- Email: [email protected]
Cold system of dipolar particles in electric field: Bose-Einstein condensation
28 July 1997
PHYSICS LETTERS A ELSEVIER
Physics Letters A 232
(1997)305-307
Cold system of dipolar particles in electric field: Bose-Einstein condensation LT. Iakubov a, A.V. Nedospasov a Science Centerfir
Applied
Problems
in Electrodynamics,
Russian Acudemy oj’Sciences.
b
I~horskayrr
Str. 13 / 19. Moscow
127412. Russian
Federution ’ lnstltute jb- High Temperatures,
Russiun Academy
of Sciences.
I:horskuya
Str. 13 / 19, Moscow
127412, Russian Federation
Received 2 I August 1996; revised manuscript received 17 March 1997; accepted for publication I4 May 1997 Communicated by J. Flouquet
Abstract The influence of an external electric field on the state of a system of molecules possessing permanent electric moments is considered. In strong fields the intermolecular attraction transverse to the field direction is suppressed by the electrostatic repulsion. In the longitudinal direction the system is stabilized due to the molecular zero-oscillations. The system cannot undergo van der Waals condensation, but Bose-Einstein condensation becomes possible. 0 1997 Published by Elsevier Science B.V. Keyrrort/.\: Dipole-dipole
interaction; Bose-Einstein
condensation
The nonideal degenerate Bose gas with a repulsive interparticle interaction has been studied actively since the classical works by Lee and Yang, see Ref. [I]. The Bose-Einstein condensate of nonideal gas was experimentally obtained and diagnosed [241. In the present paper a system of particles possessing permanent dipole moments is considered. The possibility of Bose-Einstein condensation of a molecular liquid-like system in an electric field is demonstrated. Consider a system of molecules possessing permanent dipole moments ed. If the mean interparticle distance F z=- atomic lengths one can ignore the short-range component of the interparticle interaction and consider only the long-range one. Such is a dipole-dipole interaction with the potential V(r)
=
,*r-‘[(
d, . d2) - 3r-7
d - r,)( d *
rz)] .
where d,, dZ are the vectors of the dipole moments of the interacting molecules, r is the distance between them. At high temperatures the system is an ideal gas, and the dipole moments are randomly oriented. With the temperature decreasing attractive interactions manifest themselves, the gas undergoes a phase transition of the first kind. However, if a strong electric field is applied the cooling of the system can lead to another result. In strong electric fields all the dipoles are oriented along the field. and the potential (1) takes the following form, V( 1.) = &*1--“(
where 0 is the azimuthal angle between direction and the radius-vector r, looking
(1) 0375-9601/97/$17.00
0
P/I SO775-9601(97)00390-3
1 - 3 COG O),
1997 Published by Elsevier Science B.V. All rights reserved.
(2) the field from the
306
I.T. Iukuhou,
A.V. Nedospc~.wu
/ Physics
origin. The system is axially anisotropic. The anisotropy is a result of the presence of the electric field. Under strong cooling spatial interparticle correlation will increase. It is known that the interdipolar repulsion prevails in the interaction between correlated dipoles, and the mean energy of interaction per particle U becomes positive. I/ > 0. Analysis of dimensionality results in the relation, U=A(ed)‘n, (3) where R = (4~?/3)-’ is the molecular particle density, F is the mean interparticle distance, and A is a numerical coefficient. Its value depends on the intensity of the interaction and becomes comparable to unity in regimes of strong spatial correlation when the interaction energy becomes comparable to temperature T or exceeds it, r=
(ed)??’
2 1,
(4) where r is the nonideality parameter of the dipole system. If inequality (4) is fulfilled the system becomes liquid-like. Consider as an example the hexagonal structure to estimate U. Let us guide the axis of the sixth order along the field and calculate the energy of interaction of the dipole in the origin with the whole environment assuming all the dipoles to be oriented along the field. The energy equals the sum of energies AU, which are the energies of interaction of the chosen dipole with dipoles of the subsequent coordinate spheres. Both parameters of the lattice are equal to F. It leads to the following result, I/=
CA&
z (3,‘4)“‘(ed)‘n.
