Exact Solution to Ideal Chain with Fixed Angular Momentum

PHYSICAL REVIEW E 77, 051804 ͑2008͒

Exact solution to ideal chain with ﬁxed angular momentum

J. M. Deutsch Department of Physics, University of California, Santa Cruz, California 95064, USA ͑Received 6 February 2008; revised manuscript received 7 April 2008; published 13 May 2008͒ The statistical mechanics of a noninteracting polymer chain in the limit of a large number of monomers is considered when the total angular momentum L is ﬁxed. The radius of gyration for a ring polymer in this situation is derived exactly in closed form by functional integration techniques. Even when L=0 the radius of gyration differs from that of a random walk by a prefactor of order unity. The dependence on L is discussed qualitatively and the large-L limit can be understood by physical arguments, which can also be extended to self-avoiding chains.

DOI: 10.1103/PhysRevE.77.051804 PACS number͑s͒: 82.35.Lr, 82.37.Ϫj, 82.80.Ms

I. INTRODUCTION effect on the results obtained, for a large number of degrees The statistical properties of polymers have been the sub- of freedom. Therefore what we will find is at first sight rather ject of intensive research for many decades ͓1͔. However, surprising, that a polymer chain with conserved total energy these efforts have been almost entirely confined to polymers E, total linear momentum ptot=0, and total angular momen- in liquids or solids, while in contrast their properties in a tum L has a radius of gyration that depends strongly on L,so L vacuum have received little attention. Until recently there that even when =0 the radius of gyration differs signifi- cantly from that of an ideal chain without this restriction. were not clear physical realizations of such situations, but The somewhat counterintuitive nature of this effect makes it recent developments have changed this as will now be dis- worthwhile to prove it for an exactly solvable model, which cussed briefly. Mass spectrometry techniques have been de- is the main point of this work. veloped using lasers that desorb whole proteins into a We will show in Sec. II that the microcanonical formula- vacuum where their mass can be determined by measuring tion of this problem, with all conservation laws enforced, can the time of flight ͓2͔. It might be possible to probe molecules be expressed in a canonical ensemble in the limit of large N, such as DNA using optical tweezers in a manner similar to after which it can be converted to a functional integral. This what is done routinely in solution ͓3͔, and use this to probe can then be evaluated to obtain the radius of gyration as a in detail the polymer’s dynamics. In such a case, there may function of angular momentum L. We will analyze in Sec. prove to be applications in determining the structure of bio- II A how this can be understood more physically in the limit molecular complexes. For example, DNA often has mol- of large total angular momentum, and use this method to ecules bound to it, such as regulatory proteins, that are not derive the scaling behavior of a ring with excluded volume easily observable. Probing DNA in a vacuum might prove to interactions. In Sec. II B, we analyze the relation between the be more efficacious than in a solution because of the impor- temperature and total energy. In Sec. II C we calculate the tance of inertial effects that we will see play a much more distribution of angular momentum in equilibrium. In the important role than in a solvent. Finally, in the search for derivation, we used a quadratic potential between neighbor- long hydrocarbon molecules in interstellar media ͓4͔, such ing monomers. In Sec. II D we show how the results found systems are now of experimental interest. Short chains in this should be universally true for a large class of models with situation have been modeled ͓4͔ in order to compare experi- nonquadratic potentials. Finally, in Sec. III, we conclude mental spectroscopic frequency data with theory in order to with a discussion of the importance of ergodicity in the dy- determine the chain’s structure. It is therefore useful to ex- namics and a brief comparison with numerical results. tend this to model longer chains, for the same purpose. The author ͓5͔ recently considered such systems theoreti- II. DERIVATION OF MAIN RESULTS cally and by means of computer simulation, and the purpose The statistical mechanics of a classical system of N par- of this paper is to present an exact derivation for the radius of ticles interacting via a general potential ⌽, and with con- gyration of a polymer in a vacuum with conserved angular served total energy, momentum, and angular momentum, has momentum. The solution requires periodic boundary condi- been considered previously by Laliena ͓6͔. He has shown tions for the chain, which means we are considering a ring that the conservation of linear momentum does not affect polymer ͑ring DNA being a potential experimental example͒. answers obtained in the microcanonical ensemble with con- The qualitative features found persist for linear chains as servation of angular momentum enforced ͓7͔. So we will well, as has been confirmed by computer simulations ͓5͔. write down the volume of phase space with L and energy E In reality, angular momentum conservation is weakly bro- kept constant, ken by interaction with thermal electromagnetic radiation ͓5͔ but it is still important to understand the case of conservation ͑ ͒ ͵ ␦͑ ⌽͒␦͑3͒ ϫ W E,L,N = C E − K − ͩL − ͚ ri piͪ laws properly in order to understand these more complicated i effects. Also intrachain interactions are not considered; in N other words, this is the case of an “ideal” chain ͓1͔. In stan- ϫ␦͑3͒͑ ͒ͩ 3 3 ͪ ͑ ͒ dard treatments of statistical mechanics, angular momentum rc.m. ͟ d rid pi 1 i=1 conservation is mostly ignored as it is thought to have no

