Physics
Rutherford Scattering
Rutherford scattering refers to the phenomenon where alpha particles are deflected by the positively charged nucleus of an atom, as observed in an experiment conducted by Ernest Rutherford in 1909. This experiment led to the discovery of the atomic nucleus and provided evidence for the planetary model of the atom, revolutionizing our understanding of atomic structure and leading to the development of modern nuclear physics.
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11 Key excerpts on "Rutherford Scattering"
eBook - PDFQuantum Mechanics
Foundations and Applications
- Donald Gary Swanson(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
10.1 Scattering in Three Dimensions 10.1.1 Rutherford Scattering The field of nuclear physics began with the discovery that there existed a nu-cleus inside the atom, which came about by the scattering of α -particles (from radioactive decay) by Rutherford. This scattering is simple Coulomb scatter-ing, but the details of the classical, nonrelativistic analysis is instructive. We will subsequently solve the Coulomb scattering problem quantum mechani-cally, but we begin with the classical case. The dynamics of the encounter is shown in Figure 10.1 where an incident α -particle comes in from the left with velocity v 0 in such a direction that if the particles were uncharged, they 243 244 QUANTUM MECHANICS Foundations and Applications would be a distance b apart at closest approach. The parameter b is called the impact parameter . The trajectory is curved due to the Coulomb repul-sion, and the target nucleus (gold in the Rutherford experiments) recoils as shown. The objective is first to calculate the deflection angle θ as a function of the charges, masses, incident velocity and impact parameter b and then to calculate what fraction of incident particles have a deflection angle between θ and θ + Δ θ as the impact parameter is changed from b to b + Δ b . . . . . . . . . . . . . . . . . . . . . . b • • m 1 , Z 1 e , v 0 m 2 , Z 2 e Path of recoiling nucleus • O • . . r 1 • . . r 2 . . r 12 = r 1 -r 2 Path of scattered nucleus . θ . . FIGURE 10.1 A Rutherford Scattering encounter. Labeling the location of the incident particle by r 1 and the target particle’s position by r 2 , the equations of motion are m 1 ¨ r 1 = Z 1 Z 2 e 2 ( r 1 -r 2 ) 4 π 0 | r 1 -r 2 | 3 , (10.1) m 2 ¨ r 2 = -Z 1 Z 2 e 2 ( r 1 -r 2 ) 4 π 0 | r 1 -r 2 | 3 . (10.2) These two equations may be combined into the pair of equations m 1 ¨ r 1 + m 2 ¨ r 2 = ( m 1 + m 2 )¨ r CM = 0 , (10.3) ¨ r 2 -¨ r 1 = ¨ r 12 (10.4) = m 1 + m 2 m 1 m 2 Z 1 Z 2 e 2 r 12 4 π 0 r 3 12 .
eBook - ePub- C.A. Bertulani, P. Danielewicz(Authors)
- 2019(Publication Date)
- CRC Press(Publisher)
Chapter 1Classical and quantum scattering
1.1 Experiments with nuclear particlesFigure 1.1 represents schematically a typical scattering experiment, in fact this is the sketch of E. Rutherford experiment in 1910 [1 ]. Projectiles (here, α-particles) from a source go through a collimator and collide with a target (gold foil in Rutherford’s experiment). Some projectiles are scattered by the target and reach the detector (here, a fluorescent screen). Rutherford expected them to go straight through with perhaps a minor deflection. Most did go straight through, but to his surprise some particles bounced directly off the gold sheet! What did this mean? Rutherford hypothesized that the positive alpha particles had hit a concentrated mass of positive particles, which he termed the nucleus. We shall describe the Rutherford Scattering theoretically later in this chapter.Instead of a radioactive nuclear source, as in Rutherford’s experiment, one could use particles accelerated in a beam of particles, which could be protons, electrons, positrons, pions, ions, ionized molecules, etc. According to the nature of the projectile, to the required collision energy, resolution, beam intensity, etc., a variety of accelerators may be used. Ideally, the beam should be sharp in momentum space, i.e., should have good energy resolution and be well collimated. Usually, good experiments require high count rates. This is attained with an intense beam and/or a thick target. However, if the beam it so intense that the projectiles are close enough to interact, considerable complications appear in the theoretical description. Theoretical difficulties also arise when the target is so thick that multiple scattering becomes relevant.In figure 1.2 we show the concept of the most popular of the particle accelerators: the cyclotron.
