ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES.

Transcription

1 ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. All systems with interation of some type have normal modes. One may desribe them as solutions in absene of soures; they are exitations of the system that an appear without any external stimuli; they are solutions to the homogeneous versions of the differential equations that desribe the system. Before we study the eletromagneti normal modes let us study some mehanial systems. We start with the single pendulum where a mass, m, is attahed to a string of length l. This system has one normal mode. θ l m The system is governed by Newtons nd law. For small angles this law is redued to l θ + gθ =. The Fourier transformed version is l ( i ) + g θ =, whih has non-trivial solutions if ( ) + = l i g, i.e., = gl.

2 Bo E. Sernelius : This is the frequeny of the normal mode. Let us now extend this problem to the double pendulum. θ l m ϕ For small amplitudes the equations are redued to l θ + gθ + l ϕ = l θ + l ϕ + gϕ =. Fourier transformation gives ( l + g) θ l ϕ = ( ) = l θ + l + g ϕ. This system of equations may be written on matrix form, θ l g l A A ϕ = = + ; l l + g. There are non-trivial solutions if the determinant of the matrix is zero, ( ) ( ) = l + g l, or m

3 Bo E. Sernelius :3 or ( ) ± ( ) = l + g l, ( ) = ( ) = + g l; g l. Thus, there are two normal modes in this system. In the one with larger frequeny the two masses move out of phase of eah other and in the other they are in phase. Next we study a mass in i spring x m If we express x as the displaement from the equilibrium point we have the equation of motion as x + m x =. On Fourier transformed form it is ( + mx ) =, with non-trivial solutions for = / m. For two masses and two springs we have

4 Bo E. Sernelius :4 m = m =, = m =, ; where the first (seond) frequeny is the frequeny with whih the lowest (upper) mass is osillating if we eep the upper (lower) in fixed position. When we let them both loose we have the two normal modes from the equations m mz + z z = mz + z z = ; Fourier transforming gives m + z + = m z and the ondition for modes, m + m + =, whih gives

5 Bo E. Sernelius :5 ( m + ) ( m + ) ( ) = m m = = and finally = = ; Now we study many masses attahed to strings as in the figure m m m m m m m m m m a a a a a a a a a If we only have one mass the mode frequeny is = m. For inreasing numbers we get : = m 3: +

6 Bo E. Sernelius : 5: 7: If we plot these modes they give a rather unorganized impression n

7 Bo E. Sernelius :7 If we on the other hand hange the horizontal axis q = π λ = nπ L π/a q all points ome on one and the same urve, q msin qa. Curves lie this are named dispersion urves. In the insets are shown how the masses move when these modes are exited. What is shown is the displaements at the two turning points of the osillations. The displaements we disuss here are along the springs, so the exitations are longitudinal. The system also has transverse exitations whih we do not onsider here. Next we disuss eletromagneti normal modes when two objets are present, surrounded by vauum. We start with two mirosopi objets, two atoms. Van der Waals interation between two atoms. = ( ) We will now derive the interation energy and fore between two atoms without permanent dipole moments. We will neglet retardation effets; this is allowed exept for very large separations between the atoms. We will return to this problem and inlude retardation effet later; the treatment beomes more ompliated but we feel that the piture would not be omplete without it. Johan Dideri van der Waals (837-93) graduated on his thesis: On the ontinuity of the gaseous and liquid states, in 873. He found deviations for

