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Atomic H on liquid He surface: Interaction and bound states
VoIume 60, number 2
CHEMICAL
PHYSICS
LET-ERS
ATOMIC H ON LIQUID He SURFACE: INTERACTION
1 January
1979
AND BOUND STATES
C. DE SIMONE and B. MARAVIGLIA Istincto di Fisica. Univex& di Roma. Rome 0018.5, IMy and Gruppo NazionaZe di Sttuttura de!& Mate& de1 CNR. Rome, Ifary Received 1 September 1978
The interaction energies of a hydrogen atom, H, which crosses the liquid vapour interface of ‘He and 3He are calculated in two different approaches. Liquid helium is considered non-deformabible in the fust approach whiie in the second one the H atom is supposed to be in a rigid cavity (bubble) while crossing the interface. Bound states are found for H outside liqliid 4He only. The implications of the results for obtaining stable atomic H at low temperature are discussed_
The purpose of this work is to study the potentials and the binding energies for atomic hydrogen (H) close to the surface of liquid helium under different approximations_ Besides being interesting for the wide field of bound states of impurities on liquid surfaces [l] , the solution of this problem might be useful in overcoming some of the technical difficulties encountered in obtaining stable H at low temperature. At present some laboratories are devoting much experimental effort to get atomic hydrogen at sufficient density and at a temperature low enough for it to have a Bose condensation_ This condition can possibly be reached if atomic hydrogen atoms, which do not bind when selected in the same electronic spin state, have a very low rate of spin flip and thus of recombination. The nature of the H container is in this respect quite crucial because spin inversion can occur during the collisions of H on the walls. To minimize thii effect the container should have inert gas wails. Among the inert gases helium should be the most convenient one. The H-He potential used for all the calculations is the classic Lennard-Jopes potential, V(r) = 46 [(a/r)‘2 - (G.!@] ,withE=5_34Kando=3_31a as obtained from Toennies et al. [;?I. The first interaction energy that we considered is the energy which is required for H to form a “bubbie” in liquid helium, in analogy with the bubbles formed by a free electron [3], an excited 4He atom [4] and a 3He atom [S]
in liquid 4He. The total energy f’(r) of a bubble of radius r is expressed by four terms r(r) = W(f) f Ko(‘) + 47z.y; + 2 7rPr3,
(1)
where the first term W(r) = 47r Jy V(.$)pot2 dg is the total potential energy at the helium density po_ The second term is the zero-point energy [S] Kg(r) =
h% 2?7nz(r-
0.891d)‘(r+0.713d)
(2) ’
where the hard-core parameter d is supposed to be equal to the Lennard-Jones parameter o and m is the hydrogen atomic mass. In the third and fourth terms of(l),
7 is the surface
tension
and P is the vapor pres-
sure_ The vaIue of r which minimizes r in liquid 4He at T= 0 K is ro = 49 4, if one assumes 7 = 0.36 erd cm2 and P = 0.The minimum energy of the bubble is r(rg) = 94 K. Analogous calculations for atomic hydrogen in liquid ‘He gave r. = 5.3 A and r(rO) = 52 K, with y = 0.16 erg/cm’ and P = 0. The interaction energy between au atom and the free surface of a liquid is substantially affected by the eventual deformation that the atom causes on the surface. In fact rhere is no experimental evidence of what actually happens at the surface when an extra
atom gets very close to it_ In this situation the best way is to adopt different models and to compare the variations which occur in the results. Indeed we will compare two different models, both of which assume 289
Volume 60, number 2
CHEM ICK
1 January 1979
PHYSICS LFXJ’EPS
liquid helium as a continuum with a density which varies stepwise at the surface from the liquid value to that of the vapor. In these models x is the distance of rhe H atom from the liquid surface (x = 0) with x>Ointhevaporandx
xH
Y I
VAPOlIR
energy between the H atom in the vacuum and the liquid-helium is given by: U(x) = j-I+,
x)po d%,
(3)
where V(r, x) is the Lennard-Jones potential between the H atom and the mass element p. d3r. The integration of (3) over the whoie volume of the liquid gives, for x > 0 the expression: U(x) = 47rW,a3 [‘&(o/x)g - ~(o/x)3],
(4)
which is represented in iig_ I _ In the second model the liquid surface is assumed to be deformable when f&e distance x of the H atom is ‘0 ax 2 -ro_ The deformation is supposed to be caused by a sphere of radius r. centered on the H atom [S] _ Forx >rO the potential energy is again given by (4)_ For r. >x > -r. on the other hand, by using (3) with the integration contour properly changed, we obtain U(x) = 8ireooa3 [&~j~~>9 - +(o/ro)3] - 4rreooo2 ];(o/~o)10 - f(o/ro)E’] x.
