A shadow of the repulsive Rutherford scattering in the fixed-target and the center-of-mass frame
AA shadow of the repulsive Rutherford scattering in the fixed-target and thecenter-of-mass frame
Petar ˇZugec ∗ and Ivan Topi´c Department of Physics, Faculty of Science, University of Zagreb, Zagreb, Croatia Archdiocesan Classical Gymnasium, Zagreb, Croatia ∗ Electronic address: [email protected] paper explores the shadow of the repulsive Rutherford scattering—the portion of space entirelyshielded from admitting any particle trajectory. The geometric properties of the projectile shadoware analyzed in detail in the fixed-target frame as well as in the center-of-mass frame, where boththe charged projectile and the charged target cast their own respective shadows. In both framesthe shadow is found to take an extremely simple, paraboloidal shape. In the fixed-target frame thetarget is precisely at the focus of this paraboloidal shape, while the focal points of the projectileand target shadows in the center-of-mass frame coincide. In the fixed-target frame the shadowtakes on a universal form, independent of the underlying physical parameters, when expressed inproperly scaled coordinates, thus revealing a natural length scale to the Rutherford scattering.
I. INTRODUCTION
Rutherford scattering—a scattering of electric chargesdue to the Coulomb interaction, whether it be attrac-tive or repulsive—is one of the most famous concepts inall of physics. The series of historical experiments byGeiger and Marsden [1–3], demonstrating some as yetunexpected properties in the scattering of α -particles bythin metal foils, has lead to a discovery of the atomicnucleus by Rutherford [4] and a subsequent birth of nu-clear physics. Today Rutherford scattering is a regularsubject of all textbooks on classical mechanics and theintroductory nuclear physics, and a basis of several ex-perimental techniques such as Rutherford Backscatter-ing Spectrometry (RBS) [5] and Elastic Recoil DetectionAnalysis (ERDA) [6].Throughout most of the educational sources, under-graduate or otherwise, the fact that the repulsive Ruther-ford scattering casts a shadow seems to be little known—as if it were entirely forgotten, neglected or ignored.At best, it is only tacitly recognized, whenever a plotsuch as the one from figure 1 is presented, showing sev-eral stacked trajectories for a charged projectile movingthrough a Coulomb field of a stationary target. Clearly,there is an isolated portion of space admitting no trajec-tories due to their deflection in a Coulomb field, whichcan be considered as a proverbial shadow of the repulsivescattering. The attractive scattering of opposing chargesshows no such feature, as the trajectories of a projectileof any initial energy may be continuously brought closerto a target, until they bend entirely around it, sweepingout the entire geometric space.The reality of this shadow is not only of a of great ed-ucational value, as a motivation for some beautiful cal-culations and insights, but also plays an important rolein the Low-Energy Ion Scattering Spectroscopy (LEIS)[7, 8]. It may already be intuited from figure 1 thatthe shape of the shadow in the fixed-target frame mightbe parabolic, that is paraboloidal when considered in athree-dimensional space. Indeed, the paraboloidal shapeis obtained even in a small-angle scattering (i.e. high pro- jectile energy) approximation [9], leading to the correctvalue of the paraboloid stiffness parameter (its leadingcoefficient). This result is often quoted, and its use is jus-tified by the fact that for the low-energy ions not muchwould be gained by the exact result, as the screening ofthe ‘naked’ nuclear potential plays a far greater role thanthe nature of approximation [10].However, rarely is the form of this shadow treated ac-curately, or analyzed in detail as a subject in its ownright, even in a fixed-target frame, let alone in any otherreference frame. One example is a mechanics textbookby Sommerfeld, where the topic appears as a supplemen-tal problem I.12, with a short guideline on how to obtainthe correct solution [11]. Further examples include suc-cinct works by Adolph et al. [12] and Warner and Hut-tar [13], all these efforts being decades apart. In the re-view paper by Burgd¨orfer the paraboloidal shadow form FIG. 1. Isoenergetic projectile trajectories in a repulsiveCoulomb field of the charged target (the dot), in the fixed-target frame. a r X i v : . [ phy s i c s . c l a ss - ph ] S e p is also quickly obtained, but relying on the propertiesof the Coulomb continuum wavefunctions [14]. Interest-ingly, Samengo and Barrachina [15] consider the incidentprojectile trajectories from a point source , finding in thatcase the hyperboloidal shadow. At the same time, theyrecover the more familiar paraboloidal shape in the limit R → ∞ , with R as the initial projectile-target distance.Most of these works make a point of this easily acces-sible topic being forgotten [12], generally unknown [13]and commonly omitted from the standard textbooks [15].Nothing much seems to have changed to this day.We aim to give the subject our full attention and adetailed treatment it deserves, with an intent of rekin-dling the general interest in it as a worthy educationaltopic. In this work we analyze the geometric shape ofthe Rutherford scattering shadow, as it appears in thefixed-target and the center-of-mass frame. A transitionto the laboratory frame (where the target is at rest onlyat the initial moment, i.e. when the projectile is just putinto motion) or any other comoving frame (moving witha constant velocity relative to the center-of-mass frame)is much more involved than may seem at the first glanceand will be the subject of a separate work.We will treat the scattering kinematics nonrelativisti-cally, which is a common enough approach. There is amyriad of sources offering some form of derivation of theclassical trajectories within the Coulomb field, of whichwe cite only a few classics [16–19]. The trajectories areusually determined within the context of the two-bodyKepler problem, i.e. assuming the gravitational poten-tial, while the chapters on Rutherford scattering typi-cally focus on the scattering cross section. Quite oftenthe problem is approached from the onset with an as-sumption of a large disparity between the masses, suchthat one body or particle (e.g. star or the target nu-cleus) is much heavier than the other (e.g. planet orthe charged projectile). The obvious advantage of thisapproach is that the fixed-target frame also corresponds(at least approximately) both to the center-of-mass andthe laboratory frame. As we are interested in each ofthese frames in its own right, we will follow the moregeneral approach, making no specific assumptions aboutthe masses involved.Since the fixed-target frame of a finite-mass target isaccelerated (as the target will recoil from the incomingprojectile), a savvy reader might pose a legitimate ques-tion: is the force that the target exerts upon the projec-tile in such frame purely electrostatic? Or is there also aradiative component to the target’s electromagnetic field,even in the frame where it stays at rest, due to its ac-celeration in an outside inertial frame? The question is,in fact, entirely nontrivial and has indeed been a sub-ject of a long-standing debate. The paradox has sincebeen resolved and we know now that the electric field ofa charge at rest is indeed purely electrostatic even in anaccelerated frame [20]. Therefore, we are entirely justi-fied in assuming the pure electrostatic force between thefinite-mass particles in a fixed-target frame, which willserve as a starting point for a transition into any otherreference frame. This paper is accompanied by the Supplementary note,expanding upon the main material presented herein. II. COULOMB TRAJECTORIES