8 Bo E. Sernelius :8 real gases to the ideal gas equation of state. He found empirially a p + V ( ) = V b RT, instead of the ideal equation: pv = RT, for a mole of gas. The orretion onstant b is due to the fat that the gas atoms tae up a finite fration of the volume, thus reduing the free volume. The orretion a, whih is of interest here, is due to the attrative fore between the atoms, reduing the pressure exerted on the walls of the ontainer. He was awarded the Nobel Prie in 9 for this and similar wor on the equations of state for gases and fluids. Van der Waals fore was found on empirial grounds and it was not until 93 that London gave a realisti explanation for this fore. We will go through a similar derivation here. There will be an attrative fore between two atoms or moleules even if none of them arry a permanent dipole moment. This is still true if the partiles are spherially symmetri. This is a bit ounter intuitive. One would imagine that two spherially symmetri, neutral atoms would not interat were they so far apart that their eletron wave funtions were not overlapping. We start with two atoms in vauum. We name them atom and atom. Assume that atom is polarized and has a dipole moment p. A dipole moment gives rise to an eletri field whih we in hapter 3 found is E = φp. The eletri field in position of atom is E = φ p. Now, this field will polarize atom, p = α E,

9 Bo E. Sernelius :9 where α is the polarizability of atom. We have here let open the possibility for the atom to be anisotropi by letting the polarizability be a tensor. This dipole moment gives rise to an eletri field, whih in the position of atom is E = φ p. If this field is the ause of the polarization of atom that we started with we have p = α E. Combining all these equations give p = α E = α φ p = α φ α E = α φ α φ p, or ( ) = αφ αφ p. Now we limit the treatment to isotropi atoms and let the polarizabilities be a salar times the unit tensor,. We furthermore hoose the z-axis to point along the line joining the two atoms, whih are r apart. Then φ = φ = 3 r and r αα A = ( αα ) r 4 αα ( ) ( ) r

10 Bo E. Sernelius : and from whih follows that A r αα 4 r αα. = ( ) ( ) If we have simple enough an expression for the polarizabilities we may now find the mode frequenies by putting the determinant equal to zero. London used a simple approximation for the polarizabilities, the so-alled London approximation, whih agrees with the Lorentz lassial model that we disussed in hapter, ( ) = ( ) ( ) αi αi i, where i is a harateristi frequeny for atom i. The last single fator in the expression for the determinant, above, gives rise to two modes orresponding to osillating dipoles in the z-diretion. The first squared fator gives four, in pairs degenerate, modes orresponding to dipoles osillating in the x- and y-diretions. 3 4

11 Bo E. Sernelius : 5 We study an example where both atoms are of the same element, Li...8 3,5 Li-Li Energy (Hartree)..4 4, r (a ) The interation energy an be written as { } = E = i ( r) n + i ( ) n + i ( r ) i i= i= from whih the fore an be obtained as [ ( )],

12 Bo E. Sernelius : F = d dr E ( r). The modes are so-alled mass-less bosons. At zero temperature, in absene of any external stimulation, the oupation number, n, vanishes and zero-point energy, only, ontributes. It is the shift of the zero-point energy when the interation as turned on that ontributes to the interation energy. With our approximations we end up with the following interation potential for two different atoms: VvdW ( r) = ( ) ( ) α α r +. 3 If the atoms are the same we have VvdW ( r) = α ( ). r 3 4 Potential (Hartree) Li-Li E Li =.859 a.u. α Li () = 4.8 a.u r (a ) In the figure above the dotted straight line with least (largest) slope is the van der Waals (Casimir) asymptote. The irles are from the best ab initio alulation. We see that the best result follows the van der Waals asymptote for smaller separations and the Casimir asymptote for larger separations. To get these results for the van der Waals interation we relied on a simple expression for the polarizabilities. We ould determine the modes analytially

13 Bo E. Sernelius :3 and just sum their zero-point energies. We an manage also for more ompliated polarizabilities with a mathematial treatment based on analytial funtions. However, in doing so we loose some of the physial transpareny. This method is based on the so alled generalized argument priniple We will repeatedly mae use of an extension to the so-alled Argument Priniple familiar to most of us from under-graduate mathematial ourses on analytial funtions. Let us study a region in the omplex frequeny plane. We have two funtions defined in this region; one, φ(z), is analytial in the whole region; one, f(z), has poles and zeros inside the region. The following relation holds for an integration path around the region: dzϕ z d ϕ ϕ πi ( ) ln f ( z) = ( zo ) ( z ), dz where z and z are the zeros and poles, respetively, of funtion f(z). In the Argument Priniple the funtion ϕ is replaed by unity and the right hand side then equals the number of zeros minus the number of poles, for funtion f(z), inside the integration path. For the right hand side of the equation above, to produe the interation energy, we mae the hoie ( ) = ( ) = ϕ z z ; f z A. The ontour should inlude the whole of the positive real frequeny axis. The funtion f(z) is the funtion in the defining equation for the normal modes of the system. By using this theorem we end up with an integration along a losed ontour in the omplex frequeny plane. In most ases it is fruitful to hoose the ontour shown in the figure below.

