Physicists have a habit of taking the simplest example some phenomenon and call it "physics", and it is more difficult to give examples to be torn to pieces by other sciences, say, applied mathematics, electrical engineering, chemistry or crystallography. Even solid state physics for them is only "semiphysics", because it is concerned with too many special issues. For this reason, we will omit many interesting things in our lectures. For example, one of the most important properties of crystals and most substances in general is that their electrical polarizability is different in different directions. If you apply in any direction electric field, then the atomic charges will shift slightly and a dipole moment will arise; the magnitude of this moment depends very strongly on the direction of the applied field. And this, of course, is a complication. To make life easier for themselves, physicists start the conversation with the special case where the polarizability is the same in all directions. And we leave other cases to other sciences. Therefore, for our further considerations, we will not need at all what we are going to talk about in this chapter.
The mathematics of tensors are especially useful for describing the properties of substances that change with direction, although this is just one example of its use. Since most of you are not going to become physicists, but intend to work in the real world, where the dependence on direction is very strong, sooner or later you will need to use a tensor. So, so that you don't have a gap here, I'm going to tell you about tensors, although not in great detail. I want your understanding of physics to be as complete as possible. Electrodynamics, for example, we have a completely finished course; it is as complete as any course in electricity and magnetism, even an institute one. But mechanics is not finished with us, because when we studied it, you were not yet so firm in mathematics and we could not discuss such sections as the principle of least action, Lagrangians, Hamiltonians, etc., which represent the most elegant way descriptions of mechanics. However, we still have a complete set of laws of mechanics, with the exception of the theory of relativity. To the same extent as electricity and magnetism, we have many sections completed. But here we will not finish quantum mechanics; However, you need to leave something for the future! And yet, what is a tensor, you still should know now.
In ch. 30 we emphasized that the properties of a crystalline substance are different in different directions - we say that it is anisotropic. The change in the induced dipole moment with a change in the direction of the applied electric field is only one example, but that is what we will take as an example of a tensor. We assume that for a given direction of the electric field, the induced dipole moment per unit volume is proportional to the strength of the applied field . (For many substances, at not too large, this is a very good approximation.) Let the constant of proportionality be . Now we want to consider substances that depend on the direction of the applied field, such as the tourmaline crystal you know, which gives a double image when you look through it.
Suppose we have found that for some chosen crystal an electric field directed along the axis gives a polarization directed along the same axis, and an electric field of the same magnitude with it directed along the axis leads to some other polarization also directed along axes . What happens if an electric field is applied at an angle of 45°? Well, since it will be just a superposition of two fields directed along the axes and , then the polarization is equal to the sum of the vectors and , as shown in Fig. 31.1, a. The polarization is no longer parallel to the direction of the electric field. It is not difficult to understand why this happens. There are charges in the crystal that are easy to move up and down, but which are very difficult to move sideways. If the force is applied at an angle of 45 °, then these charges are more likely to move up than to the side. As a result of such asymmetry of the internal elastic forces, the displacement does not proceed in the direction of the external force.
Fig. 31.1. Addition of polarization vectors in an anisotropic crystal.
Of course, the 45° angle is not highlighted. The fact that the induced polarization is not directed along the electric field is also true in the general case. Before that, we were simply “lucky” to choose such axes and for which the polarization was directed along the field . If the crystal were rotated with respect to the coordinate axes, then an electric field directed along the axis would cause polarization both along the axis and along the axis. In a similar way, the polarization caused by a field directed along the axis would also have both - and -components. So instead of Fig. 31.1, and we would get something similar to Fig. 31.1b. But despite all this complication, the magnitude of the polarization for any field is still proportional to its magnitude.
Let us now consider the general case of an arbitrary orientation of the crystal with respect to the coordinate axes. An electric field directed along the axis gives a polarization with components along all three axes, so we can write
By this I mean only that an electric field directed along the axis creates polarization not only in this direction, it leads to three polarization components , and , each of which is proportional to . We called the proportionality coefficients , and (the first icon indicates which component we are talking about, and the second refers to the direction of the electric field).
Similarly, for a field directed along the axis, we can write
and for the field in -direction
Further we say that the polarization depends linearly on the field; therefore, if we have an electric field with components and , then the polarization component will be the sum of two defined by equations (31.1) and (31.2), but if it has components in all three directions , and , then the polarization components should be the sum of the corresponding terms in equations (31.1), (31.2) and (31.3). In other words, it is written as
B.1 Introduction.
B.2 Cumulative particles.
B.3 Description of polarized states of particles with spin 1 5 B.4 Brief review of data on the reaction of fragmentation of deuterons into cumulative protons.
B.5 Purpose and structure of the dissertation work.
