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# 610 Review Sheet for CHEM C1260 at Purdue

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PHYSICAL REVIEW A 79 013628 2009 Multiscale theory of nitesize Bose systems Implications for collective and singleparticle excitations s Pankavich z Shreif1 Y Chen3 and P Ortoleval 1Centerfar Cell and Virus Theory Department of Chemistry Indiana University Blaamingtan Indiana 47405 USA 2Department of Mathematics University of Texas at Arlington Arlington Texas 76019 USA 3Department of Physics Purdue University West Lafayette Indiana 47907 USA Received 18 August 2008 published 29 January 2009 Boson droplets ie dense assemblies of bosons at low temperature are shown to mask a signi cant amount of singleparticle behavior and to manifest collective dropletwide excitations To investigate the balance between singleparticle and collective behavior solutions to the wave equation for a nite size Bose system are constructed in the limit where the ratio a of the average nearestneighbor boson distance to the size of the droplet or the wavelength of density disturbances is small In this limit the lowest order wave function varies smoothly across the system ie is devoid of structure on the scale of the average nearestneighbor distance The amplitude of short range structure in the wave function is shown to vanish as a power of a when the interatomic forces are relatively weak However there is residual short range structure that increases with the strength of interatomic forces While the multiscale approach is applied to boson droplets the methodology is applicable to any nite size Bose system and is shown to be more direct than eld theoretic methods Con clusions for 4He nanodroplets are drawn DOI 101103PhysRevA79013628 I INTRODUCTION Quantum clusters QCs are assemblies such as quantum dots 173 super uid droplets 476 fermion droplets 78 superconducting particles 910 and other structures 1112 They involve processes simultaneously acting over multiple scales in space and time from the interaction of individual particles to dropletwide collective dynamics There have been a number of studies presenting theories of their properties and selected citations are provided above However none appear to address the multiscale character of a QC or introduce a framework that takes full advantage of the separation of scales as a way to solve the wave equation Recently it was shown that the wave equation for a fer mion quantum nanodroplet can be cast in a form that explic itly manifests its multiscale character 13 This formulation enables a deeper understanding of the interplay of angstrom and nanometer scale processes underlying the unique behav iors of a fermion QC imposed by the exclusion principle The objective of the present study is to introduce a novel technique that builds in characteristics of lowtemperature boson droplets The central concept of our earlier approach 13 is that there are two or more distinct spatial scales of motion for the constituent fermions in a QC One is the short characteristic length ie the average nearestneighbor spacing This short scale is n 3 for average number density n within the QC A second characteristic length is the QC diameter The ratio 9 of the smallerto larger of these lengths was introduced to enable a perturbation method for solving the wave equation While the interaction potential for a condensed QC is large and thereby cannot serve as the basis of a perturbation analy sis 9 for a QC of 1037106 particles is small say 10 1 to 102 Thus 9 presents itself as a natural candidate for the basis of a perturbation theory of QCs that could yield accu rate approximations even for strongly interacting systems 1050294720097910136288 0136281 PACS numbers 0375Hh 0270Ns 0530Jp 6725dw In a boson QC there are two types of processes to be accounted for The QCwide processes are either collective eg rotations coherent density waves or shape oscilla tions or migrations of particlelike disturbances across the QC ie the coordinated motion of a given particle and a set of others responding to the rst For identical quantum par ticles the latter quasiparticle excitations are not identi able with a speci c particle In contrast to these global processes shortscale ones re ect close encounters of particles related to the interparticle potential For fermions the exclusion principle strongly affects these shortscale motions How ever bosons display the opposite tendency ie a quorum principle At low temperature all bosons tend to be in the same state ie at T0 all bosons are in the single particle like ground state for weakly interacting systems While the N atom wave function is observed to have some long range structure as expressed in the low momentum behavior of neutron scattering experimenm it is stated that only 10 of the 4He atoms are participating in this Bose condensate like behavior 14716 This seems to be a re ection of the strength of the interatomic forces in a 4He liquid In the present work we explore this