and the mean free path of UCN in rough channels exhibit a noticeable minimum at very small values of the correlation radius. This minimum sometimes has a complicated structure. The second goal is the study of UCN in Earth’s gravitational field. One of the most interesting features of ultra-cold neutrons is a possible quantization of their vertical motion by the Earth’s gravitational field: the kinetic energies are so low that they become comparable to the energy of neutrons in Earth’s gravitational field. This results in quantization of neutron motion in the vertical direction. The energy discretization occurs on the scale of several peV. In the first part of my thesis I ignore the presence of the gravitational field and look at the transport of neutrons through rough waveguides in the absence of gravity. The effects of gravity are be explored in the last part. To streamline the transition I use the common notations suitable for both types of problems. More specifically, I am studying the diffusion of ultra-cold neutrons in the context of the experiments done at ILL in Grenoble in the frame of the multinational GRANIT collaboration. The parameters used in numerical calculations are the ones most common to ILL experiments. I will be calculating the diffusion coefficient and the mean-free path (MFP) under the conditions of the quantum size effect. Specifically I look at the dependence of the diffusion coefficient and the MFP on the correlation radius of surface inhomogeneities. R. In the second and third parts of the thesis I include the study of the neutron diffusion accompanied by slow continuous disappearing of neutrons as a result of penetration into the channel walls. This includes calculating the number of neutrons N(t = τex, h, R), where τex is the experimental value of the time of flight in GRANIT experiments and h is the channel width. I look not only at the square well geometry, but will also include the effects of the Earth gravitational field. The results show that while the neutrons in the square well potential disappear almost immediately, the small perturbation near the bottom of the well caused by the presence of the Earth’s gravitational field drastically changes the results and is solely responsible for the observed exit neutron count in GRANIT experiments. The shape of the curves describing the exit neutron count on the width of the waveguide is extrememly robust. Out brute force calculations also confirms that the earlier biased diffusion approximation is quite accurate.

The number of ultra-cold neutrons available for experiment is extremely small. Therefore, a major experimental challenge is not to lose any particles while they travel from the reactor to the lab. Some of the main losses occur in the channel junctions when the neutrons disappear into the gaps between the overlapping channels. We explore the possibility of recovering some of these otherwise "lost" neutrons by making the inside surfaces of the junctions rough: scattering by the surface roughness can send some of the neutrons back out of the gap. This practical goal made us to re-examine diffusion of neutrons through rough channels which is by itself an interesting problem. We assume that the correlation function of random surface roughness is either Gaussian or exponential and investigate the dependence of the mean free path on the correlation radius R of the surface inhomogeneities. My results show that in order to ensure better recovery of the "lost" neutrons the walls of the junction should be made rough with the exponential correlation function of surface roughness with as small a correlation radius as possible. The results also show that the diffusion coefficient and the mean free path of UCN in rough channels exhibit a noticeable minimum at very small values of the correlation radius. This minimum sometimes has a complicated structure.
The second goal is the study of UCN in Earth's gravitational field. One of the most interesting features of ultra-cold neutrons is a possible quantization of their vertical motion by the Earth's gravitational field: the kinetic energies are so low that they become comparable to the energy of neutrons in Earth's gravitational field. This results in quantization of neutron motion in the vertical direction. The energy discretization occurs on the scale of several peV. and h is the channel width. I look not only at the square well geometry, but will also include the effects of the Earth gravitational field. The results show that while the neutrons in the square well potential disappear almost immediately, the small perturbation near the bottom of the well caused by the presence of the Earth's gravitational field drastically changes the results and is solely responsible for the observed exit neutron count in GRANIT experiments. The shape of the curves describing the exit neutron count on the width of the waveguide is extrememly robust. Out brute force calculations also confirms that the earlier biased diffusion approximation is quite accurate.

ACKNOWLEDGMENTS
I would like to acknowledge first and foremost my thesis adviser, Professor Alexander Meyerovich. I am deeply grateful for the guidance, advice, tremendous insight and immeasurable patience and support that you provided to me over the past 6 years.
I would also like to acknowledge Professor Leonard Kahn for the wonderful teacher that he is, for the encouragement that he provided even at times when I had doubts, and for the many informal conversations we had on a vast array of topics.
Sincere thanks go out to Professor Gerhard Muller, for all the physics tools he gave me over the past 6 years, as well as the many enjoyable and insightful conversations.
Thanks to Professor Peter Nightingale, for the crazy difficult classes that provided me with great challenges, as well as for his wit and generosity in sharing knowledge about music, history and other topics outside of physics.
Thank you to Professor Richard McCorkle for being a great support to me in my function as a TA, as well as always having a pleasant outlook on life.
Thank you to Linda Connell for always cleaning up my administrative messes, as well as being a great friend and support over the last 6 years.
Thank you to Steve Pellegrino for helping me with countless technology and computer problems, as well as for his friendship.
Thank you to David Notorianni for always fixing anything that went wrong in the labs or offices, as well as for having a kind heart behind the rough exterior.
Thank you to Professor Donna Meyer, for introducing me to engineering which has become another passion of mine, as well as for being a great role model for a woman in the hard sciences. I have great admiration for you.