I SETUP OF THE EXPERIMENT
1.1 Motivation.
1.2 Experimental setup.
1.3 Methodological measurements and modeling
1.4 Organization and principle of operation of the trigger.
II SOFTWARE
II. 1 Introductory remarks
11.2 qdpb data collection and processing system
11.3 Configurable data and hardware views
11.4 Session dependent presentations
11.5 DAQ system SPHERE.
II. 6 Data acquisition systems for polarimeters.
III EXPERIMENTAL RESULTS AND THEIR DISCUSSION
III. 1 Analysis of sources of systematic errors.
111.2 Experimental data.
111.3 Discussion of experimental data.
Recommended list of dissertations
The study of spin and isospin effects in reactions of the production of cumulative particles 2007, Doctor of Physical and Mathematical Sciences Litvinenko, Anatoly Grigorievich
Study of interactions of polarized deuterons with protons and nuclei in the momentum range 0.7-9.0 GeV/c 2006, Doctor of Physical and Mathematical Sciences Ladygin, Vladimir Petrovich
Study of the angular dependence of the analyzing abilities of the reactions -dd→3Hen and -dd→3H p at a deuteron energy of 270 MeV 2007, Candidate of Physical and Mathematical Sciences Janek, Marian
Tensor analyzing power Ayy in the reactions A(d, p)x and A(d, d)x at 9 GeV/c and the structure of the deuteron at short distances 1998, Candidate of Physical and Mathematical Sciences Ladygin, Vladimir Petrovich
Study of the analyzing abilities Ay, Ayy and Axx of the reaction of deuteron-proton elastic scattering at energies of 880 and 2000 MeV 2010, Candidate of Physical and Mathematical Sciences Kurilkin, Pavel Konstantinovich
Introduction to the thesis (part of the abstract) on the topic "Measurements of the tensor analyzing power of T20 in the reaction of fragmentation of deuterons into pions at zero angle and development of software for data acquisition systems of installations on polarized beams"
B.1 Introduction
The dissertation paper presents experimental results of measurements of the tensor analyzing power T20 in the reaction of fragmentation of tensor polarized deuterons into cumulative (sub-threshold) pions. The measurements were carried out by the SPHERE collaboration on a beam of tensor polarized deuterons at the accelerator complex of the High Energy Laboratory of the Joint Institute for Nuclear Research (LHE JINR, Dubna, Russia). The study of polarization observables provides more detailed information, compared to reactions with non-polarized particles, on the interaction Hamiltonian, reaction mechanisms, and the structure of the particles involved in the reaction. To date, the question of the properties of nuclei at distances smaller than or comparable to the size of a nucleon has not been adequately studied both from the experimental and theoretical points of view. Of all the nuclei, the deuteron is of particular interest: firstly, it is the most studied nucleus from both experimental and theoretical points of view. Secondly, for the deuteron, as for the simplest nucleus, it is easier to understand the reaction mechanisms. Third, the deuteron has a nontrivial spin structure (spin equal to 1 and a nonzero quadrupole moment), which provides wide experimental possibilities for studying spin observables. The measurement program, within which the experimental data presented in the dissertation work were obtained, is a natural continuation of studies of the structure of atomic nuclei in reactions with the production of cumulative particles in the collision of unpolarized nuclei, as well as polarization observables in the deuteron decay reaction. The experimental data presented in the dissertation work make it possible to advance in understanding the spin structure of the deuteron at small internucleon distances and supplement the information on the structure of the deuteron obtained in experiments with a lepton probe and in the study of the breakup reaction of tensor polarized deuterons, and therefore seem to be relevant. To date, the data presented in the dissertation work are the only ones, since such studies require beams of polarized deuterons with an energy of several GeV, which at present and in the next few years will be available only at the JINR LHE accelerator complex, where it is natural to continue research in this direction. The mentioned data were obtained as part of an international collaboration, were reported at a number of international conferences, and also published in peer-reviewed journals.
Further in this chapter, we give the information about cumulative particles necessary for further presentation, the definitions used in the description of polarization observables, and also give short review known in the literature results on the deuteron decay reaction.
B.2 Cumulative particles
Studies of the regularities of the birth of cumulative particles have been carried out since the beginning of the seventies of the XX century, , , , , , , , , , , , . The study of reactions with the production of cumulative particles is interesting in that it provides information about the behavior of the high-momentum (> 0.2 GeV/c) component in fragmenting nuclei. These large internal momenta correspond to small ones (< 1 ферми) межнуклонным расстояниям. На таких (меньших размера нуклона) расстояниях использование нуклонов как квазичастиц для описания свойств ядерной материи представляется необоснованным, и могут проявляться эффекты ненуклонных степеней свободы в ядрах , , , . В глубоконеупругом рассеянии лептонов упомянутый диапазон внутренних импульсов соответствует значениям переменной Бьоркена хъ >1, where the cross sections become very small.
