IC/2001/14

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

DISCRETE BOSE-EINSTEIN SPECTRA

Valentin I. VladInstitute of Atomic Physics, NILPRP- Laser Dept. and The Romanian Academy-

CASP, P.O.Box, MG-36, Bucharest, Romaniaand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

Nicholas Ionescu-PallasInstitute of Atomic Physics, NILPRP- Laser Dept. and The Romanian Academy-CASP,

P.O.Box, MG-36, Bucharest, Romania.

Abstract

The Bose-Einstein energy spectrum of a quantum gas, confined in a rigid cubic box,is shown to become discrete and strongly dependent on the box geometry (size L),temperature, T and atomic mass number, Aat, in the region of small y=AatTVlB. This behavioris the consequence of the random state degeneracy in the box. Furthermore, we demonstratethat the total energy does not obey the conventional law any longer, but a new law, whichdepends on y and on the quantum gas fugacity. This energy law imposes a faster decrease tozero than it is classically expected, for y —> 0. The lighter the gas atoms, the higher thetemperatures or the box size, for the same effects in the discrete Bose-Einstein regime.

MIRAMARE - TRIESTE

March 2001

'Regular Associate of the Abdus Salam ICTP. E-mail: vlad@ifin.nipne.ro

1. Introduction

Einstein introduced the brilliant hypothesis that the particles (atoms, moleculeswith rest mass and without limit for level occupation) of a gas are, like photons,indistinguishable from each other and calculated the gas statistics almost in the sameway as for photons. The specific feature of the particle gas is the finite number ofparticles, which leads to a non-zero chemical potential [1-4]. This model became oneof the pillars of modern physics and prompted the research of Bose-Einsteincondensation (BEC) and its challenges. An excellent review of this matter wasrecently presented by Tino and Inguscio [5]. The Bose-Einstein distribution law isdescribed by a continuous function, which is dependent on kinetic energy,temperature and chemical potential, but is independent on the container size and shape(considering the quantum gas in a large container).

Will this remain true when the box size and the temperature are smaller andsmaller? Following Einstein's line of thinking, can the results obtained with photonsin a small box be generalized to the problem of atoms in a small box?

The ideal classical box may be defined as a closed surface with a perfectlysmooth and unitary reflection interior wall. The quantum counterpart of this classicaldefinition is the concept of an infinite potential well, ensuring a vanishing probabilityfor the atom presence outside its surface.

The quantum version of the atom confinement is actually an eigenvalueproblem, the discrete spectrum of the atom energies being a direct consequence of thevolume finiteness and of the shape of limiting surface. For a free particle with restmass, the energy equation (with the corresponding quantum operators) can lead to aSchrodinger-Helmholtz equation. The rigid box introduces a Dirichlet boundarycondition. The history of these types of problems is very rich [6]. Gutierrez and Yanez[7] and Pathria [8] gave good accounts of it. The previous attempts to calculate theeffects of the container size on the boson gas thermodynamics have used the (Weyl-Pleijel) asymptotic state density corrections only. These corrections were relativelysmall, but increasing with the box size decrease. More recently, a number of paperscalculated the effect of trap dimension and size on BEC [5, 9-12].

If we refer definitely to the Bose-Einstein energy spectrum (BEES), we showthat the effect of the geometrical confinement upon the energy spectrum of theparticles stored inside the box may be assigned to an additional quantisation, similarto the case of photons in a small box. In this case, not only the atom internal energy isquantified, but also its kinetic energy, through the agency of the discrete spatialdirections of the allowed wave-vectors (as a result of the confinement) [13-18]. Wename this quantum device as double quantized box (DQB). The effect of theadditional energy quantisation is controlled by the factor y = AatTVm (Aat - atomicmass number, T - absolute temperature and V - the allowed volume), which isproportional to the adiabatic invariant and by the chemical potential (or alternatively,by the particle number). For 2.28-10"14 < y< 76-10"14 [cm2-K], we show that BEESpresents a discrete pattern (of lines with irregular intensities). For larger y, theasymptotic region is reached and the continuous BEES is obtained by averaging overmany non-resolvable spectrum lines.

Furthermore, the total energy, obtained in this case by summing up the exactcontributions of the eigenvalues and their weights, for well-defined values of y, doesnot obey the conventional law any longer. We demonstrate that the new total energylaw is depending on y and on the chemical potential (fugacity) and imposes a fasterdecrease to zero than it is classically expected, for y-» 0.

