NEW PROBLEMS. Bose Einstein condensation. Charles H. Holbrow, Editor

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1 NEW PROBLEMS Chares H. Hobrow, Editor Cogate University, Hamiton, New York 3346 The New Probems department presents interesting, nove probems for use in undergraduate physics courses beyond the introductory eve. We wi pubish worked probems that convey the excitement and interest of current deveopments in physics and that are usefu for teaching courses such as Cassica Mechanics, Eectricity and Magnetism, Statistica Mechanics and Thermodynamics, Modern Physics, or Quantum Mechanics. We chaenge physicists everywhere to create probems that show how their various branches of physics use the centra, unifying ideas of physics to advance physica understanding. We want these probems to become an important source of ideas and information for students of physics. This project is supported by the Physics Division of the Nationa Science Foundation. Submit materias to Chares H. Hobrow, Editor. Bose Einstein condensation Isaac F. Sivera Department of Physics, Harvard University, Cambridge, Massachusetts 0238 Received September 996; accepted 6 February 997 I. SCOPE After a brief description of what Bose Einstein condensation BEC is and why it is expected to occur in an unconfined three-dimensiona voume, five probems are posed. Two of them show how cod and dense a sampe must be before BEC is possibe; the soutions of the other three iustrate how the critica temperature and the temperature dependence of the condensate density depend on dimensionaity and on externa potentias. To work the probems a student shoud be acquainted with the Bose distribution function and the chemica potentia. The probems are appropriate for a statistica physics course such as might be taught out of Kitte and Kroemer. II. INTRODUCTION In 995 whie studying a very diute gas of rubidium atoms confined in space by an inhomogeneous magnetic fied, physicists observed Bose Einstein condensation in a gas of atoms for the first time. 2 The phenomenon was originay predicted in 925 by Abert Einstein shorty after Satyendra Bose brought to his attention a simpe derivation of the Panck radiation aw that treated photons as partices that obeyed the counting rues that are today caed Bose Einstein statistics. Einstein showed 3 that a coection of atoms that obeyed these counting rues bosons might at the proper temperature and density suddeny popuate the coection s ground state in observy arge numbers. This macroscopic occupation of the ground state is a phase transition and takes pace at critica temperature T c and/or number density n c, and was given the name Bose Einstein condensation BEC. Athough BEC has been a textbook exampe 4 in statistica mechanics for a ong time, and athough BEC is beieved to be responsibe for superfuidity in iquid heium, it has taken more than 70 years of increasingy intensive effort to unambiguousy observe BEC. An important experimenta goa wi be to show that Bose condensed gases possess the property of superfuidity. Other observations have been made, and there are current experiments to study a Bose gas in a two-dimensiona geometry. 5 III. BACKGROUND Partices with integra spin 0,,2,... are caed bosons. For an assemby of identica bosons there is no imit to the number of partices which can occupy a singe quantum state. Bose and Einstein derived the expression for the therma occupation of a state with energy at temperature T N exp, where k B is the Botzmann constant and is the chemica potentia. Athough interactions are extremey important in a rea gas, the probems are made tracte and the essentia physics is retained by assuming an idea gas of noninteracting partices. We aso assume that the states are non-degenerate. The chemica potentia, which is a measure of the change of interna energy of the system when a partice is added, is a very important quantity in the theory of BEC. In equiibrium the chemica potentia must be uniform athough quantities such as pressure and density may vary across an inhomogeneous system. We sha see that for TT c, 0 and partices added to the system go into the zero-energy state. The chemica potentia is determined by requiring that the sum of partices occupying every state be equa to the tota number of partices N. From Eq. we see that 0 because the occupation number must be positive. For a gas in the owdensity, high-temperature imit is arge and negative and goes to zero as temperature is decreased or density is increased to the critica vaue where BEC occurs. 570 Am. J. Phys. 65 6, June American Association of Physics Teachers 570

