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  • 1. Quantum capacitor at a metal–liquid interface Fredy R. Zypmana) Department of Physics, Yeshiva University, New York, New York 10033 ͑Received 17 March 2000; accepted 4 September 2000͒ ͓DOI: 10.1119/1.1328352͔ I. PROBLEM Consider a metal immersed in an electrolyte. As the metal electrostatic potential, V, is externally varied, a layer of sol- ute deposits on the metal surface. The charge, Q, deposited on the metal surface provides topographic information. Ex- perimentalists typically report the differential capacitance CϭdQ/dV as a function of V, C(V). ‘‘Normally’’ C is a nonconstant function of V. However, for large solute con- centrations, the C(V) curves show plateaus, that is, voltage regions in which the capacitance does not change. This ‘‘anomalous’’ behavior is presented and explained in this problem. The complete understanding of metal–electrolyte inter- faces is of great relevance in physics applied to biomedicine. For example in electrocardiograms and electroencephalo- grams, the electrodes are in contact with the skin, which plays the role of an electrolyte ͑dehydrated skin is an insu- lator͒. Another example is the connection of nerves by me- tallic wires. The nerve–wire interface is of an electrolyte– metal kind. In orthopedic implants, metal–metal interfaces are a concern. An artificial hip must be soft and squishy in the region close to the femur so as not to abrade the much softer bone, but hard everywhere else. Thus, a typical artifi- cial hip is made of ‘‘soft’’ titanium in contact with hard alloy Co–Cr–Mo. Due to differences in chemical potential, that metal interface is prone to corrosion. In the formation of adlayers of mostly organic adsorbates on metals1 such as Ag, Au, and particularly liquid Hg an interesting ‘‘cut’’ appears in the differential capacitance Cd curve as a function of potential. The first such observation was made by Lorenz, who studied the capacitance of a solu- tion of nanoic acid near a mercury electrode.2 The effect was initially attributed to film formation via lateral interactions, and theoretically studied by Rangarajan and co-workers us- ing mean field techniques3,4 and a lattice gas model for the surface. However, more recent experiments by Wandlowski and De Levie5 show a rather sharp cut in the differential capacity curve that is very difficult to understand in terms of a classical model of adlayer formation. Figure 1 shows a typical plot of differential capacitance versus voltage for an aqueous solution–metal interface, for concentrations below a critical value. The height and peak position changes with solute concentration, but the general shape remains the same. However, above the critical concentration, there appear plateaus in the C–V curve, as shown in Fig. 2. The problem in understanding this phenomenon is that independently of the suddenness of the formation of the monolayer, there is no reasonable classical explanation for the almost perfectly flat region inside the cut. Problem. Explain this behavior. Hint. Postulate the existence of a surface state of trapped electrons. A very simple model shows that indeed C ϭconstant by invoking surface trapped electrons in a con- ducting state, which completely shield the metal charge. Once this band is filled then the excess charge is screened by the solution and the C–V curve recovers its normal appear- ance. The cuts appear when the adlayer undergoes a metal– insulator transition ͑as, for example, the Cs–Au alloy͒, which is related to a percolation transition. II. SOLUTION We consider the electrons at the interface to be immersed in the step potential due to a potential difference between the electrolyte and the electrode, and a thin, confining film, de- posited on the surface ͑Fig. 3͒. We define ⌽1(x) as the wave function to the left of the interface, and ⌽2(x) as that to the right of it ͑Fig. 4͒. The total wave function is continuous across the interface, that is, ⌽1(0)ϭ⌽2(0)ϵ⌽(0), where xϭ0 is the coordinate defin- ing the plane of the interface. Since there is a delta function NEW PROBLEMS Christopher R. Gould, Editor Physics Department, Box 8202 North Carolina State University, Raleigh, North Carolina 27695 ‘‘New Problems’’ solicits interesting and novel worked problems for use in undergraduate physics courses beyond the introductory level. We seek problems that convey the excitement and interest of current developments in physics and that are useful for teaching courses such as Classical Mechanics, Electricity and Magnetism, Statistical Mechanics and Thermodynamics, ‘‘Modern’’ Physics, and Quantum Mechanics. We challenge physicists everywhere to create problems that show how contem- porary research in their various branches of physics uses the central unifying ideas of physics to advance physical understanding. We want these problems to become an important source of ideas and information for students of physics and their teachers. All submissions are peer-reviewed prior to publication. Send manuscripts directly to Christopher R. Gould, Editor. 601 601Am. J. Phys. 69 ͑5͒, May 2001 © 2001 American Association of Physics Teachers
  • 2. at xϭ0, the wave function may have a noncontinuous deriva- tive. By integrating the Schro¨dinger equation in a neighbor- hood of the origin, the derivative difference is6 ប2 2m ͓⌽2Ј͑0͒Ϫ⌽1Ј͑0͔͒ϩ␣⌽͑0͒ϭ0, ͑1͒ where, as shown in Fig. 4, ␣ represents the strength of the ␦ function at the origin, and is, for a thin layer, the product of the potential well and its spatial extension. We are interested in charge accumulation at the interface. Then, we look for electron bound eigenstates. These eigen- states ͑in this particular problem there is only one͒ must have negative total energy E. If they did not, they would be delo- calized states. As in the case of transmission and reflection through a potential step, if 0ϽEϽV0 then the electron is delocalized for xϽ0 and confined within a small region for xϾ0. If, on the other hand, V0ϽE, then the electron state is delocalized for all x. In the case of a potential step ͑that is, our problem with ␣ϭ0͒, there cannot be states completely confined around xϭ0, since for EϽ0 the only solution to the Schro¨dinger equation would be ⌽ϵ0. However, nontrivial solutions can exist when ␣ 0. Thus we search for exponen- tial solutions decaying away from the origin and with EϽ0: ⌽1͑x͒ϭAe␯x , ͑2͒ ⌽2͑x͒ϭAeϪ␮x , ͑3͒ EϭϪ ប2 ␯2 2m , ͑4͒ EϭVϪ ប2 ␮2 2m , ͑5͒ where ␮ and ␯ are real constants. We next define a new parameter, ␤, to simplify the nota- tion: ␯ϭͱ2mV ប2 sinh ␤, ͑6͒ Fig. 1. Typical differential interfacial capacitance of mercury in contact with aqueous solution of guanidinium nitrate for concentrations below 0.3 M. Fig. 2. Typical differential interfacial capacitance of mercury in contact with aqueous solution of guanidinium nitrate for concentrations above 0.3 M. Fig. 3. Diagram of the system show- ing the metal, the electrolyte, the thin layer, and the electron wave function. 602 602Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 3. ␮ϭͱ2mV ប2 cosh ␤. ͑7͒ From these definitions and Eqs. ͑4͒ and ͑5͒, one obtains the state given by ␤ϭ 1 2 log 2m␣2 ប2 V . ͑8͒ Let Q1 and Q2 be the net charges to the left and to the right of the interface, respectively. Then Q1ϭe ͵Ϫϱ 0 ͉⌽1͑x͉͒2 dxϭ eA2 2␯ , ͑9͒ where the normalization constant A is obtained by requiring that ͵Ϫϱ 0 ͉⌽1͑x͉͒2 dxϩ ͵0 ϩϱ ͉⌽2͑x͉͒2 dxϭ1: A2 ϭ2 V ␣ cosh ␤ sinh ␤. ͑10͒ In the previous expression, we have made use of the fact that Eq. ͑1͒ is equivalent to ␮ϩ␯ϭ2ma/ប2 . Then Q1ϭ e␮ ␮ϩ␯ . ͑11͒ And, similarly Q2ϭ e␯ ␮ϩ␯ . ͑12͒ The charge at the interface capacitor can now be evaluated QcϭQ1ϪQ2ϭe ␮Ϫ␯ ␮ϩ␯ ϭe ប2 V 2m␣2 . ͑13͒ The capacitance, CϭdQc /dV, is Cϭ eប2 2m␣2 , ͑14͒ which is a constant that depends only on the properties of the layer through the parameter ␣. Thus by simply adding a small confining layer at the in- terface, it is possible to explain the flat characteristics of the total capacitance. As the external voltage in the electrolytic cell changes, the thin layer may appear or disappear, thus creating ‘‘normal’’ C–V regions, and flat ones. ACKNOWLEDGMENTS I would like to thank Dr. Steven J. Eppell and Dr. Lesser Blum for useful comments. The National Cancer Institute through Grant No. CA77796-01 has financially supported this work. a͒ Electronic mail: 1 E. Budevski, G. Staikov, and W. J. Lorenz, Electrochemical Phase For- mation and Growth ͑Wiley, New York, 1996͒, pp. 200–210. 2 Wolfgang Lorenz, ‘‘The rate of absorption and of two-dimensional asso- ciation of fatty acids at the mercury-electrolyte interface,’’ Z. Elektro- chem. 62 ͑1͒, 192–199 ͑1958͒. 3 M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms for neutral organic-compounds—A hierarchy in modeling,’’ J. Electroanal. Chem. Interfacial Electrochem. 