Let us consider the Coulomb force F t → p exerted uponthe projectile by the charged target: F t → p = Z p Z t e π(cid:15) r p − r t | r p − r t | , (1)where Z p and Z t are the projectile and target charge,respectively, in units of the elementary charge e ; (cid:15) is thevacuum permittivity; r p and r t are the particle and targetposition-vectors. For each particle the Newton’s secondlaw, in combination with the third one ( F t → p = − F p → t ),states: m p ¨ r p = F t → p , (2) m t ¨ r t = − F t → p , (3)with m p and m t as the projectile and target mass, re-spectively. By introducing the center-of-mass position R : R ≡ m p r p + m t r t m p + m t (4)and summing (2) and (3): m p ¨ r p + m t ¨ r t = ( m p + m t ) ¨ R = , (5)one immediately obtains the equation of motion for thecenter of mass: ¨ R = , clearly showing that in the ab-sence of the additional external forces the system as thewhole can not accelerate, which is to say that the lin-ear momentum of the isolated system is conserved. Bysubtracting the acceleration terms from (2) and (3) anddefining the target-relative projectile position r : r ≡ r p − r t , (6)one obtains the second equation of motion:¨ r = (cid:18) m p + 1 m t (cid:19) F t → p = Z p Z t e π(cid:15) µ ˆ r r , (7)where the common definition of the reduced mass µ :1 µ ≡ m p + 1 m t (8)has been used, together with the vector norm r = | r | andthe corresponding unit direction ˆ r = r /r .In parameterizing the projectile trajectory we willmake use of the cylindrical coordinates ρ and z , along-side their spherical counterparts r and θ . Figure 2 clearlyillustrates their relation. In that, the direction of the z -axis corresponds to the projectile’s initial direction ofmotion (i.e. its initial velocity). Assuming that the pro-jectile has been put into motion as a free particle of ini-tial speed v and with the impact parameter (cid:37) —from r ρθ z FIG. 2. Geometric parameters used for describing the projec-tile trajectory (full line). The target-relative projectile posi-tion r is described both by the spherical coordinates r and θ ,and their cylindrical counterparts ρ and z . Due to the axialsymmetry the azimuthal angle ϕ bears no relevance to theproblem. the negative side of the z -axis, at the infinite distancefrom the target ( θ = π )—for the initial conditions wecan write: r ( θ = π ) = (cid:37) ˆ ρ + (cid:16) lim z →−∞ z (cid:17) ˆ z , (9)˙ r ( θ = π ) = v ˆ z . (10)Section A of the Supplementary note derives a solutionto the equation of motion (7) under these conditions.Introducing the following shorthand: χ ≡ Z p Z t e π(cid:15) µv , (11)the solution for the relative coordinate may be expressedas: r ( θ ; (cid:37) ) = (cid:37) (cid:112) χ + (cid:37) sin[ θ − arctan( χ/(cid:37) )] − χ . (12)It is well known that for the repulsive interaction, i.e.for χ > z -axis,at a distance (cid:37) from it. The other asymptote is de-fined by the famous scattering angle ϑ from a fixed-targetframe, which is easily determined from (12) as the anglefor which the expression diverges, i.e. the denominatorvanishes, leading to: ϑ = 2 arctan χ(cid:37) . (13)This also means that some particular angle θ may bereached only by those trajectories whose scattering angleis further on ( ϑ < θ ). Evidently, those are the trajectorieswhose impact parameter satisfies: (cid:37) > χ/ tan( θ/ χ , when expressed in the scaled, dimensionlesscoordinates ¯ r = r/χ and ¯ (cid:37) = (cid:37) /χ :¯ r = ¯ (cid:37) (cid:112) (cid:37) sin[ θ − arctan(1 / ¯ (cid:37) )] − , (14) which will have the same repercussions upon the laterresults. This allows us to intuit that there is a natu-ral length scale to the Rutherford scattering, which is anotion that will only be reinforced further on, and re-peatedly so. III. FIXED-TARGET FRAME
The target-relative position r , as defined by (6), im-mediately implies the fixed-target frame. In other words,for the absolute positions it holds by definition r (fix)t = and thus r (fix)p = r . In order to determine the geometricshape of the shadow we pose the following question: un-der a particular angle θ , which trajectory passes closestto the target? In other words, for a given θ , which im-pact parameter (cid:37) minimizes the distance r ( θ ; (cid:37) )? Theanswer is, of course, to be found by finding the zero ofthe derivative in respect to (cid:37) :d r ( θ ; (cid:37) )d (cid:37) (cid:12)(cid:12)(cid:12)(cid:12) ˜ (cid:37) = ˜ (cid:37) sin θ − χ (1 + cos θ )˜ (cid:37) r ( θ ; ˜ (cid:37) ) = 0 . (15)This is satisfied by a vanishing numerator, yielding thesought impact parameter:˜ (cid:37) ( θ ) = 2 χ tan θ . (16)Returning this value to (12), we find that under an an-gle θ the trajectory with an impact parameter ˜ (cid:37) comesclosest to the target, at the distance: r [ θ ; ˜ (cid:37) ( θ )] = 2 χ sin θ . (17)Since r [ θ ; ˜ (cid:37) ( θ )] determines the shadow boundary, it isthe solution to our problem: it represents the shadowequation in spherical coordinates. However, to makeshadow shape more evident, we express its cylindricalcoordinate ρ (see figure 2): ρ ( θ ) = r [ θ ; ˜ (cid:37) ( θ )] sin θ = 4 χ tan θ , (18)as well as its z -coordinate: z ( θ ) = r [ θ ; ˜ (cid:37) ( θ )] cos θ = 2 χ (cid:32) θ − (cid:33) . (19)Eliminating the term tan( θ/
2) from previous two equa-tions, the following connection is obtained: z ( ρ ) = ρ χ − χ, (20)which is the shadow equation in the cylindrical coordi-nates, and the main result of this paper. Evidently, inthe fixed-target frame—as suggested by an example fromfigure 1—all the projectile trajectories of a given energy − − z/χ ρ / χ χ χ χ χ χ χ = χ χ = 2 χ χ = 3 χ χ = 4 χ χ = 5 χ FIG. 3. Shadow examples in a fixed-target frame for severalvalues of χ , with χ as the arbitrary length scale. The chargedtarget is shown by a central dot, which is precisely at thefocus of each paraboloidal form. The shadow approaches thetarget and becomes stiffer as the initial energy of the projectileincreases ( χ decreases). form a paraboloidal shadow. As portended by (14), theshadow features a universal shape in a scaled coordinates¯ z = z/χ and ¯ ρ = ρ/χ , such that: ¯ z = ¯ ρ / −
2, thus con-firming the notion that there is a natural length scale tothe Rutherford scattering. In addition, it is easily deter-mined from the paraboloid’s leading coefficient that thefocal distance f between the shadow focus and its vertexequals f = 2 χ , exactly corresponding to its free param-eter. Therefore, in the fixed-target frame the target isprecisely at the shadow focus!Figure 3 shows shadow shapes for several arbitrary val-ues χ , i.e. for several values of the initial projectile en-ergy. From (11) it is clear that the increase in the initialenergy—i.e. in the initial relative speed v —means adecrease in χ . The shadows are perfectly in accord withexpectations: not only do the projectiles of higher energy(lower χ ) manage to come closer to the central target (asgoverned by the free parameter − χ ), but they are alsoless easily deflected than the projectiles of lower energy,meaning a stiffer paraboloid (as governed by the leadingcoefficient 1 / χ ).Section B of the Supplementary note offers some addi-tional observations in regard to (13), (16) and (18). IV. CENTER-OF-MASS FRAME