First of all, let us define what will be further understood by the term "cumulative particle" (see, for example, the references therein). Particle c, born in the reaction:
Ar + Ac.^c + X, (1) is called "cumulative" if the following two conditions are satisfied:
1. the particle c was born in a kinematic region inaccessible in the collision of free nucleons having the same momentum per nucleon as the nuclei Ai and Ac in reaction (1);
2. particle c belongs to the fragmentation region of one of the colliding particles, i.e. must be done either
St, - Yc\< \YAii - Ус| , (2) либо
YA„-Ye\ YA„ - Yc\ «- Ye\ = - Ye\ + \YAii - YAi\ . (four) It follows from the experimental data (see, for example, , , , , , , , ) that for experiments on a fixed target, the shape of the spectrum of cumulative particles weakly depends on the collision energy, starting from the energies of the incident particles Tb > 3-1-GeV. This statement is illustrated in Fig. 1, reproduced from , which shows the dependences on the energy of the incident proton: (b) the ratio of the outputs of pions of different signs 7r~/7r+ and (a) the parameter of the inverse slope of the spectrum To for the approximation Eda/dp - C exp(-Tx/To ) cross sections for the production of cumulative pions measured at an angle of 180°. This means that the independence of the shape of the spectra from the primary energy begins with the difference in the speeds of the colliding particles \YAii - YAi\> 2. Another established pattern is the independence of the spectra of cumulative particles from the type of particle on which fragmentation occurs (see Fig. 2). Since the dissertation paper considers experimental data on the fragmentation of polarized deuterons into cumulative pions, the regularities established in reactions with the production of cumulative particles (dependence on the atomic mass of the fragmenting nucleus, dependence on the type of detected particle, etc.) will not be discussed in more detail. If necessary, they can be found in the reviews: , , , . Rice. 1: Energy dependence of the incident proton (Tp) of (a) the reciprocal slope parameter To and (b) the ratio of outputs tt~/tt+ integrated starting from a pion energy of 100 MeV. Figure and data marked with circles are taken from . Data marked with triangles are cited from . B.3 Description of polarized states of particles with spin 1 For the convenience of further presentation, we give a brief overview of the concepts , , which are used in describing the reactions of particles with spin 1. Under ordinary experimental conditions, an ensemble of particles with a spin (a beam or a target) is described by a density matrix p, the main properties of which are as follows: 1. Normalization Sp(/5) = 1. 2. Hermitianity p = p+. Present experiment r Reference 6 P-1-1-1-1-S f Present experiment T ▼ Reference 6 L-S O - Si - Rb f d sh Cumulative scale variable xs Rice. Fig. 2: Dependence of the cross section for the production of cumulative particles on the cumulative scaling variable xc (57) (see Section III.2) for the fragmentation of a deuteron beam on various targets into pions at zero angle. Picture taken from work. 3. The average of the operator O is calculated as (O) = Sp(Op). The polarization of an ensemble (for definiteness, a beam) of particles with spin 1/2 is characterized by the direction and average back. As regards particles with spin 1, one should distinguish between vector and tensor polarizations. The term "tensor polarization" means that the description of particles with spin 1 uses a tensor of the second rank. In general, particles with spin I are described by a tensor of rank 21, so for I > 1 one should distinguish between the polarization parameters of the 2nd, 3rd ranks, etc. In 1970, at the 3rd International Symposium on Polarization Phenomena, the so-called Madison Convention was adopted, which, in particular, regulates the notation and terminology for polarization experiments. When recording the nuclear reaction A(a, b)B, arrows are placed over particles that enter into the reaction in a polarized state or whose polarization state is observed. For example, the notation 3H(c?,n)4He means that the unpolarized 3H target is bombarded by polarized deuterons d and that polarization of the resulting neutrons is observed. When we talk about measuring the polarization of a particle b in a nuclear reaction, we mean the process A(a,b)B, i.e. in this case, the beam and the target are not polarized. The parameters describing the changes in the reaction cross section when either the beam or the target (but not both) are polarized are called the analyzing powers of the A(a, b)B reaction. Thus, apart from special cases, polarizations and analytical abilities must be clearly distinguished, since they characterize different reactions. Reactions like A (a, b) B, A (a, b) B, etc. are called polarization transfer reactions. The parameters relating the spin moments of the particle b and the particle o are called the polarization transfer coefficients. The term "spin correlations" is applied to experiments on reactions of the form A(a,b)B and A(a,b)B, in which case the polarization of both resulting particles must be measured in the same event. In experiments with a beam of polarized particles (measurements of analyzing abilities), in accordance with the Madison Convention, the z-axis is directed along the momentum of the beam particle kjn, the y-axis is directed along k(n x kout (i.e., perpendicular to the reaction plane), and the x-axis should be directed so that the resulting coordinate system is right-handed. The polarization state of a system of particles with spin / can be completely described by (21+1)2 - 1 parameters. Thus, for particles with spin 1/2, the three parameters pi form a vector p, called the polarization vector. The expression in terms of the spin 1/2 operator, denoted by c, is the following: Pi = fa) , i = x,y,z , (5) where the angle brackets mean averaging over all particles of the ensemble (in our case, the beam). The absolute value of p is limited< 1. Если мы некогерентно смешаем п+ частиц в чистом спиновом состоянии, т.