176 ͑1-2͒, 45–64 ͑1984͒. 4 M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms— microscopic modeling,’’ J. Electroanal. Chem. Interfacial Electrochem. 176 ͑1-2͒, 119–137 ͑1984͒. 5 T. Wandlowski, G. Jameson, and R. De Levie, ‘‘Two-Dimensional Con- densation of Guadinidium Nitrate at the Mercury-Water Interface,’’ J. Phys. Chem. 97 ͑39͒, 10119–10126 ͑1993͒. 6 C. Cohen-Tannoudji, Bernard Diu, and Frank Laloe¨, Quantum Mechanics ͑Wiley, New York, Paris, 1977͒, Vol. 1, p. 87. Fig. 4. Potential function representing the metal, the interface, and the elec- trolyte. 603 603Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 4. Electromagnetically induced transparency Tao Panga) Department of Physics, University of Nevada, Las Vegas, Nevada 89154-4002 ͑Received 6 June 2000; accepted 21 September 2000͒ ͓DOI: 10.1119/1.1331303͔ I. SCOPE Imagine that you are taking a quick walk and you are in fact traveling faster than light. This is what has been achieved in a recent experiment by Lene Hau and her co-workers.1 In that experiment, the group velocity of a pulsed laser was effectively reduced to about 1 mile per hour ͑0.45 m/s͒ in a cold, laser-dressed sodium atom cloud. In an earlier experiment, the same group had successfully slowed the group velocity of light to 38 miles per hour ͑17 m/s͒ in a similar system.2 Propagation of light in a medium is a well-studied subject even though there have been some very tough questions, such as the group velocity exceeding the speed of light in vacuum.3 Five years ago, it was demonstrated for the first time that the speed of light can be reduced significantly4 in a cold atom cloud that is in a state called electromagnetically induced transparency.5,6 What has made the experimental work of Hau et al. so unique is that the orders of magnitude in the reduction of speed of light, which cannot commonly be accomplished by increasing the index of refraction, is achieved instead by the extremely rapid variation of the in- dex with the frequency, and the elimination of light absorp- tion at resonance frequency—a quantum phenomenon result- ing from the coupling and interaction between lasers and electrons at different atomic levels. Here we would like to highlight some basic understanding of this exciting phenom- enon. From the Maxwell equations for a propagating electro- magnetic wave with angular frequency ␻ and complex wave vector ␬ in a nonconducting medium, we have7 ␬2 ϭ␮⑀␻2 , ͑1͒ where ␮ is the magnetic permeability and ⑀ is the electric permittivity of the medium. If we assume that ␻ is real and ␬ϭkϩi ␣ 2 , ͑2͒ where k is the real propagating wave vector and ␣ is the absorption coefficient, we have nϭ ck ␻ ϭReͱ ␮⑀ ␮0⑀0 , ͑3͒ ␣ϭ2␻ Im ͱ␮⑀, ͑4͒ with cϭ1/ͱ␮0⑀0 being the speed of light in vacuum and ␮0 and ⑀0 the vacuum permeability and permittivity, respec- tively. In most cases, we have ␮Ӎ␮0 , a condition that is assumed here. From the above relation, we find the phase velocity of the wave, vpϭ ␻ k ϭ c n , ͑5͒ which characterizes how fast the wave changes its phase. So the field propagating along the z direction is proportional to eikzϪ␣z/2Ϫi␻t . Now let us consider that the electric field is a nondecaying (␣ϭ0) wave packet with a range of frequencies ␻ϭ␻(k): E͑z,t͒ϭ ͵dk A͑k͒eikzϪi␻t , ͑6͒ where A(k) is a narrow function peaked at kϭk0 . We can then expand ␻(k) as ␻͑k͒ϭ␻͑k0͒ϩ͑kϪk0͒ d␻ dk ͯkϭk0 ϩO͓͑kϪk0͒2 ͔, ͑7͒ if it is well behaved, that is, a smooth function around the given wave vector k0 . If only the zeroth- and first-order terms are kept in the expansion, we have E͑z,t͒Ӎei͓k0vgϪ␻͑k0͔͒t ͵dk A͑k͒eikzϪikvgt ϭei͓k0vgϪ␻͑k0͔͒t E͑zϪvgt,0͒, ͑8͒ where vgϭ d␻ dk ͯkϭk0 ͑9͒ is termed the group velocity of the wave at kϭk0 because the packet acts if it is traveling in space with such a velocity without changing its shape, with an overall phase change. Using Eq. ͑3͒, we obtain vgϭ vp 1ϩ͑␻/n͒͑dn/d␻͒ . ͑10͒ Note that we have assumed that n is a function of ␻ through k. We can consider the group velocity to be the velocity of the wave packet, that is, the velocity of the energy and in- formation contained in the packet, if the linear term in the above Taylor expansion is the dominant term. However, the meaning of the group velocity can change if the wave packet becomes incoherent. Care must be taken when the angular frequency of the wave is near a resonance or dn/d␻Ͻ0.3 Quantum mechanically, the resonance occurs when the frequency of light matches the energy difference between two allowed quantum levels in the system and is typically accompanied by strong absorption under normal circum- stances. This is why normal matter that can be well approxi- mated by a two-level model can never slow the light very much. For laser-dressed atom clouds, the coupling and inter- action between a three-level atom and two lasers can drasti- cally alter the behavior of the system, including effectively 604 604Am. J. Phys. 69 ͑5͒, May 2001 © 2001 American Association of Physics Teachers
  • 5. eliminating the absorption at the resonance frequency and therefore creating electromagnetically induced transparency, as shown in the problems given here. II. PROBLEMS A. Coherent population trapping The key to keeping the group velocity at the vicinity of a resonance frequency meaningful lies in the properties of the laser-dressed atomic cloud. Without such an effect, absorp- tion would be too strong to have any transmitted light. Consider that each atom in the medium has three levels. The presence of a coupling ͑dressing͒ laser (␻cӍ␻2Ϫ␻1) and a probe laser (␻Ӎ␻2Ϫ␻0) causes a mixing of the three levels, ͉0͘, ͉1͘, and ͉2͘. The Hamiltonian of such a system is HϭH0ϩH1 . ͑11͒ Here the unperturbed Hamiltonian H0 is given by ͗l͉H0͉lЈ͘ϭប␻l␦llЈ , ͑12͒ with l, lЈϭ0,1, 2. The perturbation H1 is restricted to be ͗l͉H1͉lЈ͘ϭ͗lЈ͉H1͉l͘*ϭប⍀llЈeϪi␻llЈt , ͑13͒ with ␻llЈϭ␻lϪ␻lЈ and ⍀llϭ⍀01ϭ⍀10ϭ0. Note that ␻2 Ͼ␻1Ͼ␻0ϭ0 and ␻l0ϭ␻l . This is a so-called ‘‘⌳’’ system with the highest level coupled to two lower levels. For the Hamiltonian given, find the time-dependent wave function ͉␺͑t͒͘ϭ͚lϭ0 2 cl͑t͉͒l͘, ͑14͒ if ͉␺(0)͘ϭc0(0)͉0͘ϩc1(0)͉1͘ with ͉c0(0)͉2 ϩ͉c1(0)͉2 ϭ1. Discuss the condition for c2(t)ϵ0 and its implication. B. Electromagnetically induced transparency If we define a density matrix ␳͑t͒ϭ͉␺͑t͒͗͘␺͑t͉͒, ͑15͒ whose diagonal elements are the probabilities of occupying specific states and off-diagonal elements represent the tran- sition rates between two given states, we have iប ‫␳ץ‬ ‫ץ‬t ϭ͓H,␳͔, ͑16͒ from the Schro¨dinger equation. The interactions between at- oms in the cloud can cause a finite linewidth and decay of each level, which can be accounted for by a relaxation ma- trix: ͗l͉⌫͉lЈ͘ϭ2␥l␦llЈ , ͑17͒ and change Eq. ͑16͒ into iប ‫␳ץ‬ ‫ץ‬t ϭ͓H,␳͔Ϫ iប 2 ͑⌫␳ϩ␳⌫͒. ͑18͒ Assuming that only the dominant decaying factor is nonzero, that is, ␥2ϭ␥ and ␥0,1ϭ0, and that the atom is in the ground state at tϭ0, show that ⑀͑␻͒ϭ⑀0ͫ1ϩ ␻d͑␻Ϫ␻2͒ ␻R 2 /4Ϫ͑␻Ϫ␻2͒2 Ϫi␥͑␻Ϫ␻2͒ͬ, ͑19͒ where ␻dϭna͉p20͉2 /ប⑀0 with ͉p20͉ being the coupling dipole strength between ͉2͘ and ͉0͘ and ␻Rϭ2͉⍀21͉ the Rabi angu- lar frequency between ͉2͘ and ͉1͘. C. The slowest light In the recent experiment, Hau and co-workers have suc- cessfully reduced the group velocity of light in a cold, laser- dressed sodium atom cloud to 1 mile per hour ͑0.45 m/s͒.1 Each sodium atom can be approximated well by a three-level system. Assume that the permittivity of such a laser-dressed atom cloud is given by Eq. ͑19͒ and the frequency of the probe laser (␻/2␲) is near the resonance frequency ͑␻2/2␲ Ӎ5.1ϫ1014 Hz for sodium atom͒. Estimate the number den- sity of the atom cloud needed in order to have vg Ӎ0.45 m/s. Assume that the Rabi angular frequency is about ␻Rϭ3.5ϫ107 rad/s and the coupling dipole strength is about ͉p20͉Ӎ2.5ϫ10Ϫ29 C m. III. SOLUTIONS A. Coherent population trapping From the time-dependent Schro¨dinger equation iប ‫ץ‬͉␺͑t͒͘ ‫ץ‬t ϭH͉␺͑t͒͘, ͑20͒ we have ic˙0͑t͒ϭ␻0c0͑t͒ϩ⍀20ei␻2t c2͑t͒, ͑21͒ ic˙1͑t͒ϭ␻1c1͑t͒ϩ⍀21ei␻21t c2͑t͒, ͑22͒ ic˙2͑t͒ϭ␻2c2͑t͒ϩ⍀02eϪi␻2t c0͑t͒ϩ⍀12eϪi␻21t c1͑t͒. ͑23͒ If we redefine the coefficients by cl͑t͒ϭeϪi␻lt bl͑t͒, ͑24͒ the equation set is simplified to ib˙ 0͑t͒ϭ⍀20b2͑t͒, ͑25͒ ib˙ 1͑t͒ϭ⍀21b2͑t͒, ͑26͒ ib˙ 2͑t͒ϭ⍀02b0͑t͒ϩ⍀12b1͑t͒. ͑27͒ Multiplying Eq. ͑25͒ with ⍀02 and Eq. ͑26͒ with ⍀12 and adding them together, and substituting the resulting equation into Eq. ͑27͒ after taking one more time derivative, we ob- tain b¨ 2͑t͒ϭϪ͉͑⍀20͉2 ϩ͉⍀21͉2 ͒b2͑t͒. ͑28͒ We have used ⍀02ϭ⍀20* and ⍀12ϭ⍀21* . So we have b2͑t͒ϭAei⍀t ϩBeϪi⍀t , ͑29͒ with ⍀ϭͱ͉⍀20͉2 ϩ͉⍀21͉2 . Taking the initial condition b2(0)ϭc2(0)ϭ0, we arrive at b2͑t͒ϭC sin ⍀t, ͑30͒ with C being a constant. Substituting this result back into Eqs. ͑25͒ and ͑26͒, we have b0͑t͒ϭ͓c0͑0͒Ϫ␣͔cos ⍀tϩ␣, ͑31͒ b1͑t͒ϭ͓c1͑0͒Ϫ␤͔cos ⍀tϩ␤, ͑32͒ where ␣ and ␤ are constants constrained by 605 605Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 6. ⍀02␣ϩ⍀12␤ϭ0. ͑33͒ We have used the initial conditions b0(0)ϭc0(0) and b1(0)ϭc1(0). The coefficient C is given by Cϭ i ⍀ ͓⍀02c0͑0͒ϩ⍀12c1͑0͔͒. ͑34͒ If c0(0) and c1(0) are such that Cϵ0, we have c2(t)ϵ0 all the time. A typical case is ͉⍀02͉ϭ͉⍀12͉ and ͉c0(0)͉ ϭ͉c1(0)͉ϭ1/&, with the total phase difference between the two terms being ␲. So the state ͉2͘ will stay empty and the atoms are trapped in the lower states. The effect of such a coherent population trapping is that the absorption or emis- sion of light is completely eliminated. B. Electromagnetically induced transparency Consider that the traveling ͑probing͒ laser is described by a time-dependent electric field E(t)ϭE0eϪi␻t with ␻ very close to ␻2 . The perturbation from such a field is ͗2͉H1͉0͘ϭϪ͗2͉p͉0͘E0eϪi␻t ϭប⍀20eϪi␻t , ͑35͒ where p is the dipole moment induced by the field. Now if we examine the density matrix elements between two states, ␳llЈϭ͗l͉␳͉lЈ͘, we have i ‫␳ץ‬20 ‫ץ‬t ϭ͑␻2Ϫi␥͒␳20ϩ⍀21eϪi␻21t ␳10 ϩ⍀20eϪi␻t ͑␳00Ϫ␳22͒, ͑36͒ i ‫␳ץ‬10 ‫ץ‬t ϭ␻1␳10ϩ⍀12eϪi␻12t ␳20Ϫ⍀20eϪi␻2t ␳12 . ͑37͒ We have used ͚lϭ0 2 ͉l͗͘l͉ϭ1 ͑38͒ in deriving the above equations. We can then replace ␳00 , ␳22 , and ␳12 by their values at tϭ0, that is, ␳00ϭ1, ␳22ϭ0, and ␳12ϭ0, and change a variable with ␨10ϭ␳10eϪi␻21t , be- cause we are only looking for the linear solution. Then we have i ‫␳ץ‬20 ‫ץ‬t ϭ͑␻2Ϫi␥͒␳20ϩ⍀21␨10ϩ⍀20eϪi␻t , ͑39͒ i ‫␨ץ‬10 ‫ץ‬t ϭ␻2␨10ϩ⍀12␳20 . ͑40͒ This equation set resembles a harmonic oscillator under damping and driving forces. The steady solutions are there- fore given by ␳20͑t͒ϭAeϪi␻t , ͑41͒ ␨10͑t͒ϭBeϪi␻t . ͑42͒ Substituting the above solutions into the equations, we obtain Aϭ ⍀20͑␻Ϫ␻2͒ ͑␻Ϫ␻2ϩi␥͒͑␻Ϫ␻2͒Ϫ͉⍀21͉2 . ͑43͒ Because ␳20 represents the dipole transition rate between ͉2͘ and 0͘, the polarization of the system is given by P ϭna␳20p02ϭ(⑀Ϫ⑀0)E(t) with p02ϭ͗0͉p͉2͘ϭp20* . Then we reach Eq. ͑19͒. C. The slowest light We know that the group velocity is given by vgϭ d␻ dk ϭ c nϩ␻͑dn/d␻͒ . ͑44͒ For all known materials, nϳO(1). So if vgӶc, we must have vgӍ c ␻͑dn/d␻͒ . ͑45͒ For vgϭ0.45 m/s, as observed in the experiment by Hau’s group,1 one must have ␻͑dn/d␻͒Ӎ6.7ϫ108 . ͑46͒ From the given permittivity, we have nϩi c␣ 2␻ Ӎ1ϩ 1 2 ␻d͑␻Ϫ␻2͒ ␻R 2 /4Ϫ͑␻Ϫ␻2͒2 Ϫi␥͑␻Ϫ␻2͒ . ͑47͒ Considering that ␻ is very close to ␻2 , we have nϩi c␣ 2␻ Ӎ1ϩ 2␻d͑␻Ϫ␻2͒ ␻R 2 ϫͫ1ϩ 4͑␻Ϫ␻2͒2 ␻R 2 ϩ i4␥͑␻Ϫ␻2͒ ␻R 2 ϩ¯ͬ, ͑48͒ which gives ␻͑dn/d␻͒Ӎ 2␻ ប⑀0 na͉p20͉2 ␻R 2 . ͑49͒ We have used ␻dϭna͉p20͉2 /ប⑀0 . With the numerical values of the quantities given, we then obtain naӍ2ϫ1020 mϪ3 , a density quite difficult to achieve experimentally. Note that the absorption coefficient ␣ is zero at the reso- nance frequency. This is the essence of the electromagneti- cally induced transparency, a condition that must be met in order to have a significant light transmission at the resonance frequency. Otherwise, the drastically slowed group velocity of light observed by Hau’s group would not have been pos- sible. a͒ Electronic mail: 1 L. V. Hau, presentation at the American Association for the Advancement of Science, February 2000, Washington, DC. 2 L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, ‘‘Light speed reduction to 17 meters per second in an ultracold atomic gas,’’ Nature ͑London͒ 397, 594–598 ͑1999͒. 3 For a recent review, see R. Y. Chiao and A. M. Steinberg, Tunneling Times and Superluminality, Progress in Optics Vol. 37, edited by E. Wolf ͑Elsevier, Amsterdam, 1997͒, pp. 347–405. 4 A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, ‘‘Electromagnetically introduced transparency: Propagation dynamics,’’ Phys. Rev. Lett. 74, 2447–2450 ͑1995͒. 5 S. E. Harris, ‘‘Electromagnetically induced transparency,’’ Phys. Today 50 ͑7͒, 36–42 ͑1997͒. 6 M. O. Scully and M. S. Zubairy, Quantum Optics ͑Cambridge U.P., Cam- bridge, 1997͒, Secs. 7.2 and 7.3. 7 D. J. Jackson, Classical Electrodynamics ͑Wiley, New York, 1999͒, 3rd ed., Secs. 7.5 and 7.8. 606 606Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 7. Negative group velocity Kirk T. McDonalda) Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 ͑Received 21 August 2000; accepted 21 September 2000͒ ͓DOI: 10.1119/1.1331304͔ I. PROBLEM Consider a variant on the physical situation of ‘‘slow light’’ 1,2 in which two closely spaced spectral lines are now both optically pumped to show that the group velocity can be negative at the central frequency, which leads to apparent superluminal behavior. A. Negative group velocity In more detail, consider a classical model of matter in which spectral lines are associated with oscillators. In par- ticular, consider a gas with two closely spaced spectral lines of angular frequencies ␻1,2ϭ␻0Ϯ⌬/2, where ⌬Ӷ␻0 . Each line has the same damping constant ͑and spectral width͒ ␥. Ordinarily, the gas would exhibit strong absorption of light in the vicinity of the spectral lines. But suppose that lasers of frequencies ␻1 and ␻2 pump both oscillators into inverted populations. This can be described classically by assigning negative oscillator strengths to these oscillators.3 Deduce an expression for the group velocity vg(␻0) of a pulse of light centered on frequency ␻0 in terms of the ͑uni- valent͒ plasma frequency ␻p of the medium, given by ␻p 2 ϭ 4␲Ne2 m , ͑1͒ where N is the number density of atoms, and e and m are the charge and mass of an electron. Give a condition on the line separation ⌬ compared to the linewidth ␥ such that the group velocity vg(␻0) is negative. In a recent experiment by Wang et al.,4 a group velocity of vgϭϪc/310, where c is the speed of light in vacuum, was demonstrated in cesium vapor using a pair of spectral lines with separation ⌬/2␲Ϸ2 MHz and linewidth ␥/2␲ Ϸ0.8 MHz. B. Propagation of a monochromatic plane wave Consider a wave with electric field E0ei␻(z/cϪt) that is incident from zϽ0 on a medium that extends from zϭ0 to a. Ignore reflection at the boundaries, as is reasonable if the index of refraction n(␻) is near unity. Particularly simple results can be obtained when you make the ͑unphysical͒ as- sumption that the ␻n(␻) varies linearly with frequency about a central frequency ␻0 . Deduce a transformation that has a frequency-dependent part and a frequency-independent part between the phase of the wave for zϽ0 to that of the wave inside the medium, and to that of the wave in the region aϽz. C. Fourier analysis Apply the transformations between an incident monochro- matic wave and the wave in and beyond the medium to the Fourier analysis of an incident pulse of form f(z/cϪt). D. Propagation of a sharp wave front In the approximation that ␻n varies linearly with ␻, de- duce the waveforms in the regions 0ϽzϽa and aϽz for an incident pulse ␦(z/cϪt), where ␦ is the Dirac delta function. Show that the pulse emerges out of the gain region at zϭa at time tϭa/vg , which appears to be earlier than when it enters this region if the group velocity is negative. Show also that inside the negative group velocity medium a pulse propa- gates backwards from zϭa at time tϭa/vgϽ0 to zϭ0 at t ϭ0, at which time it appears to annihilate the incident pulse. E. Propagation of a Gaussian pulse As a more physical example, deduce the waveforms in the regions 0ϽzϽa and aϽz for a Gaussian incident pulse E0eϪ(z/cϪt)2/2␶2 ei␻0(z/cϪt) . Carry the frequency expansion of ␻n(␻) to second order to obtain conditions of validity of the analysis such as maximum pulse width ␶, maximum length a of the gain region, and maximum time of advance of the emerging pulse. Consider the time required to generate a pulse of rise time ␶ when assessing whether the time advance in a negative group velocity medium can lead to superlumi- nal signal propagation. II. SOLUTION The concept of group velocity appears to have been first enunciated by Hamilton in 1839 in lectures of which only abstracts were published.5 The first recorded observation of the group velocity of a ͑water͒ wave is due to Russell in 1844.6 However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a Smith’s Prize examination paper.7 The early history of group velocity has been reviewed by Havelock.8 H. Lamb9 credits A. Schuster with noting in 1904 that a negative group velocity, i.e., a group velocity of opposite sign to that of the phase velocity, is possible due to anoma- lous dispersion. Von Laue10 made a similar comment in 1905. Lamb gave two examples of strings subject to external potentials that exhibit negative group velocities. These early considerations assumed that in case of a wave with positive group and phase velocities incident on the anomalous me- dium, energy would be transported into the medium with a positive group velocity, and so there would be waves with negative phase velocity inside the medium. Such negative phase velocity waves are formally consistent with Snell’s 607 607Am. J. Phys. 69 ͑5͒, May 2001 © 2001 American Association of Physics Teachers