(5)
k
It is important that taking into account a few members in the series results in its rapid converges. Therefore the procedure can be applied for the modeling of our liquid-like system. The field strength E should exceed the dipole field taken at the mean interparticle distance, E z- erh. (6) In a system of hard spheres the interaction is characterized by scattering length a = MU/4rnfi2, where hl is the particle mass. Introduction of a scattering length u is not justified for an anisotropic system. Let us consider it as the scattering length of the comparable isotropic system. It may help to estimate qualitatively’ the strength of the repulsion, 0 = ( M/4mt)d2a,
’,
(7)
Letters
A 232 f 1997)
305-307
where m is the electron mass and a, = fi”/e’m is the Bohr radius. The scattering length must be small compared with the mean interparticle distance F and thermal wave length Ar = fi( MT)- I/‘, a<
a -=XAT.
(8)
It seems that the fulfillment of inequality (4) prevents the van der Waals condensation to the liquid state as a result of cooling. However, the effects of the anisotropy acquired in the external field display themselves stronger in an anisotropy of pressure. In nonideal systems the cold pressure exceeds the thermal one. The longitudinal pressure p,, differs from the transversal one pI , P I = d( UN)/dV,
,
pII = d( UN)/dV,,
1
(9)
where N is the total number of particles, dV,, and dV, are small changes in the system volume as a result of its enlargement along the anisotropy axis and perpendicularly to it. Two parallel neighboring dipoles interact diversely depending on their relative position. One can foresee a positive sign for p I but a negative one for p,,. We calculated p ~ and p,, in the above model of the hexagonal system. The pressures were calculated enlarging the lattice parameter in longitudinal or transversal direction and using Eqs. (9) straightforwardly. This leads to pa z -(9/4)(
ec1)2n2,
p, = (9/16)(
ed)2n2. ( 10)
Such a state is unstable. Let us demonstrate that in an electric field the quasi-two-dimensional cooled system of dipolar molecules can be stabilized. Let the volume V= bL2 represent the layer, b is the width of the layer and L is the length, b <(: L. The volume holds N particles. The number N is large enough to consider the system as macroscopic. The size b must be larger than the mean interparticle distance to prevent the compression of the system to a liquid film. The field E is directed transverse to the layer. Stability of the system in the longitudinal direction to the field is guaranteed by one-dimensional zero-oscillations of molecules transverse to the short size of the layer. The oscillation quantum (11)
I.T. Iakuhou. A.V. Nerlospusov/Phy.sics
Letters A 232 (1997) 305-307
307
The stability in the transversal direction is maintained by the interdipolar repulsion. The energy of the system is given by the sum of the energy of the dipoles in the field E, the energy of interdipolar interaction UN, and the energy of the zero-oscillations,
with V, = -Ze2d’/R2; here x is the distance from the surface R. The wave function of zero-oscillations decays with the distance from the surface as where 5 = (4ZA4/m)“3( x/R) [5]. exp(-53/2), Hence, particles are localized in a layer the thickness of which is by an order of magnitude smaller than R,
g=
b= R(ZM/m)-“3.
-edEN+(ed)*nN+fiwN.
(12)
The transversal pressure is given by Eq. (10). We write the longitudinal pressure as a superposition of the expressions which are valid in the limiting cases: if b is large it is p,, from Eq. (lo), in the opposite case p,, is the pressure of the zero-oscillations, PII z - (9/4)
( cd) ‘II’ + (&‘/Mb*)n.
(13)
The thermodynamic stability is ensured by the negative sign of the derivative of the pressure with respect to volume. It imposes the following constraint on 5, b < (3/2,rr2)h2(
ed)‘( Mn)-‘.
(14)
Let us consider inequality (14). It combines with the requirement of strong nonideality (4), which ensured the interdipolar repulsion. If r= 1. inequality (14) takes the form b* < h*/MT
= A;..
(15)
Note that A, is large, so inequality (15) is not restrictive. If ?
[(m/M)n,>?d-‘I”‘.