effect seeking to show how short scale structure increases in intensity as the strength of the interatomic forces increase In particular we seek a de scription wherein the N atom wave function can be under stood as a short lengthscale factor with variability that in creases with the strength of the potential multiplied by a long scale envelope factor representing collective modes We suggest that this is a distinct picture from that wherein it is stated that 10 of the particles are in the Bose condensate BBC and 90 are in higher momentum dressed single par ticle states ie the N atom system cannot be understood in this fashion All atoms participate in the collective motion and are part of the quasiparticle response Though the multiscale analysis is demonstrated for boson droplets the approach also applies to trapped atomic Bose 2009 The American Physical Society PANKAVICH et al Einstein condensates with typical interatomic distances be tween 01 and 1 micron and system size ranging from 1071000 microns hence containing between 1037106 atoms 1718 In the atomic Bose condensate the interatomic in teraction can be tuned by a Feshbach resonance 19 The study of strongly interacting trapped BECs near a Feshbach resonance has been a subject of active investigation 171820725 Many properties such as the condensate frac tion and collective mode frequencies can be modi ed near the Feshbach resonance 222 Such strongly yet still short ranged interacting Bose condensates offer an interesting middle ground between the weakly interacting BEC well described by mean eld theory and 4He super uid with strong and longer range interaction More recently trapped BECs with longrange dipolar interactions have also gener ated much interest 2627 In the present study we attempt to rigorously determine the limiting behavior of the wave function for boson QCs as ea 0 The objective is to develop a theory of boson droplets by integrating these notions into a multiscale theory valid for strongly interacting nite Bose systems described above There is a long history of multiscale analysis for Nparticle systems 28739 Most relevant is the analysis of the classical Liouville equation wherein one identi es order parameters that characterize the slow and long lengthscale behaviors of a system and mesoscopic equations for the sto chastic dynamics of order parameters are derived 40753 These order parameters describe nanometer scale features such as the position orientation and major substructures of a nanoparticle 52755 The ansatz starting the analysis is that solutions to the Liouville equation ie the Nparticle prob ability density depend on the atomistic con guration both directly and indirectly the latter through the order param eters Other approaches for quantum systems eg truncated Laplace transformation of the interaction potential and quan tum mechanicsmolecular mechanics 56765 have been de veloped but are distinct from the elength ratio perturbation approach developed here and do not yield a rigorous meso scopic wave equation for QC dynamics The ansatz that starts our multiscale QC analysis is that the wave function 1 re ects a dual dependence on the con guration of the N bosons in the QC Let F1F2 FN be the set of positions of the N bosons assumed identical By choice of units the position F of particle i is displaced a distance of about one unit when it moves a distance If In contrast R9F changes by a distance of about one unit as particle i traverses the entire QC Denote the collection of these scaled positions by RR1R2 RN To capture the distinct types of behavior long and short scale we hypoth esize the wave function 1 has a dual dependence on con guration ie 1 R This dual dependence is not a viola tion of the number of degrees of freedom 3N Rather it is a way to express the distinct ways in which 1 depends on the droplet con guration We show that if 9 is small the two distinct dependencies of 1 can be constructed via a multi scale perturbation technique Hence multiscale analysis naturally reveals the implications of these notions for a bo son QC PHYSICAL REVIEW A 79 013628 2009 II MULTISCALE FORMULATION The behavior of a lowtemperature boson QC is now ex plored via multiscale analysis We demonstrate how the in dividuality of the particles is lost yielding QCwide coop erative dynamics The absence of a Fermi level makes it dif cult to track the number of particles in a given region of space Thus the system lapses into a collective delocalized bosonic presence without a wellde ned sense of the indi viduality of particles However strong interactions between particles as in liquid 4He are expected to induce shortscale character in the wave function In this section we show how delocalization can emerge naturally from a multiscale analy sis of the wave equation for a boson QC We rst consider the case of relatively weak interactions and then explore the effect of stronger ones and the inclusion of shortscale struc ture in the wave function A Weak interactions delocalization and residual