  • 8. law11 ͑since ␪tϭsinϪ1 ͓(ni /nt)sin ␪i͔ can be in either the first or second quadrant͒, but they seemed physically implausible and the topic was largely dropped. Present interest in negative group velocity is based on anomalous dispersion in a gain medium, where the sign of the phase velocity is the same for incident and transmitted waves, and energy flows inside the gain medium in the op- posite direction to the incident energy flow in vacuum. The propagation of electromagnetic waves at frequencies near those of spectral lines of a medium was first extensively discussed by Sommerfeld and Brillouin,12 with emphasis on the distinction between signal velocity and group velocity when the latter exceeds c. The solution presented here is based on the work of Garrett and McCumber,13 as extended by Chiao et al.14,15 A discussion of negative group velocity in electronic circuits has been given by Mitchell and Chiao.16 A. Negative group velocity In a medium of index of refraction n(␻), the dispersion relation can be written kϭ ␻n c , ͑2͒ where k is the wave number. The group velocity is then given by vgϭReͫd␻ dk ͬϭ 1 Re͓dk/d␻͔ ϭ c Re͓d͑␻n͒/d␻͔ ϭ c nϩ␻ Re͓dn/d␻͔ . ͑3͒ We see from Eq. ͑3͒ that if the index of refraction de- creases rapidly enough with frequency, the group velocity can be negative. It is well known that the index of refraction decreases rapidly with frequency near an absorption line, where ‘‘anomalous’’ wave propagation effects can occur.12 However, the absorption makes it difficult to study these effects. The insight of Garrett and McCumber13 and of Chiao et al.14,15,17–19 is that demonstrations of negative group ve- locity are possible in media with inverted populations, so that gain rather than absorption occurs at the frequencies of interest. This was dramatically realized in the experiment of Wang et al.4 by use of a closely spaced pair of gain lines, as perhaps first suggested by Steinberg and Chiao.17 We use a classical oscillator model for the index of refrac- tion. The index n is the square root of the dielectric constant ⑀, which is in turn related to the atomic polarizability ␣ ac- cording to Dϭ⑀EϭEϩ4␲PϭE͑1ϩ4␲N␣͒ ͑4͒ ͑in Gaussian units͒, where D is the electric displacement, E is the electric field, and P is the polarization density. Then, the index of refraction of a dilute gas is nϭͱ⑀Ϸ1ϩ2␲N␣. ͑5͒ The polarizability ␣ is obtained from the electric dipole moment pϭexϭ␣E induced by electric field E. In the case of a single spectral line of frequency ␻j , we say that an electron is bound to the ͑fixed͒ nucleus by a spring of con- stant Kϭm␻j 2 , and that the motion is subject to the damping force Ϫm␥jx˙, where the dot indicates differentiation with respect to time. The equation of motion in the presence of an electromagnetic wave of frequency ␻ is x¨ϩ␥jx˙ϩ␻j 2 xϭ eE m ϭ eE0 m ei␻t . ͑6͒ Hence, xϭ eE m 1 ␻j 2 Ϫ␻2 Ϫi␥j␻ ϭ eE m ␻j 2 Ϫ␻2 ϩi␥j␻ ͑␻j 2 Ϫ␻2 ͒2 ϩ␥j 2 ␻2 , ͑7͒ and the polarizability is ␣ϭ e2 m ␻j 2 Ϫ␻2 ϩi␥j␻ ͑␻j 2 Ϫ␻2 ͒2 ϩ␥j 2 ␻2 . ͑8͒ In the present problem we have two spectral lines, ␻1,2 ϭ␻0Ϯ⌬/2, both of oscillator strength Ϫ1 to indicate that the populations of both lines are inverted, with damping con- stants ␥1ϭ␥2ϭ␥. In this case, the polarizability is given by ␣ϭϪ e2 m ͑␻0Ϫ⌬/2͒2 Ϫ␻2 ϩi␥␻ ͑͑␻0Ϫ⌬/2͒2 Ϫ␻2 ͒2 ϩ␥2 ␻2 Ϫ e2 m ͑␻0ϩ⌬/2͒2 Ϫ␻2 ϩi␥␻ ͑͑␻0ϩ⌬/2͒2 Ϫ␻2 ͒2 ϩ␥2 ␻2 ϷϪ e2 m ␻0 2 Ϫ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 Ϫ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 Ϫ e2 m ␻0 2 ϩ2⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 ϩ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 , ͑9͒ where the approximation is obtained by the neglect of terms in ⌬2 compared to those in ⌬␻0 . For a probe beam at frequency ␻, the index of refraction ͑5͒ has the form n͑␻͒Ϸ1Ϫ ␻p 2 2 ͫ ␻0 2 Ϫ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 Ϫ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 ϩ ␻0 2 ϩ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 ϩ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2ͬ, ͑10͒ where ␻p is the plasma frequency given by Eq. ͑1͒. This is illustrated in Fig. 1. The index at the central frequency ␻0 is Fig. 1. The real and imaginary parts of the index of refraction in a medium with two spectral lines that have been pumped to inverted populations. The lines are separated by angular frequency ⌬ and have widths ␥ϭ0.4⌬. 608 608Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 9. n͑␻0͒Ϸ1Ϫi ␻p 2 ␥ ͑⌬2 ϩ␥2 ͒␻0 Ϸ1Ϫi ␻p 2 ⌬2 ␥ ␻0 , ͑11͒ where the second approximation holds when ␥Ӷ⌬. The electric field of a continuous probe wave then propagates according to E͑z,t͒ϭei͑kzϪ␻0t͒ ϭei␻͑n͑␻0͒z/cϪt͒ Ϸez/͓⌬2c/␥␻͑2/p͔͒ ei␻0͑z/cϪt͒ . ͑12͒ From this we see that at frequency ␻0 the phase velocity is c, and the medium has an amplitude gain length ⌬2 c/␥␻p 2 . To obtain the group velocity ͑3͒ at frequency ␻0 , we need the derivative d͑␻n͒ d␻ ͯ␻0 Ϸ1Ϫ 2␻p 2 ͑⌬2 Ϫ␥2 ͒ ͑⌬2 ϩ␥2 ͒2 , ͑13͒ where we have neglected terms in ⌬ and ␥ compared to ␻0 . From Eq. ͑3͒, we see that the group velocity can be negative if ⌬2 ␻p 2Ϫ ␥2 ␻p 2 у 1 2 ͩ⌬2 ␻p 2 ϩ ␥2 ␻p 2 ͪ2 . ͑14͒ The boundary of the allowed region ͑14͒ in (⌬2 ,␥2 ) space is a parabola whose axis is along the line ␥2 ϭϪ⌬2 , as shown in Fig. 2. For the physical region ␥2 у0, the boundary is given by ␥2 ␻p 2 ϭͱ1ϩ4 ⌬2 ␻p 2Ϫ1Ϫ ⌬2 ␻p 2 . ͑15͒ Thus, to have a negative group velocity, we must have ⌬р&␻p , ͑16͒ which limit is achieved when ␥ϭ0; the maximum value of ␥ is 0.5␻p when ⌬ϭ0.866␻p . Near the boundary of the negative group velocity region, ͉vg͉ exceeds c, which alerts us to concerns of superluminal behavior. However, as will be seen in the following sections, the effect of a negative group velocity is more dramatic when ͉vg͉ is small rather than large. The region of recent experimental interest is ␥Ӷ⌬Ӷ␻p , for which Eqs. ͑3͒ and ͑13͒ predict that vgϷϪ c 2 ⌬2 ␻p 2 . ͑17͒ A value of vgϷϪc/310 as in the experiment of Wang cor- responds to ⌬/␻pϷ1/12. In this case, the gain length ⌬2 c/␥␻p 2 was approximately 40 cm. For later use we record the second derivative, d2 ͑␻n͒ d␻2 ͯ␻0 Ϸ8i ␻p 2 ␥͑3⌬2 Ϫ␥2 ͒ ͑⌬2 ϩ␥2 ͒3 Ϸ24i ␻p 2 ⌬2 ␥ ⌬2 , ͑18͒ where the second approximation holds if ␥Ӷ⌬. B. Propagation of a monochromatic plane wave To illustrate the optical properties of a medium with nega- tive group velocity, we consider the propagation of an elec- tromagnetic wave through it. The medium extends from z ϭ0 to a, and is surrounded by vacuum. Because the index of refraction ͑10͒ is near unity in the frequency range of inter- est, we ignore reflections at the boundaries of the medium. A monochromatic plane wave of frequency ␻ and incident from zϽ0 propagates with phase velocity c in vacuum. Its electric field can be written E␻͑z,t͒ϭE0ei␻z/c eϪi␻t ͑zϽ0͒. ͑19͒ Inside the medium this wave propagates with phase velocity c/n(␻) according to E␻͑z,t͒ϭE0ei␻nz/c eϪi␻t ͑0ϽzϽa͒, ͑20͒ where the amplitude is unchanged since we neglect the small reflection at the boundary zϭ0. When the wave emerges into vacuum at zϭa, the phase velocity is again c, but it has accumulated a phase lag of (␻/c)(nϪ1)a, and so appears as E␻͑z,t͒ϭE0ei␻a͑nϪ1͒/c ei␻z/c eϪi␻t ϭE0ei␻an/c eϪi␻͑tϪ͑zϪa͒/c͒ ͑aϽz͒. ͑21͒ It is noteworthy that a monochromatic wave for zϾa has the same form as that inside the medium if we make the frequency-independent substitutions z→a, t→tϪ zϪa c . ͑22͒ Since an arbitrary waveform can be expressed in terms of monochromatic plane waves via Fourier analysis, we can use these substitutions to convert any wave in the region 0Ͻz Ͻa to its continuation in the region aϽz. A general relation can be deduced in the case where the second and higher derivatives of ␻n(␻) are very small. We can then write ␻n͑␻͒Ϸ␻0n͑␻0͒ϩ c vg ͑␻Ϫ␻0͒, ͑23͒ where vg is the group velocity for a pulse with central fre- quency ␻0 . Using this in Eq. ͑20͒, we have E␻͑z,t͒ϷE0ei␻0z͑n͑␻0͒/cϪ1/vg͒ ei␻z/vgeϪi␻t ͑0ϽzϽa͒. ͑24͒ In this approximation, the Fourier component E␻(z) at fre- quency ␻ of a wave inside the gain medium is related to that of the incident wave by replacing the frequency dependence Fig. 2. The allowed region ͑14͒ in (⌬2 ,␥2 ) space such that the group ve- locity is negative. 609 609Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 10. ei␻z/c by ei␻z/vg, i.e., by replacing z/c by z/vg , and multi- plying by the frequency-independent phase factor ei␻0z(n(␻0)/cϪ1/vg) . Then, using transformation ͑22͒, the wave that emerges into vacuum beyond the medium is E␻͑z,t͒ϷE0ei␻0a͑n͑␻0͒/cϪ1/vg͒ ϫei␻͑z/cϪa͑1/cϪ1/vg͒͒ eϪi␻t ͑aϽz͒. ͑25͒ The wave beyond the medium is related to the incident wave by multiplying by a frequency-independent phase, and by replacing z/c by z/cϪa(1/cϪ1/vg) in the frequency- dependent part of the phase. The effect of the medium on the wave as described by Eqs. ͑24͒ and ͑25͒ has been called ‘‘rephasing.’’ 