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  1. (PDF) Two introductory lectures on quantum scattering theory | Sergei Winitzki -
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The large q -divergence is removed by incorporating the center-of-mass correlation function Q in the first order calculations [45] or in the fourth order evaluations [47] or in the full series calculations [56], of the Glauber amplitude. Such consistent treatment in the later case [56] has modified the phase-shift functions to new forms, which are simply expressed in terms of the uncorrelated ones. The bars are introduced to distinguish the modified quantities from their corresponding uncorrelated ones.

Assuming the target ground state wave function has the form:. Accordingly, the modified optical phase-shift function, n , can be written in terms of the uncorrelated phase-shift function, n , as [47]. With the definition of the profile function. It is related to the pion-nucleon p N elastic scattering amplitude f j by. Assuming for simplicity that all the p N amplitudes are equal which is approximately true at high energy and neglecting further spin effects, f j can be parameterized by [56].

Here, s is the total p N cross-section, r is the ratio of the real to the imaginary parts of the forward p N scattering amplitude and ''a'' is taken to be complex; the real part is typically the slope parameter b 2 of the p N elastic scattering differential cross section while the imaginary part is a free parameter introducing a phase variation of the p N scattering amplitude, which is taken here to be zero. To perform the integrations 8 and 9 analytically, consider the approximation in which the nucleons inside the target nucleus are completely uncorrelated, then we can write.

Adopting the wave function 15 with the density 16 and using the same analysis as that utilized in our references [55, 56] with the invoking of the description mentioned above, eq. Obviously, Eq. In the method of Yin et al. This matrix D m, l m corresponds to one typical term expressing the multi-cluster elementary collision and its element is equal to ''1'' if G 1 j appears in the expansion term and is equal to ''0'' if it is absent. Here, ''Ru'' and ''sg'' are the row and column sum vectors of the corresponding orbit, respectively. Incorporating the c. The form of o is obtained by inserting the expressions of Z o and Q q into Eq.

Finally, the modified extended charge correction to the Coulomb phase-shift, , has already been derived analytically in Ref. With the results of eqs. The angular distribution of the elastic scattering is given by. The total elastic cross section, s el , is given by subtracting the total reaction cross section '' s R '' from the total cross section which is related to the forward scattering amplitude of the elastic scattering by the optical theorem as:. The reaction cross section is found to has the form [47, 70, 71]. With the results of Eqs. The total elastic cross section is evaluated for the elastic scattering of p - with 6 C 12 at incident energies , , , and The theoretical results were compared with the available experimental data.

For the above energies, we used the values of the p N parameters given in Table 1. The values of the rms radii we have used for the present nuclei are given in Table 2. The structure specific to the considered reactions and the corresponding orbits, lengths and D -matrices are exhibited in the appendix. The center-of-mass correlation is treated here in a consistent way in which its expressing function Q q is incorporated in each order of the optical phase-shift expansion.

Such function is found to have an exact form in the case of the employed single-Gaussian density [48]. To show to what extent the inclusion in-medium pN scattering amplitude affects the angular distribution, we first evaluate it with the free p N parameters s free , s free. Second, we calculate it with the effective p N ones s eff. The results obtained from these calculations for the considered reactions are shown in Figs.

We see from this figure that the predicted angular distribution obtained with the effective p N parameters referred to as solid curve is much better than that obtained using the free p N ones referred to as dashed curve. In particular, the dashed curves reproduced well the positions of the minima while the solid curves reproduce more satisfactorily the backward angles in comparison with the results shown by the dashed curves.

For p - - 6 C 12 case, Figs. As we referred before [58, 59] that the use of density distribution more suitable than that employed in our calculations may improve our results. This will be the subject of future work. To ensure the necessity of accounting for the full multiple scattering series of the Glauber amplitude, comparison has been made with the similar first-order calculations optical limit result using Franco and Varma approach [47]. The details of their analysis and the phase-shift expressions developed are quoted in their reference.

An overall picture of these figures shows that the full series results are relatively better than the optical limit results, especially at large angles. This may ensure the necessity of including the higher order terms at large momentum transfers. The results of the calculated total and total elastic cross-sections and the corresponding experimental data are listed in Tables One can easily see from these tables that the full series calculations for the total and total elastic cross-sections are much better than that calculated by the optical limit approach.

This ensures the necessity of including the full multiple scattering series of the Glauber amplitude. In the framework of Glauber's multiple scattering theory and taking into account the Coulomb contribution and a consistent treatment of the center-of-mass c.

