Probability and Energy Laws of Particle Collisions

Chapter 3

Probability and Energy Laws of Particle Collisions Chapter Outline 3.1 Probability Laws of Particle Collisions: Cross Sections of Interaction 3.1.1 The Concept of Cross Section 3.1.2 Solid Angle 3.1.3 Differential and Partial Cross Sections 3.1.4 Number of Particles That Passed a “Thick” Target 3.1.5 The Average Path of a Particle Before Collision (Mean Free Path) 3.1.6 The Momentum Transfer Cross Section 3.1.7 Classical Definition of Cross Section: Impact Parameter 3.1.8 Quantum Definition of Cross Section 3.1.9 Some Length Dimension Constants That Are Used in Cross Section Calculations 3.1.9.1 Radius of the First Bohr’s Orbit 3.1.9.2 Classical Electron Radius 3.1.9.3 Compton Wavelength 3.2 Energy Laws of Particle Collisions

63 63 64 65 66 66 67 68 69 70 70 70 71 71

3.2.1 3.2.2 3.2.3 3.2.4

Conservation Laws and Coordinate Systems Analysis of Collisions in the Laboratory Frame Analysis of Collisions in the Center of Mass Frame Correlations Between the Angles and Energy (Velocity, Momentum) of Particles 3.2.5 Interaction Potentials 3.2.5.1 Model Potentials 3.2.5.2 Screened Potentials 3.2.5.3 Combined Potentials 3.2.5.4 Change in the Acting Potential With a Change in the Particle Energy 3.2.5.5 Centrifugal Potential 3.2.5.6 Movement in the Field with the Centrifugal Potential 3.2.6 Collision of “Hard Spheres” 3.2.7 Coulomb Collisions 3.3 Relativistic Relations References

71 72 73 75 76 76 77 77 77 78 79 81 83 85 86

3.1 PROBABILITY LAWS OF PARTICLE COLLISIONS: CROSS SECTIONS OF INTERACTION 3.1.1 The Concept of Cross Section A significant part of the processes in the microcosm is the collision of microparticles. A quantitative measure that determines the collision probability is a parameter called the cross section. Let a beam of particles with a fluence rate n pass through a layer of matter with thickness dx containing atoms in the concentration of N. Then if the thickness of the layer is sufficiently low, i.e., target atoms do not shade each other and manifest themselves independently, each incident particle experiences no more than one interaction in the target, and the values of n and N vary little on the thickness dx, then it can be stated that the number of interactions per time unit and per area unit dn is expressed by dn ¼ snNdx.

(3.1)

Here s is the coefficient of proportionality. The minus sign shows that there is a decrease in the number of incident particles. It is easy to see that the proportionality coefficient s has the area dimension. It is this coefficient that is called a cross section. In the case of elastic scattering of particles in classical physics, this proportionality coefficient has a clear

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geometric interpretation. It shows the cross-sectional area of the target particle on which scattering of the bombarding particles occurs. It is also seen that the ratio dn/n is a dimensionless quantity, and it shows the probability of interaction of the incident particle. By the condition jdn=nj ¼ sNdx << 1;

(3.2)

the target is called thin. The concept of a thin target (a thin absorber) turns out to be not a geometrical concept, but a physical one. The linear size of a thin absorber depends on the concentration of the absorbing centers (target particles) and on the cross section. At the condition opposite to (3.2), such an absorber is called thick. All reasoning and conclusions are also valid for collisions in which quanta of electromagnetic radiation participate, if they are regarded as particles with zero rest mass, with velocity equal to the speed of light c, energy equal to hn, and momentum equal to hn/c.

3.1.2 Solid Angle To analyze the scattering of particles, one needs to introduce the notion of solid angle. By definition, the elementary solid angle is the ratio of the element of the surface area of the sphere to the square of the radius of the sphere. For a complete understanding of how angles are measured in determining the solid angles, the reader can represent the sphere in the form of a globe with lines of parallels and meridians drawn on it (Fig. 3.1). The angle q, which determines the position of the parallel, is called polar, and the angle 4 determining the position of the meridian is called azimuthal. The surface area element of the sphere is the area of the elementary trapezium shown in Fig. 3.1. The sides of the surface area element are evident from the figure. But note that the polar angle q is measured from the pole (unlike the globe, where this angle is measured from the equator). If the angles dq and d4 are sufficiently small, then we can neglect the difference between the trapezoid and the rectangle and assume that the area of the elementary trapezium on the surface of the sphere is R2sinq dqd4. Thus, an elementary solid angle, an angle of the second order, is given by dUð2Þ ¼ sin qdqd4.

(3.3)

The solid angle of the first order dU is enclosed between two nested cones with a common vertex in the center of the sphere. This angle is shown more clearly in Fig. 3.2, and it can be obtained by integrating dU(2) over 4 from 0 to 2p. Z dU ¼

2p

sin qdqd4 ¼ 2p sin qdq. 0

FIGURE 3.1 The definition of a solid angle.

(3.4)

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FIGURE 3.2 The scheme of the elastic scattering of bombarded particles on the target. The elementary solid angle dU(2) is evident from the figure, and the angle dU is the solid angle between two nested cones.

The total solid angle can be obtained by integrating dU over the angle q from 0 to q: Z q sin qdq ¼ 2pð1  cos qÞ. U ¼ 2p

(3.5)

0

3.1.3 Differential and Partial Cross Sections The cross section in (3.1) is the so-called total cross section, i.e., under the value of dn we mean all the particles scattered at any angles. Often it is necessary to calculate or measure the number of particles scattered at a certain angle or having a certain energy after a collision, etc. In such cases, the differential cross section is used: l l l l l

differential differential differential differential differential

scattering cross section in dependence on scattering angle, scattering cross section in dependence on the energy of scattered particles, recoil cross section in dependence on recoil angle, recoil cross section in dependence on recoil energy, momentum transfer cross section.

One of the variants of the particle scattering is shown in Fig. 3.2. The fraction of particles scattered at an angle q to the direction of the incident beam within a certain elementary solid angle dU(2) is determined by the relation d 2 n ¼ nNdx$d2 s=dUð2Þ ;

(3.6)

where d s/dU is a double differential cross section of the scattering into solid angle dU . In the same manner it is possible to determine differential cross section of the scattering into solid angle dU  ds/dU. Often one needs to use the cross section of scattering not through the angle dU, but in the energy range dE. The connection between cross sections can be found from the obvious arguments. Because the particles scattered through a solid angle dU, with the angle ranging from q to q þ dq, have energy in the range from E to E þ dE, then 2

(2)

(2)

nNsðUÞdUdx ¼ nNsðEÞdEdx. Considering that in this case dU ¼ 2p sin q dq, one can obtain  ds ds dE ¼ 2p sin q . dE dU dq

(3.7)

(3.8)

In the case of inelastic interactions, the energy of the resulting collision products is complicatedly related to the angular distribution. Then it is necessary to use the double differential cross section with respect to the angle and energy d2s/dEdq. In relation (3.1), s is the total cross section; it takes into account all the particles scattered at any angle. It is obvious that the total and differential cross sections are related by Z s ¼ dsðqÞ; (3.9) where the integration is carried out within the range of the angle q, which usually changes from 0 to p.

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Let us note once again that, nominally, the cross section is a proportionality coefficient in such relations as (3.1). Therefore, if on the left-hand side of these relations not the number of scattered particles but the number of outgoing recoil particles are recorded, then the proportionality coefficient is the recoil cross section. Obviously, in an elastic collision, the numbers of scattered particles and recoil particles are equal, but the corresponding angles are different, and therefore the differential cross sections also differ. The total cross sections that do not depend on the angles are, of course, equal. The relations of the type (3.1) can be compound not only for the number of particles but also for their energies. In this case one can keep in mind both the energy of the scattered particles and the energy of the recoil particles. In the latter case, the proportionality factor is the energy transfer cross section. In some cases, a particle in a collision can participate in one of the several possible processes. For example, an electron in a certain energy range can, with comparable probabilities, ionize an atom, excite it, or experience elastic scattering. Each of these processes can be characterized by its cross section. Such cross sections are called partial cross sections. In the above example of the collision of an electron with an atom, the ionization cross section si, the excitation cross section sex, and the elastic scattering cross section ss are partial cross sections. The total cross section is equal to the sum of the partial cross sections s ¼ si þ sex þ ss .