2 In a gas of atoms obeying Maxwe Botzmann statistics there is a arge continuum of energy or momentum states which are microscopicay popuated, i.e., the number of partices occupying any one state is a very sma fraction of the tota number of partices. Ony at T 0 is the ground state with 0 macroscopicay popuated by a of the partices. However, Einstein showed that for a gas of identica bosons, macroscopic occupation of the ground state, compare to the tota number of partices in the gas, can occur at a finite nonzero temperature. This phase change is BEC. For a uniform space no spatiay varying potentias the usua derivation of BEC is as foows. Equation shows if 0 then N 0, the occupation of the state with 0, diverges. N 0 is caed the condensate. When evauating the chemica potentia by summing a states, this term is separated out because of its possibe divergent behavior, exp NN This sum is bounded in three-dimensiona space and can be evauated by converting it to an integra, which you may do because the free partice states are very cosey spaced, so that in the semi-cassica approximation we can go over to a continuum mode. Noting that the number of states equas the phase space voume divided by h 3, we have dp 3 dr p 2 dp dr 3. Using p 2 /, we have 4& 2 3 M 3/2 d dr 3. Since is the number of states, you can recognize in Eq. 4 the number of states between and d, i.e., the density of states, 4&M 3/2 V /2 2 3, 5 where V dr 3 is the voume in a three-dimensiona space, r 3. Then Eq. 2 becomes NN 0 4& 2 3 VM 3/2 0 /2 d exp N 0 N*. For finite, negative a states are out equay, and microscopicay, popuated. But consider what happens to a system hed at constant T and V when you add more partices. As N increases must adjust by increasing toward zero. Now for 0 the second term in Eq. 6 can be evauated as 4& 2 3 VM 3/2 3/2 x /2 dx 0 e x. 7 The integra has the vaue 2.62/2. Ceary for any given V and T as approaches zero, the integra approaches its maximum vaue, and N* in Eq. 6 cannot increase further. At this point T represents a critica vaue T c at which the Fig.. The chemica potentia and condensate popuation N 0 of an idea Bose gas as a function of temperature adapted from Ref. 6. addition of more partices eads to new behavior. Writing the density n N/V, we have c M n2/3. 8 Now suppose you add more partices to the system. How can adjust further? The answer is found by expanding the expression for the condensate, N 0 exp k BT. In the thermodynamic imit N, V such that N/V remains constant, approaches zero as /N. Thus, as 0, N 0 becomes macroscopicay occupied in a very controed way; as more partices are added beow T c, they a go into N 0. The behavior of and N 0 is shown in Fig.. Likewise, N and V can be hed constant and T owered beow T c, so that ove-condensate partices N* are transferred to the condensate. The gas is partitioned into two sets of partices, N 0 the condensate and the noncondensate partices, which are sometimes referred to as the ovecondensate or norma partices. From Eq. 7 we have N* N(T/T c ) 3/2, which with Eq. 6 impies N 0 N T 3/2 T c. 9 IV. PROBLEMS A. De Brogie waveength and BEC In the semicassica approximation at ow density and high temperature, atoms are ocaized to wave packets with dimensions sma compared to the average interatomic separation. The average de Brogie waveength h/p, which is a quantum measure of deocaization of a partice, must satisfy this criterion. Here, p is the momentum spread or momentum uncertainty of the wave packet. Indeed, in the other extreme for T T c, for partices in the zero momentum eigenstate the deocaization is infinite; i.e., the packet is spread over the entire voume V. BEC occurs when the interpartice separation is of the order of the deocaization. A usefu quantity in the thermodynamics is the therma de Brogie waveength 57 Am. J. Phys., Vo. 65, No. 6, June 997 New Probems 57