2. Schrodinger-Helmholtz eigenvalue problem for bosons confined in a box

The energy levels, £„, of a particle of mass m, in a box, can be obtained fromthe energy eigenfunctions of the time-dependent Schrodinger equation [2,6,7]:

h2

\7y/(r,t) + V(r)y(r,t) ih^2m I at

(1)

with y/(r, t) = (j)(r )e 'l2nst/h , e = p212m - the particle kinetic energy. In Eq.(l), wecan take V(r ) = 0 and impose a Dirichlet condition on the box boundary. The solutionof this problem is similar to that obtained in the case of the three-dimensionalharmonic oscillator, which is better realized in the present experimental conditions[6,11]). Eq.(l) can be separated on the spatial and temporal variables. The spatial parthas the form of Helmholtz equation:

(V2 +K2)(j)(r) = 0; K2=2melh2 (2)

with Dirichlet boundary condition for a rigid (reflecting) box: y/s= 0. (3)For a cubic box with size L, Eq.(2) can be separated on the independent

variables and the corresponding energy eigenfunctions have the form [2,17]:

- , — sin2n \—Qk \xk ; (k = 1,2,3); [</>,(±L/2) = 0] (4)

where the quantum numbers, q^, are integers and zero. The allowed wave-vectors inthe cubic box are:

K2=(4n2/L2)(qf+q22+q2) (5)

and the kinetic energies in the cubic box can be written as:

e = K2 = ~(q2 +q2 +ql) jq - S0-q, (6)2m 2m L 2m L

where q is an integer state number and £o - the ground level energy.The energies of the box states, e(q), are distributed in a discrete spectrum

defined by the spatial quantisation rule:

2 , 2 , 2 /r~i\

Gl +?2 + ?3 = # • V')

The allowed triplets of integers of the Diophantine equation (7) are allnumbers which do not lead to state numbers of the form (Gauss solution) [13,16]:

q(p,l)- 4? (SI + 7), (p and / positive integers) . (8)

We have observed that the number of degenerate states in the box is stronglyand randomly fluctuating. The degeneracy occurs due to the discrete spatialorientations of the state wave-vectors with the same quantum number (q). The exactweights, g(q), resulted from Eq.(7), are given in Table 1 and are represented in Fig.las a graph with jointed points.

The weight (degeneracy) of state with a quantum number q can be found, forlarge level numbers (asymptotic case), as [13,16]:

(9)

The average of the distribution g(q) follows the asymptotic trend from Eq.(9).There are combinations of integers, which did not satisfy Eq.(7) leading to

"antiresonances" in the spectrum [13,16]. The antiresonance frequencies can beidentified in Fig.l as points on the g-axis (g(q) = 0). It is interesting to point out that(1/6) of the box energy spectrum is emptied by antiresonances.

For particle confinement in relatively small volumes and at relatively smalltemperatures (we shall define later what "relatively small" means), we have to face therandom distribution of the eigenvalue intervals and/or degeneracy. One can expectthat the selection rules imposed by the boundary conditions and eigenvalue ortho-normalization will lead to allowed states and forbidden states (antiresonances), i.e. adiscrete and irregular spectrum of the S-H operator.

We can define the quantum degeneracy factor:

(10)g

asy

which includes the spatial quantization effects. We have checked that the factor Qq)is randomly fluctuating around the value 1 by the calculation of the average numberof states on constant frequency intervals. The result from Fig.l is very convincing:although the degeneracy fluctuations are large for a box with a small number of states(and must be taken into account), the average number of states tends to the asymptoticvalue very rapidly. For a number of states larger than =100, the classical equation (9)can be safely used.

