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Positive ion structure in 3He-rich liquid helium mixtures
Volume 64A, number 2
PHYSICS LETTERS
12 December 1977
POSITIVE ION STRUCTURE IN 3He-RICH LIQUID HELIUM MIXTURES* T.J. SLUCKIN1 Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA Received 29 September 1977 The structure of positive ions in impure 3He is investigated. We predict a ~halo” of “He around the ion. We use the halo to interpret recent experiments on ion mobility in 3He. Other properties of the halo are studied.
In a recent paper, Roach et al. [1],have shown that previously inconsistent results on positive ion mobility 3He near 0.1 K [2—5] can be understood in terms in a sensitive dependence of mobility on small 4He of impurity concentrations. In this letter we discuss the structures associated with positive ions in such mixtures, and the experimental consequences thereof, Laheurte [6] has considered the piling up of 4He near a wall in 3He-rich liquid helium mixtures, due to the van der Waals potential near the wall. We shall use a similar formalism in order to study how pressure and 4He concentration change near a positive ion. We consider a mixture of 3He and 4He, at a temperature T, with chemical potentials respectively /13~/14. An ion at r = 0 provides attractive external potentials Ø 4He concen3(r), ~4(r). The local tration c(r) satisfy (see pressure e.g. [7]):p(r), and
solid 3He- and 4He-rich phases. The effect of the ion potential is to bring ~i 34(r) into regions where a 4He-rich liquid, or solid, are favored, The latter is familiar in one component systems [8],in which the ion is clothed by a solid helium “snowball” of approximately 6A radius. We consider the structure for constant concentration c as temperature is reduced. The critical temperature for phase separation of 3He—4He mixtures at zero pressure is 0.88 K. Above 0.88 K, c(r) will increase with decreasing r, but no phase separation can take place. At r 6A, the melting pressure is reached, and a snowball forms. Below 0.88 K, the system will follow a “trajectory” in (p, c) space as one approaches the ion. Both p and c increase with decreasing r, approaching the freezing curve and the phase separation curve. At higher temperatures the freezing curve is reached first, and a snowball forms, as in one component systems.
p(r) p [p 3(r), p4(r), T] c(r) c [j~i3 (r), ~i4(r),T} where
, ,
(Ia) (Ib)
p3,4(r) + ~34(r) = /13,4 (2) For our case Ø3(r) = p4(r) = y = 1.91 X l0~~ c.g.s. units. The functions p. c of eq. (1) are multi-valued, assuming different values for each phase. The equilibrium phase has the largest p, and thus the lowest grand thermodynamic potential. In ourphase, case the inte3He-rich liquid a 4Heresting phases will be a rich liquid phase, a solid mixture phase, and possibly .
*
Work partially supported by N.S.F. Grant #DMR75-08516.
At lower temperatures the phase separation curve 4He-rich fluid forms is reached first, and a “halo” of around the snowball. As the temperature is lowered the radius of the halo will grow until the critical temperature T 0(c) for phase separation at this concentration is reached. The halo radius is now limited by the interfacial tension between the two phases. As the temperature is further lowered phase separation will take place, and the system will remain on the phase separation curve. The radius of the halo will remain more or less constant. The detailed structure follows from the minimization Thus of the grand thermodynamic potential 12. RH
=
f p~(r)dr—f p4(r)dr—
Present address: Department of Physics, Case Western Reserve
R5
University, Cleveland, Ohio 44106, USA.
00
fp3(r)dr RH
2a +4irR
45+4irR~u34
,
(3a) 211
Volume 64A, number 2
PI-IYSICS LETTERS
~
12 December 1977
30
0.2
~1 •
11.4.
.
50
0.05 0 I
0
C
I
0
I
00
0.1
o.r
0.3
1Q00
mK)
04
05
IlK)
Fig. 2.R
31-le, and impure 3He with cFig. = 31.X Observed i04 (curves mobilities A, B), in and pure predicted mobility K’ (curve
11(T) at the vapor pressure for c = 3 X 10~, with T> T0(c) = 0. 108 K, and for c= c0(T) for T< 0, 108 K. at which phase separation occurs. If c(c~o)~ c
0 the C) as indicated in text. R5
=
f ~(~)dr f p3(r) dr —
+
4~R~o35,
(3b)
right hand side of eq. eq. (5) becomes c(r) = c0.