3.1
Sketch of neutron beam entering the experimental cell: the neutrons pass between rough "ceiling" and smooth " ‡oor". 45 3.2 Square well levels.  3.4 N e as a function of the cuto¤ parameter S1 for h = 8 and r = 0:65. Here we can see the initial increase. 54 3.5 Neutron count as a function of the size of the matrix S1 for h = 8 and r = 0:65. We can see how it saturates nicely, at about S1 = 300. In this plot we are looking at N e over a larger scale. 55 3.6 N e as a function of the size of the matrix S1 for h = 5 and r = 0:65. We are looking at N e closer scale, so that we can see the inital increase and gradual saturation. 55 3.7 Saturation of the neutron count as a function of the size of the matrix S1, for h = 5 and r = 0:65. 3.8 N e as a function of the matrix size S1 for h = 3 and r = 0:65.
We are looking at N e closer scale, so that we can see the inital increase and gradual saturation. 56 3.9 Neutron count as a function of the cuto¤ parameter S1 and

. Preliminary Comments
The main goal of this thesis is to provide a rigorous theoretical description for the di¤usion of ultra cold neutrons (UCN) through narrow rough channels, which is based on the theory of quantum transport in systems with rough boundaries formulated by Meyerovich et al. [1]- [12]. We look at two separate problems: diffusion of the neutrons through rough waveguides on the way from the reactor to the experimental cell and the neutron count for neutrons exiting experimental cell with absorbing walls.
We use numerical computations to investigate the e¤ect of two types of random roughness on the di¤usion coe¢ cient and use numerical methods to evaluate the neutron count using the experimental values of input parameters. We analyze two types of potentials inside the cell: one the idealized square well potential (SW) and the other the SW potential with an addition of the gravitational …eld. The experimental parameters were provided for us by our experimental collaborators at the Institute Laue-Langevin (ILL) in Grenoble, France in the frame of the GRANIT project.
The purpose of this multinational collaborative experimental and theoretical work is two-fold: to investigate the quantization of the motion of UCN by the Earth gravitational …eld and to create UCN with well-de…ned energies in the peV range necessary for studies of fundamental forces in quantum …eld theory.
Typical UCN coming out of the reactor have large wavelengths, s 100 nm.
Neutral particles with such large wavelengths exhibit nearly (locally) specular re- ‡ection when re ‡ected by the solid surfaces at almost any angle of incidence. One of the most interesting features of ultra-cold neutrons is their quantization in the Earth gravitational …eld: the particle kinetic energies can be so low ( 1 peV The purpose of the GRANIT spectrometer is to eliminate the particles in higher gravitational states and leave only the ones in the few lowest states. This allows one to achieve both goals: to study the quantization of neutron motion in the gravitational …eld and to produce neutrons with well-de…ned energies in the peV range.
The lower surface of the spectrometer is as close as possible to being perfectly smooth, in order to make it to be a perfect re ‡ector which specularly re ‡ects the UCN. The upper surface of the GRANIT cell has microscale roughness. This "rough" ceiling scatters the UCN in higher gravitational states, which can reach it. The scattered neutrons from the higher gravitational states eventually acquire large vertical velocities su¢ cient to trigger penetration through the walls and disappearance from the system. Due to this setup, only the UCN in low gravitational states, which do not reach the rough ceiling, can continue bouncing along the ‡at ‡oor and arrive at the exit neutron detector. [ [8]] showed that within the biased di¤usion approximation all the information about the surface imperfections can be accounted for in the neutron count as a single parameter , which is a complicated integral of the power spectrum.
However, in practice, it is impossible to create imperfections with a predetermined CF on real surfaces, and even if it were possible it would be highly non-trivial to identify this CF.
In order to increase the resolution of the observed quantum gravitational states of the UCN in the GRANIT spectrometer, proper identi…cation of the surface correlator is paramount. If one can establish a superior way to control the necessary random roughness of the scatterer and absorber mirror, it will contribute greatly to the optimization of results from the GRANIT experiment.
In the context of the theoretical background and numerical experiments, we designate the shape of the CF explicitly, and analyze its potential impact on phys- There is also a supplementary practical issue. The number of ultra-cold neutrons available for experiment is extremely small. Therefore, a major experimental challenge is not to lose many particles while they travel from the reactor to the lab. The main losses occur in the channel junctions when the neutrons get into the gaps between the overlapping channels. Therefore the minimization of losses in channel junctions becomes an important goal which will also be approached in this thesis.
This thesis is arranged as follows: In the remainder of Chapter 1, we will provide a fairly detailed description of the experiment and its setup used by GRANIT to observe the quantum gravitational states of the ultra cold neutrons. In particular we will describe the GRANIT cell, and introduce the important parameters that are used to describe the roughness of the surfaces of the GRANIT mirror. In section 2 we will discuss the details of the mirror used in the newer experiments including the design and providing a description of how they made the roughness. In section 3 we introduce the main parameters and dimensionless variables. And, …nally, in section 4, we will provide the main equations and the theoretical framework for the quantum transport equation and di¤usion.
In Chapter 2, we will explore the possibility of recovering these "lost" neutrons that we discussed above by making the inside surfaces of the junctions rough: scattering by the surface roughness can turn some of the neutrons back. This practical goal made us to re-examine di¤usion of neutrons through rough channels which is by itself an interesting general problem. We assume that the correlation function of surface roughness is either Gaussian or exponential (see below) and investigate the dependence of the mean free path on the correlation radius R. Our conclusion is that if ideally we could create the type of roughness we want, it would be better to use exponential roughness.
In Chapter 3, we will be discussing the exit neutron count in an idealized condition, in the square-well potential without gravity. This involves solving large sets of equations with complicated coe¢ cients which tie together neutrons in thousands of quantum states. We …rst look to investigate the exit neutron count as a function of matrix size, in order to assess the value of the possible cuto¤. The matrix size here being the number of equations we are solving. In other words, to reduce the computation time, we deduce what size of the matrix is su¢ cient for our computations to be accurate. We then cut o¤ the matrix at the cuto¤ parameter and proceed to extract the neutron count and its dependence on the width of well H.
In the case of the square well potential we will see that the neutron count should go quickly to zero.
In Chapter 4, we will be doing something very similar to Chapter 3, except this time we take into account the gravitational potential. To simplify the computations, we assume that the matrix of the interstate transition probabilities has a block structure. The …rst block contains the transitions between the lowest (gravitational) states. Since for the higher states there is practically no di¤erence between the gravitational and square well states, the other three blocks describe the transitions between the square well states and between the gravitational and square well states. From this we derive the neutron count.
In the last chapter we will summarize the results presented earlier, and discuss some suggestions for what can be done looking towards the future.   One can visualize in a simple way the observation of gravitationally induced quantum states of UCN experiment. There is a collimated beam entering an experimental cell (see Fig.1.1) consisting of a smooth " ‡oor" and a rough "ceiling".
More explicitly, we have a collimated beam of UCN with a large horizontal velocity on the order of (5 15)m/s and a small vertical velocity of a few cm/s propagating between two parallel horizontal sapphire mirrors. The bottom mirror or " ‡oor" is made as close to perfect as possible. This ensures high probability of specular re ‡ection for the bouncing neutrons. The upper mirror or "ceiling" has a rough surface which is made rough by simply scratching the surface [ [16], [61]] .
This rough mirror e¤ectively serves as a selector for the vertical component of the  The roughness of the imperfections of the ceiling mixes the gravitational states and broadens the energy levels. Below, we provide a quantitative description of the roughness parameters governing the surface inhomogeneities. Interferometry (VSI) technique. The surface was scanned using a light source that splits into two coherent light beams. One of the two beams is sent towards a mirror which is coupled with a di¤erent light beam that has been re ‡ected from a sample (amplitude of roughness of 0.5 Å). The interference patterns are then analyzed using a CCD camera and provide a surface pro…le. Unfortunately however, this technique is not perfect. For example, the measurement fails if some peak is too sharp and therefore the beam doesn't re ‡ect back onto the detector. The experimental data on the surface pro…le were analyzed numerically. It was determined that the roughness correlation function most likely has an exponential shape. [9] This technique though is more appropriate then other scanning techniques such as the Atomic Force Spectroscopy. One of the reasons that VIS is better is that the scanned surface is considerably larger than the correlation radius.