For bosons with non-zero rest mass and chemical potential, the energy densityspectrum can be deduced from the Bose-Einstein law for particle distribution [1-4]:

SE = g-q = g-q g^qSe e < - w w _ i g-*""-. «,*'«• _i A-l-exp(a-q/AatL

2T)-l'

with g - the level degeneracy, \i - the chemical potential, k - the Boltzmann constantand T - the absolute temperature of the boson quantum gas, A = exp(jU / kT) - the

fugacity, n - the chemical potential (< 0), a = h212m0pk = 9.5060 • 10"14[cm2 • K] and

Aat - the atomic mass number (m = Aat mop). Using the quantum degeneracy factordefined in (10), the boson spectrum (11) can be put in the form:

-Sid).u{q,A,AalLT) = f = \be A • exp(a • q I AatLT) -1

g(q)

150

125

100

Qq) = g(q)/2ni/q2.5

1.5

0 . 5

Fig. 1. (a) The random fluctuations of the level degeneracy, g(q), around the curve 2nVg, for statenumbers, q, including the first antiresonant doublet (111,112). (b) The random fluctuation of the weightfactor, Qq), around the unit value (graph with jointed points); the dots represent the calculated averagenumber of modes on constant frequency intervals and show that the classical (asymptotic) modedensity can be reached when q > 110, by the averaging of the actual mode density.

We define double quantized cubic cavities (DQB) as cavities with a smallnumber of states, more precisely, with a special upper limit on the highest significantstate number in the box: qT < 100. In this case, the energy density spectrum of theboson gas does depend upon the box size (volume), i.e. upon the boundary conditions,(which is a non-classical effect).

The Bose-Einstein energy spectrum, from Eq.(12), is discrete for a smallnumber of states in the cubic box, as shown in Fig.2 and Fig.3. From these graphs,one can observe that the quantum effects may occur in cubic cavities with micrometersizes, at temperatures around (xK, which are presently reached by evaporation andlaser cooling [5]. These spectra show that, for specified particles (atoms), the higherthe adiabatic invariant, L2T, the higher the number of levels in the DQB. At a certainresolution limit, the spectrum is obtained by averaging the energy lines (£—>1) andthe continuous BEES is reached (dashed graphs in Fig.2 and 3).

We can introduce a reasonable superior limit of the number of states in thebox, qT, which brings a significant contribution to the BEES. Observing that, at highenergies, in Eq.(12), the exponential term dominates and £(q) goes to 1, the totalenergy density can be brought to the form: u(x) = Bx3'2 exp(-x), withx = {a IAatL}T)q = (a/y)-q .If we consider that A < 0.99 and we neglect the levelswhich bring to BEES a contribution of less than 10~2, one can truncate Bose-Einsteindistribution at the highest significant level number (HSL) in the box:

qT =12.5-(AatL2T/a) = 12.5•(/ la) . (13)

One can observe that, in the above conditions, the fugacity plays a minor role in thistruncation and for any of its values, Eq.(13) ensures an over-evaluated value for qj.For Aa, = 87 (Rubidium), L = 10 "4 cm and T = 10 ^ K, Eq.(16) leads to: qT « 114.For Li7, one can find some more convenient conditions for DQB, namely: L = 10 (imand T~ 120 nK. For a precision of 10"3, one can take: qT ~\5{yl a) -137.

The state with the maximum total energy density can be evaluated at:

(14)

Thus, for the same parameter values as in the first example and for A ~ 0.99 (quasi-degenerate gas), one can find qmax ~ 10 and the ratio between HSL and the maximum(peak) state numbers as: qj/q^ ~ 14 (the truncation precision was taken to be 10"2).

The graphs in Figs. 2 and 3 show indeed, that the higher the parameter y, the higherthe level of the BEES peak (at constant fugacity, A). Furthermore, the closer thefugacity to 1, the lower the BEES peak coordinate (at constant y).

A=0.99; Aat*L~2*T=4*10"-13

10 20 30 40 50

A=0.99; Aat*L"2*T=6*10/v-13

10 20 30 40 50u A=0.99; Aat*L^2*T=8.7*10^-13

175

150

125

100

75

50

25

q

10 20 30 40 50

A=0.99; Aat*L/s2*T=20*10/s-13

100 150 200

Fig. 2. Some conventional Bose-Einstein spectra (dashed lines) and discrete Bose-Einstein spectra(solid lines, joining the tops of the energy spectrum lines), for A = 0.99 (quasi-degenerated gas) and

for the values of y= Aat L2 T which are shown in each graph

u

14

12

10

8

6

4

2

A=0.2 ; Aat*L / v 2*T=6*10~- i :