In order to make a transition from a fixed-target frameinto any other frame, we invert the definitions of R and r from (4) and (6), thus obtaining: r p = R + m t m p + m t r , (21) r t = R − m p m p + m t r . (22) By definition, the center-of-mass position in the center-of-mass frame is the origin of the coordinate frame: R (cm) = . Thus, introducing the shorthands: η p , t ≡ m p , t m p + m t (23)we immediately obtain both the particle and target tra-jectories: r (cm)p = η t r , (24) r (cm)t = − η p r . (25)As the projectile’s position-vector in the center-of-massframe is only scaled by the factor η t relative to the posi-tion in the fixed-target frame, the definition of the angle θ stays the same. The only effect upon the projectiletrajectory is a decreased radial distance to the center ofmass: r (cm)p = η t r , leading to the minimization condition:d r (cm)p ( θ ; (cid:37) )d (cid:37) (cid:12)(cid:12)(cid:12)(cid:12) ˜ (cid:37) = η t d r ( θ ; (cid:37) )d (cid:37) (cid:12)(cid:12)(cid:12)(cid:12) ˜ (cid:37) = 0 . (26)As the minimization procedure is unaffected in regardto (15), the same minimizing value ˜ (cid:37) ( θ ) from (16) isobtained. It is only that the minimum projectile distancefrom an origin of the center-of-mass frame is scaled bya factor η t when compared to that from (17), with thesame factor propagating into the cylindrical coordinatesfrom (18) and (19), so that: ρ p ( θ ) = η t χ tan θ , (27) z p ( θ ) = 2 η t χ (cid:16) θ − (cid:17) . (28)For brevity and clarity we have dropped the explicitframe designation (cm). Eliminating again the termtan( θ/
2) from previous two equations, we arrive at theshadow equation in the center-of-mass frame: z p ( ρ p ) = ρ η t χ − η t χ. (29)Since η t <
1, the paraboloid vertex is closer to the ori-gin of the coordinate frame (as determined by the freeparameter − η t χ ), while the paraboloid shape is stifferthan in the fixed-target frame (as determined by the lead-ing coefficient 1 / η t χ ). Both effects are due to the factthat the origin is no longer the target itself, but ratherthe center-of-mass. Being somewhere in between the twoparticles, both the origin and the z -axis of the coordi-nate frame are at each point along the particle trajectorybrought closer to the projectile, when compared to thefixed-target frame, thus constricting the shadow profile.In the center-of-mass frame the target is in motion, inan entirely symmetric manner as the projectile, so it alsocasts its own shadow. Its exact shape is easily deducedfrom the projectile shadow, as at any moment we mayinterchange the roles of the target and the projectile by asimple change in indices: p ↔ t. Additionally taking intoaccount that the target shadow points at the oppositedirection from the projectile shadow (along the negativedirection of z -axis), we may immediately write: z t ( ρ t ) = − ρ η p χ + 2 η p χ. (30)Examining the focal distances f p , t of the projectile andtarget shadows from (29) and (30), we invariably find: f p , t = 2 η t , p χ , meaning that the focal points of both shad-ows are at the origin of the selected coordinate frame.Therefore, in the center-of-mass frame the two foci co-incide, i.e. the same focus is shared between the twoshadows!If we were to examine the effect of varying masses uponthe shadow form, it would not do to naively keep χ con-stant, while varying only the ratios η t , p from products η t , p χ appearing in (29) and (30). This is because theterm χ itself, as defined by (11), inherits the mass de-pendence via a reduced mass, so that the products η t , p χ : η t , p χ ∝ η t , p µ = m p + m t m p m t m t , p m p + m t = 1 m p , t (31)are dependent only on the mass of a single particle.Therefore, in the center-of-mass frame the projectileand target shadows are determined solely by their ownmasses, being entirely independent of the other particle’smass. This needs to be held in mind, as it is in strik-ing opposition with what the expression η t , p χ deceptivelysuggests: that the shadow form should not only be sen-sitive to both masses, but that it should also be moredirectly determined by the mass of the ‘wrong’ particle.In the sense of (31), figure 3 may also be interpreted asa comparison of projectile shadows in the center-of-massframe for varying projectile masses, if the labels χ i arereplaced by the projectile mass dependence m i = m /i ( m being some arbitrary reference value).Finally, it is again interesting to take note of theshadow form in the appropriately scaled coordinates. Infact, the projectile shadow from (29) takes on exactlythe same universal form (¯ z = ¯ ρ / −
2) of (20) when ex-pressed in scaled coordinates z p /η t χ and ρ p /η t χ . How-ever, in order to reach the same form for the targetshadow, its coordinates should be scaled by a differentfactor: z t /η p χ and ρ t /η p χ . Therefore, scaling all co-ordinates by a unique factor χ (or any constant, butparameter-invariant multiple of it) remains the most sen-sible choice. The price is that in the center-of-massframe there is no parameter-independent universal formfor both the projectile and the target simultaneously.Rather, with ¯ z p , t = z p , t /χ and ¯ ρ p , t = ρ p , t /χ we haveto contend with two separate forms: ¯ z p = ¯ ρ / η t − η t and ¯ z t = − ¯ ρ / η p + 2 η p , where the ‘most generalized’shadow shapes still depend on the relation between thetwo masses, but only on them. However, the advantageof this approach is that the length scale χ is revealednot only as the most natural between the two particles(i.e. for both of them simultaneously), but also betweenmultiple frames. V. QUANTUM-MECHANICAL CASE
We present a short overview of the quantum-mechanical scattering and the appearance of the scatter-ing shadow within such framework. A starting point is,of course, a Schr¨odinger’s equation for the joint particle-target system under a repulsive Coulomb interaction. Af-ter a typical separation of variables such that the motionof the center-of-mass is decoupled from the relative mo-tion, an equation for the relative motion reads: (cid:18) − (cid:126) µ ∇ + Z p Z t e π(cid:15) r (cid:19) ψ k ( r ) = E k ψ k ( r ) , (32)as a quantum-mechanical counterpart to the classicalequation of motion (7). Just like the Newton’s equa-tion, the Schr¨odinger’s equation for the relative motionfeatures a reduced mass µ . Since the relative vector r is still the same as in (6)—its origin being at the tar-get position—equation (32) describes a projectile in thefixed-target frame. In that, we have already parameter-ized both the wavefunction ψ k ( r ) and the energy E k ofthe relative motion by the wave vector k of the initial ,asymptotically free state of the system, described by theplane wave: lim k · r →−∞ ψ k ( r ) ∝ e i k · r (33)and serving as the boundary condition for solving (32). Inorder to establish the connection with the earlier classicaltreatment—in particular with the initial relative speed v from (10), which is otherwise, just like the trajectory, anill-defined concept in quantum mechanics—we parame-terize the initial wave vector as: k = µv (cid:126) ˆ z . (34)Introducing (33) and (34) into (32) in the limit r → ∞ ,a well known parameterization of energy remains: E k = (cid:126) k / µ = µv /