(5)
can be ignored. In this case (6)
R5 12H’ &2~are grand thermodynamic potentials where in the presence or absence respectively of a halo; p(r) is given by eqs. (1) and the subscripts refer to solid, 3He-rich and 4He-rich fluid phases, and the a’s represent surface tensions at the phase boundaries. R~is the snowball radius. It can be shown [9] ~‘ that in the 3He rich phase
Solving eq. (4b) [ay/(l gives +a)kB Tr4] c(r) = c(°°)exp
(7)
It has been shown that [10]: c
312 eO~~6IT
(8)
.
0(T) = 0.85 T Thus R4H~
(9)
°~‘
dp~—n
2/3)0.56j 3[(1+c)(1+a)/(1+a+c)]d~,
kBTdc~ [—ac/(1+~+c)] d~,
(4a) (4b)
3He atoms. where n3 is the number density of The equilibrium condition for the halo radius is obtamed from minimizing eq. (3a): p 4(RH)p3(Ru)=2a34/RH (5) ‘
Exact solution of eq. (5) involves detailed numerical calculations, which we hope to carry out in the future. However good approximate solutions may be obtained in the following way. For each T, there is a critical concentration c0(T) ~ In eq. 85-5 of [1O}put ~ 0, and n 3d~i/dp 3He atom = 1/(1+a), as opposed whereto a 4Fle authors of 1101 is theatom. excessThe volume occupied bygive a a 0.47, but note their a and mine have slightly different meanings.
212
.
(l+cs)[~kBT1n(T/lA14c This diverges at c c 0, but the surface tension 034 prevents the halo radius from growing indefinitely. At c 0 we may solve eq. (5) approximately by noting that 4 (10) p34(r) i~p34(oo) + ~yn34/r ,
and ~
(11)
Combining eqs. (10), (11) with (5) immediately yields RH~[~(n4— n3)/2a34] 1/3, (12) 4He at the vapor where n4 is the number density of pressure. eqs. (9) and To (12). calculate For low RH we T [11], have 034 interpolated = 0.022 dynes between cm1, eq. (12) yields R 11 13.0 A. For higher concentrations, the critical concentration is reached at higher
Volume
64A, number 2
temperatures, where
PHYSICS LETTERS 034
is lower; here larger halo radii
can be expected. Simple theories of mobility (see e.g. [12]) give for the ion mobility K ~ = (hkf/e)n 3S(kf), (13) where S(kf) is the scattering cross section for quasiparticles from the ion. However this picture is disturbed by the necessity to include recoil effects self consistently [13]. As a crude method for comparison of our theory with experiment, therefore, we plot 3He, and in 3He within c fig. 3 X1 the mobilities in pure iO4 as measured [1], and a “predicted mobility” K’ for the impure 3He: K’
2 (14) 5/R~) Here we assume that S(kf) czR~,and RH is calculated for c =3 X l0~~ for T> T 0(c), and for c = c0(T) for T< T0(c). According to our theory the halo should exist down to T= 0. We are unable therefore to explain its sudden disappearance at 60 mK [1]. We speculate that for sufficiently lowand temperatures a phase 31-Ie-rich 4He-rich phases canseparation take placeinto solid within the snowball, and this may increase the surface tension between 4He liquid and the snowball. The halo, while remaining metastable, would then become energetically unfavorable, The halo can serve as a nucleation sphere for the 4He-rich phase when the system has been supercooled, and c > c 0, by analogy to the snowball serving as a nucleation sphere for the solid in a supercooled liquid (see e.g. [71). As the walls ofsuch a sample normally would 4He-rich phase, an experiment would nucleate be thebest carried out in space, where contact perhaps with walls is not necessary. In the presence of the ion, adapting the formulae of [7] , the supercooled mixture ceases to be even metastable if the sample reaches a concentration c, where c 0(T) 413/n (15) 1 a 3kBT[(n4 n3)y] 1/3 or, equivalently, is supercooled to a temperature =
K(pure 3He) 1< (R
~-
.