A=0 .2 ; Aat*L A 2*T=8.7*10 A -13

100 150 200

Fig. 3. Some conventional Bose-Einstein spectra (dashed lines) and discrete Bose-Einstein spectra(solid lines, joining the tops of the energy spectrum lines), for A= 0.2 (almost classical gas) and

for the values of y= Aat L2 T which are shown in each graph

3. The total energy of boson gas in the double quantified cubic box

The total energy and the total number of particles of the free boson gas (inCGS-Gauss unit system) are [3]:

e akT

kT )

d \ • (16)

e x d | l

One can find that [18]:

A -> 0, E -> - NkT[l - / , (A)] = - NkT(l - 0.1767767 • A - 0.0658001 • A2 -

- 0.0364655 • A3 - 0.0239014 • A4 - 0.0171965 • A5).(17)

Introducing the new normalized variable z = e/kT, one can write Eq. (15) and (16) inthe form:

E_ = ^ . . . , .

w-'"^T/. -iT^^feH'-^- -/.w. (19)

Solving the integral Eq.(19) to obtain the particle number in function of fugacity, wehave generalized the result from [2] as [18]:

J ^ ) -3 / 2A2+3-3 / 2A3+4-3 / 2A4+5-3 / 2A5/ 2 ( ) 2-3 / 2A2+3-3 / 2A3+4-3 / 2A4+5-3 / 2A5+.. . .

(20)One can invert this function in order to obtain the dependences of the fugacity and ofthe chemical potential on iV and y, which are shown in Fig. 4. However, the presentcooling procedure, which is based on evaporation, eliminates progressively theparticles with the highest kinetic energy. Consequently, we prefer to consider avariable particle number in the system and to calculate this number in function of agiven fugacity and y (Fig. 5). One can remark that iV increases monotonically with Aand y. The higher the fugacity in the DQB, the stronger the increase of ./V with y.

8-10-13

C h e m . p o t e n t i a l

0" 2 L2 . 5 - 1 0

2-10 - 2 1

1.5-10

1-10

5-10

5-10

- 1 - 1 0

- 2 1

b. u p _ > d o w n N a t = 200;20;1;0.2

2-10- 1 2

4-10 6-10- 1 2

Aat I? T8-10 - 1 2

Fig.4. (a) The 3D plot of fugacity, A, in function of the particle numbers in the box, N and y =AalL2T

[cm2-K]. (b) The dependence of chemical potential on y [cm2-K], for different fixed values of N.

10

In DQB, the total energy should be written by summing the state energies upto the highest significant one (characterized by qr):

kT 4ln^ [kT) %A-l-e-™lkT iW y UA-l.(21)

We can calculate the ratio of total energies of the particle gas in DQB and in aconventional (large) container:

(3/2)NkT[l-fl(A)]

4 H3 /V ^

In the asymptotic limit, C,(q) goes to 1 (by averaging over many and very closemodes), F tends to 1, and one arrives to the conventional formalism. The correctivefactor is represented in Fig.6, in function of the parameter y, for two characteristicvalues of the fugacity, one close to the maximum admissible value A = 1 and thesecond, close to the classical (Maxwell- Boltzmann) regime.

Calculating the corrective factor F from (22) with the exact degeneracyprovided by the Diophantine equation (10) and with the asymptotic degeneracy, g(q)~ 2ivlq, we found out maximum differences of the order of ~ 2 -10" , for A = 0.2,which are small differences with respect to those expected in the DQB.

We can put the total energy density law of DQB into a new form:

E =Qr(Y)

Y pi A-l-e{aly)"-l•kT (23)

and observe that the small number of states in the box (up to qj, i.e. small y = Aa,L2T)

plays the key role in the dependency E(y) and not the exact degeneracy.Thus, with specified atoms and box size, the total energy in DQCB has a

stronger dependence on temperature than was predicted by the conventional law. Asthe box is emptied of states, its total energy is strongly decreasing according a newlaw derived in Eq.(23).