(5)
To obtain an interaction energy profile continuous across the surface (and thus possiily more reahstic), the surface energy and the zero-point energy neces-
Fig. 2. The cavity (bubble) around the H atom is assumed to be rigid (radius ro) while erossing the liquid surface. sary to create the bubble were expressed as a function of x. The surface energy is given by the following e&pression
c
armstxlro)
axj=r J
2~~6 sin 0 do - &
0
- x2)
1
> (6)
whose symbols are specified in fig_ 2_ The integration of (6) gives S(x) = r?i(x - ro)2 _ As the zero-point energy varies from a ftite value in the bulk liquid to zero in the vapor, it is reasonable to assume that such a variation is proportional to the volume u(x) of the deformation at the liquid surface, when the hydrogen atom crosses the interface_ In this hypothesis the zero-point energy as a function of x (with r. > x > - r0) is given by Ko(x, ro) = Ko(‘o)tix)/4”$ =Ko(ro)[f
- ;x/ro + :(x/ro)2].
(7)
The interaction energy (which is represented in fig_ 3 j, is then expressed by (4) when x > ro, by U(x) = 8YrEPoo3 [$ (olr,)g
- $ (alr,P]
-4~~~oo2[~(o/~o)~~-_:(~/~~)4]x+yx(x +K()(‘o)r; Fii_ I _ Interactionenew between a hydrogen atom in the vacuumand Iiquid 4He, assmned to be not deformqble.
240
- $x/r0 +
- ro)2
a(x/~o)31
forro2x>-r. andbyI’forx<-ro_ We have checked the possibility of the existence of bound states for atomic hydrogen on liquid 4He
(8)
Volume 60. n-zmber2
CHEMICAL PHYSICS LFXi-ERS
VAPOUR
F+_ 3_ L7tenction enew formable liquid 4He.
between atomic hydrogen and de-
and 3He in the two models for the interaction energy. The search for bound states was performed by means of a variational calculation, with the wavefunction G(x) =
Nx--b)bl ew =o,
I-&-b)bl
,
1 January 1979
uid 4He. Their energy is so small that they can be populated only for temperatures of a few hundredth Kelvin. - There are no bound states for H on the surface of liquid 3He. - Both 4He and ‘He should be convenient wails to contain stable atomic H_ Of course if the kinetic energy of the hydrogen atoms is greater than r, they can get into the liquid and eventually scatter on the under!ying metal container. This might cause a spin flip. To avoid this possibility one can either adsorb a solid film of inert gas (e.g. Ne) on top of the metal before adsorbing the He film or keep the kinetic energy well below I’. The 3He wall may be more conI venient than the 4He one at T 4 I K. In fact the existence of bound states at the liquid surface of “He can give rise to an increase of the H-H collision rate and thus to a higher recombination.
forx >b; forx ib.
In this expression b and a are variational parameters. For 4He we obtained .smalI binding energies EO = -0.06 K and E,-,= -0.04 K for the first and the second model respectively. In the case of 3He on the other hand no bound states were found. At this stage it is possible to draw some concluSiOIlSl - Atomic hydrogen forms a “bubble” in both liquid 4He and liquid -?He. - Bound states for H exist on the surface of liq-
References [I] C. de Simone, R. Fiorani and B. Maraviglia, Riv. Nuovo Cimento 7 (1977) 379_ [2] J_P. Toennies, W. Welz and G. Wolf, Chem_ Phys. Letters 44 (1976) 5. (31 Y-W. Cole, Rev. Mod. Phys. 46 (1974) 45 I, and references therein. [4] A. Hickman and N. Lane, E’hys. Rev. Letters 26 (1071) 1216. [S] C. de Simone and B_ ~Maraviglia,Chem. Phys. Letters 43 (1976) 167.