(3.10)

Not only the total but also the differential cross sections can split into the partial cross sections. If nonmonoenergetic particles interact with the target and the cross section depends on the energy, then the relationships connecting the number of scattered particles with the fluence rate are substantially more complicated. In such cases, one can use the average cross section. The number of interaction events for incident particles in the range from E to E þ dE is equal to nNs(E)dE. The number of acts in the entire energy range is obtained by integrating this quantity. The same number of acts can be written in terms of sav. Therefore, NsðE ÞnðEÞdE ¼ sav NnðE ÞdE. From this one can obtain

R sav ¼

sðE ÞnðE ÞdE R . nðE ÞdE

(3.11)

(3.12)

3.1.4 Number of Particles That Passed a “Thick” Target In the overwhelming majority of real cases, the target particle concentration N remains practically unchanged as the incident particles (projectile particles) pass through the material layer, i.e., N can be regarded as a constant. But the fluence rate along the x-axis can vary because incident particles gradually, as they go into the depths of matter, drop out of the beam. To find the law of the change in the number of acts of interaction over the depth of the absorber, one can obtain integrating (3.1) with the obvious initial condition n ¼ n0 for x ¼ 0. Then we get n ¼ n0 expðsNxÞ ¼ n0 expðSxÞ.

(3.13)

The product sN ¼ S is called the macroscopic cross section or the absorption coefficient (for the case when the incident particles are gamma quanta, the notation sN ¼ m is usually used). The number of particles scattered in the “thick” target can be obtained by subtracting from n0 the number of particles that have passed without scattering the layer d, that is, (3.13) nscat ¼ n0 ½1  expðsNdÞ.

(3.14)

In a “thin” target, sNd  1. The exponent can be expanded in a series, and neglecting higher power terms, one can obtain a simple formula for a thin target nscat yn0 sNd ¼ n0 Sd. (3.15)

3.1.5 The Average Path of a Particle Before Collision (Mean Free Path) The probability of a particle collision in the layer dx is equal to the fraction of the particles that have experienced the collision, PðxÞ ¼ dx ¼ dn=n0 ¼ sN expðsNxÞdx.

(3.16)

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Then the average path of a particle before a collision is determined by the usual rule of finding the mean values RN xPð xÞdx 1 ¼ 1=S. (3.17) ¼ xav ¼ R0 N sN Pð xÞdx 0 The average path of a particle before a collision is called the mean free path. Usually the special notation Lav is introduced for it. Thus, xav ¼ Lav ¼ 1=S:

(3.18)

The distribution of ranges can be expressed using the concept of mean free path. The integral distribution of the free paths has the form Pðx  x0 Þ ¼ expð x0 =Lav Þ.

(3.19)

The differential distribution of the free paths equals PðxÞ ¼ ð1=Lav Þexpð x=Lav Þ.

(3.20)

3.1.6 The Momentum Transfer Cross Section As is known, particles in scattering change the direction of motion, i.e., direction of the momentum. The cross sections described above, which determine the collision probability, are insufficient for characterizing the change in the direction of the momentum. In fact, if the scattering is not isotropic and the angular distribution of the scattered particles is such that the forward scattering predominates, then a larger number of collisions are required to noticeably change the direction of the particle motion than the number of collisions in the case of isotropic scattering. If backscattering dominates, then, on average, the particle needs fewer collisions to change the direction of its motion. To characterize the change in the direction of a particle, we introduce the concept of the momentum transfer cross section sm, which is also called the diffusion cross section or the transport cross section. From the above arguments it is easy to understand that if the scattering is directed primarily forward, then sm < s, and if primarily backward, then sm > s. If a particle has a momentum p and is elastically scattered at an angle q, then the initial direction of the momentum remains equal to pcos q. And the momentum loss is dp ¼ pð1  cos qÞ

(3.21)

Let us extend the conception of cross section on the change in the direction of motion. Different particles scatter randomly and differently change the direction of motion. Many, in particular, do not dissipate at all. To take into account the change in the direction of motion for the particle flux, we multiply both sides of expression (3.21) by equal values of both sides in the following expression Z Z 2 dn ¼ d n ¼ nNdx ðds=dUÞdU (3.22) and integrate the result over all the angles to which scattering can occur, i.e., from 0 to p. As a result, we obtain Z p dðnpÞ ¼ npNdx ðds=dUÞð1  cos qÞdU ¼ sm npNdx; (3.23) 0

where

Z

p

sm ¼

ðds=dUÞð1  cos qÞdU

(3.24)

0

is the momentum transfer cross section. In the same way as before (Section 3.1.5) the mean free path before collisions was introduced, it is possible to introduce an average path for the change of momentum or the mean transport path Lm ¼ 1=Nsm .

(3.25)

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Sometimes the value of Lm is called the length of isotropization because in this way, initially parallel beam of particles becomes practically isotropic.

3.1.7 Classical Definition of Cross Section: Impact Parameter The deviation of a particle in a collision and hence the scattering cross section are determined by the potential energy of interaction between the incident particle and the target particle. This interaction depends on the distance at which the incident particle flies past the target particle, i.e., on the so-called impact parameter. If on the plane, where the target particle lies and which is perpendicular to the direction of motion of the incident particle, a ring is selected, bounded by radii b and b þ db centered at point O, where the target particle is located, then all the incident particles that cross this ring deviate into angles in the interval from q to q þ dq, as shown in Fig. 3.3. Thus, the area of the ring dsðbÞ ¼ 2pbdb

(3.26)

is the differential cross section of scattering by angle q, expressed as the function of the impact parameter b. To express the cross section as a function of the scattering angle, it is necessary to relate the impact parameter b to the scattering angle as   dbðqÞ dq (3.27) dsðqÞ ¼ 2pbðqÞ dq  or

  dbðqÞ dU   dsðUÞ ¼ bðqÞ . dq  sin q

(3.28)

In the expressions (3.27) and (3.28) the modulus of the derivative of the impact distance over the scattering angle is used because in most cases, at least for monotonically varying potentials, this derivative is negative, i.e., the impact distance decreases with increasing scattering angle. The detailed nature of the relationship between the impact parameter and the scattering angle is determined by the potential energy of the interaction. The relationship between the impact parameter and the scattering angle for some potentials and the dependence of the cross section on the scattering angle is considered in Section 3.2. In many cases, the potential energy of interaction of colliding particles depends on the distance in the form U w rn. This means that the potential energy has infinite length. Then the integral (3.9) diverges at q ¼ 0; the cross section is infinitely large. Physically, this means that in the limit of very small scattering angles that correspond to very large distances from a particle, we do not have any opportunity to establish an objective criterion for the separation of scattered and unscattered particles. Therefore, for example, it is possible to calculate the number of particles scattered by an angle greater than a certain one, but it is impossible to calculate the number of particles scattered by an angle less than the specified angle. The total cross section is finite for potentials whose extent is limited, for example, for the model potential of hard spheres or when a screened potential is introduced. In quantum mechanics, the problem is slightly easier because the total cross sections are finite for any potentials that decay faster than r2. However, for more slowly decaying potentials, the problem of the divergence of the total cross section remains also in quantum mechanics.

FIGURE 3.3 The scheme that clarifies the conception of “impact parameter.”

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3.1.8 Quantum Definition of Cross Section At present, a certain tradition of using the classical and the quantum description of processes in the microcosm has been established. The classical description is simpler and more evident than the quantum one, and it is used in all cases when the accuracy of the results obtained with its help is sufficient. But if the condition ƛ > a, where a is the characteristic size of the interaction region, is satisfied, then the quantum approach must be used. As is known, the state of classical particles is described by their coordinates and momenta (velocities), and their behavior obeys the Newton equation of motion. The state of quantum particles is described by the so-called wave j-function, and their behavior obeys the Schrodinger equation. The wave function characterizes the probability of detecting a microparticle at some point in space. The square of the modulus of the wave function has the physical meaning. The probability of detecting a particle in an element of volume dV is dw ¼ jjj dV. 2

(3.29)

where jjj is the probability density, i.e., probability per unit of volume. The total probability of finding a particle in a finite volume V is obtained by integrating over the volume. The concept of a trajectory is not applicable to quantum particle waves; therefore, one cannot use the impact parameter and determine the cross section, as was done in Section 3.1.7. Let us consider the process of scattering of particles on the basis of the concepts of quantum mechanics. Schematically, the quantum-mechanical scattering pattern is shown in Fig. 3.4. The incident particles are described by a plane wave. If the incident particles move along the z-axis, then the stationary j-function depends only on the coordinates 2

jinc ðzÞ ¼ expðikzÞ.

