arXiv:1705.02337v2 [cond-mat.soft] 6 Aug 2017 rjcie eiswith begins projectiles hsrgr,w a eto h lsi rayo gunnery Poncelet on treaty Robins. classic In the by mention applications. can we military obvi- regard, example, this has for which subject, applications, this ous studied have classics Other taindphi u oGamow pen- to for Newton. due formula to simplest is The depth use. in etration still are that depth rae nsvrlmdr ok n reviews. and books are modern media several resistant in in projectiles treated of penetration and pact entitled tcnb ple otemto of motion the to applied be can it tmcprilsaepr ftesaeevrnet as environment, space the meteoroids. sub- of fast high-energy as part well particular, are medicine, In particles or etc. atomic engineering physics, ap- nuclear state of as solid areas such to physics, physics plied particle fundamental from texts, media. oei h eeomn fmdr hsc n h the- the and established. physics well modern now important of is ory development an played the has in penetration role particle of de- years. aspects completed following Bohr and the 1913 course. along in it of theory correct recent, essentially relatively an is vised matter is in which ticles km/s, physics. 10 atomic basic about by limit, essence, upper in and variable determined, variable less a a determined, from to be space will goes which in limit, consider lower hazard medium-dependent to a projectiles constitute scopic fact, typ- in are and, km/s, engineering. space 10 in about velocities, ical larger while normal, h nltclsuyo h eitnet h oinof motion the to resistance the of study analytical The etnster frssac oue nfluids, on focuses resistance of theory Newton’s h td ftepntaino atsbtmcpar- subatomic fast of penetration the of study The 6 rjcie ihvlcte bu msaefairly are km/s 1 about velocities with Projectiles 4 h oino ois(nrssigmediums). resisting (in bodies of motion The n Resal and 2 7,8 16–20 ewrs eerto yais nryls;collisions; projecti loss; and energy density dynamics; medium penetration to Keywords: projectile of ratio are depth ffs aopoetlsi ue yasihl oie Newt modified slightly a by ruled is technology. F implantation nano-projectiles cluster fast atom of technology in and physics application T space finds sizes. in energies different example, high of for at projectiles applications; penetrating matter of with problem bodies particles the macroscopic subatomic of motion charged p fast of be the can domains: principles physical different two basic which in modeling, simple i okwscniudb Euler. by continued was work His fcus,tevrosapcso h im- the of aspects various the course, Of h eerto fafs rjcieit eitn medium resistant a into projectile fast a of penetration The ∼ .INTRODUCTION I. v h ag fvlcte ffs macro- fast of velocities of range The 2 − β 5 where , eue omlsfrpenetration for formulas deduced okTwo Book 20 eerto ffs rjcie norssatmedia: resistant into projectiles fast of Penetration 22 21 β ti tde nvroscon- various in studied is It rmmcocpct uaoi projectiles subatomic to macroscopic from h eerho h many the on research The aihsa h nes fpoetl imtr atr ess Factors diameter. projectile of inverse the as vanishes 6 fNewton’s of fast u eatiue it attributed he but rjcie nany in projectiles nvria oi´ciad Madrid, Polit´ecnica de Universidad ple hsc et,ETSIAE, Dept., Physics Applied 9–15 Principia -84 ard Spain Madrid, E-28040 Dtd uy2,2017) 21, July (Dated: 3 Later, 1 o´ Gaite Jos´e but 1 , ehooia applications. technological aint arsoi projectiles macroscopic to lation rjcie,a h eu ftemcocpcadatomic and VI). macroscopic (Sect. nano- the penetration study of of we nexus theories Finally, the as V). origin (Sect. projectiles, electrodynamic follows the dimensional resistance of a analysis of deeper by A beginning sub- that analysis. with charged projectiles, analogy of in macroscopic penetration treated of is the matter phenom- IV, in particles Sect. pressure atomic in high consider Next, and to ena. us temperature of leads analysis high The projectiles complex ana- define supersonic III). we (Sect. to to Next, resistance resistance need of velocity. We physics critical the formula a lyze projectile. Gamow’s introducing fast by with a “fast” connect analy- of to dimensional penetration us a for allows including which II, sis, Sect. in penetration size). km/s) their 2000 (for than penetrating for very larger strict are velocities also neces- less (with and are is fast particles condition very Subatomic This sarily than higher projectiles. medium. considerably streamlined and the are a density of projectile see, the those the will These when we of deeply As C. strength penetrates III depth. Sect. projectile penetration in fast the considered fragmenta- reduce projectile, effects or the Another deformation of possible significant. tion the not is are effects complication If etc) surface medium. the sputtering, the considerable, (spalling, of is surface depth the penetration on the projectile the of impact nano-projectiles. useful to and of novel applicable ranges some especially deduce full facts, unified we the and a velocities, to and problem within applies sizes the penetration that study framework projectile we conceptual Here to resistance problem. the of of aspects lar sa is uesncmotion. supersonic ebgnb etn h ai rmwr o projectile for framework basic the setting by begin We the of effects the disregard we simplicity, of sake the For ewe uaoi n arsoi rjcie,there projectiles, macroscopic and subatomic Between mesoscopic eelmtcssatal aeoverlapping have actually cases limit hese h nemdaeo eocpcdomain mesoscopic or intermediate The . otbyepoe.Ti td connects study This employed. rofitably eshape. le eei ssonta h penetration the that shown is it Here sacmlxpoesta ssial for suitable is that process complex a is hc uns h w ii ae of cases limit two the furnish which , ∗ nsieta udai oc,namely, force, quadratic inertial on’s nrssatmdaadteinteraction the and media resistant in ag of range nano-projectiles nilt penetration to ential 23 h e tde fterre- their of studies few The 24,25 nytetparticu- treat only ihimportant with , 2

Our subject encompasses two fields of physics that are those that constitute the medium, namely, electrons and studied by different communities and requires concepts of atomic nuclei. The (very large) set of equations of mo- both fields, some of which are well known by the corre- tion of these particles possesses an invariance: the mul- sponding community but probably not by the other. For tiplication of all the present masses by a common factor this reason, it is preferable to refer to basic articles and λ can be compensated by the multiplication of time by textbooks, when possible. An apology is due to expert λ1/2, while the space coordinates and electric charges are readers, who may find some concepts and references too kept constant. This operation does not change the tra- basic. jectories and, hence, it does not change D, provided that −1/2 A note on notation. We employ often two signs for the the initial velocity v0 is scaled to λ v0. Therefore, 2 asymptotic equivalence of functions: f g means that we deduce that D is proportional to Mv0 times some ∼ 2 the limit of f(x)/g(x) is finite and non-vanishing when function of mv0, where m is the mass of the particles x tends to some value (or to infinity), whereas f g of the medium that interact with the body (if there are means, in addition, that the limit is one. We also≈ use several types of them, then the corresponding terms ap- the signs and , which denote, respectively, numeri- pear). Furthermore, we deduce from Eq. (2) that v must cal proportionality∝ ≃ and approximate equality. Regarding appear in F (v) as the combination m1/2v. In a continu- 1/2 our notation for functions, we generically employ f to ous medium, described by the mass density ρm, m v is 1/2 denote any one of the many unspecified functions of non- replaced by ρm v. dimensional arguments that we introduce. This fundamental law is upheld by the concrete dimen- sional analyses for macroscopic bodies in the next section (Sect. II B) and for subatomic particles (Sect. IV A) and, II. MOTION OF FAST PROJECTILES IN A furthermore, is generally useful (e.g., in Sect. V B). MEDIUM

The penetration depth, which is easily measurable, can B. Macroscopic body penetration: dimensional be obtained by integrating Newton’s Second Law. We as- analysis sume that the motion is one-dimensional, as if the projec- tile is axisymmetric and moves along its axis (the x-axis). The minimal set of parameters for penetration in a con- Given that the resistance force F is a function of the ve- tinuous medium consists of the impact velocity v0, mass locity v, the differential equation of motion can be solved M and length l of the body, its set of non-dimensional for t(v) and hence v(t) can be obtained. One more in- shape parameters s , and the density ρm of the medium. tegration gives x(t). However, to obtain the penetration The penetration depth{ } D must be a function of these depth, t is not a relevant variable and, instead, we are magnitudes. As seen above, D is proportional to M. interested in the function v(x). Therefore, we write Therefore, we define D˜ = D/M and we have to find the function D˜(v , l, s ,ρ ). We first notice that, of the dv 0 { } m Mv = F (v), (1) base mechanical dimensions L, T and M, the dimension dx − T is only present in v0. Therefore, the dimension T can- where M is the mass of the projectile. Integrating dx not be present in a non-dimensional group and, conse- ˜ between x(v ) = 0 and x(0) = D (the penetration depth) quently, D cannot depend on v0. Since the shape param- 0 ˜ one gets: eters are non-dimensional, the dimensions of D, namely, [D˜] = LM−1, must be obtained with a combination of l v0 2 D v dv and ρm . The only possible combination is 1/(ρm l ). In = . (2) M F (v) consequence, Z0 Naturally, Eq. (1) is equivalent to the kinetic energy the- ˜ f( s ) D(v0, l, s ,ρm)= { }2 . (3) orem and the braking can be understood alternately as { } ρm l a loss of momentum or as a loss of kinetic energy. where f is a non-dimensional function of the non- The resistance force depends on the geometry of the dimensional shape parameters. body but is independent of its mass, as long as gravita- Let us compare this formula with empirical scaling for- tion plays no role in the dynamics of medium resistance. mulas, which adopt the form:11,12,15–17,27 Therefore, Eq. (2) implies that the penetration depth, D, ′ is proportional to M. α α β γ D = kρM ρm v0 l , (4)