͕ ͖ ͕ ͖ where the coordinates are ri and momenta are pi . C is a Therefore we can take k to be along the z axis, k=kzˆ, and constant that involves N and ប and is of no consequence for write ͚ 2 / the purposes here. K is the kinetic energy ipi 2m, with m the mass of each monomer, and here we are taking them all ␨͑␤ ͒ ͵ 2/ ␤ ͑ 2 2͒ ␤⌽ ,k = expͩ− mk 2 ͚ xi + yi − ͪ to be equal. The center of mass rc.m. also must be conserved i ␦ and is set to 0. We use the Fourier representation of the N functions to write this as ϫ␦͑3͒͑ ͒ 3 ͑ ͒ rc.m. ͟ d ri. 6 i=1 ͑ ͒ ϰ ͵ ␭ ␭E ͵ −␭͑K+⌽͒␦͑3͒ ϫ W E,L,N d e e ͩL − ͚ ri piͪ Because the dependence of ␨ on k is only radial, we can also C i perform the k angular integrals in Eq. ͑4͒, rewriting the k N integration in spherical coordinates and obtaining ϫ␦͑3͒͑ ͒ͩ 3 3 ͪ ͑ ͒ rc.m. ͟ d rid pi . 2 ϱ i=1 c Z͑␤,L,N͒ = ͵ k sin͑kL͒␨͑␤,k͒dk ͑7͒ As shown by Lax ͓8͔, for most purposes, as discussed below, L 0 ␭ the contour of integration can be deformed in the complex where c is a constant that plays no role in the subsequent plane using the method of steepest descents. The three con- analysis. ͑ ͒ ͑ ͒ ditions are a that a saddle point exists, b that an observ- It should be noted that, although the above derivation now ͑ ϫ ͒ ͑ ͒ able not be of order exp const N , and c that there be no involves no explicit energy conservation, this is still math- singularity in the observable in the neighborhood of the ematically equivalent, in the limit of large N, to the initial ͑ ͒ ͑ ͒ saddle point. Conditions b and c are first obtained through formulation at fixed energy, momentum, and angular mo- the canonical ensemble and then tested to see if they are mentum. We will return to this point later in Sec. II B, when ͑ ͒ satisfied. Condition a , that a saddle point exists, is satisfied we discuss the dependence of temperature on the angular because we can find a relationship between the energy and momentum, and in the discussion, Sec. III. the temperature. In the case of an athermal system, say of The potential is taken to be that of an ideal Gaussian ϰ /␤ ␤ rigid links, this would just be that E 1 , where is the chain with step length l and a ring topology, value of ␭ at the saddle point. Condition ͑b͒ is satisfied for N−1 the quantity of interest here, the average radius of gyration. 3 ͑ ͒ ␤⌽ = ͚ͩ ͉r − r ͉2 + ͉r − r ͉2ͪ. ͑8͒ Condition c is also satisfied because we will see that the 0 2l2 i+1 i N 1 average radius of gyration is a smooth function of the tem- i=1 perature for finite T=1/␤. Thus we can drop the integration We expect by the central limit theorem that many models of over ␭ and replace ␭ by the inverse temperature ␤, and con- polymer chains will all give the same results for most quan- sider the partition function Z instead of the phase space vol- tities of interest, if the overall radius of gyration is ӶN.We ume integral ͑which is simply related to the entropy͒: will establish this in more detail in Sec. II D. In order to calculate the radius of gyration, one can add an ͑␤ ͒ ϰ ͵ 3 ͵ ik·L −␤͑K+⌽͒ ϫ additional potential with a parameter ⑀, Z ,L,N d k e e expͩ− ik · ͚ ri piͪ i N N ␤⌽ ␤⌽ ⑀ ͉ ͉2 ͑ ͒ = 0 + l͚ ri , 9 ϫ␦͑3͒͑ ͒ͩ 3 3 ͪ ͑ ͒ i=1 rc.m. ͟ d rid pi . 3 i=1 so that the average radius of gyration can be written as Integrating over the p ’s we obtain N lnZץ i 1 1 R2 ͳ ͚ ͉r ͉2ʹ ͯ ͯ ͑ ͒ = = − . 10 ⑀ץ N g N i Nl ͑␤ ͒ ϰ ͵ 3 ik·L͵ −1/2␤k·I·k −␤⌽ 3 i=1 ⑀=0 Z ,L,N d ke e e ͟ d ri i=1 The integration in Eq. ͑6͒ is Gaussian and we can now take the usual limit to turn this into a functional integral, ϵ ͵ d3keik·L␨͑␤,k͒. ͑4͒ M Tmk2 ␨͑␤,k͒ = ͵ expͭ− ͵ ͩ + ⑀ͪͫx2͑s͒ + y2͑s͒ + ⑀z2 2l Here I is the moment of inertia tensor for the particles, 0 3 N + ͉r˙͉2dsͬͮ␦͑3͒͑r ͒␦r͑s͒. ͑11͒ ͑ ␯ v␦␣␥ ␣ ␥͒ ͑ ͒ 2l c.m. I␣␥ = m͚ ri ri − ri ri , 5 i=1 The functional integrations in the x, y, and z directions de- where ␣ and ␥ label the coordinates ͑1,2,3͒, and the Einstein couple and the x and y functional integrals are identical. summation convention has been used for ␯. In the last equal- Each one of these three integrals is of the form of the parti- ity of Eq. ͑4͒ we have introduced the function ␨͑␤,k͒. Note tion function of a one-dimensional quantum harmonic oscil- that this cannot depend on the direction of k but only on its lator at finite temperature except for the restriction on the magnitude, if ⌽ involves only isotropic central potentials. center of mass. If we consider the Euclidean time action for