shortscale structure To begin the development we formulate the wave equa tion in a mariner that reveals the lowenergy excitations of interest at low T While our formulation facilitates the dis covery of the nature of the hypothesized delocalized behav ior selfconsistency would preclude the drawing of false conclusions since we begin with the full wave equation Let U be the Nparticle potential E mund be the ground state QC energy and de ne the deviatoric potential V U Eg ound Introduce the characteristic kinetic energy ZmL2 for each of the N bosons where m is the particle mass and L is the size of the QC The position of particle i is denoted LR for dimensionless vector R In these variables the dimension less Hamiltonian H is de ned via 1 H v2VEKV 1 while the dimensional Hamiltonian is H 2H mLZ For this E mundshifted Hamiltonian the groundstate energy is zero and V U Egond The dimensionless wave equation takes the form H 1 E 1 where E is the dimensionless deviatoric energy Let I be If ie l is the typical nearestneighbor dis tance within the QC of number density n Then the length scale ratio is given by 221 L With the above de nitions R changes by a distance of about one unit as particle i traverses the entire QC In contrast F E e lR changes by a distance of about 2 1 as particle i traverses the entire QC and by about one unit when it traverses one nearestneighbor distance 1 Thus the R s are natural for tracking QCwide disturbances while the F are ideal for characterizing close particleparticle encounters With this our multiscale ansatz is that 1 has the dependence 1 F1F2 FNR1R2 RN9 The dual de pendence of 1 does not constitute a violation of the number 3N degrees of freedom Rather we shall show that it is a re ection of our expectation that 1 depends on the Nboson con guration in two distinct ways That both dependencies can be constructed is shown below to be achieved in the 0136282 MULTISCALE THEORY OF FlNlTESIZE BOSE small 9 limit The multiscale wave equation with f1f2fN and RIEIR2RN follows from this ansatz and the chain rule H0 8H1 SZHZW Eur 2 where E 22E and 1 2 1 2 Ho Ya H1 Y0 Yia H2 EY1V 3 Here V0 is the 5 gradient and V1 is the 8 gradient The objective of our multiscale development is to con struct an equation for the mesoscopic wave function MR which varies smoothly across the QC lt1gtR J39 d3NrAR er 1 r 4 The sampling function A is a Gaussianlike expression which in 5 space is centered at about 2 11 is unit normal ized and has a halfwidth that is much greater than one 5 distance but is much less than the QC diameter A central theme of this study is that one may derive a selfconsistent equation that is closed in I in the small 9 limit A perturbation solution to the multiscale wave equation is constructed as a Taylor expansion in 9 ie 1 220 I ne To 020 one obtains the eigenvalue problem with zero potential HO I O Eowo 5 Since we seek normalizable solutions which decay to zero as lilacs I o must be independent of r This is consistent with the physical nature of the problem ie as a QC is of nite size about one unit in R1 and H0 is a free particlelike Hamiltonian the lowest order problem only admits an independent solution The absence of smallscale structure in I o implies E0 is zero and 4 0 1003 6 for D0 to be determined in higher order While it is clear that Datbo as 8H0 it remains to show that DO can be con stricted in a selfconsistent procedure Using the 020 analysis ie E020 the 0a equation becomes Ho l l H11r0 Euro 7 Since PO is independent of HI I O vanishes Hence the righthand side of Eq 7 is independent of g and thus HO I I must be independent of 5 Using the normalizability and de cay conditions ie I l vanishes at in nity We nd that PI is independent of g and E120 To 022 one nds that since 1 0 and PI are independent of r How2 HZ I O Euro 8 To arrive at an equation for Do we a multiply both sides of Eq 8 by the sampling function A and integrate over 5 b use the divergence theorem and properties of A c neglect PHYSICAL REVIEW A 79 013628 2009 surfacetovolume terms and d use the fact that DO does not change appreciably within the sampling volume ie the region wherein A is large With this one obtains the meso scopic wave equation 1 5 V00 4 Em 9 upon noting that dgtgtdgt0 as 8H0 and de ning 7 via V03 f d3NrAR rVr 10 Even if the bare potential is short range ie independent of R 7 depends on it due to the Rcentered local averaging manifest in the sampling function A Since E is the deviatoric energy E0 ie En0 for every n must be an eigenvalue corresponding to the ground state solution of the original problem Hence E220 is an eigenvalue of the mesoscopic Eq 9 and 10 While singleparticle character in D0 is lost it was present in the original wave equation The question arises as to how it was lost This can be addressed by subtracting Eq 9 from the wave equation 8 to 022 One obtains gm Mr 7Rldgto o 11 This is a 3N dimensional Poissonlike equation with charge density equal to 2V l7lgt0 Being proportional to Do the source term is limited to the region