—
T0(c) 4/3/0.56n (16) 1+o 3 [(n4 n3)~]1/3 The halo will vibrate acoustically, with the frequency _______
___________
—
12
December
1977
w1 of the 1 th harmonic given approximately [14] * 2 by (4y1/R~)(n4 n3) + 1(1--— 1)(1 + 2)(a34/R~) 21~’ ‘1 + (11(1+1) (RS/RH) x (rn4n4 (~~_ )+m3n31 (17) 2~’ l—(Rs/RH) where rn 3He, 4He atoms respectively. 34 are the masses of For the halo at low temperature at c 0, with RH = 13 A, eq. (17) yields = 2.13 X lO10Hz, (18a) = 2.91 X 1O10Hz (18b) =
—
.
We believe these modes could be observed using microwave absorption * I should like to thank J.B. Ketterson for communicating his results before publication, M.W. Cole for useful correspondence, and A.J. Dahm and A.L. Fetter for useful discussions. During the course of this work I was the recipient of a N.A.T.O. post-doctoral fellowship Greatawarded Britain.by the Science Research Council of *2
We have followed the general method indicated here, assuming constant density within the halo, and boundary conditions which do not allow transport of material across the phase interface. The fluids are almost immiscible, so we believe this not to be a bad approximation. I am indebted to A.J. Dahm for this suggestion.
References [1] P.D. Roach, J.B. Ketterson and P.R. Roach, Phys. Lett. 63A (1977) 273. (21 A.C. Anderson, M. Kuchnir and J.C. Wheatley, Phys. Rev. 168 (1968) 261. 131 M. Kuchnir, J.B. Ketterson and P.R. Roach, J. Low. Temp. Phys. 19 (1975) 531.
[41C.N. Barber, P.V.E. McLintock, I.E. Miller and G.R. Pickett, Phys.Lett.54A(1975)241. [5] P.D. Roach, J.B. Ketterson and P.R. Roach, in: Proc. Internat. Conf. on Quantum Liquids, Sanibel Island (1977). [61 J.P. Laheurte and J.P. Romagnan, Phys. Lett. 50A (1974) 213, and references therein. 171 M.W. Cole and T.J. Sluckin, J. Chem. Phys. 67 (1977) 746.
213
Volume 64A, number 2
PHYSICS LETTERS
[8] K.R. Atkins, Phys. Rev. 116 (1959) 1139. [91 L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, London, 1958) Section 85. [10] W.F. Saam and J.P. Laheurte, Phys. Rev. A4 (1971) 1170. [11] H.M. Guo, D.O. Edwards, R. Sarwinski and J.T. Tough, Phys. Rev. Lett. 27 (1971) 1259.
214
12 December 1977
[12] A.L. Fetter, in: The physics of liquid and solid helium, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1976) Vol. 1. [13] B.D. Josephson and J. Lekner, Phys. Rev. Lett. 23 (1969) Ill. [14] L.D. Landau and E.M. Lifshitz, Fluid mechanics, (Pergamon, London, 1959) Section 61.
PHYSICS LETTERS
12 December 1977
POSITIVE ION STRUCTURE IN 3He-RICH LIQUID HELIUM MIXTURES* T.J. SLUCKIN1 Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA Received 29 September 1977 The structure of positive ions in impure 3He is investigated. We predict a ~halo” of “He around the ion. We use the halo to interpret recent experiments on ion mobility in 3He. Other properties of the halo are studied.