051804-2 EXACT SOLUTION TO IDEAL CHAIN WITH FIXED… PHYSICAL REVIEW E 77, 051804 ͑2008͒ a quantum harmonic oscillator of mass M at inverse tem- restriction on the path integral that the zero mode should not ␤ perature 0, be integrated over as a consequence of the restriction on the ␤ center of mass. Using the Fourier decomposition approach, 0 M ͵ ͑ 2 ␻2 2͒ ͑ ͒ we can easily incorporate this restriction, by not including S = x˙ + 0x dt, 12 0 2 the zero mode in the product. This amounts to multiplying ͑ ͒ ␻ Eq. 13 by 0 then the partition function Rescaling variables so that the angular momentum LЈ ϵ ͱ /͑ ͱ ͒ ͑ ͒ 1 L 12 Nl mT and using Eq. 10 gives Z = ͵ e−S␦x͑t͒ ϰ ͑13͒ 0 ͑␤ ␻ / ͒ ϱ 2 sinh 0 0 2 ͵ k͓−6+k2 +6k coth͑k͔͒csch͑k͒2sin͑kLЈ͒ dk ͑ ͒ ͑␤ ͒ 2 „ … with periodic boundary conditions on the paths x 0 =x 0 . R 1 0 g = . This is true because of the general formula relating the par- Nl2 36 ϱ tition function to the Euclidean path integral over times rang- ͵ k3 csch͑k͒2sin͑kLЈ͒dk ing from 0 to the inverse temperature ͓9͔. The partition func- 0 tion Eq. ͑13͒ can also be derived in a less elegant but more ͑14͒ direct manner by writing all paths in terms of a Fourier expansion, which then decouples the integrals, and forms an Both the numerator and denominator can be computed in inﬁnite product over all modes. This latter approach is useful closed form using contour integration. This gives the ﬁnal in the present application because we have the additional result

R2 2LЈ͑3+␲2͒ + LЈ͑␲2 −6͒cosh͑LЈ␲͒ +3͑LЈ2 −1͒␲ sinh͑LЈ␲͒ g = . ͑15͒ Nl2 36␲͓2LЈ␲ + LЈ␲ cosh͑LЈ␲͒ − 3 sinh͑LЈ␲͔͒