within the droplet if Eq 9 supports boundstate solutions Through V the charge density has shortscale ie individual particle character with variations over distances of order If This implies that although all individual particle character in 1 0 and PI is lost there is residual particle character ie oscillations in the wave function across 5 with amplitude of 022 For a 1000 boson QC 9 2101 the single particle character of the wave function is two orders of magnitude smaller than the overall pro le as expressed in D0 Since W2 is proportional to D0 and D0 is zero outside the QC particlelike character is con ned to the interior of the QC as expected We conclude that the multiscale approach to boson QCs constitutes a self consistent picture and yields insights into the nature of low temperature boson QCs when boundstate solutions to the mesoscopic equation 9 exist B Strong interactions and induced shortscale structure The above development is now revisited but for the case where the potential is strong scaling as 2 1 in particular we let the potential be e lV With this H1 is now Y1Y0V V The rationale for subtracting 7 is clari ed below We nd that l7 is small when the smoothing volume is appre ciable ie averaging the large positive core potential with the weaker longrange attractive tail leads to partial cancel lation in l7 as demonstrated for helium in Sec HI Thus we assume 7 is 0a To preserve the full potential we put 7 in H2 having denoted 7 as 2 7 0136283 PANKAVICH et al With YO I 020 and the above Eq 7 becomes HO I 1V l7 1 0 1 1 0 12 Multiplying both sides by A integrating over all 5 using integration by parts and neglecting higher order terms in a one obtains 120 Hence PI is the solution of 1 z 5Y3W1IVI VRltIgto 13 Separating variables one solution to Eq 13 is of the form W1BRdgt01 where B satis es i738 m We 14 Thus I is seen to have shortscale structure of amplitude 0a and not 022 as in the case of weaker interactions Sec II A The 022 problem now reads 1 Howz Y139 0YOB EYIdDO IBV V V1430 E2 0 15 Upon multiplying both sides of Eq 15 by A integrating over 5 using the divergence theorem and Eq 14 and ne glecting surfacetovolume ratio terms one obtains a meso scopic wave equation for D0 similar to Eq 9 i7 003 W19 4 Em 16 C13 J39 d3NrAR erlYoBrRl2 17 As seen from Eq 14 B is a response to the uctuations of the potential difference V 7 1m gradient with respect to 5 re ects shortscale structure in the derivative potential Thus one might expect that is a type of kinetic energy con tribution that adds to the potential 7 driving the dynamics of Do In the next section we consider a different analysis using numerical techniques and a calibrated potential for 4 He III APPLICATION TO 4He To explore the implications of the theory of Sec II we developed computational procedures and applied them to 4He We consider factors affecting the behavior of a 4He nanodroplet and the structure of our multiscale approach These include averaging length kinetic versus potential en ergy the effective wavelength of the bosons and droplet Size Let the distance over which a Gaussianlike function 7 is appreciable be denoted g and let 7 be unit normalized The smoothing function A of Sec II is taken to be a product of N factors 7R ef il2 N where is 7 a normalized PHYSICAL REVIEW A 79 013628 2009 Gaussianlike sampling function The smoothed potential I7 for pairwise bare interaction potential 1 takes the form V03 E 6er 6 18 K where mpg f d3r1d3rzrai 513705 swarm 19 From symmetry the smoothed potential 17 only depends on the distance RU and through 7 the smoothing parameter To investigate the character of 17 we chose vr to be the Aziz potential 66 A numerical code was written to evaluate the sixfold integral in Eq 19 taking advantage of symme try Pro les of 17R g for various values of g are seen in Fig 1 These pro les show that the smoothing parameter has a drastic effect on the position of the minimum well depth and overall shape of the potential As increases interpar ticle distance at the minimum and well depth both increase Thus as g grows larger 17 becomes strictly positive and monotonically decreasing since the repulsive core dominates the attractive tail and causes the well to disappear The bare potential 1 has a shortrange repulsive core of radius about 1 and a longrange attractive tail with range of several 1 With this pairwise interaction it is expected that if g is smaller than I then 17 is roughly the same as the bare potential 1 Hence D0 would have shortscale character in contradiction to our assumption ie D0 depends on 8 not 5 If g is large ie similar to the size of the 4He nanodrop let then 17 is small ie for most of the range of integration underlying the averaging in 17 the values of 17 are small and contributions from the repulsive core and the longerrange attractive tail tend to cancel Thus for large 7 would not support bound states and the excitation energy E2 depends strongly on the choice of g This implies that a selfconsistent procedure must be invoked for