eBook - PDF- Jan Rak, Michael J. Tannenbaum(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
4 The search for structure 4.1 Rutherford Scattering Shortly after the discovery of the radioactivity of uranium by Becquerel in 1896 [155] and its ability to ionize gases, Rutherford [156] began a study of the rate of discharge of a parallel plate capacitor in gas by placing successive layers of thin aluminum foil over the surface of a layer of uranium oxide on one plate. He concluded that “the uranium radiation is complex, and that there are present at least two distinct types of radiation: one that is readily absorbed which will be termed for convenience the α radiation, and the other of a more penetrative char- acter, which will be termed the β radiation.” In 1906, Rutherford [157] observed that α particles from the decay of radium scattered, i.e. deviated from their original direction of motion, when passing through a thin sheet of mica, but did not scatter in vacuum. He made this observation by passing α particles through narrow slits and making an image on a photographic plate. In vacuum, the edges of the image were sharp while the image of α particles that passed through the mica was broad- ened and showed diffuse edges. This observation was controversial because it was not expected that α particles would scatter [158]: “Since the atom is the seat of intense electrical forces, the β particle in passing through matter should be much more easily deflected from its path than the massive α particle.” Rutherford had Geiger [159] follow up on these measurements using scintilla- tions on a phosphorescent screen with convincing results that the deflections are on the average small, on the order of a few degrees, but, as Geiger noted “some of the α particles after passing through the thin leaves – the stopping power of one leaf corresponded to about 1mm.
eBook - PDF- Ashok Das(Author)
- 2012(Publication Date)
- WSPC(Publisher)
Chapter 15 Scattering theory Scattering is a valuable probe to study the structure of particles when we cannot directly see them. For example, it is through scattering experiments that we have learned that the hydrogen atom consists of a nucleus and an electron going around it. We know that electrons are point particles also from the results of scattering experiments. Furthermore, the scattering experiments have told us that protons consist of yet other constituents – the quarks. Therefore, we see that scattering is an essential tool in our understanding of the quantum nature of particles. We will, therefore, spend some time studying this topic. However, let me begin by recapitulating some of the features of scattering theory in classical physics. Classically, the simplest scattering that we can consider is that of a beam of particles from a fixed center of force as shown in Fig. 15.1. We can think of the fixed center of force to be a charged particle of infinite mass, if the particles that are being scattered are thought of as electrons or protons. Let us assume that a beam of particles is incident on the fixed source of force along the z -axis. The trajectories of the particles change due to the force experienced and the deflection of the trajectory, from the initial direction, is known as the scattering angle. θ z → Figure 15.1: Classical scattering of a beam of particles from a scat-tering source. 409 410 15 Scattering theory Classically, of course, we are not interested in measuring the exact trajectory of each of the particles. In fact, it is impossible if the number of particles involved is very large. We can only measure the initial velocity of the particles (which are assumed to be of the same energy) and their final velocities. Furthermore, this is done statistically. Namely, we measure how many particles scatter into a solid angle d Ω at θ .