14 Bo E. Sernelius :4 Im z x o x o x o Re z Integration ontour in the omplex z-plane. Crosses and irles are poles and zeros, respetively, of the funtion f(z). The radius of the irle is let to go to infinity. We have the freedom to multiply the funtion f(z) with an arbitrary onstant without hanging the zeros and poles. If we hoose it arefully we an mae the ontribution from the urved part of the ontour vanish and we are only left with an integration along the imaginary frequeny axis. d { V ( r) d = ln α ( ) ( ) r 4 d ' α ' 4α π '( ) α '( ) r, where the primes on the polarizabilities denote that they are to be alulated on the imaginary frequeny axis. Performing a partial integration leads to V ( r) = d ( ) ( ) r ln α 4 4 ' α' α ' ( ) α ( ) π ' r. For large separation we may expand the logarithm and obtain the van der Waals result 3 3 VvdW ( r) = dα ( ) ( ) = d ' α' π r π α' ( ) α' ( ). r

15 Bo E. Sernelius :5 Using the London approximation for the polarizabilities gives VvdW ( r) = ( ) ( ) α α r +. 3 The results we have obtained here are also valid for spheres and, at large enough distanes, for all objets. We see that the interation potential varies as r- and the fore as r-7. Retardation effets and the Casimir fore We will now extend the above treatment to be valid for suh large distanes that the finite speed of light affets the results. To do that we have to mae use of the eletri field from a time dependent dipole. It is aording to the text boo in setion 9.3 E( r) p ( p r) r = [ ˆ ˆ ˆ rˆ] p t pˆ 3( pˆ rˆ) rˆ r [ ] r + r p t p t 3. r r It ontains ontributions from the dipole, its time derivative and its seond time derivative; all funtions at the retarded time. The eletri field from atom at the position of atom is now ( ) ( )( ) + ( ) E t = φ p t r r p t r ϑ p t r r where ( )( ), ϑ ϑ = =, 3 r if we let the z-axis point along the line joining the two atoms. The Fourier transformed version is

16 Bo E. Sernelius : E φ p ( ) = i r i r ( ) e ( ir ) + p ( ) e i r ϑ e i r = p ( ) + ( ) i r ( ) e φ ir ϑ ir. p So in analogy with the previous setion we now arrive at ( ( ) + ( ) ( ) ( ) i r α e φ ir ϑ ir α i r e φ ( ir ) + ϑ ( ir ) p = ). Assuming isotropi atoms we get αα e i r φ ( ) + ϑ ( ) i r i r p =. With our hoie of axes the matrix A is diagonal with elements A A e i r = = αα 3 4 i( r ) 3( r ) + i( r ) + ( r ) and A e i r 33 = 4αα i( r ) + ( r ). We want the determinant along the imaginary frequeny axis. Here it is an even funtion in. r r r A i e = α α r r ' ' r 4e α ( ) α ( ) ( ) ( ) ; i = i i ' ' r r ( ) ( ) ( ) ( ) ( ) + ( ) + ( ) The interation potential beomes r + + r ' ( ) ( ) α α