A plot of this equation is displayed in Fig. 1. high-L configurations, where we expect that a typical con- Ј 2 /͑ 2͒ ͑ /␲2͒/ For L =0 this reduces to Rg Nl = 1+15 36 figuration of the ring will be close to a circle rotating rapidly. Ϸ0.07. For a ring without angular momentum conservation, The approximate free energy contains a kinetic energy and 2 /͑ 2͒ ͑ / ͒Ϸ Rg Nl = 1 12 0.083, which means that the restriction to an elastic term L=0 causes the rings to be smaller relative to the case where the angular momentum can take on any value. L2 k F = + C2, ͑16͒ 2I 2 A. Large-L limit where C=2␲R is the circumference, and k is the entropic 1. Ideal chains elastic spring coefficient k=3T/͑Nl2͒. The moment of inertia Ј 2 /͑ 2͒→ Ј/͑ ␲͒ 2 2 In the opposite limit of large L , Rg Nl L 12 . is approximately I=mNR . Minimizing F with respect to R 2 → /͑␲ͱ ͒ gives the above result. It is not surprising that this result is Note that in terms of L, Rg Ll 12Tm independent of chain length N. To understand this behavior, we consider exact because in the large-L limit we expect that this circular configuration will become dominant. 0.2 2. Excluded volume interactions for large L

0.15 The fact that in this limit we obtain the exact asymptotic dependence for Rg on L suggests that this can be extended to 2 ϳ ␯ ␯ chains with excluded volume satisfying Rg N , with /Nl

2 0.1 Ϸ3/5 in three dimensions. In this case the elastic free energy g

R is known ͓1͔ up to a prefactor from the entropy 0.05 R 1/1−␯ S ϳ ͩ ͪ , ͑17͒ N␯ 0 0 1 2 3 4 5 and using this in the equation for the free energy, Eq. ͑16͒, L’ and then minimizing with respect to R gives FIG. 1. Radius of gyration versus rescaled angular momentum ͑ ␯͒/ ␯ ␯ / ␯ for an ideal ring chain. R ϳ L2 1− 3−2 N2 −1 3−2 . ͑18͒

051804-3 J. M. DEUTSCH PHYSICAL REVIEW E 77, 051804 ͑2008͒

B. Lack of dependence of T on L changed by the weak coupling to electromagnetic blackbody ͓ ͔ The temperature in this model at a given energy is deter- radiation 5 . In this case the probability density function ͑ Ј͒ ͑␤ Ј ͒ mined in the usual way, by requiring that the average energy P L is proportional to the partition function Z ,L ,N . in the canonical ensemble is equal to the microcanonical The normalization requirement is that energy. However, in this case, the radius of gyration is a very ϱ sensitive function of L. To see this, note that changes occur ͵ P͑LЈ͒4␲LЈ2dLЈ =1. ͑19͒ on a scale LЈϳ1, or LϳNlͱmT. Because I is typically 0 ϳ͑ ͒ 2 ϳ͑ ͒͑ 2͒ϳ 2 2 ϳ ͱ Nm Rg Nm Nl N l m then L Nl mT. The order The normalization is straightforward to calculate using Eq. 2 of L /2I is therefore ϳT. This means that a change of order ͑7͒ and integrating over LЈ first. The LЈ integration requires 2 /͑ 2͒ one degree of freedom changes Rg Nl by a number of evaluating order unity, which has a negligible effect on the temperature ϱ ϱ d cos͑kLЈ͒ but a large effect on the radius of gyration. So for fixed LЈ,as Ј ͑ Ј͒ Ј Ј ␲␦Ј͑ ͒ →ϱ ͵ L sin kL dL =−͵ dL =− k , N , we see that in the canonical ensemble, the effect of 0 0 dk the angular momentum constraint on the energy is a fraction ͑ ͒ of order 1/N. This means that when the limit N→ϱ is taken 20 with LЈ fixed, the relation between the temperature and en- and, using this, the integral over k is now easily accom- ergy can be obtained as it would be for a polymer without plished: angular momentum conservation, and thus will not have any ϱ k␨͑␤,k͒ ץ Ј dependence on L . This is also seen more rigorously by com- ͵ Z͑␤,LЈ,N͒4␲LЈ2dL = c␲ͯ „ …ͯ . ͑21͒ ץ Ј puting the exact dependence of the partition function on L , 0 k k=0 which is done below in Sec. II C. Z was evaluated previously in the process of calculating the C. Thermal distribution of L radius of gyration and is proportional to ϱ It is useful to calculate the probability density for finding ͵ 3 ͑ ͒2 ͑ Ј͒ / Ј ͑ ͒ the polymer with a particular value of total ͑rescaled͒ angular k csch k sin kL dk L . 22 0 momentum LЈ. This should be important in the case where there is a dilute gas of such polymers. It is also important for Evaluating this integral and including the correct normaliza- a single chain for long times, since the angular momentum is tion using Eq. ͑21͒ yields