(16)
It is not obligatory to use a hollow-like trap to observe the condensation experimentally. The stabilization of the cooled system is possible provided the field E is nonuniform to retain molecules against one of walls of the trap. The field of charge Ze can orient dipoles and press them against a spherical surface of a large radius R. The thickness of the layer of matter h should be small, b +c R. Particle oscillations across the layer are described by the Schrijdinger equation with the following potential, V( X) = =,
.Y< 0,
V( _r) = V,,.v/R,
.I > 0,
(17)
(18)
We now discuss concrete conditions for the observation of Bose-Einstein condensation in the dipolar medium. The limitations on scattering length (8) require that we have to deal with a system which would be considered as a gaseous one at normal temperatures. The mean interparticle distance ? should exceed the molecular size by M/m times. However, the density decrease should not disturb the fulfillment of inequality (4). This means that the temperature has to be correspondingly lowered. It is clear that small values of dipole moments d are not interesting. They should be of the order of n, (some molecules possess d values close to lOa,). Finally, if d = 0.5u,, M = IOM,, where M, is the mass of the hydrogen atom. the wanted conditions conform to particle densities n z 3 X lo”-3 X lOI cm713 at temperatures T = O.l- 1 I.LK respectively. Then the degeneration criterion is satisfied, A, > 7. Such low temperatures were achieved by the process of evaporative cooling [2-41. Inequality (5) determines the electric field strengths. They are moderate, lower than 0.1-I V/cm. In conclusion, under deep cooling in strong electric fields a gas of dipolar molecules becomes a nonideal anisotropic system possessing peculiar properties. The system can be considered as a promising one for studying Bose-Einstein condensation. This work is supported by the Russian Foundation for Basic Researches.
References [I]
L.D. Landau and E.hl. London,
[?I M.H.
Lllshlv,
Statistal
phycxs
(Pcrgamon,
1958)
Anderson
et al.. Sclencc 269
( 199%
198.
131 CC.
Bradley
et al.. Phys. Rev. Lert. 75 (1995)
[4] K.B.
Davis ct a.. Phys. Rev. Lett. 75 (I9953
3969.
[51 L.D.
Lnndau
mcchanlcs
and E.M.
gnmon, London.
195X).
Llfshltz,
Quantum
1687. (Per-
PHYSICS LETTERS A ELSEVIER
Physics Letters A 232
(1997)305-307
Cold system of dipolar particles in electric field: Bose-Einstein condensation LT. Iakubov a, A.V. Nedospasov a Science Centerfir
Applied
Problems
in Electrodynamics,
Russian Acudemy oj’Sciences.
b
I~horskayrr
Str. 13 / 19. Moscow
127412. Russian
Federution ’ lnstltute jb- High Temperatures,
Russiun Academy
of Sciences.
I:horskuya
Str. 13 / 19, Moscow
127412, Russian Federation
Received 2 I August 1996; revised manuscript received 17 March 1997; accepted for publication I4 May 1997 Communicated by J. Flouquet
Abstract The influence of an external electric field on the state of a system of molecules possessing permanent electric moments is considered. In strong fields the intermolecular attraction transverse to the field direction is suppressed by the electrostatic repulsion. In the longitudinal direction the system is stabilized due to the molecular zero-oscillations. The system cannot undergo van der Waals condensation, but Bose-Einstein condensation becomes possible. 0 1997 Published by Elsevier Science B.V. Keyrrort/.\: Dipole-dipole
interaction; Bose-Einstein
condensation
The nonideal degenerate Bose gas with a repulsive interparticle interaction has been studied actively since the classical works by Lee and Yang, see Ref. [I]. The Bose-Einstein condensate of nonideal gas was experimentally obtained and diagnosed [241. In the present paper a system of particles possessing permanent dipole moments is considered. The possibility of Bose-Einstein condensation of a molecular liquid-like system in an electric field is demonstrated. Consider a system of molecules possessing permanent dipole moments ed. If the mean interparticle distance F z=- atomic lengths one can ignore the short-range component of the interparticle interaction and consider only the long-range one. Such is a dipole-dipole interaction with the potential V(r)
=
,*r-‘[(
d, . d2) - 3r-7
d - r,)( d *
rz)] .