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We have calculated the differential, total and total elastic cross sections for the considered reactions. These calculations are carried out by considering the followings; i The full series expansion of the Glauber amplitude with free p N parameters s free , s free , ii The first order correction of the Glauber amplitude OLA with free p N parameters s free , s free , iii The full series expansion of the Glauber amplitude with the in-medium p N parameters s eff. We compared our results with the corresponding experimental data.

It is shown that, except around the minima, the full series results of the differential cross-sections employing the in-medium p N amplitude are more comprehensive in describing the scattering data rather than that using the free p N one. Also, one can easily see that the full series calculations of the Glauber amplitude are relatively better than the optical limit result OLA , especially at backward angles.

An overall view of the results shown before, we emphasize that the full series expansion of the Glauber amplitude is not sufficient to bring the Glauber theory closer to the experimental data. It must consider some corrections like; in-medium p N amplitude, phase-variation parameter of the p N scattering amplitude which has its strongest effect around the diffraction patterns, more realistic density distribution like harmonic-oscillator density.

The last two corrections will be done in our future work. Catillon, P. Radvanyi, and M. Porneuf North-Holland, Amsterdam, Bowman, L. Kisslinger, and R. Measday and A. Thomas North-Holland, Amsterdam, Marow et al. C 30 , Ericson and T. Ericson, Ann. Blecher et al. C 20 , Preedom et al. C 23 , Dytman et al. C 19 , Johnson et al. B 43 , Dam et al. C 25 , Obenshain et al. C 28 , Lee and H.

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    (PDF) Two introductory lectures on quantum scattering theory | Sergei Winitzki -

    B 10 , Michael and C. Wilkin, Nucl. B 11 , 99 Chou, Phys. B 20 , Kisslinger, Phys. Baker, H. Byfield, and J. Rainwater, Phys. Auerbach, D. Fleming, and M. Binon et al. B 17 , Krell and S. Barmo, Nucl. Watson, Phys. Francis and K. Czyz, Adv. Glauber and G. Mathiae, Nucl. B 21 , Wilikin, Nuovo Cimento Lett. Lesniak and L. Lesniak, Nucl. B 38 , Auger and R. Lombard, Phys. Nagle et al. AIP, New York, , p. Franco, Phys. C 8 , Bassel and C. Wilikin, Phys. If we take the average of the intensity over a little region, then the cosine, which goes plus, minus, plus, minus, as we move around, averages to zero.

    So if we average over regions where the phase varies very rapidly with position, we get no interference. Another example. Suppose that the two sources are two independent radio oscillators—not a single oscillator being fed by two wires, which guarantees that the phases are kept together, but two independent sources—and that they are not precisely tuned at the same frequency it is very hard to make them at exactly the same frequency without actually wiring them together. In this case we have what we call two independent sources. Of course, since the frequencies are not exactly equal, although they started in phase, one of them begins to get a little ahead of the other, and pretty soon they are out of phase, and then it gets still further ahead, and pretty soon they are in phase again.

    In other words, in any circumstance in which the phase shift averages out, we get no interference! One finds many books which say that two distinct light sources never interfere. This is not a statement of physics, but is merely a statement of the degree of sensitivity of the technique of the experiments at the time the book was written. But most detection equipment, of course, does not look at such fine time intervals, and thus sees no interference.

    Certainly with the eye, which has a tenth-of-a-second averaging time, there is no chance whatever of seeing an interference between two different ordinary sources. Recently it has become possible to make light sources which get around this effect by making all the atoms emit together in time.

    The device which does this is a very complicated thing, and has to be understood in a quantum-mechanical way. It can be of the order of a hundredth, a tenth, or even one second, and so, with ordinary photocells, one can pick up the frequency between two different lasers. One can easily detect the pulsing of the beats between two laser sources. Soon, no doubt, someone will be able to demonstrate two sources shining on a wall, in which the beats are so slow that one can see the wall get bright and dark!

    Another case in which the interference averages out is that in which, instead of having only two sources, we have many. So we would get a whole lot of cosines, many plus, many minus, all averaging out. So it is that in many circumstances we do not see the effects of interference, but see only a collective, total intensity equal to the sum of all the intensities.

    The above leads us to an effect which occurs in air as a consequence of the irregular positions of the atoms.

    Scattering Theory

    When we were discussing the index of refraction, we saw that an incoming beam of light will make the atoms radiate again. The electric field of the incoming beam drives the electrons up and down, and they radiate because of their acceleration. This scattered radiation combines to give a beam in the same direction as the incoming beam, but of somewhat different phase, and this is the origin of the index of refraction. But what can we say about the amount of re-radiated light in some other direction? Ordinarily, if the atoms are very beautifully located in a nice pattern, it is easy to show that we get nothing in other directions, because we are adding a lot of vectors with their phases always changing, and the result comes to zero.