␲3 csch͑LЈ␲/2͒4͓2LЈ␲ + LЈ␲ cosh͑LЈ␲͒ − 3 sinh͑LЈ␲͔͒ P͑LЈ͒ = . ͑23͒ 16LЈ␲2

ln͓P͑LЈ͔͒ versus LЈ is plotted in Fig. 2. Because of the non- D. Universality of results for ideal chains constant value of the moment of inertia for the chain, this We now return to a more detailed analysis of the question distribution is decidedly non-Gaussian. In the large-LЈ limit, of the universality of these results. In particular, the exact the slope of this curve approaches a constant with a slope of results were derived for a Gaussian ideal chain. In general, ␲ − . This is in agreement with the minimization argument for the links coupling monomers together will not be quadratic ͑ ͒ large LЈ given under Eq. 16 . but some more general functional form V͑⌬͒ that can be expanded in a power series in ⌬. We know that, without the 0 constraint of angular momentum conservation, the large- -2 distance properties, such as the radius of gyration, are well -4 described by a Gaussian ideal chain. However, we should

)) -6 analyze whether this continues to be the case with the addi- L’ ( tional constraints imposed here. Consider a continuous chain P

( -8

ln r͑s͒. We can ask what happens if we add higher-order terms -10 in the expansion of the potential: -12 -14 2 ϱ n 3 Nl dr dr ⌽ ͵ ͩͯ ͯ ͯ ͯ ͪ ͑ ͒ 0 1 2 3 4 5 = + ͚ a ds, 24 0 2l ds n ds L’ 0 n=3 FIG. 2. Probability density function for ﬁnding a chain in thermal equilibrium with rescaled angular momentum LЈ for an ideal where an are determined by the coefﬁcients of the expansion ring chain. of V. Rescaling variables so that sЈ=s/͑Nl͒ and ␳=r/ͱNl2,