2. With this, (32) may be rewrit-ten using the definition of χ from (11): (cid:18) ∇ + k − χk r (cid:19) ψ k ( r ) = 0 . (35)The solution to such Schr¨odinger’s equation, satisfyingthe boundary condition from (33), is well known [21]: ψ k ( r ) = e − πχk/ Γ(1 + i χk )e i k · r M [ − i χk, , i( kr − k · r )] , (36)with Γ as the conventionally defined gamma-functionand M as the Kummer’s confluent hypergeomet-ric function, otherwise denoted as F . Thewavefunctions from (36) are normalized such that: (cid:82) ψ ∗ k ( r ) ψ k (cid:48) ( r )d V = (2 π ) δ ( k − k (cid:48) ).As opposed to classical mechanics, where all trajec-tories are strictly excluded from the shadow zone, inquantum mechanics we would always expect the wave-function tunneling into this classically forbidden regionof space. A question naturally arises if the wavefunctionexhibits any recognizable features at all, that would al-low us to identify the appearance of the classical shadow.In general case, its precise position could hardly be pin-pointed from the Coulomb continuum wavefunction, asthe quantum shadow is diffuse. However, we can makethe some observations in the opposite direction: knowingthe classical shadow, we can analyze the wavefunction be-havior in its vicinity. Burgd¨orfer notices: ‘ A (smoothed)caustic appears also in the corresponding quantum scat-tering wavefunction as an (anti) nodal surface. The lo-cus of the caustic is, in fact, most conveniently derivedfrom the nodal structure of Coulomb continuum wave-functions. Nodal surfaces are given by the argument ofthe hypergeometric function (...) ’ (quotation from [14]).In this rather ingenious insight, we only caution againstthe use of the term ‘(anti) nodal’, as it might suggestthat the shadow appears at some extremum related tothe wavefunction—presumably the extermum of modulus | ψ k ( r ) | —which we will soon disprove. The more appro-priate term would be ‘level surfaces’ (‘equipotentials’),which are indeed defined by the constancy of the argu-ment of the confluent hypergeometric function from (36): kr − k · r = C. (37)For k defined as in (34), (37) reduces to k ( (cid:112) z + ρ − z ) = C . Solving for z yields: z ( ρ ) = k C ρ − C k = ρ C/ k ) − C/ k ) (38)for the shape of level surfaces of | ψ k ( r ) | (but not of ψ k ( r )itself, due to the extra e i k · r factor). This has the sameform as (20), allowing to determine the shadow-relatedvalue of constant C as: C shadow = 4 χk (39)and, indeed, to recognize the classical scattering shadowas a particular level surface in the quantum-mechanicalprobability density of the incoming projectile.Figure 4 shows an example of the modulus | ψ k ( r ) | ofthe wavefunction from (36) for χk = 1, in a plane con-taining the z -axis, where the target rests at r = . Onecan readily appreciate by eye the fact that the level sur-faces are parabolic, as shown by (38). The thick black lineindicates the level surface corresponding to the shadowcaustic—i.e. where, along the wavefunction, the classicalshadow appears—clearly proving that it is not relatedto any extremal (antinodal) surface. The portion of thewavefunction bounded by this caustic (below the thickblack line) shows a clear case of the quantum-mechanicaltunneling into the classically forbidden zone.The fact that the classical scattering shadow may in-deed be identified within the quantum-mechanical de-scription indicates that we might again perform theappropriate coordinate scaling—such that ¯ z = z/χ and¯ r = r/χ —and express the wavefunction as: ψ k ( r ) = e − πχk/ Γ(1 + i χk )e i χk ¯ z M [ − i χk, , i χk (¯ r − ¯ z )] , (40) ρ / χ z / χ − − | ψ k ( r ) | . . . . . FIG. 4. Modulus of the projectile’s Coulomb contin-uum wavefunction for a repulsive Rutherford scattering inthe fixed-target frame, in a plane containing the z -axis,with the target at r = . The relevant wavefunction fea-tures are governed by the confluent hypergeometric function M [ − i χk, , i χk ( (cid:112) ¯ ρ + ¯ z − ¯ z )], with ¯ z = z/χ and ¯ ρ = ρ/χ ,and here selected χk = 1. The thick black line shows the clas-sical shadow caustic, beyond which the wavefunction clearlyexhibits a quantum-mechanical tunneling. where we used, for simplicity, the convention k = k ˆ z from (34). This reveals that, while the shape of the scat-tering shadow still remains scale-invariant, the detailsof the wavefunction still depend on k , but in such waythat—alongside the length scale χ —there appears an-other, dimensionless scale χk , otherwise known as Som-merfeld parameter. But how can that be, consideringthat in the classical mechanics all the spatial aspectsof the Coulomb trajectories from (14) are scaled onlyby χ ? How can another scale be admitted in quantum-mechanics, since—by the correspondence principle—atsome point both the classical and quantum descriptionmust coincide? The answer lies in the temporal aspectsof the scattering, of which the purely geometrical expres-sion (14) has been devoided. For the same spatial scaling χ , the projectile speed—entering k through (34)—maystill be varied. Thus, the time the projectile spends ina given portion of space still depends on its speed. Inconsequence, so do the details of the wavefunction, gov-erning the probability density | ψ k ( r ) | of finding the pro-jectile at a given point. This probabilistic interpretation,in combination with correspondence principle applied tothe projectile trajectories displayed in figure 1, also al-lows us to understand why the highest antinode in | ψ k ( r ) | appears just before the shadow, prior to tunneling. It isfor two reasons: for z/χ (cid:46) − z/χ (cid:38) − − − − − − z/χ . . . . . . . . | ψ k ( z , ρ = ) | χk = 0 . χk = 1 . χk = 2 . FIG. 5. Modulus of the Coulomb continuum wavefunction forthe repulsive Rutherford scattering along the axis ρ = 0, fordifferent values of the Sommerfeld parameter χk . The targetis at z = 0. The vertex of the classical shadow caustic isindicated by the vertical line at z/χ = − Wavefunction dependence upon χk is further exempli-fied by figure 5, showing the modulus | ψ k ( r ) | along thecentral axis ρ = 0, passing through a target at z = 0,for different values of χk . The position of the classi-cal shadow caustic—now corresponding to its vertex—isshown by the vertical line. There is no single point atwhich all the wavefunctions intersect, as might be falselyinferred from this specific display.Figure 5 allows us to make some additional interestingobservations. Since at ρ = 0 it holds: r = | z | , for z ≥ M ( − i χk, ,
0) = 1, so | ψ k ( r ) | is indeed constant there,as suggested by the figure. At the first glance it mightbe confusing why this value is not 0 at z = 0, where thetarget lies. In other words, how can the Schr¨odinger’sequation from (32) or (35) be satisfied by a nonvanishingwavefunction at the point where the repulsive potentialdiverges? As figure 5 shows, the wavefunction at z = 0,while continuous, is not smooth—its gradient ∇ ψ k ( r ) isdiscontinuous, so that its Laplacian ∇ ψ k ( r ) diverges,canceling the divergence from the potential energy in theSchr¨odinger’s equation. Finally, figure 5 also indicatesthat by increasing χk the wavefunction behavior aroundthe shadow caustic becomes sharper and sharper. Thissuggests that if one were to investigate the limit χk → ∞ of a very strong repulsion and/or a very slow projectile—lim χk →∞ | ψ k ( r ) | ∝ lim χk →∞ | M ( − i χk, , i χkξ ) | with, inour case, ξ = ( r − z ) /χ —one would expect a sharp dropat ξ = 4, in accordance with (39). This specific result could then be extended to a case of finite χk and takenas an agreed-upon value determining the shadow causticeven in the case of a diffuse shadow. This is how the scat-tering shadow can be determined self-consistently fromthe wavefunction itself, without reference to the classicalmechanics. VI. CONCLUSION