Notations and Dimensionless Variables
For the purpose of this work it is useful to introduce some uniform notations for the calculations in both presence and absence of gravitational …eld. Some of the parameters below will be used to make the equations dimensionless. We are looking at the e¤ects of gravity on the transport of the UCN.
1. In this case it is useful to measure all lengths is units of: This is the amplitude of the particle bouncing in the lowest quantum state in the presence of the Earth gravitational …eld.
2. The energy scale is de…ned by: This is the gravitational energy of the neutron in the ground state.
3. The velocity scale is de…ned by: The time scale is given by: This is roughly the frequency of bounces in the lowest state.
5.The width of the waveguide H in units of l 0 is: 6.The roughness correlation radius R expressed as a dimensionless variable is: Similarly, the amplitude of roughness l as a dimensionless variable is expressed as: 8. The quantized energy levels E j of the ultra-cold neutrons in the gravitational well are given by: 9. The absorption threshold U c of the mirror material is given by: where U c 100 neV, and, therefore, u c 1:4 10 5 .
10. The ‡ight time for the ultra-cold neutrons through the waveguide of the length L is given by: In experimental conditions L 2 10 2 s. In dimensionless units L = 0 26: 11. The neutron momenta are measured in units of: Ultra-cold neutrons (UCN) are longwave particles. We are looking at UCN in narrow waveguides in which the width is comparable to the wavelength and the motion across the waveguide is quantized. This QSE automatically discretizes the initially continuous equations. This quantization turns out to be very fortuitous as it helps in numerical calculations: if we were working with a continuous system, we would need to discretize the problem anyway. QSE leads to a split of the energy spectrum (p) into a set of minibands j (q) such that (p x ; q) ! j (q), where p is the 3D momentum, and q is the 2D momentum in the plane of the surface.
More explicitly, an initially parabolic spectrum, (p) = p 2 =2m becomes and the 2D momentum for miniband j becomes where E is the overall kinetic energy of particles, m is the mass of the neutrons, H is the width of the channel.
In an ideal waveguide, the quantum levels are well de…ned and the states are not mixing. Scattering by random surface inhomogeneities leads to inter-and intraband transitions and eventually mixes and broadens the quantum states.
Sometimes, as in experiments performed at ILL (Grenoble), the waveguides, or, more precisely, one of the neutron mirrors, are made rough on purpose.