    But if the objects are randomly located, then the total intensity in any direction is the sum of the intensities that are scattered by each atom, as we have just discussed. Furthermore, the atoms in a gas are in actual motion, so that although the relative phase of two atoms is a definite amount now, later the phase would be quite different, and therefore each cosine term will average out.

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    Therefore, to find out how much light is scattered in a given direction by a gas, we merely study the effects of one atom and multiply the intensity it radiates by the number of atoms. Earlier, we remarked that the phenomenon of scattering of light of this nature is the origin of the blue of the sky. From Eq.

    Rather than do this, however, we shall simply calculate the total amount of light scattered in all directions, just to save time. The total amount of light energy per second, scattered in all directions by the single atom, is of course given by Eq. We have written the result in the above form because it is then easy to remember: First, the total energy that is scattered is proportional to the square of the incident field.

    What does that mean? Obviously, the square of the incident field is proportional to the energy which is coming in per second. In other words, the total energy scattered is proportional to the energy per square meter that comes in; the brighter the sunlight that is shining in the sky, the brighter the sky is going to look. Next, what fraction of the incoming light is scattered? Now we invent an idea: we say that the atom scatters a total amount of intensity which is the amount which would fall on a certain geometrical area, and we give the answer by giving that area.

    That answer, then, is independent of the incident intensity; it gives the ratio of the energy scattered to the energy incident per square meter. This area is called a cross section for scattering ; the idea of cross section is used constantly, whenever some phenomenon occurs in proportion to the intensity of a beam. In such cases one always describes the amount of the phenomenon by saying what the effective area would have to be to pick up that much of the beam.

    It does not mean in any way that this oscillator actually has such an area. If there were nothing present but a free electron shaking up and down there would be no area directly associated with it, physically. It is merely a way of expressing the answer to a certain kind of problem; it tells us what area the incident beam would have to hit in order to account for that much energy coming off.

    Let us look at some examples. This low-frequency limit, or the free electron cross section, is known as the Thomson scattering cross section. On the other hand, if we take the case of light in the air, we remember that for air the natural frequencies of the oscillators are higher than the frequency of the light that we use. That is to say, light which is of higher frequency by, say, a factor of two, is sixteen times more intensely scattered, which is a quite sizable difference.

    This means that blue light, which has about twice the frequency of the reddish end of the spectrum, is scattered to a far greater extent than red light.

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    Thus when we look at the sky it looks that glorious blue that we see all the time! There are several points to be made about the above results. One interesting question is, why do we ever see the clouds? Where do the clouds come from? Everybody knows it is the condensation of water vapor. After it condenses it is perfectly obvious. We have just explained that every atom scatters light, and of course the water vapor will scatter light, too. The mystery is why, when the water is condensed into clouds, does it scatter such a tremendously greater amount of light?

    Consider what would happen if, instead of a single atom, we had an agglomerate of atoms, say two, very close together compared with the wavelength of the light. Then when the electric field acts, both of the atoms will move together. The electric field that is scattered will then be the sum of the two electric fields in phase, i. So lumps of atoms radiate or scatter more energy than they do as single atoms.

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    Our argument that the phases are independent is based on the assumption that there is a real and large difference in phase between any two atoms, which is true only if they are several wavelengths apart and randomly spaced, or moving. But if they are right next to each other, they necessarily scatter in phase, and they have a coherent interference which produces an increase in the scattering.

    So as the water agglomerates the scattering increases. Does it increase ad infinitum? When does this analysis begin to fail? How many atoms can we put together before we cannot drive this argument any further? Answer : If the water drop gets so big that from one end to the other is a wavelength or so, then the atoms are no longer all in phase because they are too far apart. So as we keep increasing the size of the droplets we get more and more scattering, until such a time that a drop gets about the size of a wavelength, and then the scattering does not increase anywhere nearly as rapidly as the drop gets bigger.

    Furthermore, the blue disappears, because for long wavelengths the drops can be bigger, before this limit is reached, than they can be for short wavelengths. Although the short waves scatter more per atom than the long waves, there is a bigger enhancement for the red end of the spectrum than for the blue end when all the drops are bigger than the wavelength, so the color is shifted from the blue toward the red.

    Now we can make an experiment that demonstrates this. We can make particles that are very small at first, and then gradually grow in size. We use a solution of sodium thiosulfate hypo with sulfuric acid, which precipitates very fine grains of sulfur. As the sulfur precipitates, the grains first start very small, and the scattering is a little bluish. As it precipitates more it gets more intense, and then it will get whitish as the particles get bigger.