choosing For example one solves the mesoscopic wave equation for 1003 g with cor responding excitation energy 2 and then minimizes 2 with respect to g Such a strategy is equivalent to construct ing the functional i2 whose minimum over all Do is the solution to the mesoscopic wave equation and then minimiz ing this functional with respect to both D0 and g As the pro le of the effective potential changes so does the energy i2 There are several estimates of kinetic and potential energy to be considered If L is the diameter of the nanodroplet then 7 122mL2 is the kinetic energy associated with the longest wavelength bosons In contrast the rest en ergy is wZ where wzzkm and k is the second derivative of the bare potential evaluated at its minimum A relevant potential energy is the well depth for the bare potential while another is that for 17 In Table I we present values of these energies in addition to information about the potential well for differing values of g with nanodroplet size N 103 and 106 Due to the symmetry of Eq 19 the position of the 0136284 MULTISCALE THEORY OF FINITESIZE BOSE n3s 9w ml 270 320 370 my 470 szn WEE K m 39D 4m 4m Rm son syn eon b n3n as M I as g Q02 2 s at W 01 5447 590 447 km 947 74a 7947 0 FIG 1 Graphs of the effective potential for NlO3 particles in nanodroplet with differing values of smoothing parameter a 035 b 060 c 090 respectively minimum should change by a factor of 9 as N increases from 103 to 106 At low temperature boson nanodroplets display collective behaviors wherein individual particle detail gives way to nanoscale order parameter dynamics Analysis of the meso scopic wave equation for boson nanodroplets yields implica PHYSICAL REVIEW A 79 013628 2009 tions for 4He droplet dynamics including quantized surface waves ie morphological oscillations Quantized vortices in thin lms 67 suggest that there may be related excita tions in 4He nanodroplem Table I Computations with the present theory involve construct ing the order parameter R pro le This can be accom plished via a variational principle based on a functional whose minimum is attained for D0 expressed as an Nfold p product With this p satis es 1 YivetrltpEltp 20 Nve am i f dWvaitpa lwa N MN MR 21 for functional derivative 88ltpR and threedimensional Rspace Laplacian Vi This constitutes a nonlinear eigen value problem to determine p and the energy E States in the form of an Nfold product of a single p function are not necessarily the lowest energy excitations Expanding the set of admissible functions could lead to lower energy states Furthermore that these order parameters can be imaginary is central for capturing droplet analogues of quantized vortices Table I If one adopts a quasiparticle perspective then the nano droplet consists of particles of a broad range of wavelengths The longer correspond to the droplet diameter for them all detail of the bare potential is lost ie they experience an effective potential which is small due to averaging the re pulsive core and attractive tail For shorterwavelength qua siparticles details of the bare potential are experienced and shortscale structure is induced by the potential Choosing greater than 1 but much less than the droplet diameter yields the wave function lBeIgt0 from Eq 13 which captures the range of elementary excitation wavelengths If g is suf ciently large but still much less than the nanodroplet diam eter then a mean eld approximation for D0 should suf ce ie each boson is interacting with many others so that an effective eld is an accurate description IV RESULTS AND CONCLUSIONS A mesoscopic wave equation for boson QCs was derived using a multiscale approach It was shown that properties of boson QCs can be derived via a multiscale perturbation analysis of the wave equation even when interactions be tween the bosons are strong Bosons in a droplet at low T were shown to act in a collective manner losing much of their individual character Lowlying excited state solutions of the mesoscopic wave equation have pro les without spa tial variations on the n 3 scale at which one would other wise expect to re ect the presence of individual bosons Rather individual particle features merge via the averaging imposed by the smoothness of the wave function A smoothed effective N particle interaction potential I7 0136285 PANKAVICH et al PHYSICAL REVIEW A 79 013628 2009 TABLE I Table of values describing energies of the effective for given value of g and bare potentials Harmonic Position Bare rest oscillator Ground state Kinetic Size of Smoothing of Depth of energy halfwidth energy energy of nanodroplet parameter minimum Depth of minimum k L E0 Nanodroplet N g m A minimum m K m J m J m A m J m J 0 296 108 l49gtlt10 22 1108gtlt10 22 l303226gtlt10 5 680811 gtlt10 33 206gtlt10 14 N103 035 352 557 759gtlt10 23 060 440 l60 22l gtlt10 23 090 623 003 4l4gtlt10 25 N106 035 035 557 759x1023 060 044 l60 22l gtlt10 23 090 062 003 4l4gtlt10 25 emerges that only depends on dropletwide positional infor mation ie not on any n 13 scale features