In a recent paper, Roach et al. [1],have shown that previously inconsistent results on positive ion mobility 3He near 0.1 K [2—5] can be understood in terms in a sensitive dependence of mobility on small 4He of impurity concentrations. In this letter we discuss the structures associated with positive ions in such mixtures, and the experimental consequences thereof, Laheurte [6] has considered the piling up of 4He near a wall in 3He-rich liquid helium mixtures, due to the van der Waals potential near the wall. We shall use a similar formalism in order to study how pressure and 4He concentration change near a positive ion. We consider a mixture of 3He and 4He, at a temperature T, with chemical potentials respectively /13~/14. An ion at r = 0 provides attractive external potentials Ø 4He concen3(r), ~4(r). The local tration c(r) satisfy (see pressure e.g. [7]):p(r), and
solid 3He- and 4He-rich phases. The effect of the ion potential is to bring ~i 34(r) into regions where a 4He-rich liquid, or solid, are favored, The latter is familiar in one component systems [8],in which the ion is clothed by a solid helium “snowball” of approximately 6A radius. We consider the structure for constant concentration c as temperature is reduced. The critical temperature for phase separation of 3He—4He mixtures at zero pressure is 0.88 K. Above 0.88 K, c(r) will increase with decreasing r, but no phase separation can take place. At r 6A, the melting pressure is reached, and a snowball forms. Below 0.88 K, the system will follow a “trajectory” in (p, c) space as one approaches the ion. Both p and c increase with decreasing r, approaching the freezing curve and the phase separation curve. At higher temperatures the freezing curve is reached first, and a snowball forms, as in one component systems.
p(r) p [p 3(r), p4(r), T] c(r) c [j~i3 (r), ~i4(r),T} where
, ,
(Ia) (Ib)
p3,4(r) + ~34(r) = /13,4 (2) For our case Ø3(r) = p4(r) = y = 1.91 X l0~~ c.g.s. units. The functions p. c of eq. (1) are multi-valued, assuming different values for each phase. The equilibrium phase has the largest p, and thus the lowest grand thermodynamic potential. In ourphase, case the inte3He-rich liquid a 4Heresting phases will be a rich liquid phase, a solid mixture phase, and possibly .
*
Work partially supported by N.S.F. Grant #DMR75-08516.
At lower temperatures the phase separation curve 4He-rich fluid forms is reached first, and a “halo” of around the snowball. As the temperature is lowered the radius of the halo will grow until the critical temperature T 0(c) for phase separation at this concentration is reached. The halo radius is now limited by the interfacial tension between the two phases. As the temperature is further lowered phase separation will take place, and the system will remain on the phase separation curve. The radius of the halo will remain more or less constant. The detailed structure follows from the minimization Thus of the grand thermodynamic potential 12. RH
=
f p~(r)dr—f p4(r)dr—
Present address: Department of Physics, Case Western Reserve
R5
University, Cleveland, Ohio 44106, USA.
00
fp3(r)dr RH
2a +4irR
45+4irR~u34
,
(3a) 211
Volume 64A, number 2
PI-IYSICS LETTERS
~
12 December 1977
30
0.2
~1 •
11.4.
.
50
0.05 0 I
0
C
I
0
I
00
0.1
o.r
0.3
1Q00
mK)
04
05
IlK)
Fig. 2.R
31-le, and impure 3He with cFig. = 31.X Observed i04 (curves mobilities A, B), in and pure predicted mobility K’ (curve
11(T) at the vapor pressure for c = 3 X 10~, with T> T0(c) = 0. 108 K, and for c= c0(T) for T< 0, 108 K. at which phase separation occurs. If c(c~o)~ c
0 the C) as indicated in text. R5
=
f ~(~)dr f p3(r) dr —
+
4~R~o35,
(3b)
right hand side of eq. eq. (5) becomes c(r) = c0.
(5)
can be ignored. In this case (6)
R5 12H’ &2~are grand thermodynamic potentials where in the presence or absence respectively of a halo; p(r) is given by eqs. (1) and the subscripts refer to solid, 3He-rich and 4He-rich fluid phases, and the a’s represent surface tensions at the phase boundaries. R~is the snowball radius. It can be shown [9] ~‘ that in the 3He rich phase
Solving eq. (4b) [ay/(l gives +a)kB Tr4] c(r) = c(°°)exp
(7)
It has been shown that [10]: c
312 eO~~6IT
(8)
.