051804-4 EXACT SOLUTION TO IDEAL CHAIN WITH FIXED… PHYSICAL REVIEW E 77, 051804 ͑2008͒

2 ϱ n 3 1 d␳ d␳ dimensional systems. So even for the case L=0 the noninter- ⌽ ͵ ͩͯ ͯ ͚ 1−n/2ͯ ͯ ͪ Ј ͑ ͒ 0 = Ј + anN Ј ds . 25 acting polymer is no longer described correctly as an uncor- 2 0 ds n=3 ds related random walk. ͑ ͒ ⌽ Where this derivation will fail is in cases where the sys- In Eq. 11 of the above analysis, we can generalize 0 to →ϱ tem cannot be adequately described by the microcanonical include these an’s and then can take the limit N but with Nl2 held constant and with rescaled angular momentum LЈ ensemble, or equivalently, when the system is not ergodic. ϰLl held constant. Then all the above terms in the summa- For sufficiently nonlinear interactions between neighboring ⌽ monomers and large N, ergodicity is expected to occur for tion go to zero and we recover the Gaussian form for 0 that we used in the derivation. Therefore we expect the results most three-dimensional systems, and experimentally all sys- Eqs. ͑15͒ and ͑23͒ to hold for a wide variety of local cou- tems appear to be ergodic enough to obey statistical mechan- plings between monomers. This was confirmed by computer ics. However, for one-dimensional chains with nonlinear in- ͓ ͔ simulations, as discussed below. teractions, such as studied by Fermi, Pasta, and Ulam 10 , As usual with such arguments, the value of the Kuhn step such equilibration can be quite slow and depends in detail on ͓ ͔ length l has not been determined. This argument only shows the initial modes that have been excited 11 . Computer that such a step length exists. The actual value of it will simulations for polymer molecules have recently been per- depend on the detailed molecular structure, as is the case for formed to look at Lyapunov exponents, and these can be very polymers in solution ͓1͔. small for systems close to a phase transition, such as the coil-globule transition ͓12͔. More strongly nonlinear athermal models such as the Sinai-Chernov pen case model ͓13͔ III. DISCUSSION and the random collision model ͓14͔, do not suffer from It is of interest to compare the extreme sensitivity of a these equilibration effects. Therefore, to investigate the equi- polymer to restrictions in angular momentum with what librium properties discussed in this work, simulations of a highly nonlinear athermal polymer model have been recently would be expected for other kinds of systems. The system ͓ ͔ considered here is essentially one dimensional in that inter- undertaken 5 . The model used was one of rigid links with a actions are only from nearest neighbor monomers. In, for fixed distance between monomers. The links could rotate example, a membrane or a three-dimensional gel, the system freely aside from this constraint. The simulation obeyed con- is of higher dimension. In such two- or three-dimensional servation of energy, momentum, and angular momentum to systems, a perturbation that changes the free energy by high precision. It gave the same radius of gyration for L=0 ͑ ͒ as predicted by this theoretical analysis, confirming the va- O kBT is expected to affect averages by only microscopic amounts. However, for polymers, it has a much larger effect. lidity of the ergodic hypothesis for this case, and the subse- 2 /͑ 2͒ For example, a force pulling the ends of a polymer costing quent theoretical analysis of Sec. II. In this case Rg Nl was found to be 0.071, which is the same as the exact result kBT of energy will increase its radius of gyration by a mul- Ϸ tiplicative constant of order unity. Because a polymer chain mentioned above, 0.07, to within statistical error. with the restriction L=0 reduces the number of degrees of ACKNOWLEDGMENTS freedom by at least one, it should affect the free energy by ͑ ͒ O kBT . So, by the above argument, this is expected to make The author wishes to thank Peter Young and Onuttom nontrivial changes to its statistics, unlike in higher- Narayan for useful discussions.

͓1͔ P. G. de Gennes, Scaling Concepts in Polymer Physics ͑Cor- of mass conservation, one is left with three extra Gaussian nell University Press, Ithaca, NY, 1985͒. integrations, which have no affect on the chain’s statistics. ͓2͔ MALDI MS: A Practical Guide to Instrumentation, Methods ͓8͔ M. Lax, Phys. Rev. 97, 1419 ͑1955͒. and Applications, edited by F. Hillenkamp and J. Peter- ͓9͔ R. P. Feynman, Statistical Mechanics: A Set of Lectures ͑Basic Katalinic ͑Wiley, New York, 2007͒. Books, New York, 1998͒,p.72. ͓3͔ W. J. Greenleaf, M. T. Woodside, and S. M. Block, Annu. Rev. ͓10͔ E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report No. LA- Biophys. Biomol. Struct. 36, 171 ͑2007͒. 1940, 1955 ͑unpublished͒. ͓4͔ P. Thaddeus, M. C. McCarthy, M. J. Travers, C. A. Gottlieb, ͓11͔ L. Berchialla, L. Galgani, and A. Giorgilli, Discrete Contin. and W. Chen, Faraday Discuss. 109, 121 ͑1998͒. Dyn. Syst. 11, 855 ͑2004͒. ͓5͔ J. M. Deutsch, Phys. Rev. Lett. 99, 238301 ͑2007͒. ͓12͔ A. Mossa, M. Pettini, and C. Clementi, Phys. Rev. E 74, ͓6͔ V. Laliena, Phys. Rev. E 59, 4786 ͑1999͒. 041805 ͑2006͒. ͓7͔ Alternatively, in the derivation that follows, the extra ␦ func- ͓13͔ Ya. G. Sinai and N. I. Chernov, Russ. Math. Surv. 42, 181 tion constraints on the total linear momentum can be included ͑1987͒. in their Fourier representation. This can then be absorbed into ͓14͔ J. M. Deutsch and O. Narayan, Phys. Rev. E 68, 010201͑R͒ redeﬁnitions of the pi’s by completing the square. Using center ͑2003͒; 68, 041203 ͑2003͒.

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