eBook - PDF- J.M. Walls, Robin Smith(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
In general, the distance of closest approach will not be short enough for nuclear forces to come into play and the ion will be scattered according to the well known Coulomb scattering laws and with a well defined analytical probability. This process is known as Rutherford Scattering. For higher ion energies and, particularly, for low atomic number ions, some of the larger angle collisions will enable the ion to enter the region of the nuclear forces. In these cases the analytical Coulomb scattering probability will be modified by nuclear effects and the probability of interaction can no longer be calculated in a simple manner. Non-Rutherford effects are most severe for light ion/light atom combinations, being particularly evident for protons above a few hundred keV and alpha particles above 2 MeV incident on atoms with atomic number less than 11. In addition to nuclear elastic scattering, an even smaller fraction of the ions may undergo nuclear trans-formation. In favourable cases, perhaps one ion in 10 9 will induce a nuclear reaction. In discussing the interactions of ions with materials, no con-sideration has yet been given to the structure of the slowing down medium. Very many materials are crystalline in form with highly ordered arrangements of atoms. In low-index crystallographic directions, atoms lie in long rows and in planes and the ordered electric fields produced by these rows and planes profoundly influence the behaviour of incident ions. If a parallel ion beam is incident along a major crystallographic axis or plane, the electric field acts to guide the ions between the rows or planes, thus preventing them from approaching individual nuclei and greatly 136 L G Earwaker: Rutherford backscattering and nuclear reaction analysis reducing electronic stopping and the probability of Rutherford Scattering and nuclear reactions. Up to 98% of all incident ions may be trapped in axial channels by this mechanism.
eBook - PDF- G. Deconninck, T. Braun, E. Bujdosó(Authors)
- 2016(Publication Date)
- Elsevier(Publisher)
This technique provides information on the first atomic layer by employing an electrostatic analyser for the detection of the backscattered ions, ultra high vacuum is necessary and atomic collision problems must be overcome in this technique. A recent review of ion scattering spectrometry is given in [2] page 43. It is believed that Rutherford's law is valid at energies greater than 0.05 MeV for protons and 0.2 MeV for 4 He nuclei. For lower energies the Coulomb potential is not realistic, calculations of scattering by a screened Coulomb potential: r which approximately represents the potential between two atoms in collision taking into account the screening of the atomic electrons, are given in [6]. 3.3.3 High energy limitations for heavy ions In this section we shall consider the scattering of heavy ions including those of 3 He and 4 He. In Rutherford Scattering the particles are sup-posed to follow hyperbolic trajectories determined by the repulsive Coulomb force. This classical concept is valid when the wave length λ, associated with the projectile, is smaller than the nuclear radius R which is the case for heavy ions of a few MeV scattering off heavy atoms. The classical trajectories may be characterized by the impact param-eter b and by the distance of closest approach D. When D < i2 the trajectory is perturbed by nuclear forces and the particle has a high probability of inducing a nuclear reaction. This is illustrated in Fig. 3.4(a) where trajectories corresponding to the scattering at 90° cm. of 4 He ions by 40 Ca are shown. In this picture the trajectories corresponding to an energy of more than 11.5 MeV would intersect the nuclear surface, the corresponding Rutherford cross section is shown in Fig. 3.4(a) with a cut-off at 11.5 MeV corresponding to the nuclear collision. 86 Elastic scattering (a) Rutherford's orbits 5 10 15 E, MeV (b) Light nuclei cross section Rutherford -Pi MeV Fig.
eBook - PDFFlash of the Cathode Rays
A History of J J Thomson's Electron
- Per F Dahl(Author)
- 1997(Publication Date)
- CRC Press(Publisher)
[17-83] Describing his model in some detail, he went on: You will note that the structure assumed in my atom is somewhat similar to that suggested by you in a paper some years ago. I have not yet looked up your paper; but I remember that you did write on that subject. I gave a preliminary account of my results to the Manchester Literary and Philosophical Society recently, and hope soon to be published in the Philosophical Magazine. [17-84] In the Philosophical Magazine paper that resulted, Rutherford gets to the point without further ado: the importance of charged particle scattering as a tool for probing the structure of an atom. J J Thomson's theory of compound scattering assumed that the average angle of deviation was the cumulative effect of a large number of atomic collisions, any one of which was inappreciable. In such a random scattering situation, the total deviation is the average deviation in a single collision not multiplied by the total number of collisions, but multiplied by the square root of this number. This, in fact, was not what Geiger and Marsden found. In their experiments on a-particles traversing a thin gold foil, they observed that about one in 20 000 particles suffered a deflection of 90° or more. 342 A simple calculation based on the theory of probability shows that the chance of an a particle being deflected through 90° is vanishingly small. In addition, it will be seen later that the distribution of the a particles for various angles of large deflexion does not follow the probability law to be expected if such large deflexions are made up of a large number of small deviations. It Dawning of the Atomic Age seems reasonable to suppose that the deflexion through a large angle is due to a single atomic encounter, for the chance of a second encounter of a kind to produce a large deflexion must in most cases be exceedingly small.