We have shown that the positions of the energy density peak and of HSL

depend on the product Yq = AatL2T (and in some respect, on A). Eq. (22) and Fig. 6

show that the asymptotic limit can be set for F(a/y) ~ 1, at a conventional limit of

YqmaJ oc = 8, which leads to Yqmax ~ 76 -10"14 [cm2.K] and to qrmax- 100.On the other hand, the lowest box mode (1,1,1) imposes an inferior limit to the

level number at: qTrria = 12.5 • (y la) = 3 leading to Yqmin « 2.28 -10"14 [cm2.K]. Thus,

we can define the double quantization regime of the cubic box in the range:

3 < q < 100 (24)2.28 -1014 <Y< 7.6-10"13 [cm2.K]. (25)

11

N A=0.2dashed;A=0.99cont.line

256-

209-

159-

lOfr

59-

2 - 1 0 " 1 3 4 - 1 0 " 1 3 6 -10 1 3 8 - 1 0 " 1 3A a t 1 / T

Fig.5. The particle number in the DQB in function of a given fugacity, A (shown in the graph label) andy[cm2.K]

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

A = 0.2dashed;A =0.99cont . l ine

:Y,'///////

Aiiji

1

- - -_̂ _̂_ — ~

1-10- 1 3 2-10 - 1 3

3-10- 1 3

4-10 - 1 3Aat L2 T

Fig.6. The ratio between the total energy of the particles in DQB and the total energy in a conventionalcontainer, F, in function of y [cm2.K] for two values of the fugacity, A, shown in the graph label.

The following reciprocity rule holds: for given atoms, the box size and thetemperature are reciprocal parameters in the DQB, i.e. the same effects (in thethermodynamics of the boson gas) can be obtained either by varying L2 = V2/3 or byvarying T, if their product remain constant.

We can remark that BEES are produced by a small number of particles. FromEq.(20) and (22), represented in Fig.5 and 6, respectively, it is possible to calculatethe number of particles at y = 10"13[cm2.K], where the correction factor of the totalenergy is F ~ 0.9, for A = 0.99 and F ~ 0.96, for A = 0.2. In the specified quasi-degenerate gas, there are N = 9 particles and in the quasi-classical gaz, N =1 particle.

12

It is clear that the discrete Bose-Einstein effects are stronger for the quasi-degeneratequantum gases.

Previous calculations with photons have shown that the double quantizationregime of the spherical box is qualitatively similar to that of the cubic box [16,17].However, this regime is extended to values of the principal quantum numbers, whichare almost ten times higher than those obtained for the cubic box. The calculations forthe discrete Bose-Einstein spectra are in progress and we expect more favorableconditions of observation of these effects for bosons in spherical boxes.

4. Conclusions

We have shown that the energy spectrum of a boson gas, which is confined in arigid cubic box, in the regime of small y = AatTVm, is discrete and depends stronglyon the box size and temperature. The complex aspect of the spectrum, is theconsequence of the random degeneracy distribution in the cubic box, whichintroduces an additional energy quantisation controlled by y and by gas fugacity (oralternatively, by the particle number). This quantum system was called by us doublequantised box (DQB). The discrete Bose-Einstein spectra in DQB also showforbidden energies.

Furthermore, the total energy, obtained in this case by summing up the exactcontributions of the eigenvalues and their weights, for well-defined values of y, doesnot obey the conventional law any longer. We have demonstrated that the new totalenergy law depends on y and on fugacity and imposes a faster decrease to zero than itis classically expected, for y—> 0.

We have defined DQB by the conditions: 2.2810"14 <y< 76-10"12 [cm2-K].Thus, in this regime, the box size and the temperature are reciprocal parameters in thesense that the same effects (in the boson gas) can be obtained either by varying L orby varying T, if their product remain constant. The lighter the gas atoms, the higherthe temperatures or the box size, for the same effects in DQB.

The number of particles, which create the discrete behavior of DQB, is small(not exceeding -100). The discrete effects in DQB are stronger with respect to theBose-Einstein continuous distribution for the quasi-degenerate quantum gases, inwhich the number of particles is higher (at fixed A and y).

Acknowledgements. One of the authors (V.I.V.) thanks The Abdus SalamInternational Centre for Theoretical Physics, Trieste (Italy) for the working stages atthe Centre as a Regular Associate Member. Particularly, he wishes to thank Prof.Gallieno Denardo and Prof. Giuseppe Furlan for their support in these visits, whichhave offered the optimum conditions for thinking and writing this and other papers.He wishes to acknowledge also Prof. Herbert Walther for the useful discussions andfor the privilege to be an external collaborator of Max Planck Institut furQuantenoptik.