Transport Equation
Studies on the e¤ect of random surface roughness on wave or particle scattering describe the di¤usion ‡ows of UCN along a rough waveguide. Meyerovich et al.
[[1]- [8]] developed a rigorous theoretical framework of quantum transport theory in system with random rough boundaries. This framework incorporates the boundary scattering directly into the bulk transport equation. It includes the roughness of the walls explicitly into the roughness-driven transition probabilities between quantum states. The transport equation for distribution functions n j (q) in a miniband j has the form where n j (q) is the distribution function of the particles, jq is the energy spectrum, q is the momentum in the plane parallel to the surface, and W jj 0 (q; q 0 ) are the scattering-driven probabilities of transitions between the states j (q) and j 0 (q 0 ).
The probabilities of direct transitions from the lowest states to the continuous spectrum above the threshold U c are negligible and such transitions can be disregarded. After integration over the energies, the transport equation acquires the following form: where N j is the number of neutrons in the state j, and is the angle between q j and q j 0 .
Our goal is to …nd the di¤usion coe¢ cient and the mean-free path, which is proportional to the di¤usion coe¢ cient. After standard transformations (a more detailed derivation can be found in the Appendices) the transport equation reduces to a set of linear equations for j (q j ): Here Q j is the momentum, the transition times jj 0 are given below by Eq.(1:24), and j is the …rst angular harmonic of the distribution function n [ [4], [5]]. The equations can be made dimensionless using Finally, the dimensionless transport equation acquires the form:

Transition Probabilities
The roughness-driven transition probabilities between quantized states have the following form: when the absorption threshold U c ! 1. Here j and j 0 are the miniband indices, is the correlation function of surface homogeneities (see below), j (h) is the wavefunction at the surface.
In the case of the square well potential this equation becomes the following : The transitions times in the transport equation are directly related to the angular harmonics of these transition probabilities as follows: Correlation Function of Roughness. The correlation function of surface roughness (CF) is de…ned as: where (jsj) is the exact pro…le of the wall and A is the area over which the averaging is done. The mathematical form of the CF cannot be found theoretically except in very few instances in which we have exactly solvable models of surface roughness. It is usually assumed that the CF has the following general form: with some function ' (x=R), where l and R are the average amplitude and correlation radius. However, nothing prevents the CF to acquire a more complicated form, for example, with several correlation scales R c . In calculations we assume that we know the shape of the CF. The most commonly used correlation functions have either the Gaussian forms. Sometimes people also use a CF with a power law shape. Here R is the correlation radius of surface inhomogeneities., r = R=l 0 , and the dimensionless amplitude is de…ned as = l=l 0 . There are reasons to believe that the correlation function in Grenoble experiments might be exponential, Ref. [[9]].
The angular harmonics of the Gaussian correlation function are In dimensionless variables, where the dimensionless roughness parameters and r and can be used as free parameters in the identi…cation of surface correlations. These two parameters are often su¢ cient to describe the surface roughness.
The dimensionless transition probabilities for exponential roughness can be written as The di¤usion of ultra-cold neutrons displays a strong directional upward bias in terms of the transitions between j ! j 0 . This bias is due to the rapid growth of the product of the wavefunctions on the boundary j (h) This allows a growth of roughly as j 2 j 02 , see Eq.(1.23). There are two main consequences of this bias. The …rst one being that the strong upward bias may allow one to neglect particles returning back to the lowest states. And the second consequence is that the time necessary for a neutron in one of the lowest gravitational states to di¤use upward towards the absorption barrier is spent almost entirely on the …rst transition.
In the next chapter we will examine more closely the process of di¤usion, and expand upon and develop a more detailed theoretical approach. The transition probabilities are proportional to the square of the amplitude of roughness . Therefore, the scaling of the results with the roughness amplitude is trivial and in most of the computations we simply assume = 1. The scaling of the results with the correlation radius r = R=l 0 is complicated and is not known beforehand. One of our main goals is to …nd out the dependence of the di¤usion parameters on r.
In relevant experiments the width of the channels leading to the cell is H = 50 m, and the particle energy is E = 150 neV; this makes h = 8:52, and e = 2:49 10 5 .
The highest occupied quantum level j max satis…es the inequality, Solving for j max we get which means that the transport equation in this case reduces to a set of 1352 coupled equations.