With the excep tion of highly excited states of the droplet I7 only supports coherent dropletwide dynamics The key technical achievement of the multiscale analysis is the derivation of a mesoscopic wave equation for boson droplet dynamics The methodology holds for a strongly in teracting boson droplet of nite size at low T The mesos copic wave function D0 depends on the set of N particle positions 8 which are cast in unim such that they change a distance of one unit as a boson traverses the entire droplet The result is in my 0 Em 22 me f d3NrAlt1g ezvltz 23 where V is the 3N dimensional Laplacian with respect to R and I71 is the bare N particle potential with the location of each particle averaged over a sampling volume containing a statistically signi cant number of bosons E2 is the excitation energy This mesoscopic wave equation is remarkable in that it holds for strong interparticle interaction strength as long as 9 the ratio of the average nearestneighbor spacing to drop let diameter is small ie for those with 1000 or more bosons As D0 is the lowest order wave function ie I HGDO as 8H0 it satis es boson particle label exchange symmetry PUltIgt0 10 24 for permutation operator PU As D0 is governed by the smoothed potential I7 much of the individual particle particle shortrange correlation is diminished This is essen tially a selfconsistency argument ie 10I7 Particle ex change symmetry and the averaging in I7 suggest that any one boson is not interacting with particular others ie in contrast to two body interaction This suggests that to good approximation d30ltp151ltp152 N for order parameter ltpR The single particle density pR is de ned via N pa PNRE 505 Jamar 25 171 and hence is approximately Niltp1i2 Thus p is directly re lated to the number density The above implies the boson droplet at low T is charac terized by the pro le of the order parameter ltpR which is devoid of short ie If scale features In this way all individual particle behavior is lost as 9 gt 0 Since the energy for shortrange forces as for 4He determined by Eq 1 is proportional to N the difference in energy for the N 1 and the N particle droplet the chemical potential is independent of N In this way there is no energy measure of the number of bosons in the droplet This is in sharp contrast to the situation for fermions ie due to the Fermi level The averaging in I7 suggests it is singleparticle like ie each boson evolves in a local potential eld N J7E 505 11 26 For 4He there is an attractive tail in the bare twobody po tential Thus we expect 17 to be as in Fig l for a spherical droplet More generally the threedimensional spatial pro le of 7 depends on droplet morphology and hence on p imelf This is accounted for in the local averaging embedded in I7 Thus the order parameter p is determined by a mesoscopic wave equation with mean eld character The development of the present multiscale approach suggests that this is not just a crude approximation Rather it appears to be a conse quence of a the smallness of 9 for a droplet b the ex change symmetry constraints the quorum principle for bosons and c the smooth pro le of the wave function for low temperature droplets ie that there is no n 13 spatial scale structure in 1 except as an 022 correction A promising area for future developmenm is to extend these results to account for the scattering of atoms or mol 0136286 MULTISCALE THEORY OF FINITESIZE BOSE 8 FIG 2 Schematic depiction of a boson nanodroplet in a vortex like state of quantized undamped circulation ecules from a QC or the effects of external elds In work in progress we are applying the multiscale approach to trapped BEC systems as the interaction progresses from weak to strong and short range to long range to investigate the ef PHYSICAL REVIEW A 79 013628 2009 fects of interaction and to understand the evolution from weakly interacting BBC to a strongly interacting Bose sys tem such as 4He Such studies could facilitate the design of experiments to validate predictions of the multiscale theory Other natural extensions include the analysis of more com plex droplets such as those composed of fermions and bosons ie 3He 4He mixtures those with embedded solid nanoparticles or macromolecules or those with sustained in ternal circulation patterns Fig 2 ACKNOWLEDGMENTS This project was supported in part by the Indiana Univer sity College of Arm and Sciences through the Center for Cell and Virus Theory Additional support was provided by the Lilly Endowment Inc 1 R Nepstad L Saelen and J P Hansen Phys Rev B 77 125315 2008 2 c H Liu Phys Lett A 372 888 2008 3 J T Shen and S H Fan Phys Rev A 76 062709 2007 4 J M Merritt J Kupper and R Miller Phys Chem Chem Phys 9 401 2007 5 L Lehtovaara and J Eloranta J Low Temp Phys 148 43 2007 6 J Lekner J Phys Condens Matter 12 4327 2000 7 F Stienkemeier O Bunermann R Mayol F Amcilotto and M Pi unpublished 8 R Mayol F Amcilotto M Barranco M 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#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.