We have explored the geometry of the repulsiveRutherford scattering, finding the exact shape of thescattering shadow in the fixed-target and the center-of-mass frame. In both frames the projectile shadow hasa simple paraboloidal shape. The difference betweenframes is, of course, reflected in different values of theparaboloids’ coefficients, i.e. in their stiffness and the dis-tance of their vertices from the origin of the coordinateframe. In that, the projectile shadow in the center-of-mass frame is stiffer and closer to the origin than its coun-terpart from the fixed-target frame. Since the motion ofthe target in the center-of-mass frame is—in mathemat-ical form—symmetrical to the motion of the projectile,the target also casts a paraboloidal shadow in the sameframe, such that the two shadows intersect. It was foundthat the target is precisely at the shadow focus in thefixed-target frame, while in the center-of-mass frame thefocal points of the projectile and target shadow coincidewith the center of mass itself. A somewhat surprisingfinding is that the shadow parameters in the center-of-mass frame depend only on the mass of the particle cast-ing the shadow, rather than the ratio of masses as mightat first be expected. The Rutherford scattering is re-vealed to feature a natural length scale χ , determinedby the physical parameters of the system. A quantum-mechanical treatment of the repulsive Rutherford scat-tering was addressed and the scattering shadow was alsoobserved appearing in a Coulomb continuum wavefunc-tion. Unlike the sharply defined classical shadow, quan-tum mechanics yields a diffuse shadow caustic, due tothe wavefunction tunneling into a classically forbiddenzone. Alongside a length scale χ , in quantum mechanicsanother relevant scale appears: a dimensionless Sommer-feld parameter χk . A sharp shadow caustic is recoveredin the limit χk → ∞ . The transition of the scatteringshadow to the laboratory frame—wherein the target isat rest only at the initial moment, subsequently beingrecoiled by the approaching projectile—is much more in-volved and will be the subject of the future work. ACKNOWLEDGMENTS
We are grateful to Ivica Smoli´c and Kreˇsimir Dekani´cfor the useful discussions and for the help in trackingdown the relevant literature. [1] Hans Geiger, ‘On the Scattering of α -Particles by Mat-ter,’ Proc. R. Soc. London, Ser. A , 174–177 (1908).[2] Hans Geiger, Ernest Marsden, ‘On a Diffuse Reflection ofthe α -Particles,’ Proc. R. Soc. London, Ser. A , 495–500 (1909).[3] Hans Geiger, ‘The Scattering of the α -Particles by Mat-ter,’ Proc. R. Soc. London, Ser. A , 492–504 (1910).[4] Ernest Rutherford, ‘The Scattering of α and β Particlesby Matter and the Structure of the Atom,’ Philos. Mag. , 669–688 (1911).[5] Wei-Kan Chu, James W. Mayer, Marc-A. Nicolet, Backscattering Spectrometry , 1st edition (AcademicPress, New York, 1978).[6] W. M. Arnold Bik, F. H. P. M. Habraken, ‘Elastic recoildetection,’ Rep. Prog. Phys. , 859–902 (1993).[7] Masakazu Aono, Ryutaro Souda, ‘Quantitative SurfaceAtomic Structure Analysis by Low-Energy Ion ScatteringSpectroscopy (ISS),’ Jpn. J. Appl. Phys. , 1249–1262(1985).[8] H. H. Brongersma, M. Draxler, M. de Ridder, P. Bauer,‘Surface composition analysis by low-energy ion scatter-ing,’ Surf. Sci. Rep. , 63–109 (2007).[9] Jens Lindhard, ‘Influence of crystal lattice on motion ofenergetic charged particles,’ Mat. Fys. Medd. Dan. Vid.Selsk. (14), 1–64 (1965).[10] K. Oura, V. G. Lifshits, A. A. Saranin, A. V. Zotov, M.Katayama, Surface Science: An Introduction , 1st edition(Springer-Verlag, Berlin Heidelberg, 2003) p. 114.[11] Arnold Sommerfeld,
Mechanics (Lectures on TheoreticalPhysics, Vol. 1) , 1st edition (Academic Press, New York,1952) pp. 242, 258–259.[12] John W. Adolph, A. Leon Garcia, William G. Harter, G.C. McLaughlin, Richard R. Shiffman, Victor G. Surkus,‘Some Geometrical Aspects of Classical Coulomb Scat-tering,’ Am. J. Phy. , 1852–1857 (1972).[13] R. E. Warner, L. A. Huttar, ‘The parabolic shadow of aCoulomb scatterer,’ Am. J. Phy. , 755–756 (1991).[14] J. Burgd¨orfer, ‘Atomic Collisions with Surfaces,’ in Review of Fundamental Processes and Applications ofAtoms and Ions , edited by C. D. Lin (World Scientific,Singapore, 1993), p. 544.[15] I. Samengo, R.O. Barrachina, ‘Rainbow and glory scat-tering in Coulomb trajectories starting from a point inspace,’ Eur. J. Phys. , 300–308 (1994).[16] Charles Kittel, Walter D. Knight, Malvin A. Ruderman, Mechanics (Berkeley Physics Course. Vol. 1) , 2nd edition(McGraw-Hill, New York, 1973), Chapter 9.[17] Herbert Goldstein, Charles P. Poole, John L. Safko,
Clas-sical Mechanics , 3rd edition (Addison-Wesley, San Fran-cisco, 2001), Chapter 3.[18] Lev D. Landau, Evgeny M. Lifshitz,
Mechanics (Courseof Theoretical Physics, Vol. 1) , 3rd edition (Butterworth-Heinemann, Oxford, 1976), Chapter 3.[19] Murray R. Spiegel,
Theory and Problems of Theoret-ical Mechanics (Schaum’s Outline Series) , 1st edition(McGraw-Hill, New York, 1967), Chapter 5.[20] Camila de Almeida, Alberto Saa, ‘The radiation of a uni-formly accelerated charge is beyond the horizon: A sim-ple derivation,’ Am. J. Phys. , 154–158 (2006).[21] Lev D. Landau, Evgeny M. Lifshitz, Quantum mechanics,Non-relativistic theory (Course of Theoretical Physics,Vol. 3) , 3rd edition (Pergamon Press, Oxford, 1977), p.569. upplementary note
A shadow of the repulsive Rutherford scattering in the fixed-target and thecenter-of-mass frame
Petar ˇZugec ∗ and Ivan Topi´c Department of Physics, Faculty of Science, University of Zagreb, Zagreb, Croatia Archdiocesan Classical Gymnasium, Zagreb, Croatia ∗ Electronic address: [email protected] note presents the supplementary material to the main paper. The references to figures andequations not starting with the alphabetical character—such as (1)—refer to those from the mainpaper, while those starting with the appropriate letter—e.g. (A1)—refer to those from this note.
A. COULOMB TRAJECTORIES DERIVATION
We start with the equation of motion:¨ r = Z p Z t e π(cid:15) µ ˆ r r , (A1)introduced in (7). In the central-force field the total an-gular momentum L of the system is conserved—as thetorque T vanishes due to the collinearity of the positionand force vectors ( T = d L / d t = r × F t → p = )—whichmeans that the motion of the system is constrained toa single plane. By selecting the plane with the constantazimuthal coordinate ( ϕ = const . ), the acceleration term¨ r expressed in spherical coordinates takes the form:¨ r = (¨ r − r ˙ θ )ˆ r + 1 r d( r ˙ θ )d t ˆ θ , (A2)where the spherical (spatial) coordinates r and θ nowact as the polar (planar) coordinates within the planedefined by ϕ = const . (One can obtain a sense of the rel-evant geometric parameters from figure 2.) By compar-ing (A1) and (A2), two equations of motion are readilyobtained. The first follows from the vanishing derivativeterm related to the unit ˆ θ -direction, meaning that the r ˙ θ term must be constant. As we will assume that theprojectile is put into motion from infinity on the nega-tive side of the z -axis (i.e. from θ = π ; see figure 2),the angular change rate ˙ θ will be negative. We choose toparameterize the associated constant as r ˙ θ = − (cid:96) , with (cid:96) being positive. Therefore:˙ θ = − (cid:96)r , (A3)where the value of (cid:96) will be determined later, from theinitial conditions.The second equation of motion follows from equatingthe terms by the unit ˆ r -direction in (A1) and (A2):¨ r − r ˙ θ = Z p Z t e π(cid:15) µ r . (A4) This equation is commonly solved by introducingthe substitution u = 1 /r . Together with (A3), we use itto first translate the time derivative into the associatedangular derivative:dd t = d θ d t