The Di¤usion Coe¢ cient
The main purpose of this section of the work was to …nd the di¤usion coe¢ cient and the mean free path for UCN in rough channels. We are trying to examine how the di¤usion coe¢ cient changes under di¤erent conditions. More explicitly, we are interested in its dependence on r. Di¤usion is a process that originates from random motion of particles when there is a net ‡ow from one region to another. As a result, in our case, in the presence of a concentration gradient r the di¤usion equation reduces to where is the particle density. Its concentration gradient r is a simple scaling parameter, which, in the end, cancels out from the equation for the di¤usion coe¢ cient. After this cancellation, the di¤usion coe¢ cient D becomes The dimensionless di¤usion coe¢ cient d 0 =~=m = 6:3 10 8 m=s 2 : The dimensionless distributions e j = j =l 0 are obtained from numerically solving the transport equation.

Mean Free Path
We also want to calculate the particle mean-free path (MFP) in a rough waveguide. The mean free path in very basic terms is the average distance traveled between collisions. Here we de…ne it with respect to the di¤usion coe¢ cient as where v is the velocity. The dimensionless velocity where v 0 = 1:5 10 2 m=s. The dimensionless mean free path`= L=l 0 , As one can clearly see the MFP is intimately related to the di¤usion coe¢ cient.

Numerical Results
Before presenting the results, let us summarize the dimensionless equations from above. The transport equation , contains the transition times The dimensionless harmonics of the transition probabilities for the exponential and Gaussian roughness correlators are given in explicit detail in the Appendix A.
For the overall and "partial" di¤usion coe¢ cients d and d j the dimensionless equations are as follows Finally, the MFP is Using the dimensionless equations for the transition probabilities from the previous section, we are now able to perform computations to get the di¤usion coef-      Looking at d (r) in the range of r s 20 60, we also get a good …t using the power law, with the power p = 3:5. Looking at the exponential correlation function d (r) in the ranges of r from 1 40, we see that we get a good …t using the power law, with the power p = 2:95.

Conclusions
We calculated the di¤usion coe¢ cient and the mean free path for ultracold neutrons in narrow channels with random rough walls. We have concluded that there is a complicated minimum in d (r) and L(r) for small correlation radius r 2 10 4 .
We have also concluded that the di¤usion coe¢ cient and the MFP rapidly increase as the correlation radius r increases, though at di¤erent rates depending on the surface correlation function.
The growth is not monotonic, there is more then one minimum at q j 1=r.
We compared the behavior of d (r) and L(r) for surfaces with the Gaussian and the exponential correlation functions of surface roughness.

CHAPTER 3
Neutron Beams between Absorbing Rough Walls: Square Well Approximation

Description of Problem
In this Chapter we deal with a slightly di¤erent UCN di¤usion problem which is more directly related to the GRANIT experiments in the ILL, Grenoble. In experiments the UCNs travel between rough absorbing walls and the number of UCNs exiting the cell is measured as a function of the distance between the walls.
We start from discussing the case without gravity because it is simple and will serve as a good reference point. By comparing numerical results obtained with and without gravity we will understand what part of the experimental results should be directly attributed to the Earth's gravitational …eld.
The neutrons in the cell are passing between the two mirrors, the perfectly smooth bottom mirror (" ‡oor"), and the randomly rough upper mirror ("ceiling").  The quantization of restricted motion is a well-known quantum phenomenon.
In the absence of gravity we are dealing with the simplest square well potential with the energy levels  To recap from the experiment brie ‡y, we are dealing with a collimated beam being sent between two horizontal solid plates (one which is an almost ideal mirror and the other is rough) that are at a distance of several micrometers apart. We know that the neutrons hitting the wall with the normal velocity above 4 m/s get absorbed by the plates. Below this threshold velocity the neutrons get re ‡ected.
The re ‡ection is specular locally.
First, we will neglect the presence of gravity. The e¤ects of gravity will be introduced later, in the next chapter.

Wavefunction for the Square Well
We start by introducing the equation for the wavefunction on the wall in the square well, where H is the width of the waveguide. Below we will use the same dimensionless variables as in the previous chapter.
To determine the roughness-driven transition probabilities, Eq. it is also the notation that we will be using from here on out in this thesis. Hence, we de…ne the b j in dimensionless units as follows, In the case of the square well this reduces to where j is de…ned as The values of b j 's in the gravitational potential we will get from the Airy functions, which will be discussed more in the next section. The constant 10 5 is here merely as a scaling factor to avoid dealing with very small numbers.

Transition Probabilities and Neutron Count
As mentioned above, we are dealing with the same set of transport equations as in the previous Chapter of di¤usion and mean-free path. Here again, we start with the transport equation, In this section we are looking only at the exponential correlation function of the surface inhomogeneities and are not interested in the potential Gaussian correlations. The reason for this is that recent analysis of the surface roughness of the new "rough" mirror have led us to believe that the surface roughness is exponential rather than Gaussian Ref. [9] We therefore use the transition probabilities with the exponential correlation function. Since we previously introduced the transition probabilities, we will write them directly in dimensionless variables, where (3.9) = 2r and E ( ) and K ( ) are elliptical integrals.
As above we use the transition probabilities to get the dimensionless transition we can now write the neutron exit count in terms of these relaxation times in a simple form where t L is the time of ‡ight of the UCN between the mirrors and j are the eigenvalues of Eq: (3:10).