е. полностью поляризованных в некотором this direction, and particles completely polarized in the opposite direction, the polarization will be p - , or p = N+-N- , (6) if by iV+ = and AL = n™+n we understand the fraction of particles in each of the two states. Since the polarization of particles with spin 1 is described by a tensor, its representation becomes more complicated and less visual. The polarization parameters are some observable quantities of the spin operator 1, S. Two different sets of definitions for the corresponding polarization parameters are used - the Cartesian tensor moments Pi, pij and the spin tensors tkq . In Cartesian coordinates, according to the Madison Convention, the polarization parameters are defined as Pi - (Si) (vector polarization), (7) 3 u - -(SiSj + SjSi) - 25ij (tensor polarization), (8) = 5(5 + 1) =2 , (9) r we have the connection Pxx+Pyy+Pzz = 0 . (ten) Thus, the tensor polarization is described by five independent quantities (pxx, pyy, pxy, pxz, pyz), which, together with the three components of the polarization vector, gives eight parameters for describing the polarized state of a particle with spin 1. The corresponding density matrix can be written as: P = \( 1 + + SjSi)) . (eleven) The description of the polarization state in terms of spin tensors is convenient, since they are easier than Cartesian ones, they are transformed during rotations of the coordinate system. The spin tensors are related to each other by the following relationship (see): tkq - N Y,(kiqik2q2\kq)ikiqiik2qz > (12) 9192 where q\k2q2\kq) are the Clebsch-Gordan coefficients, and N is the normalization coefficient chosen so that the condition Sp(MU) = (2S + l)6kkl6qqi . (13) The lowest spin moments are: Yu \u003d 1 5 h o - Sz, h -1 \u003d ^ (Sx - iSy) . For the spin I, the index k ranges from 0 to 21, and |d|< к. Отрицательные значения q могут быть отброшены, поскольку имеется связь tk q = (-1)Ч*к + . Для спина 1 сферические тензорные моменты определяются как Thus, the vector polarization is described by three parameters: the real tw and the complex tw, and the tensor polarization is described by five parameters: the real tw and the complex tw hi Next, we consider the situation when the spin system has axial symmetry with respect to the axis ((we will leave the designation l for the coordinate system associated with the reaction under consideration, as described above). This particular case is interesting because beams from sources of polarized ions usually have axial symmetry. state as an incoherent mixture containing the fraction N+ of particles with spins along C, the fraction AL of particles with spins along, and the fraction No of particles with spins uniformly distributed in directions in the plane perpendicular to In this case, only two polarization moments of the beam are nonzero, t\ o (or p^) and t2o (or p^). Let us direct the quantization axis along the axis of symmetry t and replace in the notation t by r and z by (. It is obvious that (5^) is simply equal to N+ - N-, and according to (15) and (7): 15) vector polarization), t2i = -^((Sx.+ iSy)Sg.+ Sg(Sx+iSy)) , t22 = f((Sx + iSy)2) tensor polarization). 17) (N+ - N-) (vector polarization). From (16) and (8) it follows that T20 = ^=(1 - 3Nq) or PCC = (1-3b) where it is used that (N+ + N-) = (1 - No). If all moments of the 2nd rank are absent (N0 = 1/3), one speaks of a purely vector beam polarization. The maximum possible values of the polarization of such a beam are r0ax- - y2/3 or (19) pmax. 2/3 (purely vector polarization). For the case of purely tensor polarization (mu = 0), from equations (17) and (18) we obtain -\/5<Т2О<-7= ИЛИ (20) л/2 2 < рсс < +1 . The lower limit corresponds to No - 1, the upper - AG+ = AL = 1/2. In the general case, the symmetry axis ξ of a polarized beam from a source can be arbitrarily oriented with respect to the coordinate system xyz associated with the reaction under consideration. Let us express the spin moments in this system. If the orientation of the axis (is given by the angles /3 (between the z and C axes) and φ (rotation by -φ around the z axis brings the C axis to the yz plane), as shown in Fig. 3, and in the C frame the beam polarizations are equal to T20, then the tensor moments in the xyz system are: Vector moments: Tensor moments: 10 = r10COS/3 , t20 = -7p(3cOS2/? - 1) , (21) itn = ^Lsin/fe4*-. t2l = sinPcosRe(f, l/2 l/2 In the general case, the invariant cross section a = Eda/dp of the reaction A(a, b)B is written as: st = ao(Etkqnq) . (22) k,q The values of Tkq are called the analyzing abilities of the reaction. The Madison Convention recommends designating tensor analyzing powers as Tkq (spherical) and A;, Lu (Cartesian). Four analyzing abilities - vector gTz and tensor Ty, T2\ and T22 Rice. 3: Orientation of the symmetry axis £ of the polarized beam with respect to the xyz coordinate system associated with the reaction, xz is the reaction plane, (3 is the angle between the z axes (direction of the incident beam) and rotation na-f around the z axis brings the £ axis into the yz plane. They are valid due to parity conservation, and Ty = 0. Taking into account these restrictions, equation (22) takes the form: sg =<70-. In Cartesian coordinates, the same section is written as: 3 1 2 1 a - one hundred tkq , (25) i.e. the vector analyzing power is equal to the vector polarization in the reverse reaction: T2l = -^r.reaction. ^(2?) For elastic scattering, when the response is identical to its inverse, the vector polarization is equal to the vector analyzing power. Therefore, in some papers on the study of the scattering of polarized particles, one speaks of polarization measurements, when, strictly speaking, the analyzing power was measured. Nevertheless, for elastic scattering of deuterons it is necessary to distinguish between the analyzing power and the polarization £21 due to the difference in sign. B.4 Brief review of data on the reaction of fragmentation of deuterons into cumulative protons Let us briefly summarize the currently known results of studying the reaction of fragmentation of deuterons into protons d(pd > 1 GeV/c) + А р(® = 0°) + X , (28) since they will be required when motivating the measurements considered in the thesis and discussing the results results. Over twenty years of research into reaction (28) with polarized and unpolarized deuterons, a large amount of experimental data has been accumulated, which initiated the emergence of a number of theoretical models aimed at describing the structure of the deuteron and the reaction mechanism. This reaction has the largest, in comparison with fragmentation into other hadrons, cross section, and a clear interpretation within the framework of the impulse approximation. In this case, the main contribution to