dd θ = − (cid:96)r dd θ = − (cid:96)u dd θ . (A5)From here it follows:¨ r = dd t (cid:18) dd t u (cid:19) = − (cid:96)u dd θ (cid:18) − (cid:96)u dd θ u (cid:19) = − (cid:96) u d u d θ . (A6)Plugging (A3) and (A6) back into (A4) leaves us with:d u d θ + u = − κ(cid:96) , (A7)where we have temporarily introduced the constant κ ≡ Z p Z t e / π(cid:15) µ defined by the intrinsic system pa-rameters (charges and masses), without depending on theinitial conditions. Equation (A7) is a well known Binetequation. Having a familiar form of the shifted harmonicoscillator equation, its solution is easily found as: u ( θ ) = U cos( θ − Θ) − κ(cid:96) , (A8)with the constants U and Θ, together with (cid:96) , to be de-termined from the initial conditions.In parameterizing the projectile trajectory we willmake use of the cylindrical coordinates ρ and z , along-side their spherical counterparts r and θ used up to thispoint. Figure 2 clearly illustrates their relation. In that,the direction of the z -axis corresponds to the projectile’sinitial direction of motion (i.e. its initial velocity). It al-ways holds: r = r ˆ r = ρ ˆ ρ + z ˆ z , regardless of the specificfunctional dependency of the coordinates and unit direc-tions, whether it be angular or temporal. Assuming thatthe projectile has been put into motion as a free par-ticle of initial speed v and with the impact parameter (cid:37) , from the negative side of the z -axis, at the infinitedistance from the target ( θ = π ), we can write: r ( θ = π ) = (cid:37) ˆ ρ + (cid:16) lim z →−∞ z (cid:17) ˆ z , (A9)˙ r ( θ = π ) = v ˆ z . (A10)2From r = r ˆ r and ˙ r = ˙ r ˆ r + r ˙ θ ˆ θ it is easy to show that theconstant (cid:96) , as we have defined it, equals to : (cid:96) = | r ˙ θ | = | r × ˙ r | = (cid:37) v . (A11)Applying the initial position condition r ( θ = π ) = ∞ —that is u ( θ = π ) = 0—to (A8), it follows:U cos( π − Θ) − κ(cid:96) = 0 . (A12)Applying (A5), it is easily shown that:d r ( θ ) / d t = (cid:96) d u ( θ ) / d θ = − U (cid:96) sin( θ − Θ), so the ini-tial speed condition ˙ r ( θ = π ) = − v (negative sign dueto the initial reduction of the radial distance, as theprojectile approaches the target) translates into:U (cid:96) sin( π − Θ) = v (A13)Using (A11), (A12) and (A13) are to be solved for U andΘ, yielding: U = 1 (cid:37) (cid:115) (cid:18) κ(cid:37) v (cid:19) , (A14)Θ = π (cid:18) κ(cid:37) v (cid:19) . (A15)As we will have to analyze the solution dependence onthe impact parameter (cid:37) , we define the following termabsorbing all the parameters save (cid:37) itself : χ ≡ κv = Z p Z t e π(cid:15) µv . (A16)Plugging (A11), (A14) and (A15) into (A8) we may writethe final solution (recall that r = 1 /u ) as: r ( θ ; (cid:37) ) = (cid:37) (cid:112) χ + (cid:37) sin[ θ − arctan( χ/(cid:37) )] − χ . (A17)Alternative expressions for r ( θ ; (cid:37) ) include: r ( θ ; (cid:37) ) = (cid:37) (cid:37) sin θ − χ (1 + cos θ )= (cid:37) θ ( (cid:37) tan θ − χ ) , (A18)having been obtained by a simple use of the trigonometricidentities. It may be shown that the constant (cid:96) is related to the total angu-lar momentum L (cm) in the center-of-mass frame: (cid:96) = | L (cm) | /µ .However, this parametrization is of limited use to this work. In a fixed-target frame the initial kinetic energy E = m p v / v , allowing the parameter χ from (A16) to be expressed as: χ (fix) = Z p Z t e m p π(cid:15) µE . However, in any other frame (moving relative to the target) theparametrization by energy becomes cumbersome, as it trans-forms between the frames, while the initial projectile energy doesnot correspond any more to the total energy of the system. Onthe other hand, the initial relative speed v remains the same inall frames, providing a frame-independent parametrization of χ . B. SOME GEOMETRIC OBSERVATIONS
In regard to (13), as well as in (16) and (18) severalvery similar quantities have appeared: χ tan θ and 2 χ tan θ and 4 χ tan θ . Taking the middle one—corresponding to the term ˜ (cid:37) from (16)—as a reference, we are dealing the equivalentsequence: ˜ (cid:37) /
2, ˜ (cid:37) , 2˜ (cid:37) . Figure B1 shows the geometricmeaning of these values. The reference ˜ (cid:37) is the impactparameter minimizing the projectile-target distance un-der a given angle θ . At this point the projectile movingalong a distance-minimizing trajectory has doubled itsradial distance from the z -axis: from the starting ˜ (cid:37) to2˜ (cid:37) . Finally, only the trajectories distant enough fromthe target—those with impact parameter greater thanhalf of the value that minimizes the projectile-target dis-tance under a given angle ( (cid:37) > ˜ (cid:37) / ˜ % ˜ % / % zθ FIG. B1. Geometric relation between several notable parame-ters. The trajectory that minimizes the projectile-target dis-tance under a given angle θ (full thick line) starts from animpact parameter ˜ (cid:37) which is double the value of the lowestone (˜ (cid:37) /
2; thick dashed trajectory) required to even reachthe same angle. At the distance-minimization point the cor-responding trajectory has doubled its radial distance from the z -axis (2˜ (cid:37) ). Thin dashed trajectories show several examplesthat either do not minimize the projectile-target distance un-der the selected angle or do not even reach it. C. THE CLOSEST APPROACH
The first, intuitive thought that may come to mind inattempting to obtain the shadow shape is that it might bedetermined by the trajectories’ points of the closest ap-proach, that are—unlike the shadow itself—often quotedin the literature. However, after a brief contemplationone is quickly disabused of that notion, as one realizesthat what is commonly quoted as the ‘point of the clos-est approach’ refers to the closest approach point froma given trajectory . On the other hand, the shadow itselfis determined by the closest approach trajectories fromall possible trajectories . It is instructive to examine this3difference in detail and to determine the actual geometricplace of the points of the closest approach. We carry outthis analysis only in the fixed-target frame.For a given trajectory, i.e. a given impact parameter (cid:37) , the closest approach is determined by minimizing thetarget-projectile distance from (A17) in respect to θ . Itis easily done just by observing that the expression isminimal when the denominator is maximal. This is ful-filled when the sine term itself is maximized, i.e. whenits argument equals π/
2, immediately yielding an angle:˜ θ ( (cid:37) ) = π χ(cid:37) , (C1)under which the trajectory comes closest to the target ,at a distance: r [˜ θ ( (cid:37) ); (cid:37) ] = (cid:37) (cid:112) χ + (cid:37) − χ . (C2)If we invert (C1) in order to find the impact param-eter corresponding to a particular minimizing angle ˜ θ : (cid:37) (˜ θ ) = − χ tan ˜ θ , we may express the minimized distancefrom (C2) as a function of the same angle : r [˜ θ ; (cid:37) (˜ θ )] = χ cos ˜ θ − θ , (C3)which is the sought geometric place of the all points ofthe closest approach, in spherical coordinates. Alreadyfrom the form of this expression one can conclude thatit can not possibly describe the shadow of the Ruther-ford scattering, as this expression for the absolute dis-tance is positive only for ˜ θ > π/
2, while the shadow is ex-pected to cover also the forward angles ( θ < π/