3 where ρM M/l is the density of the body, k is a A. A fundamental law of motion under resistance dimensional∝ constant, and the exponents α, α′ and β have been assigned various values. Since D has to be The force of resistance is, basically, a certain combina- proportional to M, α = 1 (but other values appear in tion of the Coulomb electric forces of interaction between the literature). It is often assumed that α + α′ = 0, the microscopic particles that constitute the body and that is to say, that D depends on ρM and ρm through 3 their ratio, like in Eq. (3). The exponent β is usually If we take into account that the dependence on this third smaller than one (a typical value is 2/3), so that D has variable is actually determined by basic physical princi- a weak dependence on v0. However, α usually is smaller ples, as already said, Li and Chen’s formula boils down than β, which implies that the dependence of D on the to a particular case of Eq. (6). density ratio is still weaker. Naturally, Eq. (3) indicates Eq. (6) reduces to Eq. (3) when v /v , pro- 0 c → ∞ the opposite. vided that f( , v0/vc) has a finite, non-zero limit. When Eq. (3) is the simplest scaling formula and it agrees a non-dimensional· function has a finite, non-zero limit, with Gamow’s formula,6 the asymptotics is said to be of the first kind.26 The next more complex case is asymptotic similarity, namely, ρ D = M l, (5) power-law behavior.26 A logarithmic asymptotic law is a ρm border-line case, because the logarithmic law can be un- derstood as a power law of zero exponent. In fact, the except for the shape-dependent numerical factor. Of asymptotics that corresponds to Newton’s theory is log- course, Gamow must have been aware of the possible arithmic (Sect. III A). This means that the penetration presence of this factor, because he loosely defined l as the depth has a very weak dependence on the critical veloc- “length of the projectile,” without specifying any shape, ity, so that this velocity defines the asymptotic regime and commented that Eq. (5) “is true only very approxi- and almost disappears in it. mately.” Gamow’s formula is useful for estimations. For example, a fast lead bullet will penetrate 11 times its length in water (or flesh) and 11000 times in air. It also III. MODELS OF RESISTANCE provides the rationale for high density projectiles as ar- mor penetrators, as well as for high density armor plates, which favors the use of depleted uranium.28 Gamow re- A. Empirical models marked that “the length of penetration does not depend on the initial velocity of the projectile (provided that this D(v0) can be measured and hence the function F (v) velocity is sufficiently high).” To justify it, he mentioned can be determined empirically. Newton1 already con- some experimental evidence, without references. The ab- ducted experiments on air drag and proposed the empiri- 3/2 2 sence of v0 in Eq. (3) is, in fact, a consequence of our cal law F (v)= Av+Bv +Cv . Newton’s own measure- choice of the minimal set of parameters, in which only ments imply that the middle term is negligible in com- 8 v0 includes the time dimension. In this regard, one may parison with the other two terms. The quadratic polyno- 2 5,8,12,27 question the meaning of “sufficiently high” initial veloc- mial F (v)= a1v + a2v has been much employed. ity, as long as there is no velocity with which to compare Substituting this F (v) in Eq. (2), it. v0 dv 1 a v D˜ = = ln 1+ 2 0 . (7) a + a v a a Z0 1 2 2  1  C. Critical velocity Comparing this equation with Eq. (6), we identify a2 ρ l2 and v = a /a . This v is the velocity at which∝ A critical velocity is necessary to qualify a projectile m c 1 2 c the linear and the quadratic terms of F have equal mag- as “fast.” The speed of sound, cs, is the characteristic 2 nitude. For v vc, we can just use F (v) a2v , but velocity of any medium; its fundamental role in pene- 0 the corresponding≫ integral, v dv/v, diverges≈ at its lower tration dynamics is shown in Sect. III B. But it is not 0 limit, because a purely quadratic drag is too weak at low the only option: in incompressible fluids, with cs = , R the critical velocity is given by viscosity.8 Regardless∞ its velocity to stop the projectile. Naturally, the lower limit must be set to vc, obtaining origin, let us denote the critical velocity by vc and just add it to the minimal set of parameters. Then, v0 and vc v0 1 dv 1 v0 1 v0 must appear in D˜ as the ratio v /v . Therefore, Eq. (3) D˜ = ln ln . (8) 0 c ≈ a v a v ∝ ρ l2 v is replaced by 2 Zvc 2 c m c This form is asymptotically equivalent to the exact result ˜ f( s , v0/vc) D(v0/vc, l, s ,ρm)= { } 2 . (6) (7) but is more general, because it holds whenever F (v) { } ρm l 2 ∼ v for v vc. On the other hand, if F (v) a1v, then ˜ ≫ ≈ This formula is very general yet simpler than others D v0/a1, which agrees with the asymptotic limit of (7) for≈v v . that have been proposed for particular cases. For exam- ≪ c ple, let us compare it to Li and Chen’s formula for pen- Recalling the empirical power-law form of D(v0) in etration in concrete.29 Li and Chen express D in terms Eq. (4), we can propose F = av2−β, which gives D˜ = β of a non-dimensional function of three non-dimensional v0 /(βa). To have 0 <β< 1, as usually found, the expo- variables, which we can choose as: (i) the nose factor nent of v in F must be between 1 and 2. This contradicts (one shape parameter); (ii) v0/vc, where vc is expressed the conclusions drawn from Newton’s experiments. How- 3 in terms of target material properties; and (iii) M/(ρml ). ever, it is the expected result for a function such as the 4 one in Eq. (7) that one tries to fit to a power-law in a non- specific model that yields the force dependence on the asymptotic range (as further discussed in Sect. III B). shape of the body, and, remarkably, he calculated with For penetration in solids at low-velocity, the force is that model the profile of revolution of least resistance.30 2–4,12,29 ˜ 2 actually constant and D v0 . The full quadratic Newton’s theory can be put in a general form that ∼2 polynomial, F (v)= a0 + a1v + a2v , gives rise to an inte- is independent of the type of body-medium interaction gral v dv/F that can be effected by factorizing the poly- and is applicable to subatomic particles (Sect. V A). As nomial, but the result is a complicated algebraic formula the body moves at velocity v, it crosses, in a time dt, a R in terms of a0, a1 and a2. Nevertheless, the asymptotic slab of the medium of depth v dt. This slab is formed form (8) holds. Now, vc is to be identified with a1/a2, if by mass elements dm = ρm dS (v dt), where dS is the 2 1/2 2 a1 > 4a0a2, or with (a0/a2) , if a1 4a0a2. element of surface in a transversal plane. If the body We can try a higher power of v to≤ modify the asymp- imparts to an impinging mass element dm a longitu- 2 3 totics; for example, F (v)= a1v+a2v +a3v (a1,a2,a3 > dinal velocity vk, then it transfers to it a longitudi- 0). Then we have two critical velocities: vc = a1/a2 nal momentum dp = ρm dS (v dt) vk and exerts a force ′ ˜ and vc = a2/a3 . The integral for D yields a complicated dF = dp/dt = ρmv vk dS. Therefore, the total drag force function, which, remarkably, has a large-v0 asymptotics is ′ 2 of the first kind. In particular, if vc/vc = a1a3/a2 1, ˜ ′ ≪ then D ln(vc/vc)/a2. Comparing it to (8), we see that F = ρmv vk dS. (9) ≈ ′ ln(v0/vc) is replaced by ln(vc/vc), as if the largest possi- Z ′ ble value of v0 were vc, namely, the velocity for equality of The mass elements may also acquire transversal velocity the quadratic and cubic terms. Actually, the effect of the v , but this is irrelevant to F . cubic term on a motion with v > v′ is to bring v down to ⊥ 0 c Newton assumed that v is proportional to v. This con- v′ while a distance 1/a is traversed, as easily found by k c 2 dition seems natural but is not universal (see Sects. V A making F (v) a v≈3. The subsequent motion is due to 3 and VI). If we write v = c v, where c is a geometrical the quadratic≈ drag, F (v) a v2, and traverses a longer k 2 coefficient, then distance ln(v′ /v )/a . Finally,≈ under F (v) a v, the ≈ c c 2 ≈ 1 distance traversed is 1/a2. ≈ 2 Generalizing these results, we conclude that the F = ρmv c dS. (10) 2 quadratic force, F (v) = a2v , is singled out because the Z ∞ integral 0 v dv/F is divergent at both the lower and In particular, if the mass elements placed inside the cylin- upper limits. When these divergences are cut off, the der defined by the cross section of the body undergo com- R ′ factor ln(vc/vc) arises. Any addition to the quadratic pletely inelastic collisions with the body, without adher- force that is stronger than it at high velocities makes ing to it, while the other mass elements are unaffected, the integral converge at v = and works like the cubic then c = 1 in the cross section and zero out of it. There- ∞ 2 term, by slowing down the projectile to the corresponding fore, F = ρmv A, where A is the cross-sectional area. In v′ , within a distance 1/a . Analogously, any addition c ∼ 2 Newton’s specific model, the collisions are inelastic only that is stronger than it at low velocities makes the inte- in the direction normal to the surface of the body. In gral converge at v = 0 and, like the linear term, stops the general, the collisions are partially elastic and, further- projectile from v , within a distance 1/a . Moreover, c ∼ 2 more, the body can affect mass elements close to its cross the quadratic force is singled out because a2 does not in- section. Therefore, the integral in Eq. (10) defines an ef- volve the dimension T and can be expressed in terms of fective cross-sectional area, which depends on the shape the basic variables, namely, a ρ l2. 