the cross section comes from the spectator mechanism, which is depicted by the diagram shown in Fig. four. Rice. 4: Spectator diagram for the fragmentation of a deuteron into a proton. For a two-component (S- and D-wave) deuteron wave function (hereinafter referred to as "WFD"), the differential cross section (Eda/dp) and the tensor analyzing power T20 are written as follows: E~(p)^(u2(k)+w2(k)) , . , 2u(k)w(k) -w2(k)/V2 da u2(k) + w2(k) Here p is the momentum of the detected proton, and and w are the radial components of the PFD for the S and D waves, respectively. Due to the essential role of relativistic effects, the relationship between the variable k, which plays the role of the internal momentum of the nucleon in the deuteron, and the momentum of the registered proton depends on the method of describing the deuteron. This is due to the fundamental impossibility to separate , the movement of the center of mass and the relative movement in a system of particles moving with relativistic velocities. Generally speaking, the WFD relativization method, i.e. the way in which relativistic effects are taken into account in it is one of the main differences between the theoretical models used to describe the reaction (28). Therefore, when comparing experimental data with theoretical models, the specific method of PFD relativization will be specially discussed, but here we will rely on the so-called minimum relativization scheme. The scheme of minimal relativization is the consideration of WFD in dynamics on the light front with a fixed choice of the direction of the light front (z + t = 0). This approach, apparently, was first proposed in and widely used in the description of composite relativistic systems (see, for example, , , , ). In this approach, the momentum p of the detected proton and the internal momentum k of the nucleon in the deuteron are related by the relationship: m, M are the masses of the proton and deuteron, p, d are their three-dimensional momenta. Nonrelativistic functions depending on A are used as the wave function; and multiplied by the normalization factor 1/(1 - a). The fragmentation cross section of unpolarized deuterons into protons at zero angle was studied in the range from 2.5 to 17.8 GeV/c of the momentum of primary deuterons in the works , , , , , , . On the whole, the experimental spectra obtained are well described by the spec 32) by a tator mechanism using generally accepted WFD, for example, Reid or Paris WFD. 0.0 0.2 0.4 0.6 0.8 1.0 k. GeV/c Rice. 5: Nucleon relative momentum distribution in the deuteron extracted from experimental data for various reactions involving the deuteron. Picture taken from work. So, from Fig. Figure 5 shows that the momentum distributions of nucleons in the deuteron are in good agreement, extracted from the data for the reactions: inelastic scattering of electrons on the deuteron d(e,e")X , elastic proton-deuteron backscattering p(d,p)d , and breakup deuteron.Except for the interval of internal momenta k from 300 to 500 MeV/c, the data are described by the spectator mechanism using the Paris PFD.To explain the discrepancy in this region, additional mechanisms were invoked.In particular, taking into account the contribution from pion rescattering in the intermediate state , , makes it possible to satisfactorily describe However, the uncertainty in the calculations is about 50% due to the uncertainty in the knowledge of the vertex function irN, which, moreover, in such calculations must be known off-shell. (i.e. small internucleon distances 0.4 1.2 2.0 2. Inn - 0.2/k), non-nucleon degrees of freedom may appear. In particular, in that work, an admixture of the six-quark component \6q) was introduced, the probability of which was ~ 4%. Thus, it can be noted that, on the whole, the spectra of protons obtained during the fragmentation of deuterons into protons at zero angle can be described up to internal momenta of ~ 900 MeV/c. In this case, it is necessary either to take into account the diagrams following after the momentum approximation, or to modify the PFD taking into account the possible manifestation of nonnucleon degrees of freedom. The polarization observables for the deuteron breakup reaction are sensitive to the relative contribution of the PFD components corresponding to different angular momenta, so experiments with polarized deuterons provide additional information about the deuteron structure and reaction mechanisms. At present, there are extensive experimental data on the tensor analyzing power of T20 for the breakup reaction of tensorically polarized deuterons. The corresponding expression in the spectator mechanism is given above, see (30). Experimental data for Tad, obtained in the works , , , , , , , , , are shown in Fig. 6, which shows that starting from internal momenta of the order of 0.2 - f - 0.25 GeV/c, the data are not described by conventional two-component PFDs. Accounting for interaction in the final state improves agreement with experimental data up to momenta of the order of 0.3 GeV/c. Accounting for the contribution of the six-quark component in the deuteron allows one to describe the data up to internal momenta of the order of 0.7 GeV/c. The behavior of T20 for momenta of the order of 0.9 -f-1 GeV/c is in best agreement with calculations within the framework of QCD using the method of reduced nuclear amplitudes , , which takes into account the antisymmetrization of quarks from different nucleons. So, summing up the above: 1. Experimental data for the cross section for the fragmentation of unpolarized deuterons into protons at zero angle can be described in terms of the nucleon model. 2. The data for T20 have so far been described only with the involvement of non-nucleon degrees of freedom. B.5 Purpose and structure of the thesis The purpose of this dissertation work was to obtain experimental data on the tensor analyzing ability of the T20 reaction Ta, for df *12C-> p(O") + X 0 200 400 600 800 1000 k (MeV/c) Rice. 6: Tensor analyzing power of the T2o deuteron decay reaction. Picture taken from work. 60) fragmentation of tensor polarized deuterons into cumulative (subthreshold) pions at zero angle on various targets, as well as the creation software for data acquisition systems of experimental facilities conducting polarization measurements at the LHE accelerator complex. Structurally, the dissertation work consists of an introduction, three chapters and a conclusion.