2; see fig-ure 1). Employing again the coordinate transformations z (˜ θ ) = r [˜ θ ; (cid:37) (˜ θ )] cos ˜ θ and ρ (˜ θ ) = r [˜ θ ; (cid:37) (˜ θ )] sin ˜ θ , whileeliminating the term cos ˜ θ from z (˜ θ ), one obtains the ex-plicit shape of the curve from (C3) in cylindrical coordi-nates: ρ ( z ) = z (cid:112) − z ( z + 2 χ ) z + χ . (C4)It is to be noted yet again that in scaled coordinates¯ ρ = ρ/χ and ¯ z = z/χ this expression also has a universalform: ¯ ρ = ¯ z (cid:112) − ¯ z (¯ z + 2) / (¯ z + 1). While (C3) is positivefor ˜ θ > π/ z lim = 0—both the dependence z (˜ θ ) = χ (cos ˜ θ −
1) and the fact that (C4) is nonnegative Relative to the initial angle θ = π —and due to the symmet-ric shape of the hyperbole—the scattering angle ϑ from (13) isalways double the angle ˜ θ of the closest approach from (C1): ϑ − θ = 2(˜ θ − θ ) . In other words, the hyperbole is symmetric around ˜ θ . In applying the identity (cid:112) ˜ θ = 1 / | cos ˜ θ | to arrive at(C3), one needs to take: | cos ˜ θ | = − cos ˜ θ as for π < ˜ θ < π/ − − z/χ ρ / χ r [ ˜ θ ( % ); % ] r [ θ ; ˜ % ( θ )] FIG. C1. Universal shape of the geometric place of the closestapproach points (full line) compared to the universal shapeof the scattering shadow (dashed line) in a fixed-target frame(target is the central dot). The closest approach curve has anasymptote at z/χ = −
1, so that it is fully contained within − ≤ z/χ < −
1. While ˜ θ ( (cid:37) ) minimizes the target-projectiledistance for a given projectile trajectory (of given impact pa-rameter (cid:37) ), ˜ (cid:37) ( θ ) minimizes among all possible trajectories the target-projectile distance under a given angle θ . only for − χ ≤ z < − χ reveal that the curve’s asymp-tote is actually at z lim = − χ . This is clearly shown in fig-ure C1, where the universal shape of the closest approachcurve is compared to a universal shape ¯ z = ¯ ρ / − θ ( (cid:37) ) comes closer to a target than the trajectory definedby (cid:37) , whose own distance to a target is minimized underthe same angle . The problem boils down to finding animpact parameter (cid:37) for which the distance from (A17) issmaller than for (cid:37) : r [˜ θ ( (cid:37) ) , (cid:37) ] < r [˜ θ ( (cid:37) ) , (cid:37) ] . (C5) One such trajectory is certainly be the one that comes closest toa target, whose own impact parameter is ˜ (cid:37) [˜ θ ( (cid:37) )]. Indeed, it isstraightforward to show that for a given (cid:37) the trajectory with ˜ (cid:37) always comes closer: r { ˜ θ ( (cid:37) ); ˜ (cid:37) [˜ θ ( (cid:37) )] } ≤ r [˜ θ ( (cid:37) ); (cid:37) ] or, equiv-alently: r [˜ θ ; ˜ (cid:37) (˜ θ )] ≤ r [˜ θ ; (cid:37) (˜ θ )]. Using (17) and (C3)—and keep-ing in mind in manipulating the inequality that for ˜ θ > π/ θ is negative—this boils down to showing:2 χ sin (˜ θ/ ≤ χ cos ˜ θ − θ ⇔ (1 + cos ˜ θ ) ≥ . Since the left inequality is equivalent to the right one, and theright one is true for any ˜ θ , so is true the initial claim. . . . . . . . % /χ . . . . . . . % / χ % + /χ % − /χ ˜ % [ ˜ θ ( % )] /χ FIG. C2. Range of impact parameters (cid:37) (shaded area) forwhich, under an angle ˜ θ ( (cid:37) ), the projectile trajectory in afixed-target frame comes closer to the target than the trajec-tory with (cid:37) , whose own distance to the target is minimizedunder the same angle. The impact parameters ˜ (cid:37) [˜ θ ( (cid:37) )] thatminimize the projectile-target distance are also shown (dashedline), being a harmonic mean between (cid:37) + and (cid:37) − . With the help of trigonometric identities (C5) reducesto a quadratic inequality: (cid:37) (cid:113) χ + (cid:37) − (cid:37) (cid:37) (cid:112) χ + (cid:37) − χ + χ(cid:37) < (cid:37) . There are two boundaries to thisinequality: (cid:37) + = (cid:37) , (C7) (cid:37) − = χ(cid:37) (cid:112) χ + (cid:37) , (C8)meaning that, under an angle ˜ θ ( (cid:37) ), any trajectory withan impact parameter (cid:37) such that (cid:37) − < (cid:37) < (cid:37) + , comescloser to the target than the trajectory with the im-pact parameter (cid:37) . The trajectory yielding the ab-solute minimum distance among all trajectories—theone with an impact parameter ˜ (cid:37) [˜ θ ( (cid:37) )]—is certainlyto be found within this range. In fact, plugging(C1) into (16), with the help of a very useful identitytan( π/ x/
2) = tan x + 1 / cos x , it can be shown thatits impact parameter is precisely the harmonic mean be-tween the two boundaries:1˜ (cid:37) [˜ θ ( (cid:37) )] = 12 (cid:18) (cid:37) + + 1 (cid:37) − (cid:19) . (C9) Trigonometric identities in question ultimately yield:sin (cid:20) π χ(cid:37) − arctan χ(cid:37) (cid:21) = χ + (cid:37) (cid:37) (cid:113) ( χ + (cid:37) )( χ + (cid:37) ) . z ˜ θ ( % ) % + ( % )˜ % ( ˜ θ ) % − ( % ) % FIG. C3. Several trajectories related to the reference one(with an impact parameter (cid:37) ), which under an angle ˜ θ ( (cid:37) )reaches the point of the closest approach. See the main textfor a full description. The vertical and horizontal scales arenot equal, causing the connecting line, depicting ˜ θ ( (cid:37) ), not toappear orthogonal to the reference trajectory at an intersec-tion point. Since (cid:37) + = (cid:37) , any trajectory satisfying (C5) has asmaller impact parameter than (cid:37) , hence: ˜ (cid:37) [˜ θ ( (cid:37) )] ≤ (cid:37) .Figure C2 shows an exact range of impact parame-ters from which one may select sought trajectories. Bynow it should not be at all surprising that these bound-aries also take the universal form in scaled coordinates¯ (cid:37) ± , = (cid:37) ± , /χ so that: ¯ (cid:37) + = ¯ (cid:37) and ¯ (cid:37) − = ¯ (cid:37) / (cid:112) ¯ (cid:37) + 1.One may also note that for large values of (cid:37) the nontriv-ial boundary (cid:37) − saturates: lim (cid:37) →∞ (cid:37) − = χ . Therefore,for the reference trajectories sufficiently distant from thetarget—which are barely deflected and whose point of theclosest approach is roughly under ˜ θ ( (cid:37) ) ≈ π/ (cid:37) < (cid:37) )come closer to a target, except for those with (cid:37) (cid:47) χ , thatare deflected backwards even before reaching an angle θ ≈ π/ (cid:37) . Alongside the trajectory with (cid:37) + ( (cid:37) ),which is identical to (cid:37) , the one with (cid:37) − ( (cid:37) ) features thesame target-projectile distance under an angle ˜ θ ( (cid:37) ) asthe reference trajectory. The trajectory minimizing thetarget-projectile distance, under the same angle, is alsoshown: the one with ˜ (cid:37) (˜ θ ). Dashed trajectories are ex-amples of those that come closer to the target than thereference trajectory, yet do not minimize that distance.One of them is purposely taken from the range (cid:104) (cid:37) − , ˜ (cid:37) (cid:105) ,the other from (cid:104) ˜ (cid:37) , (cid:37) + (cid:105) , i.e. each one from a differentside of the distance-minimizing one. It should be notedthat for purposes of managing the figure dimensions, thehorizontal and vertical scale are not equal. Consequently,the connecting line (depicting an angle ˜ θ ) does not ap-pear to be orthogonal to the reference trajectory ( (cid:37) ) atthe intersection point (the point of the closest approach),as it should appear if the scales were the same.5 D. INTERSECTION OF SHADOWS
As seen from figure 1, in a fixed-target frame an un-limited portion of space is shielded from admitting eitherthe projectile or target trajectories. In a center-of-massframe each of the two shadows also shields an unlimitedportion of space, but only from a corresponding particle.As opposed to that, only a limited portion of space isshielded from both the projectile and the target. Thisvacuous portion of space is limited by the intersectionof their respective shadows and provides an opportunityfor some instructive geometric calculations. Figure D1shows an example of this intersection for η p = 0 .
25 (i.e. η t = 0 .