2 ∝ m of the body and its interaction with the medium. In fluid Finally, let us notice that the coefficients a0, a1 and mechanics (see below), the force is normally written as 2 a2 are necessarily positive, but we can have a3 < 0 or, F = ρmv (CD/2)A, including a drag coefficient CD that in general, that the growth of F with velocity is lower depends on the shape of the body and is of the order of than quadratic for very large velocities (Sects. V and VI). unity for bluff bodies but much smaller for streamlined Some recent experimental results about the relevant ve- bodies.31 locity ranges are cited in Sects. III B, III C and VI. Following Newton, Gamow6 argued that the projectile pushes aside the medium, boring a tunnel, and that “it is easy to see that the sidewise velocity of the medium B. Physical origin of the force of resistance is about the same as the velocity of the advancing pro- jectile,” that is to say, that c = 1 in the cross section. Newton’s argument is, in essence, very simple: the re- However, Gamow’s conclusion, namely, “the projectile sistance force results from the total backward force of will stop when the mass of the medium moved aside is reaction of the particles of the medium to the forward of the same order of magnitude as its own mass,” is a forces exerted on them by the moving body. In modern non sequitur, because it is necessary to solve the differ- terms, the force is the consequence of momentum conser- ential equation of the motion.8 As seen in (8), the inte- vation in the continuous collision of the body with the gration demands the cutoff vc and produces the factor particles of the medium. Newton actually considered a log(v/vc). To be precise, employing fluid mechanics no- 5 tation, namely, a2 = ρm(CD/2)A, one obtains and cs (for bodies of generic shape). The latter is a char- acteristic of the fluid whereas the former also depends on M v ρ l v D = ln 0 = M ln 0 , (11) the length of the body. The value of l such that they ρm(CD/2)A vc ρm(CD/2) vc are equal, namely, η/(cs ρm), is a characteristic of the fluid: it is, approximately, the molecular mean free path where, for the second equality, the volume of the pro- λ, which is an important length in molecular kinetics.33 It jectile has been set to lA (which is exact for a cylin- is a microscopic length in standard conditions and then cs der). Both the two factors that correct Gamow’s formula, is the larger critical velocity. However, molecular kinet- −1 namely, the logarithm and (CD/2) , make D grow, and ics is relevant to the motion of macroscopic projectiles the latter can easily be larger than the former. Therefore, in rarefied gases and to the motion of nano-projectiles a streamlined shape is more important than a high ve- (Sect. VI). Normal macroscopic projectiles, with l λ, locity for a projectile to be penetrating, and streamlined can undergo subsonic or supersonic inertial resistance,≫ projectiles can widely surpass Gamow’s D. However, the the former due to viscous transfer of momentum and the asymptotic form (11) is then less applicable, because vc latter due to compressional processes that raise the value increases as a2 decreases. The reason is that vc is defined of CD. 2 as the value of v such that a2v equals the remainder of For penetration in solid media, the only critical veloc- F (v), which is less sensitive to shape (see below). ity is the speed of sound, which is indeed the standard In incompressible fluids, the origin of drag is viscos- reference velocity for fast projectile impacts onto solid ity, namely, internal friction or resistance to the relative targets. The resistance force to a hypersonic projectile motion of contiguous fluid layers. For a fluid character- 2 must be, like in fluids, F = a2v , where a2 ρm A and ized by its density ρm and viscosity coefficient η, the di- is shape dependent. At low velocity, a constant∝ force mensional analysis requires that F = ρ l2v2f( s , Re), m { } results from tangential surface-friction forces. Therefore, where f is a non-dimensional function and Re = vlρm/η 2 26,31 the force is normally modeled as F = a0 +a2v , although is the Reynolds number. The force is best written as it can also have a linear term.4,12,29 A popular model ∗ 2 of cylindrical projectile penetration sets a2 = N BρmA, F = ρ A v C ( s , Re)/2, ∗ m D { } where the “nose factor” N is the only shape parameter and B 1 is an extra parameter of the target material.29 with a velocity-dependent drag coefficient. This coeffi- ≃ cient must have a limit when Re , so that η dis- In solids as well as in fluids, a supersonic body gener- appears from F , as corresponds to→ inertial ∞ motion. One ates a bow wave, which is a type of shock wave (discon- could think that there should be no friction when the vis- tinuity of the velocity field). In a bow wave, there is a cosity vanishes, but the transfer of momentum does not sharp rise of pressure at the bow of the body that weakens 31 vanish: it concentrates on a boundary layer of vanishing far from it and becomes an ordinary sound wave. The width around the body that gives rise to a fluid wake. wave carries away momentum of the body and generates For slow body motion, namely, for Re 1, the inertia resistance to its motion. A hypersonic bow wave front of the body is negligible in comparison≪ with the effect lies very close to the forward surface of the body, where the velocity field has a sharp discontinuity, such that the of viscosity, so that ρm disappears from F . This implies streamlines are bent almost tangentially to the surface, that CD 1/Re and ∼ like in Newton’s specific model of resistance.34,35 The F = ηlvf( s ), general characteristics of a shock wave discontinuity are { } given by the Hugoniot relation, in terms of the thermo- which is the general form of the Stokes drag law. In this dynamic properties of the medium.31,34 As regards these case, F is not very sensitive to shape, so f( s ) > 1 even properties, we can distinguish gases, in which the pres- when A l2 (being l the length of the body).{ } 32∼ sure has thermal origin, from condensed media, in which In compressible≪ fluids, we have an additional param- its origin is the short-range atomic repulsion. Gases are eter, the compressibility coefficient, or its inverse, the usually considered to be polytropic, with bulk modulus bulk modulus K. We neglect viscosity to simplify the proportional to pressure, namely, K = γP , where the analysis and characterize the fluid just by ρ and K. index γ is characteristic of the gas (1 < γ 5/3). In m ≤ Given that [K] = L−1 T−2 M, we must have F (v) = condensed media, K is large and is non null for vanish- 2 2 1/2 ρm l v f( s ,v/cs), where cs = (K/ρm) is the ve- ing P . The equation of state of solids is a subject of locity of{ sound} and plays the role of critical velocity. study.11,34 Some basic shock-wave physics is introduced When v cs (hypersonic motion), the resistance is in- in Sect. III C. ≫ 2 ertial and we can write F = (CD/2)ρmA v . At low ve- As already noticed, the inertial resistance to penetra- locity, we should obtain a v-independent force, namely, tion can be greatly reduced by shape optimization, with F = f( s )Kl2, which would result from the balance of the effect of raising the value of the critical velocity. In the fluid{ pressures} around the body. However, the fluid condensed media and, in particular, in solids, with large 31 is incompressible at low v and this force is null. values of cs, the critical velocities of optimized projec- Let us compare the critical velocities in incompressible tiles can be exceedingly large. For example, tests of the ∗ and compressible fluids, which are, respectively, η/(lρm) cylindrical projectile model with a2 = N BρmA against 6 experiments in which N ∗ < 0.2 and B =1.0 find a rather stresses can deform the projectile. Furthermore, its ma- 29 strong dependence of D on v0, while the asymptotic terial becomes plastic for stresses much smaller than its ∗ law (11) (with CD/2 = N B) predicts a weak depen- own K0 value (assumed to be very large). This pro- ∗ dence. The reason is that N is so low and hence vc so cess can involve fracturing, so the projectile can break in large that the asymptotic regime is not approached even fragments.9–15,38 The analysis of deep penetration into at the highest velocities tested. This situation is surely low-density materials agrees with the law (11) for im- β common and suggests that the empirical law D v0 and pact velocities below some critical velocity for fragmen- the corresponding force F v2−β, with β 2/∼3, result tation (a few km/s).38 Naturally, projectiles with optimal from numerical fits in velocity∼ ranges below≃ the asymp- shapes undergo much smaller stresses and their critical totic inertial regime. velocities for fragmentation are higher. Condensed media are strongly heated by strong shock waves and become, under the corresponding extreme C. The fate of a hypersonic projectile pressures, high-density gases. After a hypervelocity im- pact, the medium at the front of the projectile is com- The Hugoniot relation implies that mechanical energy pressed, heated and, if solid, fluidified, while it expands is dissipated in a shock wave, that is to say, is irreversibly and cools at its back. Simultaneously, a shock wave prop- lost. The ultimate reason for this irreversibility is