291
CHEMICAL
PHYSICS
LET-ERS
ATOMIC H ON LIQUID He SURFACE: INTERACTION
1 January
1979
AND BOUND STATES
C. DE SIMONE and B. MARAVIGLIA Istincto di Fisica. Univex& di Roma. Rome 0018.5, IMy and Gruppo NazionaZe di Sttuttura de!& Mate& de1 CNR. Rome, Ifary Received 1 September 1978
The interaction energies of a hydrogen atom, H, which crosses the liquid vapour interface of ‘He and 3He are calculated in two different approaches. Liquid helium is considered non-deformabible in the fust approach whiie in the second one the H atom is supposed to be in a rigid cavity (bubble) while crossing the interface. Bound states are found for H outside liqliid 4He only. The implications of the results for obtaining stable atomic H at low temperature are discussed_
The purpose of this work is to study the potentials and the binding energies for atomic hydrogen (H) close to the surface of liquid helium under different approximations_ Besides being interesting for the wide field of bound states of impurities on liquid surfaces [l] , the solution of this problem might be useful in overcoming some of the technical difficulties encountered in obtaining stable H at low temperature. At present some laboratories are devoting much experimental effort to get atomic hydrogen at sufficient density and at a temperature low enough for it to have a Bose condensation_ This condition can possibly be reached if atomic hydrogen atoms, which do not bind when selected in the same electronic spin state, have a very low rate of spin flip and thus of recombination. The nature of the H container is in this respect quite crucial because spin inversion can occur during the collisions of H on the walls. To minimize thii effect the container should have inert gas wails. Among the inert gases helium should be the most convenient one. The H-He potential used for all the calculations is the classic Lennard-Jopes potential, V(r) = 46 [(a/r)‘2 - (G.!@] ,withE=5_34Kando=3_31a as obtained from Toennies et al. [;?I. The first interaction energy that we considered is the energy which is required for H to form a “bubbie” in liquid helium, in analogy with the bubbles formed by a free electron [3], an excited 4He atom [4] and a 3He atom [S]
in liquid 4He. The total energy f’(r) of a bubble of radius r is expressed by four terms r(r) = W(f) f Ko(‘) + 47z.y; + 2 7rPr3,
(1)
where the first term W(r) = 47r Jy V(.$)pot2 dg is the total potential energy at the helium density po_ The second term is the zero-point energy [S] Kg(r) =
h% 2?7nz(r-
0.891d)‘(r+0.713d)
(2) ’
where the hard-core parameter d is supposed to be equal to the Lennard-Jones parameter o and m is the hydrogen atomic mass. In the third and fourth terms of(l),
7 is the surface
tension
and P is the vapor pres-
sure_ The vaIue of r which minimizes r in liquid 4He at T= 0 K is ro = 49 4, if one assumes 7 = 0.36 erd cm2 and P = 0.The minimum energy of the bubble is r(rg) = 94 K. Analogous calculations for atomic hydrogen in liquid ‘He gave r. = 5.3 A and r(rO) = 52 K, with y = 0.16 erg/cm’ and P = 0. The interaction energy between au atom and the free surface of a liquid is substantially affected by the eventual deformation that the atom causes on the surface. In fact rhere is no experimental evidence of what actually happens at the surface when an extra
atom gets very close to it_ In this situation the best way is to adopt different models and to compare the variations which occur in the results. Indeed we will compare two different models, both of which assume 289
Volume 60, number 2
CHEM ICK
1 January 1979
PHYSICS LFXJ’EPS
liquid helium as a continuum with a density which varies stepwise at the surface from the liquid value to that of the vapor. In these models x is the distance of rhe H atom from the liquid surface (x = 0) with x>Ointhevaporandx
xH
Y I
VAPOlIR
energy between the H atom in the vacuum and the liquid-helium is given by: U(x) = j-I+,
x)po d%,
(3)
where V(r, x) is the Lennard-Jones potential between the H atom and the mass element p. d3r. The integration of (3) over the whoie volume of the liquid gives, for x > 0 the expression: U(x) = 47rW,a3 [‘&(o/x)g - ~(o/x)3],
(4)
which is represented in iig_ I _ In the second model the liquid surface is assumed to be deformable when f&e distance x of the H atom is ‘0 ax 2 -ro_ The deformation is supposed to be caused by a sphere of radius r. centered on the H atom [S] _ Forx >rO the potential energy is again given by (4)_ For r. >x > -r. on the other hand, by using (3) with the integration contour properly changed, we obtain U(x) = 8ireooa3 [&~j~~>9 - +(o/ro)3] - 4rreooo2 ];(o/~o)10 - f(o/ro)E’] x.