where d,, dZ are the vectors of the dipole moments of the interacting molecules, r is the distance between them. At high temperatures the system is an ideal gas, and the dipole moments are randomly oriented. With the temperature decreasing attractive interactions manifest themselves, the gas undergoes a phase transition of the first kind. However, if a strong electric field is applied the cooling of the system can lead to another result. In strong electric fields all the dipoles are oriented along the field. and the potential (1) takes the following form, V( 1.) = &*1--“(
where 0 is the azimuthal angle between direction and the radius-vector r, looking
(1) 0375-9601/97/$17.00
0
P/I SO775-9601(97)00390-3
1 - 3 COG O),
1997 Published by Elsevier Science B.V. All rights reserved.
(2) the field from the
306
I.T. Iukuhou,
A.V. Nedospc~.wu
/ Physics
origin. The system is axially anisotropic. The anisotropy is a result of the presence of the electric field. Under strong cooling spatial interparticle correlation will increase. It is known that the interdipolar repulsion prevails in the interaction between correlated dipoles, and the mean energy of interaction per particle U becomes positive. I/ > 0. Analysis of dimensionality results in the relation, U=A(ed)‘n, (3) where R = (4~?/3)-’ is the molecular particle density, F is the mean interparticle distance, and A is a numerical coefficient. Its value depends on the intensity of the interaction and becomes comparable to unity in regimes of strong spatial correlation when the interaction energy becomes comparable to temperature T or exceeds it, r=
(ed)??’
2 1,
(4) where r is the nonideality parameter of the dipole system. If inequality (4) is fulfilled the system becomes liquid-like. Consider as an example the hexagonal structure to estimate U. Let us guide the axis of the sixth order along the field and calculate the energy of interaction of the dipole in the origin with the whole environment assuming all the dipoles to be oriented along the field. The energy equals the sum of energies AU, which are the energies of interaction of the chosen dipole with dipoles of the subsequent coordinate spheres. Both parameters of the lattice are equal to F. It leads to the following result, I/=
CA&
z (3,‘4)“‘(ed)‘n.
(5)
k
It is important that taking into account a few members in the series results in its rapid converges. Therefore the procedure can be applied for the modeling of our liquid-like system. The field strength E should exceed the dipole field taken at the mean interparticle distance, E z- erh. (6) In a system of hard spheres the interaction is characterized by scattering length a = MU/4rnfi2, where hl is the particle mass. Introduction of a scattering length u is not justified for an anisotropic system. Let us consider it as the scattering length of the comparable isotropic system. It may help to estimate qualitatively’ the strength of the repulsion, 0 = ( M/4mt)d2a,
’,
(7)
Letters
A 232 f 1997)
305-307
where m is the electron mass and a, = fi”/e’m is the Bohr radius. The scattering length must be small compared with the mean interparticle distance F and thermal wave length Ar = fi( MT)- I/‘, a<
a -=XAT.
(8)
It seems that the fulfillment of inequality (4) prevents the van der Waals condensation to the liquid state as a result of cooling. However, the effects of the anisotropy acquired in the external field display themselves stronger in an anisotropy of pressure. In nonideal systems the cold pressure exceeds the thermal one. The longitudinal pressure p,, differs from the transversal one pI , P I = d( UN)/dV,
,
pII = d( UN)/dV,,
1
(9)
where N is the total number of particles, dV,, and dV, are small changes in the system volume as a result of its enlargement along the anisotropy axis and perpendicularly to it. Two parallel neighboring dipoles interact diversely depending on their relative position. One can foresee a positive sign for p I but a negative one for p,,. We calculated p ~ and p,, in the above model of the hexagonal system. The pressures were calculated enlarging the lattice parameter in longitudinal or transversal direction and using Eqs. (9) straightforwardly. This leads to pa z -(9/4)(
ec1)2n2,
p, = (9/16)(
ed)2n2. ( 10)
Such a state is unstable. Let us demonstrate that in an electric field the quasi-two-dimensional cooled system of dipolar molecules can be stabilized. Let the volume V= bL2 represent the layer, b is the width of the layer and L is the length, b <(: L. The volume holds N particles. The number N is large enough to consider the system as macroscopic. The size b must be larger than the mean interparticle distance to prevent the compression of the system to a liquid film. The field E is directed transverse to the layer. Stability of the system in the longitudinal direction to the field is guaranteed by one-dimensional zero-oscillations of molecules transverse to the short size of the layer. The oscillation quantum (11)
I.T. Iakuhou. A.V. Nerlospusov/Phy.sics
Letters A 232 (1997) 305-307
307
The stability in the transversal direction is maintained by the interdipolar repulsion. The energy of the system is given by the sum of the energy of the dipoles in the field E, the energy of interdipolar interaction UN, and the energy of the zero-oscillations,
with V, = -Ze2d’/R2; here x is the distance from the surface R. The wave function of zero-oscillations decays with the distance from the surface as where 5 = (4ZA4/m)“3( x/R) [5]. exp(-53/2), Hence, particles are localized in a layer the thickness of which is by an order of magnitude smaller than R,
g=
b= R(ZM/m)-“3.