Numerical Results
Initially, we want to start o¤ by de…ning and discussing the parameter that we introduce, S1. The S1 parameter is a cuto¤ parameter. If we were to numerically solve the above equations for the whole system, we would be solving more than 10 3 coupled linear equations with complicated coe¢ cients. Solving a system of that many equations is computationally very expensive time-wise even with modern computers, as each value takes approximately 30 min, and we do 900 iterations over values of h from 9 to 1 for each r. Therefore, we wanted to examine if there were a reasonable cuto¤, which would keep enough equations to not lose much accuracy in the calculations, while also being much less expensive time-wise computationally. To identify the S1 cuto¤, we simply run the computations using more and more equations until we see that there is a saturation in the results. The saturation point becomes our cuto¤ point. We de…ne that parameter as S1: In this section we are presenting some of our numerical results.
As we see in the …gures below, Fig.[3.4]- Fig.[3:9], we are plotting the number of neutrons exiting the waveguide as a function of the size of the matrix S1. As one can see from all these …gures presenting the exit neutron count as a function of S1 for various values of r and h, in all the cases S1 300 can serve as a good cuto¤ parameter. From this point onward, we choose in all computations S1 300 and just occasionally check the results for larger matrices.
As a next step, we compute the dependence of the exit neutron count N e on r  For r = 1, Fig.[ 3:11], we see that the depletion of the total neutron count to zero is slightly less rapid, though it also goes to zero very quickly around h = 9:4.      In the last two plots, we are computing the total neutron count N e (h) over the full range of h for r = 10 ( Fig.[3:13]) and r = 30 (Fig.[3:14]). As with all the results that we presented previously for varying r, we observe again that as we increase the radius of roughness the total neutron count goes to zero slower which here means at a smaller width size h.  Again only for large r does the neutron count become non-zero. The last two …gures, Fig.[ 3:17]. and Fig.[3 :18]., show that, as in the previous ones, the neutron count becomes non-negligible only for large r for h = 7 and h = 9, respectively. This is consistent with the theory that, for large enough r or in the limit that r ! 1 we will not have any roughness and therefore all the neutrons will make it to the detector.

Conclusions
The main conclusions for this section are as follows. Irrespective of the well width and the correlation radius, a good cuto¤ parameter for all computations is around S1 = 300.
As we increase the radius of roughness r, the total neutron count goes to zero for at a slower rate, meaning for smaller and smaller values of h.
However, as we can see from the plots, the numbers that we get for N e (h) are always extremely small, so essentially all the neutrons die almost immediately.
As we will see in the following section, this is not the case for the gravitational well. Hence, we can say that the square well approximation is very     It is clear from this …gure that for small h there is hardly any di¤erence between

Results from the Preceding Work: the Biased Di¤usion Approximation
The preceding work used what the authors called the biased di¤usion approximation. Since the transitions j ! j 0 show a strong upward bias due to the factor b j b j 0 in the transition probabilities W jj 0 (essentially the factor j 2 j 02 in Eq.(1:23)), the probabilities for the neutrons to return back to the lower states j after they jump to a higher state j 0 appear to be small and can be neglected. Then the absorption times j are where is the angle between q j and q j 0 . Note, that since the absorption threshold The justi…cation for biased di¤usion is that the transition rates ( 1 jj 0 ) between the states (j and j 0 ) rapidly increase with both of these quantum numbers. Since the rates of the direct absorption processes also rapidly increase as j gets larger, this means that the neutron lifetimes in the higher states are orders of magnitude shorter then the lifetimes in the lower states. Therefore the di¤usion of a neutron between energy levels has a strong upward bias. The increase in the jump rate from j to j 0 is moderated only by the correlation function, which is determined by the correlation radius r and starts rapidly decreasing at large jj j 0 j r. This is why w jj 0 acquires a narrow peak centered at j 1 >> j Fig.[ 4:5] The bias is so strong that almost all the time j spent for the neutron in a low gravitational state to transition up to higher states and over the absorption barrier is spent in the …rst transition upwards.
The values of j determine the depletion time of each quantum state. The overall exit neutron count is where N j (0) is the number of neutrons in a state j entering the waveguide of length L. Additionally, for the lowest levels the velocities v j are almost the same, v j p "v 0 . The equation above can be rewritten using v j p "v 0 and we can directly get all the j . Then it is easy for us to get the total neutron count which is just a sum over all j. In the end, in the biased di¤usion approximation all the pertinent parameters of roughness and the waveguide entering the exit neutron count collapse into a single variable Ref [8] and we get an analytical solution, where is a complicated weighted integral of the correlation function that is dependent on the correlation radius. If the roughness is two-dimensional, where 2 (y 1 ; y) is the dimensionless zeroth harmonic of the correlation function (jq j q j 0 j) over the angle between the vectors q j and q j 0 , and y 1 = r p ", y (z) = We can write the relaxation time 1 for the lowest gravitational state as where is the angle between the vectors q 1 and q j 0 . Finally, after replacing the summation by the integration. This concludes the description of the preceding work, which provides the approximate analytical expression to the exit neutron count. We will now describe how we deal with the same problem computationally without relying on the biased di¤usion approximation.