(5)
To obtain an interaction energy profile continuous across the surface (and thus possiily more reahstic), the surface energy and the zero-point energy neces-
Fig. 2. The cavity (bubble) around the H atom is assumed to be rigid (radius ro) while erossing the liquid surface. sary to create the bubble were expressed as a function of x. The surface energy is given by the following e&pression
c
armstxlro)
axj=r J
2~~6 sin 0 do - &
0
- x2)
1
> (6)
whose symbols are specified in fig_ 2_ The integration of (6) gives S(x) = r?i(x - ro)2 _ As the zero-point energy varies from a ftite value in the bulk liquid to zero in the vapor, it is reasonable to assume that such a variation is proportional to the volume u(x) of the deformation at the liquid surface, when the hydrogen atom crosses the interface_ In this hypothesis the zero-point energy as a function of x (with r. > x > - r0) is given by Ko(x, ro) = Ko(‘o)tix)/4”$ =Ko(ro)[f
- ;x/ro + :(x/ro)2].
(7)
The interaction energy (which is represented in fig_ 3 j, is then expressed by (4) when x > ro, by U(x) = 8YrEPoo3 [$ (olr,)g
- $ (alr,P]
-4~~~oo2[~(o/~o)~~-_:(~/~~)4]x+yx(x +K()(‘o)r; Fii_ I _ Interactionenew between a hydrogen atom in the vacuumand Iiquid 4He, assmned to be not deformqble.
240
- $x/r0 +
- ro)2
a(x/~o)31
forro2x>-r. andbyI’forx<-ro_ We have checked the possibility of the existence of bound states for atomic hydrogen on liquid 4He
(8)
Volume 60. n-zmber2
CHEMICAL PHYSICS LFXi-ERS
VAPOUR
F+_ 3_ L7tenction enew formable liquid 4He.
between atomic hydrogen and de-
and 3He in the two models for the interaction energy. The search for bound states was performed by means of a variational calculation, with the wavefunction G(x) =
Nx--b)bl ew =o,
I-&-b)bl
,
1 January 1979
uid 4He. Their energy is so small that they can be populated only for temperatures of a few hundredth Kelvin. - There are no bound states for H on the surface of liquid 3He. - Both 4He and ‘He should be convenient wails to contain stable atomic H_ Of course if the kinetic energy of the hydrogen atoms is greater than r, they can get into the liquid and eventually scatter on the under!ying metal container. This might cause a spin flip. To avoid this possibility one can either adsorb a solid film of inert gas (e.g. Ne) on top of the metal before adsorbing the He film or keep the kinetic energy well below I’. The 3He wall may be more conI venient than the 4He one at T 4 I K. In fact the existence of bound states at the liquid surface of “He can give rise to an increase of the H-H collision rate and thus to a higher recombination.
forx >b; forx ib.
In this expression b and a are variational parameters. For 4He we obtained .smalI binding energies EO = -0.06 K and E,-,= -0.04 K for the first and the second model respectively. In the case of 3He on the other hand no bound states were found. At this stage it is possible to draw some concluSiOIlSl - Atomic hydrogen forms a “bubble” in both liquid 4He and liquid -?He. - Bound states for H exist on the surface of liq-
References [I] C. de Simone, R. Fiorani and B. Maraviglia, Riv. Nuovo Cimento 7 (1977) 379_ [2] J_P. Toennies, W. Welz and G. Wolf, Chem_ Phys. Letters 44 (1976) 5. (31 Y-W. Cole, Rev. Mod. Phys. 46 (1974) 45 I, and references therein. [4] A. Hickman and N. Lane, E’hys. Rev. Letters 26 (1071) 1216. [S] C. de Simone and B_ ~Maraviglia,Chem. Phys. Letters 43 (1976) 167.
291