(3.30)

Here k ¼ 2p/l is the wave number. In the vector form it is the wave vector that shows the direction of the wave propagation. The fluence rate of incident particles is equal to Finc ¼ vjjj2 ;

(3.31)

where v is the velocity of wave propagation that is equal to the particle velocity v ¼ p/m ¼ Zk/m. Scattering of the incident wave at the force center leads to the appearance of a spherical wave jscat ¼ ½f ðcÞ=rexpðik 0 rÞ;

(3.32)

where c is the scattering angle in the center of mass (center-of-momentum, COM) frame (Section 3.2.3), r is the distance from the center, k’ is the wave number of the scattered wave, and f(c) is the amplitude of the scattered wave that depends only on the polar angle c, measured with respect to the direction of motion of the incident wave. In considering only spherically symmetric potentials, the problem turns out to be axially symmetric and no azimuth angle is required. The function f(c) has the dimension of length, and it contains all the information on the scattering process.

FIGURE 3.4 The scheme of the scattering of a particle wave.

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The radial fluence rate of scattered particles through a surface element dS ¼ r2 dU ¼ r2 2psin cdc equals  2 Fscat ¼ jf ðcÞj v0 dS r 2 ;

(3.33)

where v0 is the velocity of the scattered particle. In the case of elastic scattering in the COM frame, v ¼ v0 . The differential scattering cross section in a solid angle dU is the ratio of the fluence rate of scattered particles to the fluence rate of the incident particles dsðUÞjf ðcÞj dU. 2

(3.34)

The differential cross section within the range of angles from c to c þ dc equals dsðcÞ ¼ 2pjf ðcÞj2 sin cdc

(3.35)

In quantum analysis of particle collisions, it is often possible to limit the problem to the following conditions: 1. 2. 3. 4.

only stationary states are considered, only elastic scattering is analyzed, only central potentials are taken into account, the spins of the particles are neglected.

At large distances from the scattering center, i.e., for r / N, where the role of the potential energy is practically negligible, i.e., U(r) / 0, the wave function is equal to the sum of two waves, the incident one, which is plane, and the scattered wave, which is spherical. jðrÞ/expðikzÞ þ f ðcÞexpðikrÞ=r

(3.36)

This function must be a solution of the Schrödinger equation. Solving the equation, one can find the amplitude of the scattered wave and, consequently, the scattering cross section. The solution of the problem in the general case is very laborious. There is no single general method for solving the Schrödinger equation. We indicate here several variants of such a solution. First, if the variables of the wave function j in spherical coordinates can be divided, then a solution could be found out by this method. It is called the method of partial waves and is applicable, mainly, to the case of low energies of incident particles. Second, if the kinetic energy of the incident particles is larger than potential one, then the latter can be regarded as a small perturbation and thus the problem can be solved by a perturbation theory method. This method, called the Born approximation, is applicable to high energies of incident particles and naturally complements the method of partial waves. Third, if the appropriate criteria allow (Section 1.1.4), one can make the transition from quantum to classical analysis within the framework of the so-called semiclassical approximation.

3.1.9 Some Length Dimension Constants That Are Used in Cross Section Calculations 3.1.9.1 Radius of the First Bohr’s Orbit According to the Bohr theory of the atom the radius of the first orbit equals  rB ¼ ð4pε0 ÞZ2 me e2 ¼ 0:53$108 cm.

(3.37)

3.1.9.2 Classical Electron Radius Classical electron radius is the distance where electrostatic energy of interaction of two elementary particles is equal to the rest energy of an electron  (3.38) e2 4pε0 re ¼ me c2 . From here

 re ¼ e2 4pε0 me c2 ¼ 2:82$1013 cm.

(3.39)

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The name has a historical origin. In old electron models some of its properties were tried to be explained by the effect of one of its parts on the other ones, for which the electron size had to be of the order of re. At present, this value has lost its former meaning, the classical electron is considered a point particle, and the classical radius of the electron re remained a convenient constant, which appears in the description of many electromagnetic processes.

3.1.9.3 Compton Wavelength The Compton wavelength of a particle is equal to the wavelength of a photon whose energy is the same as of the particle mass E ¼ hc=l ¼ mc2 .

(3.40)

lC ¼ h=mc;

(3.41)

Solving this equation for l, one can find

where h is the Planck constant, m is the particle’s mass, and c is the speed of light. The value of the Compton wavelength of an electron is lC ¼ 2.43$1010 cm. The name of the Compton wavelength of an electron arose from the formula for changing the wavelength of a gamma quantum in Compton scattering by an electron obtained by A. Compton in 1922. Dl ¼ ðh=mcÞð1  cos qÞ ¼ lC ð1  cos qÞ.

(3.42)

Sometimes the so-called reduced Compton wavelength is used. ƛC ¼ lC =2p ¼ 3:86$1011 cm

(3.43)

The position of a specific particle may be determined only with the accuracy of the order of the Compton wavelength of the particle. The Compton wavelength also determines the radius of operation of short-range forces. Thus, the radius of action of nuclear forces is a quantity of the order of the Compton wavelength of pion (pi-meson).

3.2 ENERGY LAWS OF PARTICLE COLLISIONS 3.2.1 Conservation Laws and Coordinate Systems In this section, elastic collisions are considered, in which the particles exchange kinetic energy and momentum, and their internal state remains unchanged. To find the probability of transmission during a collision of a certain portion of energy, or deviation to a certain angle, which is uniquely bound with such a transfer, knowledge of the interaction potential is required. But some important relationships between the masses of the colliding particles or some energy characteristics (velocity, momentum, or energy) and angles of scattering can be obtained without specifying the nature of the forces acting between the particles, but only on the basis of the laws of conservation of energy and momentum. True is the fact that in this case one can obtain some information on the particle parameters only at large distances from the collision site. In other words, on the basis of the laws of conservation of energy and momentum, only an asymptotic analysis can be carried out. The results obtained in this section are equally valid for both classical and quantum collision analysis; the trajectory of motion is not taken into account, and the conservation laws considered here are valid in both cases. The processes of particle collisions can be considered in several different coordinate systems. The most obvious, so-called laboratory frame of reference (LFR), is associated with the equipment in which the measurements are made, i.e., with the laboratory. Usually, experimental results are presented in LFR. In this system they are most obvious. However, in many cases, the use of the center of mass (COM) frame can significantly simplify the corresponding transformations and analyzes of the results. In this system, the fixed point chosen as the origin of coordinates is the common COM of the colliding particles. Then the problem of motion of a system consisting of two interacting particles can be substantially simplified by decomposing motion of the system into the motion of the center of mass and the motion of particles relative to the latter. Therefore, the theoretical analysis, as a rule, is carried out in the COM. When comparing experimental results and theoretical calculations, it is necessary to make a transition from COM to LFR and vice versa. In this section we obtain formulas that allow us to make the corresponding transitions. In the books of Refs. [1,2] one can find information about these and some other coordinate systems used in the analysis of collisions.