the loss agates through the body backwards, until it reaches the of adiabaticity at the shock front, because the medium end and unloads. If the specific kinetic energy of the body cannot adjust to the rapid mechanical change (adiabatic is about ten times the heat of vaporization of its material, 36 then the unloading of the shock wave releases as much invariance is studied in mechanics ). As the medium is 34 set in motion, it gains kinetic and internal energy,37 but energy as to completely vaporize the body. This hap- the dissipation is measured by the increase of entropy. Its pens at impact velocities over 10 km/s. In general, a fast magnitude allows one to classify shock waves as weak or projectile must be considered as a bunch of individual strong.31,34 Weak shock waves involve small compressions atoms when its kinetic energy is much greater than the and are close to sound waves. Hypersonic penetration cohesive energy of those atoms. Heats of vaporization for generates strong shock waves, in principle. However, in solids are of several eV per atom, equivalent to velocities of several km/s [1 eV/atom = 9.65 104 J/mol, J/kg = the case of a slender projectile that disturbs the medium 2 · only slightly, the bow wave is strong only in the very (m/s) ]. Besides, hypersonic penetration can induce in- hypersonic limit. ternal changes in molecules or atoms, such as molecular excitation or dissociation, chemical reactions and even The rises of temperature and pressure in a bow wave 34,35 can be high enough to alter the projectile. In gases, the ionization. At much larger velocities, we can neglect main effects are thermal.34,35 For example, it is known the binding of electrons to atomic nuclei and consider the that a large fraction of the mass of a meteorite is lost penetration of high-energy charged particles. by ablation while traversing the atmosphere. A simple model of ablation9 predicts that the fraction of the mass of a meteorite lost while braking in air only depends on IV. MOTION OF FAST CHARGED PARTICLES 2 IN MATTER the initial velocity and is 1 exp( σv0 /2), where σ 10−8 kg/J. Therefore, the mass− loss− is considerable for≃ v0 = 10 km/s but negligible for v0 = 1 km/s. The crucial difference between the interactions of In a condensed medium, even a weak shock wave can macroscopic bodies and of charged subatomic particles generate high pressures, in spite of being almost adia- while moving through matter is the difference in inter- batic and isothermal. The process is ruled by the “cold” action range, which combines with the difference in size. equation of state of the medium, which can be derived Actually, the size of a material body is defined by the rel- from the intermolecular potential through the virial the- ative positions of its neutral atoms or molecules, which orem. For example, the power-law potential V (r) r−s are kept in place by low-energy and short-range electric (e.g., the repulsive part of the Lennard-Jones potential)∼ interactions with the nearby atoms or molecules similar s/3+1 gives P/P0 = (ρ/ρ0) and K = (s/3+1)P . This is to the interactions that they have with the particles of a polytropic equation with a large index (equal to 5 for the medium, whereas, at high energy, the charged sub- s = 12). Naturally, the attractive part of the potential is atomic constituents interact directly through the long- necessary to have non-null K = K . A suitable law range Coulomb force. 0 |P =0 is K = K0 + nP , in which n empirically goes from 4, for Therefore, the basic parameters change: the body size metals, to 7, for water, and K0 is given by the standard l is substituted by the charge Q, while the other two pa- speed of sound (cs = 1.5 km/s and K0 = 2.2 GPa for rameters of the object, namely, M and v0 stay the same. water, and larger for solids).34 For P K /n, K can Besides, we need some electrical magnitude to charac- ≪ 0 be taken constant (Hooke’s law). Even for P = K0/n, terize the medium. In an electrically neutral medium, a the compression ratio is small (10% for n = 7, e.g., for charged moving particle can only induce a relative dis- water), and so is the rise of temperature. placement of negative and positive charges, namely, it However, in the absence of notable thermal effects, the must induce currents or polarization. In fact, the dis- 7 placed charges are, mostly, mobile electrons. However, where f is an arbitrary non-dimensional function of its we do not have to specify the nature of the mobile parti- non-dimensional argument (chosen as Π1/2, to make it cles yet and we treat them as constituting a charged con- proportional to v0). tinuum defined only by its charge density, denoted by ρe. Naturally, e will eventually refer to the electron charge. The mechanical parameter of the medium, namely, its B. Physical meaning of the variables and mass density, is now ambiguous, because it may refer preliminary analysis to the total mass density of the neutral medium or just to the mass density of the particles associated with the We have found, in our simple model, a built-in charac- charge density ρe. We denote it by ρm, meaning the mass teristic velocity; namely, density of particles of mass m, without specifying their 1/3 2/3 nature yet. Q ρe vc = 1/2 . (18) In summary, we assume in our modeling that the char- ρm acteristic magnitudes are: M, Q, and v0, belonging to Therefore, a first-kind v0 asymptotics of Eq. (17) would the moving particle, plus ρm and ρe, belonging to the 2/3 2/3 give an even simpler model, namely, D˜ ρe /(Q ρ ). medium. All are positive, by default. We have one more ∼ m magnitude than in the case of macroscopic bodies. This formula is analogous to Eq. (3), with the length of the projectile replaced by the characteristic length lc = 1/3 (Q/ρe) (and no shape parameters). lc is the linear A. Dimensional analysis dimension of the volume of the medium that contains a total charge equal to Q. If we assume that the density of charge ρ corresponds to mobile electrons, then ρ = We redefine the electric charge to absorb the factor e e en, where e and n are the electron charge and number 4πǫ , namely, q/√4πǫ q, as in the Gaussian or 0 0 density, respectively. So we can write electrostatic unit systems,→ so that [q]=L3/2 T−1M1/2 1/3 −1/3 and we only have mechanical dimensions. To find lc = z n , D˜(Q, v ,ρ ,ρ ), we employ the standard method. Let 0 e m where z = Q/e and the length n−1/3 is the linear di- us write mension of the volume per electron or, loosely speaking, ˜ α β γ δ the interelectronic distance. For a subatomic particle, D(Q, v0,ρe,ρm)= Q v0 ρe ρm , (12) z > 1, and lc is not much larger than the interelectronic with unknown exponents. From [D˜]=LM−1, we obtain: distance.∼ ˜ 2/3 2/3 3α 3γ We can write the relation D ρe /(Q ρm) in a L: 1= + β 3δ; (13) more convenient form: ∼ 2 − 2 − D M e M e/m T: 0= α β γ; (14) = = , (19) − − − 3 α γ lc ∼ ρm l Q m Q/M M: 1= + + δ. (15) c 2 2 −1 − where m = ρm n is the mass of the volume that con- This is a linear system of 3 equations with 4 unknowns. tains one mobile electron. Assuming that ρm is just the The matrix of coefficients has maximal rank, as proved mass density of mobile electrons, m is the electron mass by computing the determinant of the first 3 3 matrix, and the quotient D/lc is the quotient of the electron corresponding to α, β and γ. Therefore, the× system is charge-to-mass ratio (the largest possible) by the inci- soluble and has an infinite number of solutions, given dent particle charge-to-mass ratio. There are two es- by a particular solution plus the general solution of the sentially distinct situations. For an incident electron, −1/3 homogeneous system. Since the matrix for α, β and γ is D lc = n , as is natural, because one electron is non-singular, we solve the homogeneous system for them, likely≃ to lose a large fraction of its momentum in the finding α = 2δ/3, β = 2δ, γ = 4δ/3. Therefore, the collision with another electron. In contrast, an incident non-dimensional− parameter (pi-group)− is atomic nucleus has a much smaller charge-to-mass ratio, so D lc. Indeed, the nucleus transfers a small fraction v2 ρ ≫ Π= 0 m . (16) of its momentum in the interaction with one electron and 2/3 4/3 Q ρe has to interact with many electrons to lose its initial mo- mentum. Therefore, its motion is smooth and can be de- As a particular solution of system (13,14,15), we prefer scribed by the differential equation (1). This description the one with β = 0, to obtain an expression of D˜ that constitutes the continuous slowing down approximation does not contain v0. The corresponding determinant is (CSDA). non-vanishing and we find: α = 2/3, γ =2/3, δ = 1. − − To obtain an estimate of D from (19), we need to know In conclusion, lc, that is to say, the interelectronic distance. In con- 2/3 1/2 densed media, this distance is given by the atomic di- ρe v0 ρm mensions. Even assuming that D can be several thou- D˜(Q, v0,ρe,ρm)= f , (17) 2/3 1/3 2/3 Q ρm Q ρe ! sand times larger than lc, the penetration depth of a fast 8 atomic particle in a condensed medium should be macro- density n of conducting media (on Earth) varies in a wide scopically negligible. This conclusion invalidates the hy- range, from 109/m3 (in the ionosphere) to 1029/m3 (in pothesis of a first-kind asymptotics for large v0 (even with metals). Although we are not only concerned with con- the allowance for a logarithmic correction). ducting media, there is no definite distinction between Let us instead assume asymptotic similarity, for v conductors and dielectrics in their response to the rapidly 0 ≫ vc; namely, varying electric field produced by a fast particle, because all electrons respond as free electrons to very high fre- 2/3 1/2 β quency fields. To analyze the mechanism of energy loss, ˜ ρe v0 ρm D(Q, v0,ρe,ρm) (20) we need a detailed electrodynamic model. 2/3 1/3 2/3 ∼ Q ρm Q ρe !