4 C. Fourier analysis and ‘‘rephasing’’ The transformations between the monochromatic incident wave ͑19͒ and its continuation in and beyond the medium, ͑24͒ and ͑25͒, imply that an incident wave E͑z,t͒ϭf͑z/cϪt͒ϭ ͵Ϫϱ ϱ E␻͑z͒eϪi␻t d␻ ͑zϽ0͒, ͑26͒ whose Fourier components are given by E␻͑z͒ϭ 1 2␲ ͵Ϫϱ ϱ E͑z,t͒ei␻t dt, ͑27͒ propagates as E͑z,t͒Ϸ Ά f͑z/cϪt͒ ͑zϽ0͒ ei␻0z͑n͑␻0͒/cϪ1/vg͒ f͑z/vgϪt͒ ͑0ϽzϽa͒ ei␻0a͑n͑␻0͒/cϪ1/vg͒ f͑z/cϪtϪa͑1/cϪ1/vg͒͒ ͑aϽz͒. ͑28͒ An interpretation of Eq. ͑28͒ in terms of ‘‘rephasing’’ is as follows. Fourier analysis tells us that the maximum ampli- tude of a pulse made of waves of many frequencies, each of the form E␻(z,t)ϭE0(␻)ei␾(␻) ϭE0(␻)ei(k(␻)zϪ␻tϩ␾0(␻)) with E0у0, is determined by adding the amplitudes E0(␻). This maximum is achieved only if there exist points ͑z,t͒ such that all phases ␾͑␻͒ have the same value. For example, we consider a pulse in the region zϽ0 whose maximum occurs when the phases of all component frequencies vanish, as shown at the left of Fig. 3. Referring to Eq. ͑19͒, we see that the peak occurs when zϭct. As usual, we say that the group velocity of this wave is c in vacuum. Inside the medium, Eq. ͑24͒ describes the phases of the components, which all have a common frequency- independent phase ␻0z(n(␻0)/cϪ1/vg) at a given z, as well as a frequency-dependent part ␻(z/vgϪt). The peak of the pulse occurs when all the frequency-dependent phases van- ish; the overall frequency-independent phase does not affect the pulse size. Thus, the peak of the pulse propagates within the medium according to zϭvgt. The velocity of the peak is vg , the group velocity of the medium, which can be nega- tive. The ‘‘rephasing’’ ͑24͒ within the medium changes the wavelengths of the component waves. Typically the wave- length increases, and by greater amounts at longer wave- lengths. A longer time is required before the phases of the waves all become the same at some point z inside the me- dium, so in a normal medium the velocity of the peak ap- pears to be slowed down. But in a negative group velocity medium, wavelengths short compared to ␭0 are lengthened, long waves are shortened, and the velocity of the peak ap- pears to be reversed. By a similar argument, Eq. ͑25͒ tells us that in the vacuum region beyond the medium the peak of the pulse propagates according to zϭctϩa(1/cϪ1/vg). The group velocity is again c, but the ‘‘rephasing’’ within the medium results in a shift of the position of the peak by the amount a(1/c Ϫ1/vg). In a normal medium where 0Ͻvgрc the shift is negative; the pulse appears to have been delayed during its passage through the medium. But after a negative group ve- locity medium, the pulse appears to have advanced! This advance is possible because, in the Fourier view, each component wave extends over all space, even if the pulse appears to be restricted. The unusual ‘‘rephasing’’ in a negative group velocity medium shifts the phases of the fre- quency components of the wave train in the region ahead of the nominal peak such that the phases all coincide, and a peak is observed, at times earlier than expected at points beyond the medium. As shown in Fig. 3 and further illustrated in the examples in the following, the ‘‘rephasing’’ can result in the simulta- neous appearance of peaks in all three regions. Fig. 3. A snapshot of three Fourier components of a pulse in the vicinity of a negative group velocity medium. The component at the central wavelength ␭0 is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wavelength component is lengthened. Then, even when the incident pulse has not yet reached the medium, there can be a point inside the medium at which all components have the same phase, and a peak appears. Simultaneously, there can be a point in the vacuum region beyond the medium at which the Fourier components are again all in phase, and a third peak appears. The peaks in the vacuum regions move with group velocity vgϭc, but the peak inside the medium moves with a negative group velocity, shown as vgϭϪc/2. The phase velocity vp is c in vacuum, and close to c in the medium. 610 610Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 11. D. Propagation of a sharp wave front To assess the effect of a medium with negative group ve- locity on the propagation of a signal, we first consider a waveform with a sharp front, as recommended by Sommer- feld and Brillouin.12 As an extreme but convenient example, we take the inci- dent pulse to be a Dirac delta function, E(z,t)ϭE0␦(z/c Ϫt). Inserting this in Eq. ͑28͒, which is based on the linear approximation ͑23͒, we find E͑z,t͒Ϸ Ά E0␦͑z/cϪt͒ ͑zϽ0͒ E0ei␻0z͑n͑␻0͒/cϪ1/vg͒ ␦͑z/vgϪt͒ ͑0ϽzϽa͒ E0ei␻0a͑n͑␻0͒/cϪ1/vg͒ ␦͑z/cϪtϪa͑1/cϪ1/vg͒͒ ͑aϽz͒. ͑29͒ According to Eq. ͑29͒, the delta-function pulse emerges from the medium at zϭa at time tϭa/vg . If the group ve- locity is negative, the pulse emerges from the medium before it enters at tϭ0! A sample history of ͑Gaussian͒ pulse propagation is illus- trated in Fig. 4. Inside the negative group velocity medium, an ͑anti͒pulse propagates backwards in space from zϭa at time tϭa/vgϽ0 to zϭ0 at time tϭ0, at which point it ap- pears to annihilate the incident pulse. This behavior is analogous to barrier penetration by a rela- tivistic electron20 in which an electron can emerge from the far side of the barrier earlier than it hits the near side, if the electron emission at the far side is accompanied by positron emission, and the positron propagates within the barrier so as to annihilate the incident electron at the near side. In the Wheeler–Feynman view, this process involves only a single electron which propagates backwards in time when inside the barrier. In this spirit, we might say that pulses propagate backwards in time ͑but forward in space͒ inside a negative group velocity medium. The Fourier components of the delta function are indepen- dent of frequency, so the advanced appearance of the sharp wave front as described by Eq. ͑29͒ can occur only for a gain medium such that the index of refraction varies linearly at all frequencies. If such a medium existed with negative slope dn/d␻, then Eq. ͑29͒ would constitute superluminal signal propagation. However, from Fig. 1 we see that a linear approximation to the index of refraction is reasonable in the negative group velocity medium only for ͉␻Ϫ␻0͉Շ⌬/2. The sharpest wave front that can be supported within this bandwidth has char- acteristic rise time ␶Ϸ1/⌬. For the experiment of Wang et al. where ⌬/2␲Ϸ106 Hz, an analysis based on Eq. ͑23͒ would be valid only for pulses with ␶տ0.1 ␮s. Wang et al. used a pulse with ␶Ϸ1 ␮s, close to the minimum value for which Eq. ͑23͒ is a reason- able approximation. Since a negative group velocity can only be experienced over a limited bandwidth, very sharp wave fronts must be excluded from the discussion of signal propagation. How- ever, it is well known12 that great care must be taken when discussing the signal velocity if the waveform is not sharp. E. Propagation of a Gaussian pulse We now consider a Gaussian pulse of temporal length ␶ centered on frequency ␻0 ͑the carrier frequency͒, for which the incident waveform is Fig. 4. Ten ‘‘snapshots’’ of a Gaussian pulse as it traverses a negative group velocity region (0ϽzϽ50), according to Eq. ͑31͒. The group velocity in the gain medium is vgϭϪc/2, and c has been set to 1. 611 611Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 12. E͑z,t͒ϭE0eϪ͑z/cϪt͒2/2␶2 ei␻0z/c eϪi␻0t ͑zϽ0͒. ͑30͒ Inserting this in Eq. ͑28͒ we find E͑z,t͒ϭ Ά E0eϪ͑z/cϪt͒2/2␶2 ei␻0͑z/cϪt͒ ͑zϽ0͒ E0eϪ͑z/vgϪt͒2/2␶2 ei␻0͑n͑␻0͒z/cϪt͒ ͑0ϽzϽa͒ E0ei␻0a͑n͑␻0͒Ϫ1͒/c eϪ͑z/cϪa͑1/cϪ1/vg͒Ϫt͒2/2␶2 ϫei␻0͑z/cϪt͒ ͑aϽz͒. ͑31͒ The factor ei␻0a(n(␻0)Ϫ1)/c in Eq. ͑31͒ for aϽz becomes e␻p 2 ␥a/⌬2c using Eq. ͑11͒, and represents a small gain due to traversing the negative group velocity medium. In the experi- ment of Wang et al., this factor was only 1.16. We have already noted in the previous section that the linear approximation to ␻n(␻) is only good over a fre- quency interval about ␻0 of order ⌬, and so Eq. ͑31͒ for the pulse after the gain medium applies only for pulse widths ␶տ 1 ⌬ . ͑32͒ Further constraints on the validity of Eq. ͑31͒ can be ob- tained using the expansion of ␻n(␻) to second order. For this, we repeat the derivation of Eq. ͑31͒ in slightly more detail. The incident Gaussian pulse ͑30͒ has the Fourier de- composition ͑27͒, E␻͑z͒ϭ ␶ ͱ2␲ E0eϪ␶2 ͑␻Ϫ␻0͒2/2 ei␻z/c ͑zϽ0͒. ͑33͒ We again extrapolate the Fourier component at frequency ␻ into the region zϾ0 using Eq. ͑20͒, which yields E␻͑z͒ϭ ␶ ͱ2␲ E0eϪ␶2 ͑␻Ϫ␻0͒2/2 ei␻nz/c ͑0ϽzϽa͒. ͑34͒ We now approximate the factor ␻n(␻) by its Taylor ex- pansion through second order: ␻n͑␻͒Ϸ␻0n͑␻0͒ϩ c vg ͑␻Ϫ␻0͒ ϩ 1 2 d2 ͑␻n͒ d␻2 ͯ␻0 ͑␻Ϫ␻0͒2 . ͑35͒ With this, we find from Eqs. ͑26͒ and ͑34͒ that E͑z,t͒ϭ E0 A eϪ͑z/vgϪt͒2/2A2␶2 ei␻0n͑␻0͒z/c eϪi␻0t ͑0ϽzϽa͒. ͑36͒ where A2 ͑z͒ϭ1Ϫi z c␶2 d2 ͑␻n͒ d␻2 ͯ␻0 . ͑37͒ The waveform for zϾa is obtained from that for 0ϽzϽa by the substitutions ͑22͒ with the result E͑z,t͒ϭ E0 A ei␻0a͑n͑␻0͒Ϫ1͒/c eϪ͑z/cϪa͑1/cϪ1/vg͒Ϫt͒2/2A2␶2 ϫei␻0z/c eϪi␻0t ͑aϽz͒, ͑38͒ where A is evaluated at zϭa here. As expected, the forms ͑36͒ and ͑38͒ revert to those of Eq. ͑31͒ when d2 (␻n(␻0))/d␻2 ϭ0. So long as the factor A(a) is not greatly different from unity, the pulse emerges from the medium essentially undis- torted, which requires a c␶ Ӷ 1 24 ⌬2 ␻p 2 ⌬ ␥ ⌬␶, ͑39͒ using Eqs. ͑18͒ and ͑37͒. In the experiment of Wang et al., this condition is that a/c␶Ӷ1/120, which was well satisfied with aϭ6 cm and c␶ϭ300 m. As in the case of the delta function, the centroid of a Gaussian pulse emerges from a negative group velocity me- dium at time tϭ a vg Ͻ0, ͑40͒ which is earlier than the time tϭ0 when the centroid enters the medium. In the experiment of Wang et al., the time ad- vance of the pulse was a/͉vg͉Ϸ300a/cϷ6ϫ10Ϫ8 s Ϸ0.06␶. If one attempts to observe the negative group velocity pulse inside the medium, the incident wave would be per- turbed and the backwards-moving pulse would not be de- tected. Rather, one must deduce the effect of the negative group velocity medium by observation of the pulse that emerges into the region zϾa beyond that medium, where the significance of the time advance ͑40͒ is the main issue. The time advance caused by a negative group velocity medium is larger when ͉vg͉ is smaller. It is possible that ͉vg͉Ͼc, but this gives a smaller time advance than when the negative group velocity is such that ͉vg͉Ͻc. Hence, there is no special concern as to the meaning of negative group ve- locity when ͉vg͉Ͼc. The maximum possible time advance tmax by this tech- nique can be estimated from Eqs. ͑17͒, ͑39͒, and ͑40͒ as tmax ␶ Ϸ 1 12 ⌬ ␥ ⌬␶Ϸ1. ͑41͒ The pulse can advance by at most a few rise times due to passage through the negative group velocity medium. While this aspect of the pulse propagation appears to be superluminal, it does not imply superluminal signal propaga- tion. In accounting for signal propagation time, the time needed to generate the signal must be included as well. A pulse with a finite frequency bandwidth ⌬ takes at least time ␶Ϸ1/⌬ to be generated, and so is delayed by a time of order of its rise time ␶ compared to the case of an idealized sharp wave front. Thus, the advance of a pulse front in a negative group veloc- ity medium by Շ␶ can at most compensate for the original delay in generating that pulse. The signal velocity, as defined by the path length between the source and detector divided by the overall time from onset of signal generation to signal detection, remains bounded by c. As has been emphasized by Garrett and McCumber13 and by Chiao,18,19 the time advance of a pulse emerging from a gain medium is possible because the forward tail of a smooth pulse gives advance warning of the later arrival of the peak. The leading edge of the pulse can be amplified by the gain medium, which gives the appearance of superluminal pulse 612 612Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 13. velocities. However, the medium is merely using information stored in the early part of the pulse during its ͑lengthy͒ time of generation to bring the apparent velocity of the pulse closer to c. The effect of the negative group velocity medium can be dramatized in a calculation based on Eq. ͑31͒ in which the pulse width is narrower than the gain region ͓in violation of condition ͑39͔͒, as shown in Fig. 4. Here, the gain region is 0ϽzϽ50, the group velocity is taken to be Ϫc/2, and c is defined to be unity. The behavior illustrated in Fig. 4 is per- haps less surprising when the pulse amplitude is plotted on a logarithmic scale, as in Fig. 5. Although the overall gain of the system is near unity, the leading edge of the pulse is amplified by about 70 orders of magnitude in this example ͓the implausibility of which underscores that condition ͑39͒ cannot be evaded͔, while the trailing edge of the pulse is attenuated by the same amount. The gain medium has tem- porarily loaned some of its energy to the pulse permitting the leading edge of the pulse to appear to advance faster than the speed of light. Our discussion of the pulse has been based on a classical analysis of interference, but, as remarked by Dirac,21 classi- cal optical interference describes the behavior of the wave functions of individual photons, not of interference between photons. Therefore, we expect that the behavior discussed above will soon be demonstrated for a ‘‘pulse’’ consisting of a single photon with a Gaussian wave packet. ACKNOWLEDGMENTS The author thanks Lijun Wang for discussions of his ex- periment, and Alex Granik for references to the early history of negative group velocity and for the analysis contained in Eqs. ͑14͒–͑16͒. a͒ Electronic mail: 1 L. V. Hau et al., ‘‘Light speed reduction to 17 metres per second in an ultracold atomic gas,’’ Nature ͑London͒ 397, 594–598 ͑1999͒. 2 K. T. McDonald, ‘‘Slow light,’’ Am. J. Phys. 68, 293–294 ͑2000͒. A figure to be compared with Fig. 1 of the present paper has been added in the version at 3 This is in contrast to the ‘‘⌳’’ configuration of the three-level atomic system required for slow light ͑Ref. 2͒ where the pump laser does not produce an inverted population, in which case an adequate classical de- scription is simply to reverse the sign of the damping constant for the pumped oscillator. 4 L. J. Wang, A. Kuzmich, and A. Dogariu, ‘‘Gain-assisted superluminal light propagation,’’ Nature ͑London͒ 406, 277–279 ͑2000͒. Their website,, contains additional material, including an animation much like Fig. 4 of the present paper. 5 W. R. Hamilton, ‘‘Researches respecting vibration, connected with the theory of light,’’ Proc. R. Ir. Acad. 1, 267,341 ͑1839͒. 6 J. S. Russell, ‘‘Report on waves,’’ Br. Assoc. Reports ͑1844͒, pp. 311– 390. This report features the first recorded observations of solitary waves ͑p. 321͒ and of group velocity ͑p. 369͒. 7 G. G. Stokes, Problem 11 of the Smith’s Prize examination papers ͑2 February 1876͒, in Mathematical and Physical Papers ͑Johnson Reprint Co., New York, 1966͒, Vol. 5, p. 362. 8 T. H. Havelock, The Propagation of Disturbances in Dispersive Media ͑Cambridge U.P., Cambridge, 1914͒. 9 H. Lamb, ‘‘On Group-Velocity,’’ Proc. London Math. Soc. 1, 473–479 ͑1904͒. 10 See p. 551 of M. Laue, ‘‘Die Fortpflanzung der Strahlung in Dispergier- enden und Absorbierenden Medien,’’ Ann. Phys. ͑Leipzig͒ 18, 523–566 ͑1905͒. 11 L. Mandelstam, Lectures on Optics, Relativity and Quantum Mechanics ͑Nauka, Moscow, 1972͒; in Russian. 12 L. Brillouin, Wave Propagation and Group Velocity ͑Academic, New York, 1960͒. That the group velocity can be negative is mentioned on p. 122. 13 C. G. B. Garrett and D. E. McCumber, ‘‘Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,’’ Phys. Rev. A 1, 305– 313 ͑1970͒. 14 R. Y. Chiao, ‘‘Superluminal ͑but causal͒ propagation of wave packets in transparent media with inverted atomic populations,’’ Phys. Rev. A 48, R34–R37 ͑1993͒. Fig. 5. The same as Fig. 4, but with the electric field plotted on a logarith- mic scale from 1 to 10Ϫ65 . 613 613Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 14. 15 E. L. Bolda, J. C. Garrison, and R. Y. Chiao, ‘‘Optical pulse propagation at negative group velocities due to a nearby gain line,’’ Phys. Rev. A 49, 2938–2947 ͑1994͒. 16 M. W. Mitchell and R. Y. Chiao, ‘‘Causality and negative group delays in a simple bandpass amplifier,’’ Am. J. Phys. 68, 14–19 ͑1998͒. 17 A. M. Steinberg and R. Y. Chiao, ‘‘Dispersionless, highly superluminal propagation in a medium with a gain doublet,’’ Phys. Rev. A 49, 2071– 2075 ͑1994͒. 18 R. Y. Chiao, ‘‘Population Inversion and Superluminality,’’ in Amazing Light, edited by R. Y. Chiao ͑Springer-Verlag, New York, 1996͒, pp. 91–108. 19 R. Y. Chiao and A. M. Steinberg, ‘‘Tunneling Times and Superluminal- ity,’’ in Progress in Optics, edited by E. Wolf ͑Elsevier, Amsterdam, 1997͒, Vol. 37, pp. 347–405. 20 See p. 943 of R. P. Feynman, ‘‘A Relativistic Cut-Off for Classical Elec- trodynamics,’’ Phys. Rev. 74, 939–946 ͑1948͒. 