eBook - PDFPhysics 1942 – 1962
Including Presentation Speeches and Laureates' Biographies
- Sam Stuart(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
Physics 19 61 ROBERT HOFSTADTER «jor his pioneering studies of electron scattering in atomic nuclei andfor his thereby achieved discoveries concerning the structure of the nucléons» RUDOLF LUDWIG MÖSSBAUER «for his researches concerning the resonance absorption of γ-radiation and his discovery in this connection of the effect which bears his name» This page intentionally left blank Physics 1961 Presentation Speech by Professor I. Waller, member of the Swedish Academy of Sciences Your Majesties, Your Royal Highnesses, Ladies and Gentlemen. Since Rutherford's discovery of the atomic nucleus fifty years ago, one of the most fundamental problems in physics has been to investigate how it is constituted. The ideas on this question could be firmly founded when, shortly after 1930, a neutral particle called the neutron was discovered which has almost the same mass as the hydrogen nucleus i.e. the proton. A theory for the atomic nuclei was proposed according to which they are composed of protons and neutrons which are together called the nucléons. A few years later, Yukawa gave a theory of the forces which keep the nucléons together. It could according to this theory be expected that the nucléons have themselves a complicated inner structure. Professor Robert Hofstadter has developed a new experimental method for the investigation of the inner structure of the composite atomic nuclei and also of the single nucléons. His method is to bombard the atomic nuclei with electrons of very high energy. The electrons can penetrate the atomic nuclei and are then deviated by the strong electric and magnetic forces inside the nuclei. By separating the scattered electrons of different energies in magnetic spectrometers and by measuring afterwards the number of electrons which have been deviated to each particular direction, Hofstadter has succeeded in obtaining detailed knowledge of the distribution of the electric charge in the nuclei.
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
This is also evidence that, even at this moderate scattering angle, single scattering is much more important than multiple scattering. (In a random statistical theory of multiple scattering, the probability for scattering at a large angle would be proportional to the square root of the number of single scatterings, and we would expect N() ∝ t 1∕2 . Figure 6.9 shows clearly that this is not true.) 186 Chapter 6 The Rutherford-Bohr Model of the Atom This result emphasizes a significant difference between scattering by a Thomson model atom and a Rutherford nuclear atom: In the Thomson model, the projectile is scattered by every atom along its path as it passes through the foil (see Figure 6.3), while in the Rutherford nuclear model the nucleus is so tiny that the chance of even a single significant encounter is small and the chance of encountering more than one nucleus is negligible. (b) N() ∝ Z 2 . In this experiment, Geiger and Marsden used a variety of different scattering materials, of approximately (but not exactly) the same thickness. This proportionality is therefore much more difficult to test than the previous one, since it involves the comparison of different thicknesses of different materials. However, as shown in Figure 6.10, the results are consistent with the proportionality of N() to Z 2 . (c) N() ∝ K −2 . In order to test this prediction of the Rutherford Scattering formula, Geiger and Marsden kept the thickness of the scattering foil constant and varied the speed of the alpha particles. They accomplished this by slowing down the alpha particles emitted from the radioactive source by passing them through thin sheets of mica. From independent measurements they knew the effect of different thicknesses of mica on the velocity of the alpha particles. The results of the experiment are shown in Figure 6.11; once again we see excellent agreement with the expected relationship.
eBook - PDF- Herbert Uberall(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Due to the wide range of energies 92 BASIC ELECTRON SCATTERING THEORY [Ch. 2 considered, they included both the situation where screening just starts to be felt (ΤΊ = 100 keV) and where it gives a large effect on the cross section (7 ^ 1 keV). This can be seen in Fig. 2.21, taken from their paper, in which the ratio R of exact to relativistic Rutherford Scattering is plotted. Note the large diffraction effects appearing in the keV-region, and the gradual disappearance of screening effects above 33 or at least 121 keV except at the small forward angles. 3 2 R 1 0 3 R 2 1 0 2 R 1 0 0 60 120 180 60 120 180 ιϊ, deg FIG. 2.21. Ratio R of Mott to Rutherford Scattering for 7 between 0.87 keV and 121 keV, plotted versus ft for gold (Z = 79). ( ) Screened one-term field, ( ) screened two-term field, (· · ·) screened Thomas-Fermi field, ( ) Coulomb field of mercury (Z = 80) after Bartlett (40). (a) T x = 0.87 keV, k x a H = 8, (b) T x = 1.95 keV, k x a n = 12, (c) T x = 5.4 keV, k x a H = 20, (d) 7 = 12.2 keV, M H = 30, (e) 7 = 33.0 keV, M H = 50, (f) T x = 121 keV, k x a^ = 100. [Figure taken from C. B. O. Mohr and L. J. Tassie, Proc. Phys; Soc. London, Ser. A 67, 711 (1954), reproduced by per-mission of The Institute of Physics and The Physical Society.] Sec. 2.8] RECENT WORK IN POINT-CHARGE SCATTERING 93 The most recent theoretical work on electron scattering very often employs numerical integration of the Dirac equation, as mentioned. One may start with Eqs. (2-95) and note first that, at least for purposes of computation, the screened potential may be assumed to vanish for r beyond some distance ö max , unlike the Coulomb potential. As a conse-quence, the terms y In 2kj will be absent from asymptotic forms of the wave functions and from equations of phase shifts, such as Eq. (2-113h). Furthermore, the solutions for r > a mRX will then just be given by spherical Bessel functions: From the required asymptotic behavior for g x (r) y Eq.
eBook - PDF- Stephen Thornton, Andrew Rex, Carol Hood, , Stephen Thornton, Stephen Thornton, Andrew Rex, Carol Hood(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
We will see in Chapter 12 that aluminum’s nuclear radius is about twice as large as that of 4 He, and our approximate result here is in fair agreement with mod- ern data. We now know that nuclear radii vary from 1 3 10 215 to 8 3 10 215 m. Thus when a particles scatter from aluminum, an a particle may approach the nucleus closely enough to be affected by the nuclear force (see Chapter 12). CONCEPTUAL EXAMPLE 4.4 How can we find the distance of closest approach between an incoming particle and a target scatterer of like charge? Solution We can find this distance of closest approach for a given kinetic energy K and impact parameter b. The minimum separation occurs for a head-on collision. The incoming particle turns around and scatters backward at 180°. At the instant the particle turns around, the entire kinetic energy has been converted into Coulomb potential energy. By setting the original (maximum) kinetic energy equal to the Coulomb potential energy when r 5 r min , we can then solve the resulting equation for r min . Let K be the original kinetic energy of the incoming particle. K 5 sZ 1 edsZ 2 ed 4pe 0 r min (4.14) We solve this equation to determine r min . r min 5 Z 1 Z 2 e 2 4pe 0 K (4.15) 4.3 The Classical Atomic Model After Rutherford presented his calculations of charged-particle scattering in 1911 and the experimental verification by his group in 1913, it was generally conceded that the atom consisted of a small, massive, positively charged “nu- cleus” surrounded by moving electrons. Thomson’s plum-pudding model was definitively excluded by the data. Actually, Thomson had previously considered a planetary model resembling the solar system (in which the planets move in elliptical orbits about the sun) but rejected it because, although both gravita- tional and Coulomb forces vary inversely with the square of the distance, the planets attract one another while orbiting around the sun, whereas the electrons would repel one another.
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