17 Bo E. Sernelius :7 r Vr () = d ln e ' ( ) ' 4 α ( ) + 4 α r r π r + ( ) r e α ' α ( ) ( ) + r + ( ) ' r r r + r 4 ( ) or r Vr () d ln e ' ' = ( ) ( ) + r 4 α α r ( π r ) + ( ) r e ' ' α ( ) α 3 4 ( ) + r ( ) + 3 r ( ) + r ( ) + r ( ) r For large distanes the logarithm may be expanded and only the lowest order term be ept. With large distanes we here mean that they are large enough for the interation to be wea. Then we find: r VCP () r = dα ( i ) α ( i ) e + ( r ) 3 + ( ) 5 r πr 3 4 r r + ( ) + ( ) This is the Casimir-Polder interation and it gives the van der Waals result for intermediate separations and the retarded result for large separations. We will now demonstrate that this is the ase. Let us first start with the van der Waals limit. Assume that r/ is small ompared to unity. The expression in the square braets redues to 3 and the exponential prefator to unity: 3 VCP () r dα ( i ) α ( i ) r π r This is the van der Waals result. To find the other limiting result we mae the substitution u=r/. Then we have:

18 Bo E. Sernelius :8 V r du i u CP () = α r r α 7 π i u r u e u + 5u + u + u The exponential fator guarantees that only small u values ontribute to the integral. If r is big enough we an replae the polarizabilities with the stati ones and move them outside the integral. Then we have: α ( ) α ( ) u 3 4 VCP () r due 3 + u + 5 u + u + u r 7 πr α ( ) α ( ) 3 3 = = 7 πr 4 4π α [ ( ) α ( )] 7 r whih is the Casimir result. Thus we see that for intermediate separations the potential goes as r- and for large separations as r-7. Next we study marosopi objets. Eletromagneti normal modes are solutions to the homogeneous versions of Maxwell s equations, i.e., the equations when the soures are absent. The system we are onerned with here onsists of two objets surrounded by vauum. We may divide the modes into three groups: vauum modes, i.e., modes of the surrounding vauum; bul modes, i.e., modes inside the objets far away from the surfaes; surfae modes, i.e., modes bound to the surfaes of the objets. The vauum modes we have already obtained in hapter 9. They are transverse plane waves haraterized by a 3D wave vetor, q, where the E and B fields are both mutually perpendiular and perpendiular to q; their amplitudes are equal; they are in phase. The dispersion urve is = q. The energy and momentum for a mode q is E = q ; p q = q. Also the bul modes we have already touhed upon in hapter. There are both longitudinal and transverse types. The longitudinal eletri modes are

19 Bo E. Sernelius :9 obtained from the relation ( q ) =, ε q, the longitudinal magneti modes from ( q ) =, µ q, and the transverse eletromagneti from ( ) ( )( ) q = ε q, q µ q, q q. The dispersion relations are found impliitly from the above relations. In most ases we have non-magneti materials. Then the two type of modes are obtained from the relations ( q ) =, ε q, and ( q q )( q ) q =, ε, respetively. The bul modes are important for the stability of the objet. The interations among the partiles within the objet and the interation energies an be expressed in terms of these modes. The longitudinal modes are most important here. The third group of modes, the surfae modes, we have not yet disussed. These modes are solutions to Maxwell s equations with the proper boundary onditions; they are loalized to the surfaes and interfaes. They play an important role in many situations. One set of effets derive from the fat that these modes ontain energy; this means that the modes ontribute to the surfae energy and surfae tension of the objets, whih means that they have effet on the stability and shape of the objet itself; they also give rise to fores between objets, fores that are of partiular importane if the objets are small. The fores are always there but sometimes their effets are mased by the