[which is just (12) with all the exponents expressed in V. ELECTRODYNAMIC MODEL OF THE terms of β by solving Eqs. (13,14,15)]. We can place re- RESISTANCE FORCE strictions on β by reasoning on how D˜ should depend on Q, v ,ρ ,ρ . First, it is necessary that β 0, if D is 0 e m ≥ The theory of the stopping of fast charged particles in not to shrink as v0 grows. Furthermore, β > 1, if we assume that D is to shrink when ρ grows and so does matter is well established. In the macroscopic theory, the e energy of the particle is lost in polarizing the medium, the interaction. The behavior of D with respect to ρm depends on whether β is smaller or larger than two, but whereas in microscopic models the particle interacts with a distribution of mobile electrons. We rely on two classic this is a moot point. A definite value of β is obtained 41 by assuming that ρ corresponds to a given type of con- textbooks of electrodynamics: Landau and Lifshitz’s, e which concisely explains the macroscopic theory, and stituent charges, which we naturally take as electrons: 42 given that the electric force exerted by every electron on Jackson’s, which focuses on the microscopic processes. the charge Q is proportional to eQ and ρ = en, D˜ must For a recent and comprehensive reference on the subject, e see Sigmund.22 be a symmetric function of Q and ρe, which implies that 2/3 β/3=2/3 2β/3. Therefore, β = 4. This is the Before looking into the details, let us confirm that β = −right value− for a fast− moving particle, as shown in detail 4 in the asymptotic law (20). Let us consider that the resistance force is just a self force, namely, a force exerted in Sect. V. 41 To find the slow-motion asymptotics, we can try keep- on the particle by its own electric field. Because the ing the power law form (20) and looking for a first-kind electric field of the particle is proportional to Q, F must be proportional to Q2. From this condition and Eq. (2), asymptotics in some variable other than v0. The condi- ˜ 2 tion β 0 forbids that Q disappears from D˜ but we can we deduce that D is inversely proportional to Q and, ≥ try ρm or ρe. The former disappears if β = 2, and the according to (20), latter if β = 1, making D˜ quadratic or linear in v , re- 0 4 spectively. These two cases correspond to stopping forces ˜ ρm v0 D 2 2 . (23) constant or linear in velocity, as shown in Sect. III A. A ∼ Q ρe slow-motion linear force, like in fluids, seems more plausi- 1/2 Correspondingly, ble. Substituting β = 1 in (20), we have D˜ v /(Qρm ), ∼ 0 which implies 2 2 2 4 Q ρe z e n F 2 = 2 . (24) F Qρ1/2v. (21) ∼ ρmv mv ∼ m In Sect. V B, we explain how such force arises. These asymptotic laws are valid for fast particles, namely, for v v = l ω , which is related to Landau To unveil the physical meaning of vc, let us note that ≫ c c p it is, according to Eq. (18), the quotient of the length and Lifshitz’s condition for their macroscopic treatment, 1/3 1/2 namely, v aω0, where ω0 is “some mean frequency cor- lc = (Q/ρe) by the time ρm /ρe. Instead of this time, ≫ 1/2 responding to the motion of the majority of the electrons we can consider its inverse, ρe/ρm , as a characteristic in the atom” and a is the interatomic distance. frequency of a uniform charge distribution. This fre- quency corresponds to the plasma frequency ωp of the dis- 39,40 tribution of free electrons in a conducting medium, A. Calculation of the resistance force and the namely, penetration depth 4πe2n 2 41 ωp = , (22) Landau and Lifshitz express the self force as an in- m tegral in Fourier space. For a point-like particle in the as seen by putting ρe = en and ρm = mn. The plasma vacuum, this integral diverges for large wave numbers, frequency characterizes collective oscillations of the elec- but the result must vanish by symmetry (isotropy). This trons about their rest state, which can indeed absorb holds for a particle moving in a medium with constant energy from a traversing particle.39,40 The free electron electric permittivity, but, when the permittivity depends 9 on the frequency (as it must), the divergent integral can long compared to the characteristic period of its mo- be regularized to yield:41 tion, the response of the medium is adiabatic, that is to say, reversible, and no energy is transferred (however, 4πz2e4n k v F (v)= ln 0 , (25) see Sect. VB). In the opposite case, the electron can be m v2 ω¯ considered free and at rest. Therefore, we can take −1 where k0 is a transversal wave-number cutoff (k0 a bmax v/ω.¯ (29) for the macroscopic treatment to be valid), and≪ω ¯ is a ≃ mean electronic frequency, defined in terms of the per- If we further makeω ¯ me4/¯h3 (the Bohr frequency), ≃ 2 2 mittivity. Eq. (25) agrees with the law (24), up to the the ratio of bmax to the Bohr radius, a0 = ¯h /(me ), is soft v-dependence in the logarithmic factor. F is actu- approximately the ratio of v to the Bohr velocity e2/¯h. ally due to the imaginary part of the permittivity, which This ratio is bounded above by ¯hc/e2 = α−1, the inverse represents the absorption of electromagnetic energy.39–42 of the universal constant α = e2/(¯hc)=1/137 (the fine- This process is different in conductors and in dielectrics structure constant). Therefore, bmax < a0/α = 137a0 at low frequencies but we can assume, for very fast parti- (neglecting relativistic effects, of course). However, in cles, that ωp ω¯ and that they both are close to the basic a condensed medium, an atomic monolayer of linear di- ≃ 4 3 4 atomic frequency, namely, the Bohr frequency me /¯h . mension 100a0 contains of the order of 10 atoms and Jackson42 considers the stopping force as the result more mobile electrons. of the collisions of the incident particle with individual Let us now turn to bmin in Eq. (28). For soft collisions, electrons. Assuming the CSDA to be valid, the stopping such that the momentum transfer to every electron is force produced by a uniform distribution of particles is small, that is to say, for bmin =1/k0 b0, ≫ equal to the energy transferred to them per unit length. 2 4 2 4 The energy transferred to the particles in an element of z e bmax 4πz e n k0 v F 4πn 2 ln = 2 ln , volume, if the particles are independent, is given by the ≈ mv bmin m v ω¯ energy transfer T to one particle times the number of which coincides with Eq. (25). If we keep b = v/ω¯ but particles, n dS dx. Therefore, max integrate down to bmin = 0, including hard collisions, the dE argument of the logarithm is larger, namely, F = = n T dS. (26) dx 2 4 2 4 3 Z z e bmax 4πz e n mv F 4πn 2 ln = 2 ln 2 . (30) We can identify this formula with Eq. (9). Indeed, if ∆p ≈ mv b0 m v ze ω¯ denotes the momentum transfer to one particle, Hence, we obtain the penetration depth: 2 2 Mv (Mv ∆p) v0 v0 3 T = − v ∆p = m v ∆vk . ˜ v dv m v dv 2 − 2M ≈ · D = 2 4 3 2 . (31) 0 F (v) ≈ 4πz e n 0 ln[mv /(ze ω¯)] It is to be remarked that collisions with free electrons Z Z are elastic and T fully transforms into kinetic energy, so The integration over v cannot be carried out analytically, 2 2 2 unless one ignores the logarithmic factor, whence the in- that T = (∆p) /(2m) = m[(∆vk) + (∆v⊥) ]/2, unlike in Sect. III. tegration is trivial and agrees with (23). The logarith- The collision is best studied in the rest frame of the in- mic factor actually makes the integral divergent, because 1/3 cident particle.42 The energy transfer for Coulomb scat- of the pole at v = ze2ω/m¯ . After identifyingω ¯ tering is with ωp, this velocity is essentially the vc in Eq. (18),
as shown by making Q = ze, ρe = en, ρm = mn, and 2 4 2z e 1 using Eq. (22). So let us also identify vc with the value T (b)= 2 2 2 , (27) mv b + b0 of v at the pole. Of course, Eq. (30) is only valid for v v , which means that the pole is irrelevant. To cal- 2 2 43 c where b is the impact parameter and b0 = ze /(mv ). culate≫ the integral, let us first make the change of variable Because T (b) = c(b) mv2, the total stopping force, as 4 4 r = v /vc . Then, given by Eq. (26),6 is not proportional to v2, unlike in Newton’s theory, Eq. (10). Writing dS = 2πbdb and v0 v3 dv v4 r0 dr = c , carrying out the integration over b, one finds that ln(v3/v3) 3 ln r Z0 c Z0 bmax 2 4 2 2 4 4 z e bmax + b0 where r0 = v0/vc 1. The integral over r is the logarith- F = n T (b)2πbdb =2πn 2 ln 2 2 . (28) ≫ bmin mv bmin + b0 mic integral. Its asymptotic expansion for r0 1 can be Z obtained by repeated integration by parts, but≫ it is only In this context, the Coulomb interaction is of long range justified to take the first term, because (30) is already an because F diverges if b . max → ∞ asymptotic formula. Therefore, we write: The value of bmax is determined by Bohr’s adiabatic- 22,42 4 4 ity condition: when the time of passage of the in- ˜ m vc r0 mv0 D 2 4 = 2 4 . (32) cident particle by one electron, of the order of b/v, is ≈ 4πz e n 3 ln r0 48πz e n ln(v0/vc) 10

This is the best result achievable, but it is sufficiently In turn, the force produced by the electrons with velocity accurate. We can compare Eq. (32) with the general ve is given by an integral over the relative velocity with form in Eq. (17) and deduce that f(x) x4/ln x. respect to the particle. Gryzi´nsky calculated the latter So far, we have neglected quantum effects,∼ which ap- integral and obtained a complex but explicit formula.44 pear when the distance of closest approach is smaller However, this formula is not suitable for integrating over than the de Broglie wave-length of the electron. If b0 < ve. One can proceed by splitting the integral over ve h/(mv), we should replace bmax/b0 by bmax/[h/(mv)] = into the regions ve < v and ve > v and using approx- 2 mv /(hω¯) in Eq. (30). The exact value of the argumentof imations of the integrand for, respectively, ve v and the logarithm can be obtained with a semiclassical treat- v v (obtained from Gryzi´nsky’s formula or≪ directly e ≫ ment, including momentum transfers up to ¯hqmax =2mv, from the integral over relative velocities). If v is beyond 41,42 which yields: the range of ve [the support of n(ve)] or, at any rate, is much higher thanv ¯e (the rms ve), then the integral in 2 4 2 4πz e n 2mv the region ve > v is negligible while the integral in the F (v)= 2 ln . (33) m v ¯hω¯ region ve < v recovers Eq. (30). Of course, we are now interested in the opposite case, If we define the new characteristic velocity vq = namely, v v¯e . It yields a force proportional to v, in ac- [¯hω/¯ (2m)]1/2, then the argument of the logarithm is ≪ 3 2 cord with (21), but it contains the factor f(¯ve/vc), with (v/vc) in the classical case and (v/vq) in the quantum- f(x) x−3 ln x. Notice that the exponent 3 implies mechanical case. The transition from the first to the sec- that F∝ is actually proportional to Q2ρ2, in spite− of ap- 3 > 2 e ond takes place when (v/vc) (v/vq) , namely, when pearing to be just proportional to Q in (21). The value > 2 ∼ v ze /¯h. ofv ¯ is given by the properties of the medium, in terms ∼ e The calculation of the penetration range corresponding of the type of electronic distribution. For the Maxwell- to Eq. (33) yields: Boltzmann distribution of a classical plasma,v ¯e is given by the temperature. For the Fermi-Dirac distribution mv4 D˜ = 0 . (34) of a degenerate electron gas, which is more relevant in 32πz2e4n ln(v /v ) 0 q our context,v ¯e can be replaced by the Fermi velocity, given by the total electron number density n. Then F The maximal penetration, for v0 close to c but before follows the Fermi-Teller formula,44 which, curiously, is es- relativistic effects take hold, is sentially independent of n. Other formulas for F , namely, 22,23 Mmc4 the Firsov and the Lindhard-Scharff formulas, de- Dmax = rived in terms of low-v particle-atom collisions, yield pro- 16πz2e4n ln[2mc2/(¯hω¯)] portionality to n, but they qualitatively agree with the M −3 = . Fermi-Teller formula if n a0 . 2 4 2 2 ≃ 16πz m n α a0 ln[2mc /(¯hω¯)] The argument of the logarithm for high v, namely, v/v , is replaced byv ¯ /v for low v, which suggests that Using the Bohr frequency forω ¯, ln[2mc2/(¯hω¯)] = c e c v¯ /ω is the effective range in the low-v limit. In fact, this ln(2α−2) = 10, which is the maximal value of the log- e p length reproduces the screening length of a static electric arithmic factor. With this value, we obtain charge, namely, either the Debye screening length of a