Exact Calculation of the Absorption Time
In our case we are not using the biased di¤usion approximation, but actually using the brute force technique to solve the full set of transport equations numerically.
As a result, we do not get a nice analytical solution to the problem. Instead we use the full matrix, meaning the matrix with transitions upwards and downwards, for which we can only get a numerical solution. The structure of the diagonal and o¤-diagonal elements in the matrix transport equations is di¤erent. The diagonal elements have the structure de…ned below in Eq. (4:12), where for an element in row j we are summing over all the elements j 0 . The o¤-diagonal elements are simpler: these are simply w jj 0 . We rewrite the transition probabilities W jj 0 de…ned in Eq.(1:22) that represent the diagonal elements of the matrix, in a notation that is closer to the one that was used in the most recent papers in this …eld. Therefore speci…cally for diagonal elements of our matrices, when j = j 0 , we write, where 2 (y 1 ; y j 0 ) here also represents the dimensionless zeroth harmonic of the correlation function over the angles between the vectors q j and q j 0 . Since we are not working under the context of biased di¤usion, we can only perform our computations numerically.
We have o¤-diagonal elements in our matrix for square well are de…ned in dimensionless units as where (4.14) = 2r Hence our total matrix, which we will call M jj 0 looks like More detailed mathematics regarding the F jj 0 = F j (r; h) for the discrete case will be available in Appendix C.
We do not have a simple analytical description of the gravitational states similar to that for the square well states in the previous chapter. However, for the states with high index j, and especially at small h, the di¤erence between the gravitational and square well states is negligible. Therefore, for higher states we can replace the gravitational states by the square well states.
As a result, our square matrix of transition probabilities acquires a block structure. One block representing the transitions between gravitational states, two of the blocks represent the transitions between the lower gravitational states and higher square well states, and the third block represents the transitions between higher square well states. Once we have the total matrix with all the block components, we are numerically computing the eigenvalues and eigenvectors of the total matrix. We can write the neutron exit count in terms of these absorption times j in quite a simple form, where t L is the time of ‡ight of the UCNs between the mirrors. Note that the above equation was introduced in the previous section and chapter as it is a general equation that can be used with and without the biased di¤usion approximation.
It is interesting to note that the center of the peak of the transition probabilities W jj 0 for transitions from j to j 0 is located at some j 1 >> j, see  The peak is very high and relatively narrow.

Numerical Results
In this section we present the main results of the thesis as pertaining to the Grenoble experiments. As in the previous section regarding the square well potential, we are now looking at the exit neutron count in the gravitational potential.
The gravitational potential was introduced in a previous chapter. However, we will reproduce a schematic …gure here. In this …gure, the particles on the lowest three Though here we do not present a …gure to show the saturation of the total neutron count as a function of the cuto¤ parameter S1, after doing many numerical simulations, it was determined from the results that the cuto¤ parameter was more or less the same as in the case of the square well potential, meaning that S1 = 300 is a good cuto¤ size to maintain high accuracy in the calculations, while simultaneously keeping the computation time su¢ ciently short.   small width h. This is happening as well in the case of the gravitational potential.
We see that as the radius of roughness gets larger and larger, the neutrons survive for smaller and smaller well width sizes. large r. This of course is due to the fact that the neutrons are not scattering and dying but instead can easily make it to the end of the cell and to the detector since they are not in ‡uenced by roughness for large r. The ideal conditions for scattering are at qr 1. From this condition we know that as r gets larger and larger and goes to in…nity, we will have specular re ‡ection.
The last …gures, Fig.[18 20] show the neutron exit count for …xed widths h = 9; 5; 3 respectively. As we can see for all three widths, the neutron exit count increases as the correlation radius increases. This is because as we increase the correlation radius, the ceiling becomes smoother, and therefore there is less scattering and absorbtion by the rough wall.