3 db 22 M, which is a measure of the thermodynamica uncertainty in the ocaization of a partice of mass M with the average therma momentum. a Determine the interpartice separation in terms of db at the onset of BEC. b Use the Heisenberg uncertainty principe and Maxwe Botzmann statistics to show that db is, to within a factor of 2, just the uncertainty in the position of a partice r r 2 r 2. B. Vaue of T c for rubidium vapor Recenty, BEC has been reported to have been produced in a vapor of 87 Rb atoms at a number density of n cm 3. Cacuate T c for such a gas in a uniform 3-D space and compare it with the temperature of 70 nk reported by the experimenters. 2 C. BEC in an inhomogeneous magnetic fied Atoms with magnetic moments m in an externa magnetic fied B wi have an interaction energy U m B. 7,8 Assume that the magnetic fied varies quadraticay and is of the form B 0 (z/a) 2 B 0 (/b) 2, with 2 x 2 y 2, so that Um z B 0 z 2 a m B 0 b 2 a z 2 b 2. 0 Find the critica temperature for BEC and the temperature dependence of the condensate. Simpify the probem by simpifying the potentia to Uz a z 2. Notice that in an inhomogeneous potentia the density is inhomogeneous and peaked at the potentia minimum. In an idea gas the condensate forms ony in the center of the coud and is ocaized in space. In a quantum mechanica description, the ground harmonic osciator state is macroscopicay occupied and condensate atoms ony occupy this state as the temperature is owered. Hint: Note that the kinetic energy of a partice now depends on position so that the integra of the energy over dr 3 see Eq. 4 must be executed. D. Does BEC occur in a uniform 2-D space? Show that BEC wi not occur in a 2-D gas in a uniform space, i.e., one in which there is no externa potentia. BEC or the separation of the partices into a condensate and ove-condensate part takes pace ony if the sum or integra in Eq. 2 does not diverge when 0. If it does diverge faster than N 0, the occupation of N 0 does not become favored and macroscopic. 9 It is convenient to show the divergence of Eq. 2 by rewriting the sum in terms of z e, where /. E. Show that BEC occurs for a 2-D gas in a quadratic potentia An appied externa potentia can change the resuts of the previous probem. 0, To see that this is so, redo the previous probem using, for simpicity, a one-dimensiona potentia in the x direction, Uu x2 0 a 2, 2 where u 0 is the depth of the potentia we and a is a measure of its extent in the x direction. The resuts are easiy extended to a 2-D potentia. V. SOLUTIONS A. De Brogie waveength and BEC a It is straightforward from Eq. 8 to show that the interpartice spacing is n /3 db.38. b From the uncertainty principe find r, r p p 2 p. 2 From Maxwe Botzmann statistics you know that p 2 3M, p 2 8Mk BT, from which it foows that r0.59 db db. B. Vaue of T c for rubidium vapor This answer is obtained by substituting the appropriate numbers into Eq. 8, T c / K34 nk. Recent measurements show better agreement with theory. 2 To cacuate the critica temperature for the actua experimenta conditions one shoud use an expression for an inhomogeneous fied ike that given in Probem 4.3. However, in an inhomogeneous fied BEC sti takes pace at the peak density region where n c 3 db, the condition given by Eq. 8. C. BEC in an inhomogeneous magnetic fied The kinetic energy of a partice in the potentia is p 2 / U(r) so that the density of states function, Eq. 5, becomes 3/2 4&M 2 3 Uzdr 3, V* where V* is the accessibe voume. A partice with a given energy can have an excursion in the potentia to an extreme where p 0orU(z), yieding the imits on the integra z z m a, so that 572 Am. J. Phys., Vo. 65, No. 6, June 997 New Probems 572

4 3/2 4&M 2 3 4&M3/2 2 3 S 2 a, wheres dx dy. Now Eq. 7 becomes z S m zm 2 z2 /2 dz a 3/2 4&M N 2 3 S 2 ak BT 2 0 xdx e x. The definite integra has the vaue 2 /6 so that T c 3/2 N 2 3 /2 3/ S & 4 k 2 B am ak B M 3/4 /2, where N/S is the area density in the unconfined pane. Foowing the procedure eading to Eq. 9 we find N 0 N T 2 T c. If we sove for the potentia of Eq. 0, wefind T c.33 k B M N 2 /3 /2. D. BEC in a uniform 2-D space Rewritten in terms of z e, Eq. 2 becomes ze N ze e ze 0 ze, since /( x) 0 x. Next, coect terms as foows: N 0 ze ze z e. Up to this point the resut is independent of dimension. Now in 2-D so h 2 2p dp dr 2, S N z 2 pdpexp h 2 0 p2. The momentum integra is easiy evauated to give M or N z S 2. db The sum can be recognized z nz, so N S 2 db n exp. N diverges when 0. Therefore, BEC does not take pace in 2-D. Note that when you went over from the sum to the integra, you coud have separated out the condensate fraction. In 3-D the integra representing the ove-condensate partices is bounded, so that for T T c partices fow into the condensate, and the tota number of partices remains bounded. In 2-D if you partition the partices into zero-momentum and norma parts, you find that both parts diverge at T c and there is no phase transition. E. BEC in 2-D in a quadratic potentia Substituting Eq. 2 into Eq. 2 gives the foowing integra to be evauated: N dp 2 dx dy exp. Rather than evauate the integra directy, make a transformation to the dummy varie p z, x p za u 0 so that p x 2 2 p y U p x 2 2 p y u 0 u 0x 2 a 2 becomes p x 2 p 2 2 y p z u 0, and the integra becomes N dp u 0 3 px exp 2 p 2 2 y p z dp u 0 3 px exp 2 p 2 2 y p z u 0 where b is a measure of the size of the system aong y, and we have defined u 0. The integra is now identica to the integra for BEC in 3-D that was carried out to give Eq. 8. BEC takes pace when 0oru 0. The resut is that with a confining potentia BEC wi take pace in 2-D. C. Kitte and H. Kroemer, Therma Physics Freeman, San Francisco, 980, 2nd ed. 2 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Corne, Observation of Bose Einstein condensation in a diute atomic vapor, Science 269, A. Einstein, Quantentheorie des einatomigen ideaen Gases, Sitzsungber-, 573 Am. J. Phys., Vo. 65, No. 6, June 997 New Probems 573

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