20 Bo E. Sernelius : presene of stronger fores. Another effet is that their presene modifies the optial properties of the objets; with the term optial properties we do not mean to limit ourselves to the visible range of the spetrum. The energy of eah mode is sensitive to small hanges at the surfae lie the presene of other atoms. These hanges an be deteted by optial means or other. Thus, the modes an be utilized, and are so, in sensors lie gas sensors. The fores between objets are used in the atomi fore mirosope and similar instruments. In small objets the modes on different parts of the surfae affet eah other and the geometry has to be taen into aount when Maxwell s equations are solved. For larger objets the surfae an loally be onsidered flat and one may use the results for planar interfaes. The surfae modes at a planar interfae are haraterized by the D wave-vetor,. We will here only onsider planar interfaes. Solving Maxwell s equations on both sides of the interfae and using the boundary onditions that the tangential omponent of the E- and H-fields and the normal omponent of the D- and B-fields are ontinuous one arrives at the following ondition for modes ( ) ( )( ) + ( ) ( )( ) = ε ε ε ε We note that a neessary ondition is that the dieletri funtion has different sign on the two sides of the interfae. Negleting retardation effets, i.e., treating the speed of light as infinite gives ( ) + ( ) =. ε ε. Metal-vauum interfae We are interested in a metal surfae in vauum. In this ase we have for small momenta ε pl ; ε =, whih gives for the surfae plasmon energy, the energy of the surfae mode

21 Bo E. Sernelius : obtained negleting retardation effets, s pl =. The equation determining the modes when retardation effets are taen into aount has the solution = ( pl + ) pl + 4 = ( s + ) + s This result gives the surfae plasmon dispersion. It is an important result if we want to alulate surfae energies of metals. The mode approahes the nonretarded result for large momentum. The deviation from this result only ours for very small momentum, near the light dispersion urve. We will find later that the retardation effets are important for the fore between objets at large separation only. The surfae plasmon dispersion is shown in the figure below..5 / s / s Figure The surfae mode for a metal-vauum interfae. The solid horizontal line is the plasmon energy and the dotted is the surfae plasmon energy. The diagonal straight line is the light dispersion urve in vauum and the urved solid urve is the surfae plasmon dispersion. We see that the modes are lose to the non-retarded result for large momentum and are pushed below the light dispersion urve for small

22 Bo E. Sernelius : momentum. They are only present in the frequeny range where the dieletri funtion of the metal is negative and they are onfined to the region to the right of the light dispersion urve for the medium outside the metal surfae. The modes are by some named surfae polaritons for general materials, or surfae plasmon polaritons in this speial ase, sine they ontain a photon omponent; some reserve the name polariton to the solution near the light dispersion urve and all them surfae plasmons for larger momentum; some all them surfae plasmons in the whole momentum range. Semiondutor-vauum interfae In the present ase we have ε = ε L ; ε =. T The frequenies L and T are the long-wavelength limiting frequenies of the longitudinal optial and transverse optial vibration modes of the rystal. The ondition for modes beomes ( ) ± ( + ) = + 4 q L q L q sph, where ε q = + ( ) ε ε ε L = T ε ε + + sph = T. The non-retarded result is sph. The result is shown in the figure below

23 Bo E. Sernelius :3 3 L.5 sph / T T / T Figure Surfae phonon mode in a polar semiondutor. We find that the surfae phonon mode only exists in the frequeny range between the transverse and longitudinal phonon modes. It approahes the nonretarded result for high momentum and is pushed down below the light dispersion urve for small momentum. The modes are alled surfae phonons or surfae phonon polaritons.

24 Bo E. Sernelius :4 Fore between two halfspaes To find the modes in the ase of two halspaes, separated a distane d from eah other with vauum between, one first solves Maxwell s equation in the three regions and uses the standard boundary onditions at the two interfaes. We first neglet retardation effets, i.e., we let the speed of light in vauum be infinite. This is muh simpler than doing the full alulation. Maxwell s equations in absene of external harge densities then loo lie D = B = E = H =. One arrives at the following ondition for modes: [ + ( )] ( ) [ ] = d ε e ε, where d is the distane between the two halfspaes of material having dieletri funtion ε ( ). For two metal halfspaes the dispersion urves for the two resulting modes are shown in the figure below.5 No retardation / s d

25 Bo E. Sernelius :5 These two modes derive from the surfae modes on the two surfaes. There are no vauum modes involved; when retardation is negleted there are no vauum modes. For two polar semiondutor halfspaes the dispersion urves are: L 3.5 No retardation sph / T T The interation energy per unit area, when we hoose our energy referene to be at infinite separation is d ( ) = V d d d d e [ ( )] + ln ε' ε' ( ) π [ ] ( π ) d π d d = ln e π [ ε' ( )] [ + ε' ( )] ( π ) d = d d ln e [ ε' ( )] [ + ε' ( )] 4π x = d xdx e ε ε ln [ ' ( )] [ + ' ( )] 4π d. We see that the interation energy goes as d - and hene the fore as d -3.