Conclusions
It is interesting to compare these results with the previous results without gravity in the square well section.
The dependence of the exit neutron count on r was much more signi…cant in the case of the square well potential: N e (r) at …xed h changed by many orders of magnitude.  It is interesting to note that the ratio of the relaxation times for the diagonal case (in which we do not take into account the transitions between the states, meaning the cases where j 6 = j 0 ) and the case described in this section (where we allow all the transitions, and not just the biased ones, to be taken into account) diag = f ull is close to 1. This was rather unexpected.
It is also interesting to note that the ratio of the results for the matrix in the square well potential and the matrix in the gravity potential goes to one as h goes to 0, jSqwell = jgrav ! 1 as h ! 0 since all the levels are        The main content of Chapter 2 is the calculations of the di¤usion coe¢ cient and the mean free path for ultra-cold neutrons in narrow channels with random rough walls. We determined that if one wants to e¤ectively turn back the neutrons which got into the gaps in the channel junctions, one should make the correlation radius of surface roughness as small as possible. We compared the behavior of d (r) and l(r) for surfaces with the Gaussian and the exponential correlation functions of surface roughness. We found that there is a complicated minimum in d (r) and l(r) for small correlation radius r 2 10 4 . Additionally we found that the di¤usion coe¢ cient and the MFP rapidly increase as the correlation radius r increases, though at di¤erent rates depending on the surface correlation function.
The growth of the di¤usion coe¢ cient and MFP is not monotonic, there is more then one minimum at q j 1=r. We saw that at large r the function d (r) behaves roughly as r 3 , where the exponent slightly drifts with r. The computations were done for realistic values of the channel width h = 8:52. At di¤erent values of h the results were qualitatively the same. The growth of d (r) and l(r) for the Gaussian surface correlation function is much faster than for the exponential correlation function. As a result, it is preferable to have the junction walls with exponential correlation of inhomogeneities.
The conclusions for Chapter 3 regarding the UCN in the square well potential are as follows: Irrespective of the well width and the correlation radius, a good cuto¤ parameter for all computations is around S1 = 300. We see that as the radius of roughness r increases, the total neutron count goes to zero for smaller and smaller values of h. However, as we can see from the plots, the numbers that we get for N e (h) are always extremely small, so essentially all the neutrons die almost immediately. As we will see in this Chapter, this is not the case for the gravitational well. Hence, we can say that the square well approximation is very poor for this particular problem: though the weak Earth gravitational …eld introduces only a small distortion near the bottom of the potential well, its e¤ect on the neutron survival rate is very large.
The main conclusion for Chapter 4 is the exit neutron count in the presence of gravity. It illustrates the total neutron count N e (h) as a function of the width of the channel h. When comparing this result to the work done previously using the biased di¤usion approximation, we see that the curves are similar. This means that the biased di¤usion approximation is a very robust approximation. This result was a somewhat surprising result. Other conclusions include the following. The dependence of the exit neutron count on r was much more signi…cant in the case of the square well potential where N e (r) at …xed h changed by many orders of magnitude. It is interesting to note that the ratio of the relaxation times for the diagonal case (in which we do not take into account the transitions between the states, meaning the cases where j 6 = j 0 ) and the case described in this Chapter (where we allow all the transitions, and not just the biased ones to be taken into account) diag = f ull is close to 1. This was rather unexpected. It is also interesting to note that the ratio of the results for the square well potential and the gravitational potential goes to one as h goes to 0, jSqwell = jgrav ! 1 as h ! 0 since all the levels are being squeezed up and the di¤erence in potentials near the bottom of the well loses its signi…cance, see Figure above.

Recommendations for Future Work
Using rough mirrors as a quantum state selector can be extended beyond this series of GRANIT experiments. Some of the more exciting experiments include the observation of quantum gravitational states for other ultra-cold particles and anti-particles in the context of the GBAR experiment at CERN.
One of the main goals of the GBAR experiments is to measure the acceleration in free fall of ultra-cold neutral anti hydrogren atoms in the Earth's gravitational …eld. The experiment entails using anti hydrogen ions, which consist of one antiproton supplied by the ELENA deceleration ring at CERN and two positrons created by the linac, and cooling them below 10 K with Beryllium plus ions.
Their positive charge makes them easier to manipulate. Using lasers, their velocity can be reduced to half a meter per second. Once they are trapped by an electric …eld, one of their positrons will be removed with laser, which will make it neutral.
Hence, at this point the Earth's gravitational …eld will be the only force acting upon them, and they will be able to free fall a given distance, and their time of fall could be measured. The results of this experiment are much anticipated, because it could potentially mean that gravity might have a di¤erent e¤ect on antimatter then it does on matter.
For the past 60 years, there has also been an ongoing search for the neutron electric dipole moment (nEDM). Over the course of the decades the accuracy of (negative) results has been improved by many orders of magnitude. The nEDM potentially violates CP symmetry, meaning that it violates the presumption that if a particle and a respective anti particle are interchanged, while their spatial coordinates are inverted, then the laws of physics should remain the same. The goal of these ongoing and future nEDM experiments is to improve the sensitivity for detection nEDM by orders of magnitude. One of the experiments being done at Oak Ridge National Laboratory is to create a three-component ‡uid described as isotopically puri…ed Helium-4, a trace amount of spin-polarized Helium-3, and spinpolarized ultra-cold neutrons. Then once that ‡uid has been created, it should be exposed to a small but homogeneous magnetic …eld and a large electric …eld.
The nEDM could then be measured by looking at the neutron precession frequency which is linearly dependent on the magnitude of the electric …eld strength, and whose sign is dependent on the alignment between the magnetic and electric …elds.  where we substitute z = j N and dj N = dz and then we can write z 2 = j 2 N 2 . The above equation now becomes,