21 P. A. M. Dirac, The Principles of Quantum Mechanics ͑Clarendon, Ox- ford, 1958͒, 4th ed., Sec. 4. Forces in complex fluids Bruce J. Ackersona) and Anitra N. Novyb) Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078-3072 ͑Received 25 August 2000; accepted 18 December 2000͒ ͓DOI: 10.1119/1.1351151͔ I. PROBLEM Ferrofluids1 are stable suspensions of magnetic particles having linear dimension on the order of 10 nm. Due to vig- orous Brownian motion the magnetic particles assume ran- dom orientations rendering the suspension as a whole para- magnetic. These complex fluids show a variety of phe- nomena and instabilities that amuse and delight students and teachers alike.2 Because these fluids are used in a variety of applications including rotary seals, sensors, and actuators,3 they are commercially available.4 Figure 1 shows the experimental apparatus for viewing and recording the response of a ferrofluid film trapped at an air–water interface. Figure 2 shows recorded images for a drop ͑ϳ40 ␮l͒ of mineral-oil-based ferrofluid5 introduced to the surface of clean, filtered, de-ionized ͑18 M⍀͒ water. The hydrophobic ferrofluid spreads uniformly over the surface of water contained in a Petri dish. We gently stir the surface to emulsify the film, creating a collection of dark flat circular drops of ferrofluid as recorded in Fig. 2͑a͒. Figure 2͑b͒ shows the film 1 min after a cylindrical magnet having a radius of 1 cm is introduced with the axis of symmetry ver- tical and the lower end 3.3 cm above the ferrofluid film. The ferrofluid film clears from directly beneath the magnet but moves radially inward at large distances, forming tear- shaped drops with the clearer regions streaming outward. The ferrofluid collects in a ring structure at a finite radius ͑which is most dense at radius ϳ1.0 cm͒ from the center of the magnetic field symmetry axis. As the ferrofluid builds up, clumps or cone-shaped structures develop. As the cones grow, they become unstable and migrate one at a time into the central region. Figure 2͑c͒ taken at 31 4 min shows the clumping in the ring-shaped structure with one cone at two o’clock escaping to the central region. Finally in Fig. 2͑d͒, taken 21 min after introducing the magnet, a regular ‘‘crys- talline’’ array of well-separated ferrofluid cones has formed. Yet there remains a ferrofluid film ring surrounding this crys- talline structure. How is it possible that the ferrofluid is both attracted to ͑cones͒ and repelled from ͑film͒ the region directly below the cylindrical magnet? II. SOLUTION Magnetic body forces, surface tension, viscous drag, and gravity combine to produce the peculiar behavior observed here. We focus on the magnetic body forces as the primary explanation for the question posed above. Figure 3 shows a side view of one of the ‘‘cones’’ of ferrofluid which form the two-dimensional crystalline array beneath the magnet. These clumps have a nearly ellipsoidal shape above the water sur- face with the symmetry axis parallel to the applied field. Other studies show that ferrofluid droplets submerged in an immiscible fluid deform into ellipsoids and align parallel to the direction of a uniform external magnetic field.6 Solutions have long existed for the magnetic ͑electric͒ field of an el- lipsoid of permeability ␮ ͑dielectric constant ⑀͒ subjected to a uniform external field in a surrounding medium of perme- ability ␮0 ͑dielectric constant ⑀0͒.7 When the symmetry axis of the ellipse is aligned parallel to the external field direction, the field inside the ellipse is uniform and parallel to the ap- plied field ͑at large distances͒. Thus, to first order the ‘‘ellip- soidal cones’’ behave like little magnets oriented parallel to the external field and so move along the water surface to the strongest field regions located directly beneath the cylindri- cal magnet. However, the magnetic field induced in each of these cones is aligned parallel with the neighboring cone fields; therefore, the cones repel one another in the plane of the interface just like parallel oriented permanent magnets. A crystal lattice results.8 These results are qualitative, but in- tuitive, given our experience playing with permanent mag- nets. How do we understand the quite different behavior of the film? Rosensweig1 gives a general derivation of the body force f or force per unit volume, which reduces for ferrofluid suspensions to fϭ␮0͑M"ٌ͒Hϭ␮0M“H, ͑1͒ where ␮0 is the vacuum permeability, H is the magnetic field strength, and M is the magnetization in the film volume el- ement. This functional form suggests the Kelvin force den- sity on an isolated body, except that the local field H re- places the applied field H0 . Intuitively, we understand this body force to be like the force acting on a magnetic dipole. 614 614Am. J. Phys. 69 ͑5͒, May 2001 © 2001 American Association of Physics Teachers
  • 15. The force depends on the orientation of the dipole ͑repre- sented by M͒ in the local field ͑represented by H͒ and on the gradient of the local field in the vicinity of the dipole. There is no force, only a torque, acting on a dipole in a uniform field. The derivation of the last equality in Eq. ͑1͒ reasonably assumes that “ÃHϭ0 and that the magnetic flux density B is linearly related to the magnetic field strength and to the magnetization as follows: Bϭ␮Hϭ␮0͑HϩM͒ ͑2͒ with ␮ the permeability of the film. Using Eq. ͑2͒ to repre- sent the body force in terms of the magnetic flux density in the second equality of Eq. ͑1͒ gives fϭ 1 ␮ ͩ1Ϫ ␮0 ␮ ͪB“Bϭ 1 2␮ ͩ1Ϫ ␮0 ␮ ͪ“B2 . ͑3͒ Consequently, the negative square of the magnetic flux den- sity within the film corresponds to a ‘‘potential field’’ to which the film responds. Consider the film to be of constant thickness and spread uniformly over the surface of the water when in the presence of the external magnetic field. In the absence of magnetic charge and surface current density, consideration of the Max- well equations leads to the following boundary conditions for the magnetic field strength and magnetic flux density: ͑BϪB0͒"nϭ0, ͑HϪH0͒Ãnϭ0, ͑4͒ where the terms with a subscript refer to the air ͑or vacuum͒ side of the film boundary and the terms without a subscript to the ferrofluid. The vector n is a unit normal to the film sur- face. Note that these are the same boundary conditions and field relationships used to find the field inside an ellipsoid subjected to a uniform external field. For the film problem, we take the normal to the film, the zˆ direction, to be parallel to the magnet axis and assume azimuthal symmetry. The unit vector ␳ˆ designates the radial direction with respect to the Fig. 3. Ellipsoidal ‘‘clump’’ or cone of ferrofluid. Fig. 1. Experimental setup for viewing the dynamics of a ferrofluid film. Fig. 2. Ferrofluid film before the magnet was introduced ͑a͒, after 1 min ͑b͒, after 3 1 4 min ͑c͒, and after 21 min ͑d͒. Note that any dark spots that appear the same in all frames are due to imperfections in the optical train. 615 615Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 16. magnet axis. Using the relationships given in Eqs. ͑2͒ and ͑4͒, the magnetic flux density and magnetic field strength just inside the film are represented in terms of the magnetic flux density just outside the film as BϭB0zzˆϩ ␮ ␮0 B0␳␳ˆ, ͑5͒ Hϭ 1 ␮ B0zzˆϩ 1 ␮0 B0␳␳ˆ. Since the film is thin, we assume these boundary results to represent the field within the film. Next we approximate the external field at the surface of the film by a dipole field with moment m directed parallel to zˆ, B0ϭ ␮0 4␲r3 ͑3͑m"rˆ͒rˆϪm͒ ϭ ␮0m 4␲ ͫ 3␳z ͑␳2 ϩz2 ͒5/2 ␳ˆ ϩ 2z2 Ϫ␳2 ͑␳2 ϩz2 ͒5/2 zˆ ͬ, ͑6͒ where z measures the distance from the film to the dipole source ͑cylindrical magnet͒, ␳ is the radial position in the film, rϭͱ␳2 ϩz2 is the position vector magnitude in spheri- cal coordinates, and the vectors with a caret are unit vectors. This magnetic flux density, when plugged into the right-hand side of Eq. ͑5͒, gives an estimate of the magnetic flux density inside the film. Directly beneath the dipole the magnetic flux density is normal to the film and has the same magnitude on both sides of the film surface. As ␳ increases from zero, however, there is a component of the magnetic flux density which is parallel to the surface and larger inside the film than outside for ␮/␮0Ͼ1. While the magnitude of the external magnetic flux density decreases monotonically with increas- ing ␳ for this dipole field, the magnetic flux density inside the film will increase for sufficiently large permeability ratio, ␮/␮0 , before decreasing to zero. The increasing magnetic flux density results in a force ͓see Eq. ͑3͔͒ that pushes the film outward radially to collect in a ring where the flux den- sity reaches a maximum. Note that we neglected contribu- tions from the film to the external magnetic field on the assumption that this field is small for a very thin film. While an exact solution to this problem is beyond the scope of a typical undergraduate course in electromagnetism, we checked our results using the method of images to find the magnetic flux density produced by a dipole near a thin plane of material with permeability ␮. In the limit that the thick- ness goes to zero, the fields inside and outside the film be- come identical to those approximated here. There is one ca- veat. The body force must be calculated using the second equality in Eq. ͑1͒. The first