Dmax M thermal plasma or the Thomas-Fermi screening length =7 105 . (35) 40 a · z2 m (na3) of a Fermi gas. If the electron velocity and frequency 0 0 take the Bohr values, then the screening length is the 3 Bohr radius. Therefore, a slow atomic nucleus in a cold In a dense medium, na0 1. Then, for a proton, with ≃ 9 medium actually becomes a positive ion and eventually a z = 1 and M/m = 1800,Dmax 1.3 10 a0 0.1 m. For a relativistic particle, ignoring≃ the small· change≃ of the neutral atom. This occurs in a range of v given by the or- logarithmic factor, F = 40πz2e4n/(mc2) and penetration bital speeds of the electrons in the ion (Bohr’s screening 22,23 depth is proportional to kinetic energy. criterion). A slow-moving particle transfers energy not only to the electrons but also to the atomic nuclei. At high ve- 2 B. Low-velocity particle locity, F is inversely proportional to mv , where m is the mass of the interacting particles of the medium, so F is certainly due to the electrons. But F is maximal A particle with v < vc, that is to say, with bmax < 2 1/3 b , adiabatically decouples from the electrons in the at v vc = ze ω/m¯ , and becomes proportional to 0 ≃ medium. However, there is a residual “viscosity” force m1/2v for lower v (F always depends on m1/2v, in accord
of the form (21), due to the interaction with the least with Sect. IIA). For decreasing v < vc, the much smaller bound electrons. The precise form of F (v), according to force due to the nuclei eventually takes over, because it Gryzi´nsky,44 is given by the force produced by the elec- is still inversely proportional to mv2, where m is now the trons with velocity v on a particle with velocity v in- nucleus mass. This occurs at v 500 km/s. However, e ≃ tegrated over the electron velocity distribution n(ve) dve. in that range of v, the charge is screened, turning the 11 long-range Coulomb interaction into a short-range inter- the maximum of the short-range atomic force and the action. absolute maximum of the total force is complex. A projectile that consists of a cluster of atoms (or even a single atom) is called a nano-projectile. If we assume VI. NANO-PROJECTILES that the cluster is spherical and its collisions with the atoms of the medium are elastic, we can relate the col- As seen above, the velocity of the particle determines lision potential to the atom-atom collision potential. In s −s the type and range of its interaction with the medium. particular, we can use V (r)= V0 a (r R) for r > R − In particular, as a slow atomic nucleus becomes a neutral and for r R, where R is the cluster radius. As is nat- ∞ ≤ atom, its initial charge Q vanishes and its initial negligi- ural, the collisions are harder for larger R, in a relative ble size grows to the atomic size, which is not negligible. sense. To obtain a quantitative result, we can approxi- The variable ρe loses significance and, at the same time, mate the preceding V (r) by a power law, by means of a the total mass density of the medium becomes the rele- linear expansion of ln V at r = R + a in terms of ln r, vant density, because most of the energy and momentum which yields of the projectile are transferred to the nuclei rather than s(1+R/a) to the electrons. Moreover, the reduced interaction range V (r)/V0 (a/r) makes the interaction similar to a macroscopic contact ∝ repulsive interaction. Therefore, the parameters of a suf- (the error is less than 10% for 0.2 < V/V0 < 5). There- ficiently slow nucleus are the same macroscopic param- fore, for R a, ≫ eters employed in Sect. II. There must be a transition, F v2−β, with β 1/R. ruled by the screening process, from the linear force law ∼ ∼ for v < vc to a quadratic force for small v. In fact,∼ the force produced by atomic hard-sphere col- This model can be generalized to non-spherical clusters, lisions is quadratic in v, according to the CSDA and provided that the cluster size is much larger than the Eq. (26), because T = (∆p)2/(2m) and ∆p mv for atomic force range, and can also be generalized to colli- a hard-sphere collision, although it depends∝ on b (m sions with molecules instead of atoms. is the reduced mass). However, the real atomic repul- However, the assumption of elastic collisions is ques- sion has short but non-vanishing range, and the de- tionable: the collisions can induce relative motions of the pendence of ∆p on v deviates from linearity. A bet- atoms in the cluster. For small clusters, this process is ter model of atomic repulsion is the power-law potential similar to the excitation of molecular vibrations by the s collisions of molecules in a hot gas.34,35 A vibration of fre- V (r) = V0 (a/r) , already employed in Sect. III C. In 2 general, ∆p =2p cos ψ, where ψ is the angle between the quency ω is not excited while mv /2 < ¯hω 1 eV. Even for mv2 ¯hω, the probability of excitation≪ is small, as direction of the incoming particle and the direction of the ≫ closest approach point.36 This angle is given by an inte- long as the collision is quasi-adiabatic, namely, as long as v aω (which is a few percent of the Bohr veloc- gral over the distance r, which can be expressed for the ≪ power-law potential in terms of the nondimensional vari- ity). At larger velocities, the atomic configuration of the s 2 2 2 cluster can change and, when the collision energy reaches able u = (a/b) V0/(mv ), so that T = 2mv cos ψ(u). The change of integration variable from b to u in Eq. (26) the cohesive energy per cluster atom, one of them can be gives the form of F , namely, knocked off. For clusters with many atoms and many possible vibra- 2 1−2/s 2/s 2 tion modes, the collision is best described as an impact F n (mv ) V0 a . ∝ on a quasi-macroscopic cluster, although the velocity and Notice that F is linear in the number density n but not “density” of the projectile are insufficient for penetration in ρm; but, as s grows and the potential gets harder, and the result is just inelastic scattering. As the force be- F tends to be linear in ρm and actually becomes the tween cluster and atom of the medium, we can employ quadratic Newton force. Let us consider a realistic interatomic potential, f = V ′(r) r/r g(r)v, namely, the “universal” ZBL potential, which provides − − a good description in a broad range of interaction with a friction term to account for the inelasticity [g(r) > energies.23 Although it can only be handled numerically, 0]. To find the effect of a collision ruled by this force, we an accurate analytical fit of the resulting F (v) is avail- need to calculate the longitudinal transfer of momentum, 2 1−0.21 able. For small v, F (mv ) , which agrees with ∆pk = m ∆vk. Unfortunately, we cannot do this analyt- the result for the power-law∼ potential with s =9.4. F (v) ically, except in very simplified cases, e.g., in a head-on has a maximum for a value of v that depends on the atom collision and with restrictions.45 At any rate, this simpli- types and is in the range of hundreds of km/s. For large fied calculation shows that ∆vk is not proportional to the v, F v−2 ln v, due to the nuclear repulsion. However, initial relative velocity v and Newton’s theory fails. In then∼ there are, in addition to nuclear interactions, the general, the friction term is negligible for low v and the nucleus-electron interactions that give rise to electronic collision is elastic; whereas, during a collision at high v, viscosity. The dynamics in the velocity range between the friction dominates until the relative velocity becomes 12 low enough for the conservative part of f to come into molecular mean free path is macroscopically large, ρm is action, but by then most of the kinetic energy has been so small that the heating is not significant. Nevertheless, dissipated. For even higher v, the collision becomes, in cold sputtering can be significant in long duration space- 20 practice, totally inelastic, and ∆vk v, in accord with craft missions. Of course, thermal ablation is impor- Newton’s theory (see Sect. IIIB). However,≈ energetic col- tant at lower altitudes, for meteorites or for spacecraft lisions cause atom displacements (plasticity) and sputter- at reentry (its major effect on meteorites is mentioned ing (ablation). in Sect. IIIC). In the case of atom clusters, the energy Although we have considered inelasticity as a sink of transfer is also of thermal nature, provided that local kinetic energy, it can also be a source, because a col- thermal equilibrium holds inside the cluster (a condition lision can transform cluster internal energy into kinetic less strict than that it hold outside). Needless to say, energy. For example, molecular vibrations in a hot gas a cluster penetrating in a dense medium can reach high are in thermodynamic equilibrium with the Maxwell- temperatures, although the effects of stress are surely Boltzmann velocity distribution.34,35 In our case, the more important (Sect. III C). cluster sees an essentially unidirectional velocity distri- bution. Nevertheless, in a stationary situation, the tran- sitions between vibrational states are ruled by a master A. Critique of reports of molecular dynamics equation, and therefore the distribution of these states simulations evolves towards a stationary distribution.33 In this dy- namical equilibrium, the internal energy of the cluster Most studies of atom cluster penetration in solids concentrates on its frontal part. The kinetic energy