13

References

1. A.Einstein, Quantentheorie des einatomiger idealen Gases, Sitzungsber. Kgl.Preuss. Akad. Wiss., 261(1924); 3(1925)

2. M.Born, Atomic Physics, 8th Edition, Blackie Ltd., London, 19723. D.Landau, E.M.Lifschitz and L.P.Pitaevskii, Statistical Physics, 3rd Ed.,

Pergamon Press, 1980.4. K. Huang, Statistical Mechanics, J. Wiley, N.Y., 2d Ed., 19875. G.M. Tino and M.Inguscio, Experiments on Bose-Einstein condensation, Riv.

Nuovo Cimento, 22(4), 1(1999)6. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol.1, J. Wiley,

N.Y.,1989, pp.314 and 445.7. G. Gutierrez and J.M.Yanez, Am. J. Phys. 65(8), 739(1997)8. R. K. Pathria, Am. J. Phys. 66, 1080(1998)9. V. Bagnato and D. Kleppner, BEC in low-dimension traps, Phys. Rev. A, 44,

7439(1991)10. J.R.Ensher, D.S. Jin, M.R. Matthews, C.E. Wieman and E.A. Cornell, BEC in a

dilute gas: measurements of energy and ground-state occupation, Phys. Rev.Lett.,77, 4984(1996)

11. W. Ketterle and N. J. van Druten, BEC of a finite number of particles trapped inone or three dimensions, Phys. Rev. A, 54, 659(1996)

12. R. Napolitano, J. De Luca, V. Bagnato and G.C. Marquez, Effect of a finitenumber of particles in the BEC of a trapped gas, Phys. Rev. A, 55, 3954(1997)

13. V. I. Vlad and N. Ionescu-Pallas, Ro. Repts. Phys. 48(1), 3(1996)14. V. I. Vlad and N. Ionescu-Pallas ICTP Preprint No. IC/97/28, Miramare-Trieste,

1997; Proc. SPIE, 3405, 375(1998)15. V. I. Vlad and N. Ionescu-Pallas, ICTP Preprint No.IC/99/27, Miramare-Trieste,

199916. V. I. Vlad and N. Ionescu-Pallas, Fortschritte der Physik, 48(5-7), 657(2000)17. V. I. Vlad and N. Ionescu-Pallas, Discrete Planck spectra, ICTP Preprint

No.IC/2000/154, Miramare-Trieste, 2000.18. N.Ionescu-Pallas, V.I.Vlad, paper submitted for publication in Ro. Repts. Phys.

14

Table 1. The weights g(q) in the box-shaped box for state numbers up to 350

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

0

6

24

48

0

96

48

72

0

102

48

168

0

78

120

96

0

192

120

120

0

144

72

192

0

144

168

216

0

240

96

96

0

240

72

8

1

12

8

24

12

48

24

96

36

48

0

48

0

120

48

48

72

108

24

144

8

72

72

144

24

132

0

72

48

96

48

144

48

96

72

336

2

8

24

0

48

24

72

0

48

72

48

0

96

48

48

0

144

24

168

0

48

96

96

0

144

104

48

0

96

72

216

0

96

96

120

216

3

6

48

24

48

24

96

6

120

48

96

72

96

0

168

30

96

96

144

48

240

48

72

48

216

72

192

96

144

0

168

24

312

102

144

0

4

24

0

30

48

72

0

96

56

48

0

96

48

144

0

96

96

96

0

192

96

96

0

150

48

168

0

96

120

192

0

192

144

168

0

120

5

24

6

72

30

48

48

96

24

120

24

72

72

144

48

192

0

120

24

144

54

240

96

96

72

144

6

240

96

144

120

240

0

264

48

96

6

0

48

32

24

0

48

24

96

0

48

72

120

0

96

56

72

0

48

48

120

0

96

120

144

0

192

48

72

0

192

72

120

0

96

120

7

12

36

0

72

8

24

48

48

24

108

32

72

12

96

24

96

48

96

0

120

24

120

48

96

96

96

24

168

36

72

96

144

48

180

120

8

30

24

72

0

54

72

96

0

144

72

72

0

144

72

168

0

78

120

192

0

240

96

120

0

144

96

264

0

102

192

144

0

288

144

0

9

24

24

48

24

84

0

48

24

20

30

44

48

48

48

20

24

144

72

48

84

96

0

240

0

144

96

192

48

240

56

96

24

96

4

168

15