75 or, equivalently, m t = 3 m p ).In order to calculate the geometric parameters of thisintersection, we first need to identify several relevant ref-erence points. The first two are the positions of theshadow vertices along the z -axis. From (29) and (30)they are trivially read out as: z p (0) = − η t χ and z t (0) = 2 η p χ. (D1)The remaining values are the paraboloids’ intersectioncoordinates ρ ∗ and z ∗ , which are easily determined bysolving the equation z p ( ρ ∗ ) = z t ( ρ ∗ ), thus obtaining: ρ ∗ = 4 χ √ η p η t and z ∗ = 2 χ ( η p − η t ) . (D2)Let us first consider a Rutherford scattering in a two-dimensional space. This space is equivalent to any planecontaining the z -axis of a full three-dimensional space,and is exactly represented by an example from figure D1.We are interested in calculating the perimeter P and thearea A of this two-dimensional intersection of parabolic shadows. Returning to a full three-dimensional space,we are interested in calculating two additional, analogousvalues: the surface area S and he volume V of the three-dimensional intersection of paraboloidal shadows.Starting with the perimeter P , we firstnote that an infinitesimal length element d l p , t along any of the two parabolas is given by(d l p , t ) = (d ρ p , t ) + (d z p , t ) , so that depending onthe choice of an independent integration variable, wemay either write d l p , t = d ρ p , t (cid:112) z p , t / d ρ p , t ) ord l p , t = d z p , t (cid:112) ρ p , t / d z p , t ) . In order to calculatethe newly appearing derivatives as functions of ap-propriate coordinates, one may use already availableexplicit forms z p , t ( ρ p , t ) from (29) and (30). Alterna-tively, one may first wish to calculate inverse relations ρ p , t ( z p , t )—which is done easily enough—and proceedwith an integration over z p , t . Therefore, we have theseequivalent approaches to calculating P : P = 2 (cid:88) i =p , t (cid:90) ρ ∗ (cid:115) (cid:18) d z i d ρ i (cid:19) d ρ i = 2 (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z i (0) z ∗ (cid:115) (cid:18) d ρ i d z i (cid:19) d z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (D3) z p (0) z ∗ z t (0) ρ ∗ ρ ∗ FIG. D1. Intersection of the projectile (from left) and thetarget (from right) shadows in a center-of-mass frame, for η p = 0 .
25, i.e. η t = 0 .
75. The portion of space correspondingto a shaded area is entirely shielded from any particle tra-jectory. The focal points of both shadows coincide and aresituated at the origin of the coordinate frame (the dot). and it is an instructive exercise to show that both pro-cedures indeed yield (as they must!) the same result. Incalculating the area A closed by two parabolas, one mayagain select an independent integration variable, whichis equivalent to selecting a particular order of integrationwhen a full double integral is written down. Immediatelysolving the first, trivial of the two integrations, we mayagain select one of the two finishing procedures: A = 2 (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ρ ∗ [ z i ( ρ i ) − z ∗ ]d ρ i (cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z i (0) z ∗ ρ i ( z i )d z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (D4)These integrals may very easily be set in place just byobserving the geometry from figure D1. Similarly, for asurface area S of a three-dimensional intersection, the ba-sis for an integration is 2 πρ p , t d l p , t (where the first of thetwo integrals goes over the azimuthal angle ϕ p , t , yielding2 π ), so that: S = 2 π (cid:88) i =p , t (cid:90) ρ ∗ ρ i (cid:115) (cid:18) d z i d ρ i (cid:19) d ρ i = 2 π (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z i (0) z ∗ ρ i ( z i ) (cid:115) (cid:18) d ρ i d z i (cid:19) d z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (D5)6Finally, for he volume V we have: V = 2 π (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ρ ∗ ρ i [ z i ( ρ i ) − z ∗ ]d ρ i (cid:12)(cid:12)(cid:12)(cid:12) = π (cid:88) i =p , t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) z i (0) z ∗ ρ i ( z i )d z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (D6)Though it is instructive to carry out both types of inte-grations and for both particles separately, in reality it isentirely sufficient not only to perform just one type of in-tegration (either over ρ p , t or z p , t ), but just for one of theparticles (either for the projectile or the target). The rea-son is the symmetry between the projectile and the targetin the center-of-mass frame, as they roles can always beinterchanged by a simple interchange of indices p ↔ tthroughout all the expressions. Therefore, if one calcu-lates any of the integrals for one particle, the integral forthe other one immediately follows by consistently switch-ing all the indices: p ↔ t. As a consequence, the finalresults—as the sums of these two contributions—must befully symmetric in respect to both particles and invari-ant under the same index interchange. Indeed, whichevercourse of calculation is selected, one invariably arrives at: P = 4 χ (cid:18) √ η p + √ η t + η p arsinh (cid:114) η t η p + η t arsinh (cid:114) η p η t (cid:19) , (D7) A = 323 χ √ η p η t , (D8) S = 323 χ π (cid:0) √ η p + √ η t − η − η (cid:1) , (D9) V = 16 χ πη p η t . (D10)As a closing note, it is worth examining the limit-ing case of an infinitely heavy target, as in that casethe center-of-mass frame coincides with the fixed-targetframe. Thus the target stays at rest, its shadow spanningan entire geometric space , in a sense of not admitting anytarget trajectory. We already know from an open projec-tile shadow (figure 1) that an infinite portion of spaceis then shielded from admitting any particle. Therefore,it is a useful exercise to confirm whether or not (D7)–(D10) reproduce that limit, as the very limiting case is,strictly speaking, outside their domain . In that, it isnot sufficient just to set the mass ratios to η t = 1 and η p = 0. Rather, one needs to insist that the target massbe infinite ( m t → ∞ ), as opposed to allowing the van-ishing projectile mass instead ( m p →
0, leading to thesame set of mass ratios). This is because the Ruther-ford scattering is sensitive to absolute masses, which is The reason for the limiting case of the finite-mass tar-get ( m t → ∞ ) not corresponding to the ‘exact’ case of theinfinite mass target ( m t = ∞ ) is the following: the ini-tial offset of the finite-mass target from the center-of-massframe z (cm)t ( m t < ∞ ; t = 0) = ∞ cannot reproduce, in the limit m t → ∞ , the exact offset (at any moment) of the infinite-masstarget z (cm)t ( m t = ∞ ; t ) = 0. realized through the appearance of the reduced mass µ within the definition of χ from (A16), i.e. from the factthat the reduced mass can not be expressed solely as afunction of the mass ratios, but depends on the particu-lar masses: µ = η p η t ( m p + m t ). In examining the limitswe are again faced with an issue from (31), consistingin a caution against naively manipulating just η p and η t as they explicitly appear in (D7)–(D10). Taking into ac-count also their presence within χ , the relevant portionsof the expressions to be examined take the form: P ∝ √ m t µ + 1 √ m p µ + 1 m t arsinh (cid:114) m t m p + 1 m p arsinh (cid:114) m p m t , (D11) A ∝ µ √ m t m p , (D12) S ∝ (cid:112) m t µ + 1 (cid:112) m p µ − m − m , (D13) V ∝ m t m p µ . (D14)Since lim m t →∞ (arsinh (cid:112) m t /m p ) /m t = 0 (which is easilyshown by the L’Hˆopital’s rule) and lim m t →∞ µ = m p , wesee that in the limit m t → ∞ most of the terms vanish,except for the second term from (D11), together with thesecond and the fourth term from (D13). As their place-ment corresponds to that of the terms from (D7) and(D9)—with η t = 1 in the same limit—we may immedi-ately write:lim m t →∞ P = 4 χ, (D15)lim m t →∞ A = lim m t →∞ S = lim m t →∞ V = 0 . (D16)These results are easy to understand, as in the limitingcase of (30) the target shadow in the center-of-mass framebecomes an infinitely narrow paraboloid, intersecting theprojectile shadow before it even detaches from the z -axis.Thus the intersection of such shadows forms a geometricshape akin to a one-dimensional line segment, of van-ishing volume and any areal property, but still of finiteone-dimensional features such as the perimeter. As wehad set to investigate, we see now that this limiting casedoes not coincide with the ‘exact’ case of the infinitelymassive target, since the qualitative change in dynamicstakes place—the target stays at rest, exempting an en-tire geometric space from its trajectories. Therefore, thetrue value for each of the calculated geometric propertiesΓ ∈ { P, A, S, V } diverges, in a sense Γ( m t = ∞ ) = ∞ , sothat we have an incongruity: Γ( m t → ∞ ) (cid:54) = Γ( m t = ∞∞