rely on molecular dynamics simulations. Employing this can grow in an individual collision ruled by the dissi- method, some authors47 conclude that D is quadratic in pative force f because f must be complemented with a 48 v0 and others that it is linear, corresponding to a con- stochastic or fluctuating component, in accord with the stant or linear force, respectively (Sect. IIIA). However, fluctuation-dissipation theorem. Therefore, the result of their data fits are questionable (e.g., Fig. 3 of Ref. 47 a collision is not determined by the initial conditions, and and Fig. 3 of Ref. 48). Moreover, the penetration depths ∆vk is a random variable, with a stationary probabil- only reach a few nm. The simulations by Anders and ity distribution in the dynamical equilibrium. However, Urbassek49,50 of the stopping of heavy clusters in soft if ∆vk v for the corresponding elastic collision, also targets (Au clusters in solid Ar) reach depths of tens of ∆v ∝v. h ki∝ nm. These simulations span a range of energy per clus- At any rate, the hypothesis of binary encounters be- ter atom between 10 and 1000 eV, equivalent to a range tween the cluster and the atoms of the medium only ap- of mv2/2 between 2 and 200 eV. Measures of the initial plies to small clusters, in the following sense. We recall force find that it is nearly linear in v2. To be precise, from Sect. III B that the transition from the continuum the power-law exponent of v2 grows with cluster size, be- 49 to the molecular-kinetic description of penetration in a ing, e.g., 0.87 for Au43 and 1.01 for Au402. For given v, fluid is determined by the ratio of the body length l to the the force depends on the number of atoms in the cluster molecular mean free path λ. Given that a solid medium as N 2/3; that is to say, it is proportional to the cross- responds as a fluid in penetration at high v, the same cri- sectional area (although the exponent of N grows with v terion applies. Notice that the mean free path is indepen- and tends to 1 for large v). dent of the temperature and only depends on the number The simulations of self-bombardment (with cluster and density and interaction cross-section of the molecules of target atoms of the same type) by Anders et al51 ob- the medium.33 When a l λ, the binary-encounter tain an F versus v2 exponent α(N) that starts at 0.6 ≪−1 ≪ model and the law β l hold. However, when l λ, for N = 1 and tends to 1 for large N. Although self- ∼ ≫ the cluster affects simultaneously many atoms, justify- bombardment does not produce deep penetration, it is ing an approach based on the hypothesis of local ther- fine for studying the initial force, and these results for mal equilibrium.33 Then, the nano-projectile generates a α agree with the preceding results in Ref. 49. Anders mesoscopic version of the macroscopic bow wave studied et al51 state that “no theoretical argument exists regard- in Sect. III C. In solids, it is normal to describe meso- ing why the interaction of a cluster with a solid can be scopic penetration in terms of a thermal spike,23 but it described by an energy-proportional stopping for large can also be described in terms of a shock wave.46 cluster sizes.” Our model predicts precisely that the ex- Macroscopic projectiles in rarefied gases can also be ponent α(N) is, for N = 1, sensibly smaller than one “small” in comparison with the corresponding molecu- (α < 0.8) and tends to one for large N. Furthermore, lar mean free path; e.g., meteorites or spacecraft at high the∼ model (in a certain approximation) predicts that the altitudes. In this context, the preceding issues, namely, difference, 1 α, decreases as N −1/3. − type of drag, conversion of kinetic to internal energy, ab- Finally, let us comment on a theoretical analysis of lation, etc, have been much studied.9,20 In steady hyper- cluster implantation made by Benguerba.52 This analysis sonic motion, F is quadratic in v and the heat trans- has broad scope, in spite of being restricted to graphite fer to the body, as a fraction of the power F v, is pro- targets. Benguerba assumes that the dynamics is ruled 3 portional to ρmv . However, at altitudes such that the by the displacement of target atoms caused by a “gener- 13 ated wave,” so that, if ct is the wave speed, the medium related to the plasma frequency. Assuming asymptotic mass dm = ρmActdt is displaced with the velocity v of similarity for fast motion and considering the stopping the projectile, and hence F = v dm/dt = Aρmctv; that is force as an electric self force, we can determine its general to say, the force is linear. Although Benguerba does not form, in particular, that it is inversely proportional to v2, specify the type of generated wave, it must be a shock and hence that the penetration depth is proportional to 4 wave, and his model indeed corresponds to the genera- v0. These velocity dependences miss a logarithmic factor, tion of a weak plane shock wave by an impulsive load and which can be derived only with detailed electrodynamic its propagation at a speed close to the speed of sound.34 models. A macroscopic model that includes the electric However, a penetrating supersonic projectile generates permittivity of the medium yields the logarithmic factor, a bow wave with a shock front that propagates at the with a mean frequencyω ¯ that replaces ωp. An atom- same velocity of the projectile. Therefore, ct must be istic model, which considers the transfer of momentum identified with v in Benguerba’s formula and F is actu- to electrons down to the smallest scales, only produces a ally quadratic in v. slightly larger logarithmic factor (generally smaller than 10 for non-relativistic particles). 2 1/3 The characteristic velocity vc = (ze ω/m¯ ) is, in VII. SUMMARY AND CONCLUSIONS practice, of the order of magnitude of the Bohr veloc- ity (e2/¯h = 2200 km/s). Projectiles with velocities in We have presented a unified formalism for projectile the range 10 – 2000 km/s are in the transition from penetration that encompasses the full ranges of projectile the inertial fluid-like resistance described by the parame- sizes and velocities. As regards macroscopic projectiles, ters M,l,ρm, and caused by short-range interactions with Newton’s inertial force of resistance, with its quadratic the atoms of the medium to the electric resistance de- dependence on velocity, constitutes a universal law, be- scribed by the parameters M,Q,ρe,ρm, and caused by cause it only contains the basic mechanical variables, long-range interactions with the electrons of the medium. namely, the size and velocity of the projectile and the In both cases, the resistance force arises from the con- density of the medium. In correspondence with New- tinuous transfer of momentum from the projectile to the ton’s force, the universal formula for penetration depth medium, and Eq. (9) is always valid. The initial manifes- is Gamow’s formula. However, Gamow’s formula does tation of the long-range electronic interaction, for veloci- not follow from Newton’s force. The formula that actu- ties in the range 100 – 2000 km/s, is as an “electronic vis- ally follows from Newton’s force contains a logarithm of cosity” force. Naturally, when v approaches 2000 km/s, the velocity and, therefore, introduces a critical velocity, the projectile can only consist of one bare atomic nu- showing that Newton’s inertial force is only valid for fast cleus. Indeed, an atom cluster at that velocity rapidly projectiles. fragments and the resulting atoms are fully ionized. The The critical velocity must involve some variable that force on an atom cluster in the range 1 – 100 km/s is contains the time dimension and is associated to the basically given by the inertial Newton’s law, as long as mechanism of resistance at low velocity. The resistance of the cluster is stable. However, for nano-projectiles with incompressible fluids turns from viscous to inertial as the diameter l smaller than the mean free path of the par- Reynolds number grows. Compressible fluids or solids ticles of the medium, the force is not exactly quadratic but is F v2−β, with β small and vanishing as l−1. offer inertial resistance to supersonic motion, which in- ∼ volves bow waves. At velocities of some km/s, the in- From a fundamental standpoint, the resistance to a tegrity of macroscopic projectiles can be seriously com- projectile traversing a medium takes place because of the promised by the effects of high pressures, in condensed irreversible transfer of momentum (or kinetic energy) to media, or by the effects of high temperatures, at some- the medium. The main cause of it is a non adiabatic in- what larger velocities, in any media. The maximal veloc- teraction. For example, non-adiabaticity manifests itself ity depends on several factors, such as the strength and as an increase of entropy in the bow wave generated by shape of the projectile, but it is generally lower than 10 a supersonic projectile. The non-adiabatic interaction of km/s. The lower end of the quadratic force range is very a fast charged particle manifests itself in the irreversible variable, especially, in fluids, which offer subsonic iner- transfer of energy to the high-frequency modes of elec- tial resistance. The key factors for deep penetration are, tronic motion, which results in a resistance force that is first, a large ratio of projectile density to medium density large but is such that it decreases with velocity, allowing and, second, a streamlined shape. The impact velocity for considerable penetration depths. For velocities below is not important in the inertial regime, but the projec- the Bohr velocity, the electronic interactions become adi- tile must be strong enough to withstand the generated abatic, but there can be non-adiabatic transfer of energy pressure and temperature. to low-frequency atomic motions. Nevertheless, there is The stopping of fast charged subatomic particles in also “electronic viscosity.” This type of viscosity as well matter is due to the long-range Coulomb force. The di- as the ordinary viscosity in fluid motion are forms of irre- mensional analysis of the most basic model unveils, be- versible transfer of momentum that belong to the general sides a characteristic length, a built-in characteristic ve- class of transport processes. locity, and hence a characteristic frequency, which can be A few final remarks of practical nature. Gamow’s for- 14 mula is, in fact, a sensible and quick rule for estimat- etration depth for a given velocity v0 is obtained from it 4 ing the penetration depth of a fast projectile of generic by multiplying it by (v0/c) (ignoring the small change shape. For non-relativistic subatomic particles, formula of the logarithmic factor). The penetration depth of rel- (35) gives the maximum penetration depth, and the pen- ativistic particles is proportional to their kinetic energy.