equality involves a derivative with respect to z, which must be performed before the film thickness is taken to zero. Figure 4 shows the reduced potential, estimated for our ferrofluid film using the dipole field given in Eq. ͑6͒, as a function of ␳ for zϭ1, and different values of the permeabil- ity ratio ␮/␮0 . The functional form of this reduced potential is ⌽ϭϪ B2 ͑␮0m/4␲͒2 ϭϪͩ␳4 ϩ͑Ϫ4ϩ9͑␮/␮0͒2 ͒␳2 z2 ϩ4z4 ͑␳2 ϩz2 ͒5 ͪ. ͑7͒ For permeability ratios greater than ͱ8/3Ϸ1.63 a potential minimum obtains at finite radius. The minimum position ␳min obeys the following linear relationship with respect to z: ␳min /zϭͱϪ6͑␮/␮0͒2 ϩ3ϩͱ36͑␮/␮0͒4 Ϫ33͑␮/␮0͒2 ϩ1. ͑8͒ This ratio ranges between zero and one half as the perme- ability ratio increases from 1.63 to infinity. Since the initial magnetic susceptibility of our ferrofluid is ␹ϭ1.9, the per- meability ratio is 2.9, and Eq. ͑8͒ predicts ␳min /zϭ0.43. Ap- proximating the magnet by a dipole placed at the lower end of the magnet, the end closest to the ferrofluid, gives ␳min /z ϳ1.0 cm/3.3 cmϳ0.30 or placing the dipole at the physical center of the magnet gives ␳min /zϳ1.0 cm/5.8 cmϳ0.17. Considering that the magnet is an extended source rather than a point dipole, the agreement is quite good. The response of a ferrofluid film to a nonuniform external field is complex, but a fairly straightforward undergraduate boundary value calculation explains puzzling observations. The large susceptibility of the magnetic fluid and fluid sur- face orientation beneath a magnet determines the net force on the ferrofluid, giving attraction for the cones and repul- sion for the film. a͒ Author to whom correspondence should be addressed; electronic mail: b͒ This communication originated as an honors thesis project of A.N., who is presently a graduate student in physics at Colorado State University. 1 R. E. Rosensweig, Ferrohydrodynamics ͑Cambridge U.P., Cambridge, 1985͒, p. 110 ff. 2 R. E. Rosensweig, ‘‘Magnetic Fluids,’’ Sci. Am. 247 ͑4͒, 136–145 ͑1982͒. 3 B. M. Berkovsky, V. F. Medvedev, and M. S. Krakov, Magnetic Fluids Engineering Applications ͑Oxford U.P., Oxford, 1993͒, Chap. 6. 4 Ferrofluidics Corporation, 40 Simon Street, Nashua, NH 03060-3075. 5 Ferrofluidics: Catalog No. EMG 905, Lot No. F8193A. 6 V. I. Arkhipenko, Yu. D. Barkov, and V. G. Bashtovoi, ‘‘Study of a magnetized fluid drop shape in a homogeneous magnetic field,’’ Magn. Gidrodin. ͑3͒, 131–134 ͑1978͒. 7 G. Arfken, Mathematical Methods for Physicists ͑Academic, New York, 1970͒, p. 603; S. D. Poisson, ‘‘Seconde me´moire sur la the´orie du mag- ne´tisme,’’ Mem. Acad. R. Sci. Inst. France 5, 488–533 ͑1821–1822͒. 8 A. T. Skjeltorp, ‘‘One- and two-dimensional crystallization of magnetic holes,’’ Phys. Rev. Lett. 51 ͑25͒, 2306–2309 ͑1983͒. Fig. 4. Reduced potential for an element of ferrofluid film for zϭ1 as a function of the distance ␳ from the magnet axis and shown for different values of the permeability ratio ␮/␮0ϭ1, 2, 2.9, and 5 from top to bottom. 616 616Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
  • 17. A mechanical model that exhibits a gravitational critical radius Kirk T. McDonalda) Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 ͑Received 7 July 1999; accepted 21 December 2000͒ ͓DOI: 10.1119/1.1351152͔ I. PROBLEM A popular model at science museums ͑and also a science toy1 ͒ that illustrates how curvature can be associated with gravity consists of a surface of revolution rϭϪk/z with z Ͻ0 about a vertical axis z. The curvature of the surface, combined with the vertical force of Earth’s gravity, leads to an inward horizontal acceleration of kg/r2 for a particle that slides freely on the surface in a circular, horizontal orbit. Consider the motion of a particle that slides freely on an arbitrary surface of revolution, rϭr(z)у0, defined by a con- tinuous and differentiable function on some interval of z. The surface may have a nonzero minimum radius R at which the slope dr/dz is infinite. Discuss the character of oscillations of the particle about circular orbits to deduce a condition that there be a critical radius rcritϾR, below which the orbits are unstable. That is, the motion of a particle with rϽrcrit rapidly leads to excursions to the minimum radius R, after which the particle falls off the surface. Give one or more examples of analytic functions r(z) that exhibit a critical radius as defined above. These examples provide a mechanical analogy as to how departures of gravi- tational curvature from that associated with a 1/r2 force can lead to a characteristic radius inside which all motion tends toward a singularity. II. SOLUTION We work in a cylindrical coordinate system (r,␪,z) with the z axis vertical. It suffices to consider a particle of unit mass. In the absence of friction, there is no torque on a particle about the z axis, so the angular momentum component J ϭr2 ␪˙ about that axis is a constant of the motion, where the dot ͑•͒ indicates differentiation with respect to time. For motion on a surface of revolution rϭr(z), we have r˙ϭrЈz˙, where the prime ͑ Ј͒ indicates differentiation with respect to z. Hence, the kinetic energy can be written Tϭ 1 2͑r˙2 ϩr2 ␪˙ 2 ϩz˙2 ͒ϭ 1 2͓z˙2 ͑1ϩrЈ2 ͒ϩr2 ␪˙ 2 ͔. ͑1͒ The potential energy is Vϭgz. Using Lagrange’s method, the equation of motion associated with the z coordinate is z¨͑1ϩrЈ2 ͒ϩz˙2 rrЉϭϪgϩ JrЈ r3 . ͑2͒ For a circular orbit at radius r0 , we have r0 3 ϭ J2 r0Ј g . ͑3͒ We write ␪˙ 0ϭ⍀, so that Jϭr0 2 ⍀. For a perturbation about this orbit of the form zϭz0ϩ⑀ sin ␻t, ͑4͒ we have, to order ⑀, r͑z͒Ϸr͑z0͒ϩrЈ͑z0͒͑zϪz0͒ϭr0ϩ⑀r0Ј sin ␻t, ͑5͒ rЈϷr0Јϩ⑀r0Љ sin ␻t, ͑6͒ 1 r3 Ϸ 1 r0 3 ͩ1Ϫ3⑀ r0Ј r0 sin ␻tͪ. ͑7͒ Inserting ͑4͒–͑7͒ into ͑2͒ and keeping terms only to order ⑀, we obtain Ϫ⑀␻2 ͑1ϩr0Ј2 ͒sin ␻t ϷϪgϩ J2 r0 3 ͩr0ЈϪ3⑀ r0Ј2 r0 sin ␻tϩ⑀ r0Љ sin ␻tͪ. ͑8͒ From the zero’th-order terms we recover ͑3͒, and from the order-⑀ terms we find that ␻2 ϭ⍀2 3r0Ј2 Ϫr0r0Љ 1ϩr0Ј2 . ͑9͒ The orbit is unstable when ␻2 Ͻ0, i.e., when r0r0ЉϾ3r0Ј2 . ͑10͒ This condition has the interesting geometrical interpretation ͑noted by a referee͒ that the orbit is unstable wherever ͑1/r2 ͒ЉϽ0, ͑11͒ i.e., where the function 1/r2 is concave inwards. For example, if rϭϪk/z, then 1/r2 ϭz2 /k2 is concave outwards, ␻2 ϭJ2 /(k2 ϩr0 4 ), and there is no regime of insta- bility. We give three examples of surfaces of revolution that sat- isfy condition ͑11͒. First, the hyperboloid of revolution defined by r2 Ϫz2 ϭR2 , ͑12͒ where R is a constant. Here, r0Јϭz0 /r0 , r0ЉϭR2 /r0 3 , and ␻2 ϭ⍀2 3z0 2 ϪR2 2z0 2 ϩR2 ϭ⍀2 3r0 2 Ϫ4R2 2r0 2 ϪR2 . ͑13͒ The orbits are unstable for z0Ͻ)R, ͑14͒ or equivalently, for r0Ͻ 2) 3 Rϭ1.1547Rϵrcrit. ͑15͒ As r0 approaches R, the instability growth time approaches an orbital period. Another example is the Gaussian surface of revolution, 617 617Am. J. Phys. 69 ͑5͒, May 2001 © 2001 American Association of Physics Teachers
  • 18. r2 ϭR2 ez2 , ͑16͒ which has a minimum radius R, and a critical radius rcrit ϭRͱ4 eϭ1.28R. Our final example is the surface rϭϪ k zͱ1Ϫz2 ͑Ϫ1ϽzϽ0͒, ͑17͒ which has a minimum radius of Rϭ2k, approaches the sur- face rϭϪk/z at large r ͑small z͒, and has a critical radius of rcritϭ6k/ͱ5ϭ1.34R. These examples arise in a 2ϩ1 geometry with curved space but flat time. As such, they are not fully analogous to black holes in 3ϩ1 geometry with both curved space and curved time. Still, they provide a glimpse as to how a particle in curved space–time can undergo considerably more com- plex motion than in flat space–time. ACKNOWLEDGMENTS The author wishes to thank Ori Ganor and Vipul Periwal for discussions of this problem. a͒ Electronic mail: 1 The Vortx͑tm͒ Miniature Wishing Well, Divnick International, Inc., 321 S. Alexander Road, Miamisburg, OH 45342, AWESTRUCK SCIENTISTS The second feature of science is that it shows that the world is simple. Even many scientists do not appreciate that they are hewers of simplicity from complexity. They are often more deluded than those they aim to tell. Scientists are often overawed by the complexity of detecting simplicity. They look at the latest fundamental particle experiment, see that it involves a thousand kilograms of apparatus and a discernible percentage of a gross national product, and become thunderstruck. They see the complexity of the apparatus and the intensity of the effort needed to construct and operate it, and confuse that with the simplicity that the experiment, if successful, will expose. Some scientists are so awestruck that they even turn to religion! Others keep a cool head, and marvel not at an implied design but at the richness of simplicity. P. W. Atkins, ‘‘The Limitless Power of Science,’’ in Nature’s Imagination—The Frontiers of Scientific Vision, edited by John Cornwell ͑Oxford University Press, New York, 1995͒. 618 618Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems