1972
/
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.
A deuteron is a nucleus consisting of one proton and one neutron. By studying the properties of this simplest nuclear system (deuteron binding energy, spin, magnetic and quadrupole moments), one can choose a potential that describes the properties of the nucleon-nucleon interaction.
The deuteron wave function ψ(r) has the form
is a good approximation for the entire range of r. 
Since the spin and parity of the deuteron are 1 + , the nucleons can be in the s-state (L = 0 + 0), and their spins must be parallel. The absence of a bound state with spin 0 in the deuteron says that the nuclear forces depend on the spin.
The magnetic moment of the deuteron in the S-state (see Magnetic moment of the nucleus) μ(S) = 0.8796μ N , is close to the experimental value. The difference can be explained by a small admixture of the D state (L = 1 + 1) in the deuteron wave function. Magnetic moment in the D-state
μ(D) = 0.1204μ N . The D-state impurity is 0.03.
The presence of an admixture of the D-state and a quadrupole moment in the deuteron testify to the non-central character of nuclear forces. Such forces are called tensor forces. They depend on the magnitude of the projections of the spins s 1 and s 2 , nucleons on the direction of the unit vector , directed from one deuteron nucleon to another. The positive quadrupole moment of the deuteron (prolonged ellipsoid) corresponds to the attraction of nucleons, the flattened ellipsoid corresponds to repulsion.
The spin-orbit interaction manifests itself in the features of the scattering of particles with nonzero spin on non-polarized and polarized targets and in the scattering of polarized particles. The dependence of nuclear interactions on how the orbital and spin moments of the nucleon are directed relative to each other can be found in the following experiment. A beam of unpolarized protons (spins with the same probability are directed conventionally "up" (blue circles in Fig. 3) and "down" (red circles)) falls on the 4 He target. Spin 4 He J = 0. Since the nuclear forces depend on the relative orientation of the vectors of the orbital momentum and spin , protons are polarized during scattering, i.e. protons with spin "up" (blue circles), for which ls, are more likely to scatter to the left, and protons with "down" spin (red circles), for which ls, are more likely to scatter to the right. The number of protons scattered to the right and to the left is the same, however, upon scattering at the first target, beam polarization occurs - the predominance of particles with a certain spin direction in the beam. Further, the right beam, in which protons with spin "down" predominate, falls on the second target (4 He). Just as in the first scattering, protons with spin "up" mostly scatter to the left, and those with spin "down" mostly scatter to the right. But since in the secondary beam, protons with spin "down" predominate; upon scattering on the second target, the angular asymmetry of the scattered protons relative to the direction of the beam incident on the second target will be observed. The number of protons that are registered by the left detector will be less than the number of protons that are registered by the right detector.
The exchange nature of the nucleon-nucleon interaction manifests itself in the scattering of high-energy neutrons (several hundreds of MeV) by protons. The differential neutron scattering cross section has a maximum for backscattering in the cm, which is explained by the charge exchange between a proton and a neutron.
Properties of nuclear forces
- Short range of nuclear forces (a ~ 1 fm).
- Large value of the nuclear potential V ~ 50 MeV.
- Dependence of nuclear forces on spins of interacting particles.
- Tensor character of interaction of nucleons.
- Nuclear forces depend on the mutual orientation of the spin and orbital moments of the nucleon (spin-orbit forces).
- Nuclear interaction has the property of saturation.
- Charge independence of nuclear forces.
- Exchange character of nuclear interaction.
- Attraction between nucleons at large distances (r > 1 fm) is replaced by repulsion at short distances (r< 0.5 Фм).
The nucleon-nucleon potential has the form (without exchange terms)
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 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