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3.2.2 Analysis of Collisions in the Laboratory Frame To analyze the collisions, we make some simplifying assumptions. Because we consider collisions of mainly bombarding particles, i.e., possessing high energy, with particles of matter, whose relatively slow motion can be neglected in comparison with the motion of the incident particles, we consider the target particles to be at rest. It is shown below that such a simplification can be done without diminishing the generality of the consideration. We also assume that the particles move with velocities much less than the speed of light, i.e., in the nonrelativistic case. The collision scheme required for the analysis is shown in Fig. 3.5. We consider only central interactions, i.e., those in which the potential energy of the particles depends only on the distance to a certain point, called the center. The scheme shown in Fig. 3.5B is the cut by the figure plane through the axially symmetric collision pattern. Then (Section 3.2.5.5) it is shown that the particles trajectories lie in the same plane. In Fig. 3.5B the particle momentum vectors are shown. In principle, the same scheme can be constructed on the particle velocity vectors. Note that some quantitative ratios for pulses and velocities are different. The incident particle has a mass m1, velocity, momentum, and kinetic energy before the collision v0, p0 ¼ m1v0, and E0 ¼ m1v20 / 2. The target particle has a mass of m2 and before the collision is in rest, i.e., its velocity, momentum, and kinetic energy equal zero (Fig. 3.5A). After the collision, the incident particle m1 has a velocity v1 and is scattered by an angle q1 to the direction of the initial motion, and the target particle m2 flies off with the velocity v2 at the angle q2 (Fig. 3.5B). The laws of conservation of energy and momentum in scalar form have the view m1 v20 ¼ m1 v21 þ m2 v22

(3.44a)

m1 v0 ¼ m1 v1 cos q1 þ m2 v2 cos q2

(3.44b)

m1 v1 sin q1 ¼ m2 v2 sin q2

(3.44c)

Here, Eq. (3.44b) is the projection of the momenta on the axis, which coincides with the direction of the incident particle motion, and Eq. (3.44c) is the projection of the momenta perpendicular to this direction. The solution of the system of Eq. (3.44) leads to a quadratic equation with respect to the velocity v1  2 n1 v1 m1 m1  m2 2 cos q1 þ ¼ 0: (3.45) n0 v0 m 1 þ m 2 m1 þ m2 The solution of this equation performs the value of the velocity v1 as a function of the scattering angle q1 h 1=2 i  v0 cos q1  ¼ m22  m21 sin2 q1 v1 ¼ m1 þ m2

(3.46)

For m1 < m2, in formula (3.46), only the plus sign before the root makes sense because with the minus sign, negative values of v1 are obtained. For m1 > m2, both signs can be used. From the same system of Eq. (3.44) one can find the velocity v1 as a function of the angle q2 " # 4m1 m2 cos2 q2 2 2 v1 ¼ v0 1  ; (3.47) ðm1 þ m2 Þ2

FIGURE 3.5 The scheme of the particle collisions in the laboratory frame (A) before collision and (B) after collision.

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as well as the value of the velocity v2 v2 ¼

v0 2m1 cos q2 . m1 þ m2

(3.48)

Analyzing Eqs. (3.44b) and (3.44c) of the momentum conservation law and comparing the triangles in Fig. 3.5B, which are built on the vectors p1, p2, and p0, it is easy to see that the momenta p1 and p2 form a parallelogram which diagonal is the momentum p0. We note that in a similar velocity vector diagram, the velocity v0 is not a diagonal of a parallelogram built on the vectors v1 and v2. Because the vectors p0, p1, and p2 form a triangle, all known triangle formulas can be used to obtain the necessary relations. Applying the cosine theorem to this triangle, one finds p22 ¼ p20 þ p21  2p0 p1 cos q1

(3.49)

p1 ¼ p0 sin q2 =sinðq1 =q2 Þ

(3.50)

Using the sine theorem one obtains As p1 ¼ m1v1 and p0 ¼ m1v0, then m1 can be shortened and then for v1 v1 ¼ v0 sin q2 =sinðq1 þ q2 Þ

(3.51)

In the same way the formula for momentum p2 can be obtained p2 ¼ p0 sin q1 =sinðq1 þ q2 Þ;

(3.52)

although now the masses cannot be shortened because momentum p2 depends on mass m2 and the momentum p0 on mass m1. On the basis of formulas (3.45) and (3.47) one can get the relation for the energy of particles after collision h 1=2 i2  m1 cos q1  m22  m21 sin2 q1 . (3.53) E1 ¼ E0 ðm1 þ m2 Þ2 The use of signs before the radical is the same as for the velocity; see formula (3.46) and the indication after it. E2 ¼ E0

4m1 m2 cos2 q2 ðm1 þ m2 Þ

2

.

(3.54)

3.2.3 Analysis of Collisions in the Center of Mass Frame Now, let us consider the collision in the center of mass frame. The problem of the mutual motion of two particles in a center of mass frame is reduced to two problems: (1) the motion of one particle with a mass equal to the reduced mass m1 m2 =ðm1 þ m2 Þ ¼ m;

(3.55)

in the field of a stationary center and (2) the motion of the mass center. The reduced mass is less than the lightest of the two colliding particles. The scheme of the particle collision in the center of mass frame is shown in Fig. 3.6. As the target particle is at rest, the center of mass moves rectilinearly in a direction parallel to the direction of the initial motion of the incident particle. The center of mass is located between the particles at a distance r1 from the particle m1 and r2 from the particle m2. The distances r1 and r2 are in the ratio r1m1 ¼ r2m2. The mass of the center of mass is m1 þ m2, and its velocity is denoted by vc. The velocity of the center of mass can be found from the law of conservation of momentum ðm1 þ m2 Þvc ¼ m1 v0 ;

(3.56)

vc ¼ m1 v0 =ðm1 þ m2 Þ.

(3.57)

where The momentum of the center of mass is equal to the momentum of the incident particle pc ¼ ðm1 þ m2 Þvc ¼ m1 v0 ¼ p0 .

(3.58)

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FIGURE 3.6 The scheme of the particle collision in the center of mass frame (A) before collision and (B) after collision.

The total kinetic energy of the entire system is determined, in our case, only by the incident particle. It equals E0 ¼ m1v20 /2. The center of mass carries away part of the kinetic energy, viz  Ec ¼ ðm1 þ m2 Þv2c 2 ¼ E0 m1 =ðm1 þ m2 Þ. (3.59) The remaining part of the energy is the energy of the relative motion of the particles  m2 ¼ mv20 2; Erel ¼ E0  Ec ¼ E0 ðm1 þ m2 Þ

(3.60)

where m is a reduced mass (3.55). The relative motion is determined by the relative energy and only it changes during the collision. Theoretical analysis of collision processes is usually carried out by taking into account only the relative motion. In inelastic collisions, only the relative energy can change the state of the system. In the center of mass frame, the two colliding particles before the collision move toward each other with equal by absolute magnitude and oppositely directed momenta, as shown in Fig. 3.6; hence the total momentum of the system before collision is zero. The velocities of particles in the center of mass frame both before and after the collision are equal: v1c ¼ v0  vc ¼ m2 v0 =ðm1 þ m2 Þ ¼ mv0 =m1 ;

(3.61)

v2c ¼ vc ¼ m1 v0 =ðm1 þ m2 Þ ¼ mv0 =m2 .

(3.62)

Particle momenta both before and after collision are equal p1c ¼ p2c ¼ m1 m2 v0 =ðm1 þ m2 Þ ¼ mv0 .

(3.63)

Because the collision is elastic, after collision the total momentum is zero, and hence the momenta of the particles, which fly apart after the collision, remain equal in magnitude and opposite in direction, and according to the law of energy conservation their absolute value also does not change. However, the direction of their movement after the collision no longer coincides with the initial one. In other words, the result of an elastic collision is the rotation of the particle velocities by the same angle c, which is called the scattering angle in the center of mass frame. If the vector schemes of collision in the velocity space in the laboratory frame and the center of mass frame are combined on the same graph, we obtain the vector diagram as shown in Fig. 3.7. It is seen from the diagram that the velocity vectors v1 and v2 in the laboratory frame after the collision are equal to the vector sums of the velocities v1c and v2c in the center of mass frame and the velocity of the center of mass vc, respectively. The lower triangle in the figure 3.7, constructed at the velocities of the particle m2, is isosceles because v2c ¼ vc; see the formulas (3.57) and (3.62). From here we immediately obtain a connection between the angles. q2 ¼ ðp  cÞ=2:

(3.64)

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FIGURE 3.7 Vector diagram of velocities.