-edEN+(ed)*nN+fiwN.
(12)
The transversal pressure is given by Eq. (10). We write the longitudinal pressure as a superposition of the expressions which are valid in the limiting cases: if b is large it is p,, from Eq. (lo), in the opposite case p,, is the pressure of the zero-oscillations, PII z - (9/4)
( cd) ‘II’ + (&‘/Mb*)n.
(13)
The thermodynamic stability is ensured by the negative sign of the derivative of the pressure with respect to volume. It imposes the following constraint on 5, b < (3/2,rr2)h2(
ed)‘( Mn)-‘.
(14)
Let us consider inequality (14). It combines with the requirement of strong nonideality (4), which ensured the interdipolar repulsion. If r= 1. inequality (14) takes the form b* < h*/MT
= A;..
(15)
Note that A, is large, so inequality (15) is not restrictive. If ?
[(m/M)n,>?d-‘I”‘.
(16)
It is not obligatory to use a hollow-like trap to observe the condensation experimentally. The stabilization of the cooled system is possible provided the field E is nonuniform to retain molecules against one of walls of the trap. The field of charge Ze can orient dipoles and press them against a spherical surface of a large radius R. The thickness of the layer of matter h should be small, b +c R. Particle oscillations across the layer are described by the Schrijdinger equation with the following potential, V( X) = =,
.Y< 0,
V( _r) = V,,.v/R,
.I > 0,
(17)
(18)
We now discuss concrete conditions for the observation of Bose-Einstein condensation in the dipolar medium. The limitations on scattering length (8) require that we have to deal with a system which would be considered as a gaseous one at normal temperatures. The mean interparticle distance ? should exceed the molecular size by M/m times. However, the density decrease should not disturb the fulfillment of inequality (4). This means that the temperature has to be correspondingly lowered. It is clear that small values of dipole moments d are not interesting. They should be of the order of n, (some molecules possess d values close to lOa,). Finally, if d = 0.5u,, M = IOM,, where M, is the mass of the hydrogen atom. the wanted conditions conform to particle densities n z 3 X lo”-3 X lOI cm713 at temperatures T = O.l- 1 I.LK respectively. Then the degeneration criterion is satisfied, A, > 7. Such low temperatures were achieved by the process of evaporative cooling [2-41. Inequality (5) determines the electric field strengths. They are moderate, lower than 0.1-I V/cm. In conclusion, under deep cooling in strong electric fields a gas of dipolar molecules becomes a nonideal anisotropic system possessing peculiar properties. The system can be considered as a promising one for studying Bose-Einstein condensation. This work is supported by the Russian Foundation for Basic Researches.
References [I]
L.D. Landau and E.hl. London,
[?I M.H.
Lllshlv,
Statistal
phycxs
(Pcrgamon,
1958)
Anderson
et al.. Sclencc 269
( 199%
198.
131 CC.
Bradley
et al.. Phys. Rev. Lert. 75 (1995)
[4] K.B.
Davis ct a.. Phys. Rev. Lett. 75 (I9953
3969.
[51 L.D.
Lnndau
mcchanlcs
and E.M.
gnmon, London.
195X).
Llfshltz,
Quantum
1687. (Per-