26 Bo E. Sernelius : Now we inlude retardation effets. The alulations beome muh more elaborate. One finds two types of mode, TM (Transverse Magneti) and TE Transverse Eletri). One arrives at the following onditions for modes: TM: ε ( ) + ε ( )( ) ( ) e ( ) d ε ( ) ε ( )( ) ( ) =, TE: ε ( )( ) + ( ) e d ( ) =. ε ( )( ) ( ) We show the results for simple Drude metals in the figure below..5 s d/=9 / s.5... TM exponentially deaying TM standing-wave-type TE standing-wave-type.5.5 / s The two surfae modes are modified and one even rosses the light dispersion urve. New standing wave type of modes appear originating from the vauum modes.

27 Bo E. Sernelius :7 This leads to V d d d e d ( ) ( ) ( ) = + +( ) 4 π ε ln ' ( )( ) ( ) ( ) ( ε ε ε ' ' ' )( ) + π d d 4 e d ln ' + + +( ) ( )( ) ( ) ε ( )( ) ( ) ε '. Here it is diffiult to use a transformation to find the distane dependene sine now both momentum and frequeny appear in the exponent. For large separations however we an only have ontributions for both and / small. This is taen are of by the exponential fator in the integrands. This means that the dieletri funtion may be replaed by its stati value: V d d d e d ( ) ( ) = + ( ) dlarge 4 π ε ln ' ( ) ( )( ) ( ) ( ) ε ε ' ' ( )( ) ε ' π ε d d e d ln ' + + +( ) ( )( ) ( ) ( )( ) ( ) ε '. Now we may mae the substitutions d d ; and find

28 Bo E. Sernelius :8 ( ) V d dlarge ( ) = d d e + ln 3 3π d + 3 3π d ddln e ε ' ε +( ) ( ) + ( ) + ε ' ( )( ) ' ( ) + ( ) + + ε ' ( )( ) ( )( ) ( ) + ε ' + + ε ' + + ( )( ) ( ). The separation dependene of the interation energy is now d -3 and from this follows that the Casimir fore goes as d-4. Thus the fore at large separations drops off faster when retardation is inluded. For a metal the dieletri funtion diverges for zero frequeny. This may be used to find the result: ( ) = V d dlarge +( ) d d e 3 ln 3π d + 3 ddln e 3π d = dd e 3 ln π d + x = dxd e 3 ln π d y + x = 3 dxdyy ln e. π d +( ) +( ) Treating the x- and y-oordinates as Cartesian oordinates and hanging to polar oordinates gives

29 Bo E. Sernelius :9 ( ) = V d dlarge for the energy and π drr d r { e r } π d 3 θ osθln r π = 3 drr ln { e } =, 3 π d 7d ( ) = F d dlarge π 4 4d, for the fore. This is what Casimir found for perfet metals, in that ase for all separations. For perfet metals there are no van-der-waals region. Now, the results of the full alulation for two gold halfspaes, using experimentally obtained dieletri funtions, are shown in the figure, below.. Potential (Joule/m ) T= K T = 3K vdw asymptote Casimir asymptote Experiments Gold plates d (µm) I next figure we have expanded the length sale.

30 Bo E. Sernelius :3 - Potential (Joule/m ) T= K T = 3K vdw asymptote Casimir asymptote Experiments Gold plates -3. d (µm) The solid urve with filled irles is the result using the equation above. It is valid for zero temperature. The open irles are the orresponding result at room temperature. The red solid irles are the result from modern experiments by Lamoreaux using a torsion pendulum.