From the upper triangle ABC in Fig. 3.7, which is built on the velocities of the particle m1, one can find tgq1 ¼ BC=AC ¼

½v1c cosððp=2Þ  cÞ sin c ¼ ½v1c sinððp=2Þ  cÞ þ vc  ½cos c þ ðvc =v1c Þ

(3.65)

Substituting here the values vs and v1s from (3.57) and (3.61), one can finally find tgq1 ¼

sin c . cos c þ ðm1 þ m2 Þ

(3.66)

Using expression (3.64), one finds formulas, connecting angles q1 and q2 tgq1 ¼

sin 2q2 ðm1 =m2 Þ  cos 2q2

(3.67)

or cos 2q2 þ ctgq1 sin 2q2 ¼ m1 =m2

(3.68)

3.2.4 Correlations Between the Angles and Energy (Velocity, Momentum) of Particles Let us analyze the relationship between the angles and velocities of the colliding particles with different masses. In the case m1 < m2, the angle q1 can vary from 0 to p, the angle q2 from p/2 to 0, and the angle c from 0 to p. q1 > c=2;

q1 þ q2 > p=2

(3.69)

The case q1 ¼ p and q2 ¼ 0 corresponds to the so-called frontal collision. In this case, the particle m1 loses, while the particle m2 receives the maximum energy. They are equal E1;min ¼ E0 E2;max ¼ E0

ðm2  m1 Þ

2

ðm1 þ m2 Þ

2

4m1 m2 ðm1 þ m2 Þ

2

(3.70) .

(3.71)

In the limiting case m1  m2 czq1 ;

q1 þ q2 zðp þ cÞ=2

(3.72)

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the laboratory and center of mass frames are practically identical. In this case E1;min wE0 ;

E2;max wE0 4m1 =m2

(3.73)

It can be seen that a light incident particle is capable of transmitting to a heavy target particle only a negligible fraction of its energy. In the case of m1 > m2, the angle q1 can vary from 0 to q1,max, the angle q2 from p/2 to q1,min, and the angle c from 0 to cmax. The value of q1,max can be found under the condition that in expression (3.46) for v1 the radicand must be positive. Then q1max ¼ arcsinðm2 =m1 Þ  p=2

(3.74)

 2 1=2 m1  m22 m1 ðcos q1max Þ n1 ¼ n0 ¼ n0 . m1 þ m2 m1 þ m2

(3.75)

q2min ¼ arcsin½ðm1  m2 Þ=2m1 1=2 ;

(3.76)

cmax ¼ arccosðm2 =m1 Þ;

(3.77)

q1 þ q2 < p=2:

(3.78)

In this case

Similarly one obtains

In the range of changes in the angle q1 from 0 to q1,max, the plus sign must be used in the formulas for the velocity (3.46) and the energy (3.53) of the incident particle. The presence of a limiting scattering angle means that the incident particle, if it is heavier than the target particle, cannot  be scattered back. For example, in scattering by hydrogen, alpha particles cannot deviate by more than 14 300 . After reaching the limit value, the angle q1 decreases back to 0, and the angle c increases further to p. In this range, in the formulas for the velocity (3.46) and the energy (3.53) of the incident particle, the minus sign has to be used. In the limiting case m1 [ m2, the limiting angle q1,max is very small and the incident particle practically does not change its motion direction. In a head-on collision E1min wE0 ;

E2max wE0 4m2 =m1

(3.79)

It can be seen that a heavy incident particle can only transmit a small fraction of its energy to a light target particle. Finally, for m1 ¼ m2 q1 ¼ c=2;

q1 þ q2 ¼ p=2;

(3.80)

i.e., the angle of scattering of the particles after collision in the laboratory frame is p/2 and there is no scattering into the “back” hemisphere. Besides E1min ¼ 0;

E2max ¼ E0

(3.81)

In this case, the incident particle at a frontal collision stops, transmitting all its energy and its entire momentum to the target particle.

3.2.5 Interaction Potentials To calculate the scattering cross sections both in classical physics and in quantum theory, the knowledge of the real form of the potential energy of interaction of two colliding particles is required. As this does not cause ambiguity, we will, traditionally, use the shortened “potential” instead of “potential energy.” To date, a large number of different potentials have been invented to describe real interactions more or less accurately.

3.2.5.1 Model Potentials One of the simplest potentials is the interaction potential of “hard spheres.” The potential energy of interaction of “hard spheres” has the form U ¼ N

at r  R;

U ¼ 0

at r > R

(3.82)

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This potential is of interest as a model and can also be used to describe the collision of two neutral atoms, and in some cases a charge and a neutral atom, a neutral particle, and a nucleus. Here is a series of power potentials of the form U(r) w rn: 1. n ¼ 1, the Coulomb interaction, describing the interaction of two charged particles, charges of the same namedthe interaction of repulsion, and the oppositely charged onesdthe interaction of attraction. The potential energy of the Coulomb interaction has the form UðrÞ ¼ k=r 2. 3. 4. 5. 6.

(3.83)

Here the minus sign corresponds to attraction and the plus sign to repulsion. n ¼ 2, potential energy of the chargeedipole interaction; n ¼ 3, potential energy of chargeequadrupole interaction; n ¼ 4, polarization interaction describing the collision of a charged and neutral polarizing particle; n ¼ 6, the so-called dispersion intermolecular interaction; n ¼ 12, intermolecular repulsion in the LennardeJones potential.

3.2.5.2 Screened Potentials All the listed power law potentials have an infinite range of action, which can be true only in the idealized conditions of interaction of two bodies in a vacuum. Because the interacting bodies are surrounded by other bodies, for various reasons the interaction is screened at large distances. In addition, there are short-range interactions. 1. The Yukawa potential, the first potential proposed for the description of nuclear forces; in this case the carrier of interaction is a massive particle. U ðr Þ ¼ ðg=rÞexpðr=lC Þ

(3.84)

where lC is the Compton wavelength for the field mediator (pion). 2. A significant number of different atomic potentials, the BorneMayer’s, and other potentials. They all contain in one form or another exponential factor, similar to an exponential in the Yukawa potential. 3. The atomic potentials closest to reliable are the potential calculated by the HartreeeFock method and the ThomaseFermi potential. The HartreeeFock method makes it possible to obtain a near-real distribution of the potential in light atoms. For atoms with large Z, the ThomaseFermi method offers better results.

3.2.5.3 Combined Potentials The actual picture of the interaction of particles is usually more complex and cannot be described by detailed analysis of one of the above potentials. For example, a situation is often encountered when attraction takes place at large distances and repulsion at small distances. To describe such situations in intermolecular interactions, the Sutherland model (an impenetrable sphere at the center and power attraction at the periphery), the LennardeJones potential, known as the potential 6e12 e (a repulsive power term with exponent n ¼ 12 and an attracting power term with exponent n ¼ 6), and others are used. It should be taken into account that, as a rule, the better the potential takes into account the real picture of the interaction, the more complex are the calculations using this potential.

3.2.5.4 Change in the Acting Potential With a Change in the Particle Energy Note that in fact the situation is even more complicated and even the most intricate potential cannot fully describe the real interaction. Let us explain this with an example of the scattering of electrons by neutral atoms. While electrons have a small energy of the order of the thermal energy, they are in the field of an attractive polarization potential and cannot approach close to the atom because of the centrifugal barrier (Section 3.6.5). With increasing energy, electrons are able to approach the electron shell closely and the attraction is replaced by repulsion according to the law of interaction of solid spheres. It can be shown that if the interaction potential changes as U w rn, then the dependence of the cross section on the velocity of the particle in the center of mass frame in classical mechanics has the form s w v4/n. Then for the polarization

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potential it is (n ¼ 4) s w v1. Consequently, the mean free path is proportional to the velocity Lav ¼ 1/sN w v, and the mean free time does not depend on the velocity tav ¼ Lav/v w const. In the solid sphere model, the collision cross section does not depend on the velocity (Section 3.6.4); therefore, in this case the mean free path is constant. Therefore, books often characterize collisions of electrons with atoms as a model of a constant average mean free time (slow electrons) or as a model of a constant mean free path (faster electrons). With a further increase in the electron energy, they are able to penetrate through the electron shell, to feel the structure of the shell, and to experience the influence of the positive charge of the nucleus, but for the time being as a whole. At even greater energy, the electrons penetrate into the nucleus and allow measuring the distribution of electric charge along the nucleus. It was by investigating the scattering of fast electrons (hundreds of MeVetens of GeV) on nuclei that the American physicist Robert Hofstadter showed the existence of point formations inside nucleons, later called quarks. For these studies Hofstadter was awarded the Nobel Prize in Physics in 1961. It is clear that the intraatomic and intranuclear potentials are very different.

3.2.5.5 Centrifugal Potential When the particle moves in the central field, a centrifugal force appears that prevents the particle from approaching the center. It is obvious that when moving toward the center, the particle accelerates and then, after passing the point of closest approach, slows down. Therefore, instead of the momentum, we need to introduce the angular momentum and use the angular momentum conservation law. Because we are considering a field that depends only on the distance between the particles r, it is advisable to choose a polar coordinate system in which the analyzed relationships will look most simply. We first obtain an expression for the energy in polar coordinates. The connection of polar coordinates r and 4 and Sartesian coordinates x and y is given by the evident relations x ¼ r cos 4;

y ¼ r sin 4

The kinetic energy in Cartesian coordinates for a motion in the xy plane is given by 2 2  E ¼ m ðdx=dtÞ þ ðdy=dtÞ 2

(3.85)

(3.86)

where m is the reduced mass (3.55). The values of the derivatives can be found by differentiating the expressions (3.85)

Then

dx=dt ¼ ðdr=dtÞ ¼ cos 4  ðd4=dtÞr sin 4

(3.87a)

dy=dt ¼ ðdr=dtÞsin 4 þ ðd4=dtÞr cos 4

(3.87b)

2 2  E ¼ m ðdr=dtÞ þ ðd4=dtÞ r 2 2.

(3.88)

Because the analysis is carried out in the center of mass frame, where the center is at rest, there is no term in the expression for energy that describes the kinetic energy of the center of mass. Now let us turn to an angular momentum. The angular moment is a vector perpendicular to the plane in which the momentum and trajectory radii along which the particle moves are numerically equal to the vector product of the radius per momentum, ! L ¼ m½! r! v (3.89) The position of the vectors is shown in Fig. 3.8.

FIGURE 3.8 Angular momentum L of the particle that moves relative to the point O with the momentum p.

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We assume that the vectors ! r and ! v are in the xy plane and then the expression for the product of the vectors in Cartesian coordinates has the form (3.90) ½! r! v  ¼ xvy  yvx ¼ xðdy=dtÞ  yðdx=dtÞ As has been known, the angular momentum in a closed system is preserved, not only in magnitude but also in direction. To proof this proposition, we refer the reader to any manual on mechanics, in particular Ref. [3]. Consequently, if the invariable position in space preserves a moment vector perpendicular to the plane in which the particle moves, then the position of this plane is also unchanged. Thus, the orbit of the particle is within a plane. According to (3.89) and (3.90), the angular momentum in Cartesian coordinates is given by L ¼ m½xðdy=dtÞ  yðdx=dtÞ

(3.91)

From the expressions (3.85) and (3.87a and b) one can find xðdy=dtÞ  yðdx=dtÞ ¼ r 2 ðd4=dtÞ

(3.92)

Substituting (3.92) by (3.91) one obtains L ¼ mr 2 ðd4=dtÞ Thus, the laws of conservation of energy and angular momentum in polar coordinates have the form 2 2  m ðdr=dtÞ þ ðd4=dtÞ r 2 2 þ UðrÞ ¼ E mr 2 ðd4=dtÞ ¼ L where E and L are constants. Expressing d4/dt in terms of L from (3.94b) and substituting it into (3.94a), we can obtain 

2

E ¼ mðdr=dtÞ 2 þ L2 2mr 2 þ UðrÞ

(3.93)

(3.94a) (3.94b)

(3.95)

The first term on the right-hand side of (3.95) is the kinetic energy of the particle motion, and, consequently, the other two terms are the potential energy. Thus, it can be seen that the particle moves in a field with effective potential energy

 (3.96) Ueff ¼ L2 2mr 2 þ UðrÞ; where the first term corresponds to the centrifugal energy. Note that this term has a positive sign, which means that the manifestation of the centrifugal effect is equivalent to repulsion. In quantum mechanics, the angular momentum has the form L ¼ Z½‘ð‘ þ 1Þ1=2

(3.97)

It is important that the main relations of this section are also valid in quantum mechanics when expressing the angular momentum in the form (3.97).

3.2.5.6 Movement in the Field with the Centrifugal Potential Let us define the potential energy in the general form

Then

UðrÞ ¼ k=r n

(3.98)

 Ueff ¼ L2 2mr 2  k=r n

(3.99)

In the case of repulsion, the effective potential energy decreases monotonically from þN to zero when r varies from 0 to N. In the field of the repulsive center, a particle with energy E > 0 cannot approach the center by a distance less than r0 for which E ¼ Ueff (r0). In the case of an attractive potential, the second term in (3.99) is negative and the form of the effective potential energy depends essentially on the value of the parameter n in (3.98). For n < 2, the potential energy changes with distance more slowly than the centrifugal energy, which depends on the distance as r2. Therefore, for r / 0, centrifugal energy plays the dominant role and Ueff (r / 0) tends to þN.

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FIGURE 3.9 Effective potential energy of the particle in an attractive field with the exponent in the expression (3.99): (A) n < 2 and (B) n > 2.

The effective potential energy curve is shown in Fig. 3.9A. It is essential that the effective potential energy, even for an attractive potential, has a repulsive branch for small r. In the classical case, when E > 0, the particle goes to infinity along the hyperbola. When E ¼ 0, the particle also goes to infinity, but the motion occurs along a parabola. When E < 0, the particle is captured by the center, and in the particular case E ¼ (Ueff)min the particle moves along the circle. Reasoning about possible trajectories for cases with E < 0 is given only for generality of consideration. In the collision of particles, when incident particles arrive from infinity, such situation cannot be realized. For particles arriving from infinity, the limiting case is such when, at infinity, the particle has a velocity v0 ¼ 0 and falls on the attracting center, accelerating under the influence of its field. The particle, as mentioned above, moves along a parabolic trajectory. In all cases with v0 > 0, hyperbolic trajectories are realized. Cases with E < 0 are possible only for particles born in the central field, or arriving from infinity, but nearing the center they somehow lose some of the energy, e.g., in a collision with a third body or due to radiation. If, in expression (3.98), for potential energy of the attracting center U(r) ¼ k/rn, the exponent n > 2, then for small r the dominant role belongs to the negative term, which varies more rapidly with changing r, in accordance with the potential energy of the center. The graph of the effective potential energy now looks as shown in Fig. 3.9B. The allowed areas are the areas above the curve. The center is surrounded by a potential barrier. For generality, let us note that if the exponent in (3.99) is n ¼ 2, then in the case of the attractive potential, the form of the effective potential energy depends on the ratio of the numerators in (3.99), as shown in Fig. 3.10. If k > L2/2m, then the negative term value is greater and attraction dominates (Fig. 3.10A), and if k < L2/2m, then the positive term value is larger and repulsion dominates (Fig. 3.10B). Note that the centrifugal potential enters the expression for the effective potential energy only for L s 0. Equality to zero of the orbital angular momentum corresponds to a “frontal” collision. Now we are ready to obtain expressions for the collision cross sections for some model potentials.

FIGURE 3.10 The effective potential energy of the particle in the field of attractive center with the exponent in the expression (3.106): n ¼ 2: (A) k > L2/2m and (B) k < L2/2m.

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3.2.6 Collision of “Hard Spheres” In Fig. 3.11 the collision scheme of two solid spheres is shown. Because all the parameters of the collision depend on the distance between the centers of the balls, the collision analysis of two balls with radii r1 and r2 can be replaced by collision analysis of an incident particle with a resting body with radius R ¼ r1 þ r2. As in the idealized case of an absolutely hard sphere the angular momentum is not transmitted, and the scattering in the center of mass frame is simply a reflection of the incident particle from the tangent plane to the fixed sphere at the collision point. Two balls fly in opposite directions, not along the same straight line but preserving the impact distance. In the laboratory frame, the trajectory of scattered particle consists of two straight lines arranged symmetrically with respect to the line connecting the centers of the balls. The recoil particle flies in the direction of this line at an angle q2, which does not depend on the mass ratio of the particles. The scattering angle q1 depends on the mass ratio in accordance with Eq. (3.66). The basic relationships between the angles and the impact distance are obvious from the geometric constructions of Fig. 3.11: In the center of mass frame b ¼ R cosðc=2Þ

(3.100)

b ¼ R sin q2

(3.101)

In the laboratory frame For the connection between the impact distance and the scattering angle q1 in the laboratory frame, it is difficult to obtain an expression in the analytical form, but this does not hinder the calculation of the corresponding cross section, for which this link is actually searched. The differential cross section is related to the impact parameter by the relation (3.26) ds ¼ 2pbdb. Using the relationship between the impact distance and the scattering angle in the center of mass frame for the collision of “hard spheres” (3.100), one can find the collision cross section. By differentiating (3.100), we obtain dbðcÞ ¼ ðR=2Þsinðc=2Þdc

(3.102)

Here the minus sign means that in this case (as in many others), as the impact parameter b increases, the scattering angle c decreases. To calculate the cross sections, we use the modulus of the derivative rdb/dcr. Substituting (3.102) and (3.100) into (3.26), we find that    dsðcÞ ¼ pR2 2 sin cdc (3.103)

FIGURE 3.11 Collision of hard spheres (A) in the laboratory frame and (B) in the center of mass frame.

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In practice, when carrying out measurements, often the result is expressed not in the function of the plane scattering angle c, but in the function of the solid angle dUc ¼ 2psin c dc. After simple transformation one can obtain    dsðUc Þ ¼ R2 4 dUc . (3.104) We draw attention to the fact that in the center of mass frame the cross section, referred to the solid angle element, does not depend on the angle, i.e., the scattering in the center of mass frame is isotropic. The total cross section is obtained by integrating the differential cross section over the angles in the range from 0 to p Z p Z p  2  s ¼ dsðcÞ ¼ (3.105) pR 2 sin cdc ¼ pR2 : 0

0

Eq. (3.105) shows that the total cross section is the target area equal to the cross-sectional area of the ball. The target particle must fall into this target to scatter. For the transition to the laboratory frame, we can use the relation b ¼ Rsin q2 (3.101) as the relation between the impact distance b and the target particle emission angle q2. By differentiating (3.101) and substituting in (3.26), we obtain dsðq2 Þ ¼ ds2 ¼ pR2 sin 2q2 dq2

(3.106)

ds2 ¼ R2 cos q2 dU2 .

(3.107)

or

Obviously, the same result is obtained if we use the connection between the angles c and q2 (3.64), where q2 ¼ (p  c)/2, and formula (3.102). The derivation in the laboratory frame of the cross section, expressed in terms of q1dthe scattering angle of the incident particle, is more complicated. For this we use (3.66). Solving it with respect to cos c, we obtain

1=2 2 cos c ¼ ðm1 =m2 Þsin2 q1  ðcos q1 Þ 1  ðm1 =m2 Þ sin2 q1 . (3.108) In the case of q1 ¼ 0, cos c ¼ þ1 and c ¼ 0 if there is a “þ” sign in front of the root, and cos c ¼ 1 and c ¼ p if there is a “” sign in front of the root. For m1 < m2 (a light particle hits a heavy target), in formula (3.108), the “plus” sign has to be used in front of the root (so that c ¼ 0 for q1 ¼ 0), while c and q1 are uniquely related. By differentiating (3.108), we obtain (

) 1 þ ðm1 =m2 Þ2 cos 2q1 2 dðcos cÞ ¼ sin cdc ¼ dq1 sin q1 2ðm1 =m2 Þ cos q1 þ (3.109)

1=2 . 1  ðm1 =m2 Þ2 sin2 q1 By substituting (3.109) into (3.103), we obtain the formula for scattering cross section of incident particles into a solid angle dU1 for m1 < m2 (light particle hits a heavy target) (

) 2  1 þ ðm1 =m2 Þ cos 2q1 2 2 (3.110) ds1 ¼ dsðU1 Þ ¼ R 4 2ðm1 =m2 Þ cos q1 þ

1=2 dU1 2 1  ðm1 =m2 Þ sin2 q1 For m1 > m2 (a heavy particle hits a light target), it is necessary to take into account both possible connections of the angles c and q1. As one of the corresponding values of c grows with increase of q1 (with the “plus” sign in front of the root in (3.108)) and the other decreases (with the “minus” sign before the root in (3.108)), then we need to take use of the difference and not of the sum of the expressions d(cos c) with two signs before the root in (3.108). As a result, we obtain    1 þ ðm1 =m2 Þ2 cos 2q1 ds1 ¼ dsðU1 Þ ¼ R2 2

1=2 dU1 . 2 1  ðm1 =m2 Þ sin2 q1

(3.111)

ds1 ¼ R2 jcos q1 jdU1

(3.112)

For m1 ¼ m2

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Because the relationship between the scattering angle in the center of mass frame and the emission angle of the initially resting particle in the laboratory frame does not depend on the particle masses, it is described by the expression (3.64) q2 ¼ (p  c)/2, so we find ds2 ¼ dsðU2 Þ ¼ R2 jcos q2 jdU2

(3.113)

It can be seen from formulas (3.110e3.113) that the scattering of solid spheres in laboratory frame is not isotropic and that the scattering probability decreases with increasing angle. Let us recall that in the center of mass frame this scattering is isotropic (3.104). However, this could be foreseen in advance. Indeed, to be scattered by a certain angle q1, the particle must fall into a certain ring, whose area, equal to 2pb (Fig. 3.3), grows with increasing b, and, the greater b, the less q1. The probability of getting into a certain ring is proportional to the area of the ring. Consequently, larger values of b and, hence, smaller values of q1 are more probable. It is also useful to analyze the energy distribution of particles, i.e., to find the cross section as a function of energy. Here we must distinguish the energy of the incident particle and the energy lost in collision. The energy lost in collision by the incident particle is obviously equal to the energy acquired by the target particle. In accordance with (3.54), it is equal to  2 E2 ¼ E0 4m1 m2 cos2 q2 ðm1 þ m2 Þ . (3.114) This formula can be presented in the form E2 ¼ E0max cos2 q2 .

(3.115)

E2 ¼ E0max sin2 ðc=2Þ.

(3.116)

dE2 ðE0max =2Þsin cdc.

(3.117)

Using (3.64), we obtain

By differentiation, (3.116) becomes Now we express sin c from (3.117), then substitute it into (3.103), and find    dsðE2 Þ ¼ pR2 E0max dE2 .

(3.118)

Thus, the distribution of the scattered particles in terms of the energy lost by them and the distribution of the recoil particles over the energy obtained by them turn out to be homogeneous and not dependent on energy. This is a characteristic property of the collision of hard spheres.

3.2.7 Coulomb Collisions From the laws of conservation of energy and angular momentum, one can obtain an exact expression for the trajectory of particle motion in a field. Solving the relation (3.88) with respect to dr/dt, we obtain 

1=2  dt (3.119) dr ¼  ð2=mÞ E  UðrÞ  L2 2mr 2 In this expression, when a particle moves from infinity, it is necessary to use the minus sign, and after passing the vertex of the trajectory, i.e., the point of closest rapprochement, to use the plus sign. Now we express d4 from (3.94b)   (3.120) d4 ¼ L=mr 2 dt; we substitute here dt from (3.119) and obtain  1=2   d4 ¼ Ldr r 2 2m½E  UðrÞ  L2 r 2

(3.121)

By integrating (3.121), we find the value of the polar angle q, depending on the chosen limits of integration. If the integration is from r0 to N, then the limiting value of the angle 40 Z N  1=2   40 ¼ Ldr r 2 2m½E  UðrÞ  L2 r 2 (3.122) r0

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The limiting values of the polar angle 40 and the scattering angle c according to (3.64) are related by c þ 240 ¼ p: Consequently

Z c ¼ p2

N

(3.123)

 1=2   Ldr r 2 2m½E  UðrÞ  L2 r 2

(3.124)

r0

By integrating we obtain a relation between the scattering angle c and the impact distance b tgðc=2Þ ¼ k=mv20 b

(3.125)

For a Coulomb interaction of a particle with a charge ze and a particle with a charge Ze, the value of k equals  k ¼ zZe2 4pε0 (3.126) and then

From the formula (3.127) we find

 tgðc=2Þ ¼ zZe2 mv20 bð4pε0 Þ

(3.127)

 b ¼ Zze2 mv2 tgðc=2Þð4pε0 Þ

(3.128)

By differentiating the resulting expression (3.128) and substituting it into (3.26), we find  2 Zze2 cosðc=2Þdc dsðcÞ ¼ p 2 mv ð4pε0 Þ sin3 ðc=2Þ

(3.129)

or  dsðUc Þ ¼

Zze2 2mv2 ð4pε0 Þ

2

dUc . sin4 ðc=2Þ

(3.130)

The resulting expression is usually called the Rutherford formula. It was obtained by E. Rutherford in 1911 when analyzing the collisions of alpha particles with the atoms of matter. As it is known, it was this analysis that led to the nuclear model of the atom. However, strictly speaking, Rutherford obtained a formula for scattering in the laboratory frame. It coincides with formula (3.130) for m1  m2 and then m z m1. The differential cross section for the Coulomb interaction in any form sometimes is called Rutherford formula. To calculate the total cross section, we must integrate (3.129) with respect to c from 0 to p. It is easy to see that in this case an infinite cross section value is obtained and the integral diverges into the lower limit. This is a natural consequence of the assumption of an infinite length of potential. In general, the divergence of the total cross section is a common problem for unlimited potentials of the form U(r) w 1/rn. Collisions of “solid” balls are one of the few cases when the total cross section is finite because in this case the length of the interaction is limited by the size of the balls. In quantum mechanics, the total cross sections are finite for any potentials that decay faster than r2. However, for more slowly decreasing potentials, the problem still remains. In both quantum and classical theories, the difficulty associated with the divergence of the total cross section is eliminated by introducing a screened potential. The transition from the scattering formulas in the center of mass frame (3.129) and (3.130) to the formulas in the laboratory frame yields hard-to-analyze relationships. However, for certain limiting cases, simple and useful relations are obtained. The transition from the center of mass frame to the laboratory frame is related to the recalculation of the angles, both planar and solid, and with the transition from the reduced mass in the center of mass frame to the corresponding mass value in the laboratory frame. In the case m1  m2 (a very light particle strikes a very heavy target), c z q1, m z m1 and dscmf ¼ dslf. dU1 ¼ 2p sin q1 dq1  2 Zze2 dU1 . (3.131) dsðU1 Þ ¼ 2 m1 v ð4pε0 Þ 4 sin4 ðq1 =2Þ In this form, the formula is usually used to analyze the scattering of particles.

Probability and Energy Laws of Particle Collisions Chapter | 3

85

In the case m1 [ m2 (a very heavy particle strikes a very light target), we use the relations c ¼ p  2q2, and m z m2, dU2 ¼ 2p sin q2 dq2  2 Zze2 dU2 . (3.132) dsðU2 Þ ¼ 2 m2 v ð4pε0 Þ cos3 ðq2 Þ In this form, the formula is used to analyze the emission of delta electrons. In the case m1 ¼ m2 ¼ m c ¼ 2q1, q1 þ q2 ¼ p/2, m ¼ m/2, dUc ¼ 4 cos q1 dU1  2 Zze2 cos q1 dU1 dsðU1 Þ ¼ . mv2 ð4pε0 Þ sin4 q1

(3.133)

3.3 RELATIVISTIC RELATIONS In the three-dimensional Euclidean space, where the laws of Newton’s mechanics and Galileo’s relativity principle operate, the transformation of spatial and temporal coordinates for different inertial reference frames moving with respect to each other with speed v  c, where c is the speed of light, is carried out by the formulas called Galilean transformations. If the ordinates are arranged so that the motion occurs along the x-axis, then the Galileo transformation formulas have the form X 0 ¼ x  vt; y0 ¼ y; z0 ¼ z; t 0 ¼ t:

(3.134)

Here the coordinates with a “prime” refer to a system moving with velocity v relative to another inertial system. In an implicit form, we used these transformations in Section 3.1, when we recorded the velocities of the center of mass in the laboratory frame and the velocity of the particles in the center of mass frame, i.e., when moving from one system to another. In the relativistic case, i.e., at v w c, when the Einstein relativity principle operates, the situation changes significantly. The speed of light is the velocity limit, as it is the same in all inertial reference frames. Then the transformation of the spatial and temporal coordinates in transition from one inertial system to another, which moves relative to the first one with the speed v w c, is carried out by the formulas called the Lorentz transformations. Addressing the reader to the literature (see, for example, Ref. [4]) to find the derivation of the Lorentz transformation formulas, we offer here the formulas for the one-dimensional motion along the x-axis x0 ¼ 

x  vt 1=2 ; 1  b2

y0 ¼ y;

z0 ¼ z;

t  xv=c2 t0 ¼  1=2 1  b2

(3.135)

For nonrelativistic particles, i.e., particles whose velocity of motion is much less than the speed of light v  c, the relations between the velocity, momentum, and kinetic energy are determined by the known expressions p ¼ mv;

E ¼

mv2 ; 2

p2 ; 2m

E ¼

p ¼

pffiffiffiffiffiffiffiffiffi 2mE .

(3.136)

For relativistic particles these relations vary noticeably. We introduce some notation and definitions v=c ¼ b;  1=2 1  b2 ¼ g.

(3.137) (3.138)

The quantity g is called the Lorentz factor. Both these quantities determine the velocity of the particle and, speaking of velocity, we can henceforth mean any of the quantities v, b, or g. In relativistic mechanics, the analog of the energy of a nonrelativistic particle is the total energy, defined by expression W ¼ 

mc2

 2 1=2

1b

¼ mc2 g.

(3.139)

It is seen that for a free particle the total energy does not vanish at v ¼ 0 but is equal to W ¼ mc2

(3.140)

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PART | I Fundamentals

This quantity is called the rest energy. It follows from relation (3.139) that the Lorentz factor g is the total energy expressed in units of rest mass. On the other hand, the total energy is equal to the sum of the rest energy and kinetic energy W ¼ mc2 þ E

(3.141)

The particle momentum can be written in the form . 1=2 ¼ mvg p ¼ mv 1  b2

(3.142)

Now combining formulas (3.139), (3.141), and (3.142) and using the notations (3.137) and (3.138), we obtain a number of useful relativistic relations  1=2  (3.143) pc ¼ E 2mc2 þ E pc ¼ gbmc2  1=2 E ¼ p2 c 2 þ m 2 c 4  mc2 i h 1=2  1 ¼ mc2 ðg  1Þ E ¼ mc2 1  b2

(3.144)

E ¼ pcðg  1Þ=gb   1=2 .  b ¼ E E þ 2mc2 E þ mc2

(3.147)

b ¼ pc

.

p2 c2 þ m2 c4

1=2

(3.145) (3.146)

(3.148) (3.149)

b ¼ pc=W  1=2  b ¼ 1  m 2 c4 W 2    b ¼ 1  1 g2

(3.152)

 1=2 W ¼ p2 c2 þ m2 c4

(3.153)

(3.150) (3.151)

In the ultrarelativistic case, E [ mc , b w 1, g [ 1, and pc ¼ E ¼ W. The laws of conservation of energy and momentum in the relativistic case do not differ from laws for nonrelativistic particles, unless expressions for energy and momentum are presented in accordance with relativistic relations. As a rule, the law of conservation of energy records not the kinetic energy, but the total energy. In the case of elastic collisions this is not necessary because the rest mass of the particles remains unchanged in such collisions. In the case of inelastic collisions, it is possible to change the kinetic energy due to rest energy; therefore, in the energy conservation law, the total energy must necessarily be written. The principal feature of the collision of particles in the relativistic case is that the transition from one inertial system to another, e.g., from laboratory frame to center of mass frame, is accomplished by means of the Lorentz transformations. 2

REFERENCES [1] C. Lehmann, Interaction of Radiation with Solids and Elementary Defect Production, North-Holland Publ. Co., 1977. [2] A.M. Baldin, V.I. Goldanskii, I.L. Rosenthal, Kinematics of Nuclear Reactions, Translated from Russian by R.F. Peierls, Oxford University Press, New York, 1961, 223 p. [3] R. Fitzpatrick, Newtonian Dynamics, 2011. Updates 2011-03-31, http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/Newtonhtml.html. [4] L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, fourth ed., vol. 2, Butterworth-Heinemann, 1975.