Similar theses in the specialty "Physics of the atomic nucleus and elementary particles", 01.04.16 VAK code
Study of the angular dependence of the analyzing abilities of the reaction dd→3Hp at an energy of 200 MeV 2010, Candidate of Physical and Mathematical Sciences Alexey Konstantinovich Kurilkin
Measurement of tensor and vector analyzing abilities of inelastic scattering of polarized deuterons on protons in the region of Roper resonance and delta isobar excitation energies 2001, candidate of physical and mathematical sciences Malinina, Lyudmila Vladimirovna
Mass Spectrum of the Bethe-Salpeter Equation and Relativistic Effects in Proton-Deuteron Scattering 2001, candidate of physical and mathematical sciences Semikh, Sergey Sergeevich
Study of the analyzing abilities of the reactions dd→pX and d12C→pX at intermediate energies 2011, Candidate of Physical and Mathematical Sciences Kiselev, Anton Sergeevich
Creation of a polarized hydrogen-deuterium gas target for the ANKE experiment on the internal beam of the storage ring of the COZY accelerator 2007, candidate of physical and mathematical sciences Grigoriev, Kirill Yurievich
Dissertation conclusion on the topic "Physics of the atomic nucleus and elementary particles", Isupov, Alexander Yurievich
CONCLUSION
Let us formulate the main results and conclusions of the dissertation work:
1. For the first time, the value of the tensor analyzing power Т2о was measured in the reaction d + А -7Г±(@ = 0°) + X fragmentation of tensor polarized deuterons into cumulative pions at zero angle in two formulations:
For a fixed pion momentum pn = 3.0 GeV/c for deuteron momentum pd in the range from 6.2 to 9.0 GeV/c;
For a fixed deuteron momentum pa = 9.0 GeV/c for pion momenta Pt in the range from 3.5 to 5.3 GeV/c.
2. The measured value of the tensor analyzing power T20 does not depend on the atomic mass A of the target nucleus in the interval A = 1->-12.
3. The measured value of T2o does not depend on the sign of the registered pion.
4. The measured value of T20 is not even qualitatively described by currently known theoretical calculations in the momentum approximation in the nucleon model of the deuteron.
5. A distributed data collection and processing system qdpb has been created, which provides the basis for building data collection systems for experimental installations.
6. Based on the qdpb system, a data acquisition system DAQ SPHERE was created, which has been used so far in 8 sessions on the extracted beam of the Synchrophasotron and Nuclotron LHE.
7. On the basis of the qdpb system, data collection systems were created for LHE polarimeters: high-energy on the extracted beam, as well as on the internal target of the Nuclotron - the vector polarimeter and subsequently - the vector-tensor polarimeter.
In conclusion, I would like to thank the leadership of the High Energy Laboratory and personally A.I. Malakhov, as well as the staff of the accelerator complex and the POLARIS source, who for many years provided the opportunity to conduct experimental work, the results of which formed the basis of the presented dissertation work.
I express my deep gratitude to my supervisors - A. Glitvinenko, without whose help this dissertation work would not have been completed in work and life, and L. S. Zolin, who initiated both the setting of the described experiments and many of the technical developments included in this work.
I consider it a pleasant necessity to express my sincere gratitude to I.I. Migulina for moral support, which cannot be overestimated, as well as for many years of work as part of the SPHERE collaboration, the results of which greatly facilitated the preparation of the dissertation work.
I consider it my duty to thank my colleagues K.I. Gritsai, S.G. Reznikov, V.G. Olshevsky, S.V. Afanasiev, A.Yu. on professional (and not only) topics, as well as all participants of the SPHERE collaboration over the past decade, because without them it would have been absolutely impossible to obtain the results presented in this paper.
Special thanks to the author - L.S. Azhgirey and V.N. Zhmyrov, employees of the LHE high-energy polarimeter, and also to the late G.D. Stoletov for fruitful cooperation, which led to the creation of modern polarimetric software.
I am grateful to Yu.K. Pilipenko, N.M. Piskunov and V.P. Ladygin, who at different times initiated some of the developments included in the dissertation work.
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Please note that the scientific texts presented above are posted for review and obtained through original dissertation text recognition (OCR). In this connection, they may contain errors related to the imperfection of recognition algorithms. There are no such errors in the PDF files of dissertations and abstracts that we deliver.
1972
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June
Modern state of physics and technology for obtaining beams of polarized particles
Contents: Introduction. Spin state of the particle. Principles of obtaining polarized ions. Atomic beam method. Dissociation of hydrogen molecules. Formation of a free atomic beam. Hydrogen and deuterium atoms in a magnetic field. Separating magnet. RF transitions. RF transitions in a weak field. RF transitions in a strong field. Operating installations. Ionization of an atomic beam. Ionizer with a weak magnetic field. Ionizer with a strong magnetic field. Obtaining negative ions by recharging positive polarized ions. Ionization by heavy particles. Lamb method. Energy levels of hydrogen and deuterium atoms with n= 2 in a uniform magnetic field. Times of life. Polarization in the metastable state. recharge processes. Getting negative ions. Getting positive ions. Methods for increasing beam polarization. Source of negative polarized ions. Measurement of ion polarization. fast ions. slow ions. Sources of polarized helium-3 and lithium ions. Polarized singly charged helium-3 ions. Sources of polarized lithium ions. Magnetized single crystal as a polarization donor. Injection of polarized ions into the accelerator. Cockcroft-Walton accelerator and linear accelerator. Van de Graaff accelerator. Tandem accelerator. Cyclotron. Accumulation of polarized ions. Acceleration of polarized ions. Cyclotron. Synchrocyclotron. Phasotron with spatial variation of the magnetic field. Synchrotron. Achievements of individual laboratories. Berkeley, California. Los Alamos. Conclusion. Cited Literature.
If the applied field E0 has an arbitrary direction, then the induced dipole moment can be easily found from the superposition
Where, are the field components with respect to the principal axes of the ellipsoid. In scattering problems, the coordinate axes are usually chosen to be fixed with respect to the incident beam. Let x" y" z" be such a coordinate system where the propagation direction is parallel to the z-axis". If the incident light
x" is polarized, then from the optical theorem we have:
To carry out calculations using formula (2.2), it is necessary to write out the p components with respect to the axes drawn by dashed lines. Equality (2.1) can be written in matrix form:
We write column vectors and matrices in a more compact form in accordance with the following notation:
With this notation, 2.3 takes the following form:
The components of an arbitrary vector F are transformed in accordance with the formula:
Where, etc. As a result, from (2.5) and transformation (2.6) we have:
where, due to the orthogonality of the coordinate axes, the matrix inverse to is the transposed matrix. Thus, the polarizability of an ellipsoid is a Cartesian tensor; if its components in the principal axes are given, then its components in the rotated coordinate axes can be determined by formula (2.8). The absorption cross section for incident - polarized light is determined simply by the formula:
Where. Similarly, if the incident light is polarized, then
If the vector scattering amplitude
for a dipole illuminated by -polarized light, substitute into the cross section equation, then we obtain the scattering cross section
Where we used the matrix identity. A similar expression holds for the scattering cross section and for incident polarized light.
Application.
Polarized light was proposed to be used to protect the driver from the blinding light of the headlights of an oncoming car. If film polaroids with a transmission angle of 45o are applied to the windshield and headlights of a car, for example, to the right of the vertical, the driver will clearly see the road and oncoming cars illuminated by their own headlights. But for oncoming cars, the polaroids of the headlights will be crossed with the polaroid of the windshield of this car, and the headlights of oncoming cars will go out.
Two crossed polaroids form the basis of many useful devices. Light does not pass through crossed polaroids, but if you place an optical element between them that rotates the plane of polarization, you can open the way for light. This is how high-speed electro-optical light modulators are arranged. They are used in many technical devices - in electronic rangefinders, optical communication channels, laser technology.
The so-called photochromic glasses are known, darkening in bright sunlight, but not able to protect the eyes with a very fast and bright flash (for example, during electric welding) - the darkening process is relatively slow. Polarized glasses have an almost instant "reaction" (less than 50 microseconds). The light of a bright flash enters miniature photodetectors (photodiodes), which supply an electrical signal, under the influence of which the glasses become opaque.
Polarized glasses are used in stereo cinema, which gives the illusion of three-dimensionality. The illusion is based on the creation of a stereo pair - two images taken at different angles, corresponding to the angles of view of the right and left eyes. They are considered so that each eye sees only the image intended for it. The image for the left eye is projected onto the screen through a polaroid with a vertical transmission axis, and for the right eye with a horizontal axis, and they are precisely aligned on the screen. The viewer looks through polaroid glasses, in which the axis of the left polaroid is vertical, and the right one is horizontal; each eye sees only “its own” image, and a stereo effect arises.
For stereoscopic television, the method of rapidly alternating dimming of glasses is used, synchronized with the change of images on the screen. Due to the inertia of vision, a three-dimensional image arises.
Polaroids are widely used to dampen glare from glass and polished surfaces, from water (the light reflected from them is highly polarized). Polarized and light screens of liquid crystal monitors.
Polarization methods are used in mineralogy, crystallography, geology, biology, astrophysics, meteorology, and in the study of atmospheric phenomena.
Physicists have a habit of taking the simplest example of a phenomenon and calling it “physics,” and leaving more difficult examples to other sciences, such as applied mathematics, electrical engineering, chemistry, or crystallography. Even solid state physics for them is only "semiphysics", because it is concerned with too many special issues. For this reason, we will omit many interesting things in our lectures. For example, one of the most important properties of crystals and most substances in general is that their electrical,
polarizability is different in different directions. If you apply an electric field in any direction, then the atomic charges will shift slightly and a dipole moment will arise; the magnitude of this moment depends very strongly on the direction of the applied field. And this, of course, is a complication. To make life easier for themselves, physicists start the conversation with the special case where the polarizability is the same in all directions. And we leave other cases to other sciences. Therefore, for our further considerations, we will not need at all what we are going to talk about in this chapter.
The mathematics of tensors are especially useful for describing the properties of substances that change with direction, although this is just one example of its use. Since most of you are not going to become physicists, but intend to study real in a world where the dependence on direction is very strong, sooner or later you will need to use a tensor. So, so that you don't have a gap here, I'm going to tell you about tensors, although not in great detail. I want your understanding of physics to be as complete as possible. Electrodynamics, for example, we have a completely finished course; it is as complete as any course in electricity and magnetism, even an institute one. But mechanics is not finished with us, because when we studied it, you were not yet so firm in mathematics and we could not discuss such sections as the principle of least action, Lagrangians, Hamiltonians, etc., which represent mostelegant way descriptions of mechanics. However, the full set laws mechanics, with the exception of the theory of relativity, we still have. To the same extent as electricity and magnetism, we have many sections completed. But here we will not finish quantum mechanics; However, you need to leave something for the future! And yet, what is a tensor, you still should know now.
In ch. 30 we emphasized that the properties of crystalline matter are different in different directions - we say that it anisotropically. The change in the induced dipole moment with a change in the direction of the applied electric field is only one example, but that is what we will take as an example of a tensor. We will assume that for a given direction of the electric field, the induced dipole moment per unit volume P is proportional to the strength of the applied field E. (For many substances at not too large E, this is a very good approximation.) Let the constant of proportionality be α
. Now we want to consider substances that have α
depends on the direction of the applied field, for example the tourmaline crystal you know, which gives a double image when you look through it.
Suppose we have found that for some selected crystal, the electric field E 1 directed along the axis X, gives polarization P 1 directed along the same axis, and alonethe same size as him electric field E 2 directed along the axis y, leads to some other polarization P 2 , also directed along the axis y. What happens if an electric field is applied at an angle of 45°? Well, since it will just be a superposition of two fields directed along the axes X and y, then the polarization P is equal to the sum of the vectors P 1 and P 2 , as shown in FIG. 31.1, a. The polarization is no longer parallel to the direction of the electric field. It is not difficult to understand why this happens. There are charges in the crystal that are easy to move up and down, but which are very difficult to move sideways. If the force is applied at an angle of 45 °, then these charges are more likely to move up than to the side. As a result of such asymmetry of the internal elastic forces, the displacement does not proceed in the direction of the external force. Of course, the 45° angle is not highlighted. That induced polarization not directed along the electric field, just and in general. Before that, we were just “lucky” to choose such axes X and y, for which the polarization P was directed along the field E. If the crystal were rotated with respect to the coordinate axes, then the electric field E 2 directed along the y axis would cause polarization both along the axis y, as well as along the axis X. Similarly, the polarization P caused by a field directed along the axis X, would also have X-, so are the y-components. So instead of Fig. 31.1, a we would get something similar to Fig. 31.1b. But despite all this complication, magnitude polarization P for any field E is still proportional to its magnitude.
Let us now consider the general case of an arbitrary orientation of the crystal with respect to the coordinate axes. Electric field directed along the axis X, gives a polarization P with components in all three axes, so we can write
By this I only want to say that the electric field directed along the axis X, creates polarization not only in this direction, it leads to three components of polarization R x,RU and Pz, each of which is proportional E x. We called the coefficients of proportionality a xx, a yx and a zx(the first icon indicates which component is involved, and the second refers to the direction of the electric field).
Similarly, for a field directed along the axis y, we can write
and for the field in the z-direction
Further we say that the polarization depends linearly on the field; so if we have an electric field E with components X and y, then the x-component of the polarization P will be the sum of two R x, defined by equations (31.1) and (31.2), but if E has components in all three directions x, y and z, then the polarization components P must be the sum of the corresponding terms in equations (31.1), (31.2) and (31.3). In other words, P is written as
The dielectric properties of a crystal are thus completely described by nine quantities (α xx, α xy,α xz, α yz , ...), which can be written as a symbol α ¡j . (Indices i and j replace one of three letters: x, y or z.) An arbitrary electric field E can be decomposed into components E x, E y and Ez. Knowing them, you can use the coefficients α ¡j and find P x, P y and P z , which together give the complete polarization of P. A set of nine coefficients α ¡j called tensor- in this example polarizability tensor. Just like the three quantities (E x, E y,Ez) "form a vector E", and we say that nine quantities (α xx, α hu,...) "form a tensor α ¡j ».