∗ Electronic address: [email protected] 20 A.C. Tribble, The Space Environment. Implications for 1 I. Newton, Principia Mathematica, Book 2, translated into Spacecraft Design (Princeton Univ. Press, 2003). English by Andrew Motte and revised by Florian Cajori 21 N. Bohr, “The penetration of atomic particles through (University of California Press, 1934) [The original was matter,” Mat. Fys. Medd. Dan. Vid. Selsk. 18 (8), 1–144 published in Latin in 1687]. (1948). 2 B. Robins, New Principles in Gunnery, second edition 22 P. Sigmund, Particle Penetration and Radiation Effects (Wingrave, London, 1805) [First printed in 1742]. (Springer, 2006). 3 L. Euler, The true principles of gunnery, translated into 23 M. Nastasi, J.W. Mayer and J.K. Hirvonen, Ion-solid inter- English by Hugh Brown (Nourse, London, 1777) [The orig- actions: Fundamentals and applications (Cambridge Uni- inal was published in German in Berlin in 1745]. versity Press, 1996). 4 J.V. Poncelet, Introduction a la M´ecanique Industrielle, 24 J. Samela and K. Nordlund, “Atomistic Simulation of second edition (Thiel, Metz, 1839) [in French]. the Transition from Atomistic to Macroscopic Cratering,” 5 M.H. Resal, “Sur la p´en´etration d’un projectile dans les Phys. Rev. Lett. 101, 027601 (2008). semi-fluides et les solides,” Comptes rendus 120, 397–401 25 C. Anders et al, “Why Nanoprojectiles Work Differently (1895) [in French]. than Macroimpactors: The Role of Plastic Flow,” Phys. 6 G. Gamow, The Great Physicists from Galileo to Einstein Rev. Lett. 108, 027601 (2012). (New York, Dover, 1988) [reprint of The Biography of 26 G.I. Barenblatt, Scaling, self-similarity, and intermediate Physics (New York, Harper & Row, 1961)], pp 65–66. asymptotics (Cambridge University Press, 1996). 7 Wikipedia’s entry on Impact depth features Gamow’s for- 27 D.I. Goldman and P. Umbanhowar, “Scaling and dynamics mula but also attributes it to Newton. Saslow and Lu no- of sphere and disk impact into granular media,” Phys. Rev. tice in Ref. 8 that Newton did not actually obtain the E 77, 021308 (2008). penetration depth and show that Gamow’s formula does 28 See, for example, Wikipedia’s entry on Depleted Uranium. not follow from Newton’s results. 29 Q.M. Li and X.W. Chen, “Dimensionless formulae for 8 W.M. Saslow and H. Lu, “Newton on objects moving in a penetration depth of concrete target impacted by a non- fluid—the penetration length,” Eur. J. Phys. 29, 689–696 deformable projectile,” Int. J. Impact. Eng. 28, 93–116 (2008). (2003). 9 V.A. Bronshten, Physics of meteoric phenomena (Dor- 30 See the Scholium to Proposition 34, Sect. 7 of Newton’s drecht, Reidel, 1983). Principia, Ref. 1. It is the earliest solution of a problem by 10 J.A. Zukas, High Velocity Impact Dynamics (Wiley, 1990). the calculus of variations, but Newton did not present the 11 R. Kinslow, High Velocity Impact Phenomena (Elsevier, details of the calculation there, although he did it later in 2012). private correspondence [S. Chandrasekhar, Newton’s Prin- 12 G. Ben-Dor, A. Dubinsky and T. Elperin, High-speed pen- cipia for the Common Reader (Clarendon Press, Oxford, etration dynamics (World Scientific, 2013). 1995), chapter 26]. 13 M. Iskander, S. Bless and M. Omidvar, Rapid Penetration 31 L.D. Landau and E.M. Lifshitz, Fluid Mechanics, vol. 6 into Granular Media (Elsevier, 2015). of Course of Theoretical Physics, 2nd edition (Pergamon 14 J.C. Ruiz-Su´arez, “Penetration of projectiles into granular Press, 1987). targets,” Rep. Prog. Phys. 76, 066601 (2013). 32 E. Loth, “Drag of non-spherical solid particles of regu- 15 Y. Sun, C. Shi, Z. Liu, and D. Wen, “Theoretical Research lar and irregular shape,” Powder Technol. 182, 342–353 Progress in High-Velocity/Hypervelocity Impact on Semi- (2008). Infinite Targets,” Shock and Vibration 2015 Article ID 33 F. Reif, Fundamentals of Statistical and Thermal Physics 265321 (2015), doi:10.1155/2015/265321. (McGraw-Hill, 1965). 16 K.B. Hayashida and J.H. Robinson, “Single wall penetra- 34 Ya.B. Zel’dovich and Yu.P. Raizer, Physics of Shock Waves tion equations,” NASA Technical Memorandum, NASA and High-Temperature Hydrodynamic Phenomena, Vol. I TM-103565 (1991). and II (Academic Press, New York, 1967). 17 M. Landgraf, R. Jehn, W. Flury and V. Dikarev, “Haz- 35 J.D. Anderson, Hypersonic and High Temperature Gas Dy- ards by meteoroid impacts onto operational spacecraft,” namics (McGraw-Hill, 1989). Advances in Space Research 33, 1507–1510 (2004). 36 L.D. Landau and E.M. Lifshitz, Mechanics, vol. 1 of Course 18 NASA, “Micrometeoroids and Orbital Debris,” of Theoretical Physics, 3rd edition (Pergamon Press, 1976). 37 . ditions at the shock front (Ref. 31) imply that the medium 19 ESA, “Hypervelocity impacts and protecting spacecraft,” behind it moves perpendicular to it and that its maximum . Therefore, the minimum fraction of energy converted to 15

− 2 2 internal energy is 1 (vk + v⊥)/(2vvk) = 1/2, for any α. particle collisions,” Phys. Rev. E 55, 1940–45 (1997). 46 38 T. Kadono, “Hypervelocity impact into low density mate- Z. Insepov and I. Yamada, “Molecular dynamics study of rial and cometary outburst,” Planetary and Space Science shock wave generation by cluster impact on solid targets,” 36 Nucl. Instr. and Meth. B 112, 16–22 (1996). , 305 (1999). 47 39 R.P. Feynman, R.B. Leighton, and M. Sands, The Feyn- S.J. Carroll, P.D. Nellist, R.E. Palmer, S. Hobday and R. man Lectures on Physics, Volume II (Addison-Wesley, Smith, “Shallow Implantation of ‘Size-Selected’ Ag Clus- ters into Graphite,” Phys. Rev. Lett. 84, 2654 (2000). 1964). 48 40 C. Kittel, Introduction to Solid State Physics, 7th edition S. Pratontep et al, “Scaling Relations for Implantation (Wiley, 1996). of Size-Selected Au, Ag, and Si Clusters into Graphite,” 41 Electrodynamics of Con- Phys. Rev. Lett. 90, 055503 (2003). L.D. Landau and E.M. Lifshitz, 49 tinuous Media, vol. 8 of Course of Theoretical Physics, 2nd C. Anders and H.M. Urbassek, “Nonlinear stopping of edition (Pergamon Press, 1984), ch. XIV. heavy clusters in matter: A case study,” Nucl. Instr. and 42 Meth. B 258, 497–500 (2007). J.D. Jackson, Classical Electrodynamics, 3rd edition (John 50 Wiley, 1999), ch. 13. C. Anders and H.M. Urbassek, “Dependence of cluster 43 In the collision of charges with the same sign, b0 = ranges on target cohesive energy: Molecular-dynamics 2 2 ze /(mv ) is half the distance of closest approach in a study of energetic Au402 cluster impacts,” Nucl. Instr. and Meth. B 266, 44–48 (2008). head-on collision with velocity v. 51 44 M. Gryzi´nski, “Stopping Power of a Medium for Heavy, C. Anders, E.M. Bringa, G. Ziegenhain and H.M. Ur- Charged Particles,” Phys. Rev. 107, 1471 (1957). bassek, “Stopping of hypervelocity clusters in solids,” New 45 J. Phys. 13, 113019 (2011). Part of the problem is to specify the potential V and the 52 function g. The choice of power laws facilitates the task. M. Benguerba, “Mechanism of implantation of size- In the related case of granular particle collisions, a specific selected clusters into graphite,” Eur. Phys. J. D 66, 208 power-law form of f appears in: W.A.M. Morgado and I. (2012). Oppenheim, “Energy dissipation for quasielastic granular