0(T) = 0.85 T Thus R4H~
(9)
°~‘
dp~—n
2/3)0.56j 3[(1+c)(1+a)/(1+a+c)]d~,
kBTdc~ [—ac/(1+~+c)] d~,
(4a) (4b)
3He atoms. where n3 is the number density of The equilibrium condition for the halo radius is obtamed from minimizing eq. (3a): p 4(RH)p3(Ru)=2a34/RH (5) ‘
Exact solution of eq. (5) involves detailed numerical calculations, which we hope to carry out in the future. However good approximate solutions may be obtained in the following way. For each T, there is a critical concentration c0(T) ~ In eq. 85-5 of [1O}put ~ 0, and n 3d~i/dp 3He atom = 1/(1+a), as opposed whereto a 4Fle authors of 1101 is theatom. excessThe volume occupied bygive a a 0.47, but note their a and mine have slightly different meanings.
212
.
(l+cs)[~kBT1n(T/lA14c This diverges at c c 0, but the surface tension 034 prevents the halo radius from growing indefinitely. At c 0 we may solve eq. (5) approximately by noting that 4 (10) p34(r) i~p34(oo) + ~yn34/r ,
and ~
(11)
Combining eqs. (10), (11) with (5) immediately yields RH~[~(n4— n3)/2a34] 1/3, (12) 4He at the vapor where n4 is the number density of pressure. eqs. (9) and To (12). calculate For low RH we T [11], have 034 interpolated = 0.022 dynes between cm1, eq. (12) yields R 11 13.0 A. For higher concentrations, the critical concentration is reached at higher
Volume
64A, number 2
temperatures, where
PHYSICS LETTERS 034
is lower; here larger halo radii
can be expected. Simple theories of mobility (see e.g. [12]) give for the ion mobility K ~ = (hkf/e)n 3S(kf), (13) where S(kf) is the scattering cross section for quasiparticles from the ion. However this picture is disturbed by the necessity to include recoil effects self consistently [13]. As a crude method for comparison of our theory with experiment, therefore, we plot 3He, and in 3He within c fig. 3 X1 the mobilities in pure iO4 as measured [1], and a “predicted mobility” K’ for the impure 3He: K’
2 (14) 5/R~) Here we assume that S(kf) czR~,and RH is calculated for c =3 X l0~~ for T> T 0(c), and for c = c0(T) for T< T0(c). According to our theory the halo should exist down to T= 0. We are unable therefore to explain its sudden disappearance at 60 mK [1]. We speculate that for sufficiently lowand temperatures a phase 31-Ie-rich 4He-rich phases canseparation take placeinto solid within the snowball, and this may increase the surface tension between 4He liquid and the snowball. The halo, while remaining metastable, would then become energetically unfavorable, The halo can serve as a nucleation sphere for the 4He-rich phase when the system has been supercooled, and c > c 0, by analogy to the snowball serving as a nucleation sphere for the solid in a supercooled liquid (see e.g. [71). As the walls ofsuch a sample normally would 4He-rich phase, an experiment would nucleate be thebest carried out in space, where contact perhaps with walls is not necessary. In the presence of the ion, adapting the formulae of [7] , the supercooled mixture ceases to be even metastable if the sample reaches a concentration c, where c 0(T) 413/n (15) 1 a 3kBT[(n4 n3)y] 1/3 or, equivalently, is supercooled to a temperature =
K(pure 3He) 1< (R
~-
.
—
T0(c) 4/3/0.56n (16) 1+o 3 [(n4 n3)~]1/3 The halo will vibrate acoustically, with the frequency _______
___________
—
12
December
1977
w1 of the 1 th harmonic given approximately [14] * 2 by (4y1/R~)(n4 n3) + 1(1--— 1)(1 + 2)(a34/R~) 21~’ ‘1 + (11(1+1) (RS/RH) x (rn4n4 (~~_ )+m3n31 (17) 2~’ l—(Rs/RH) where rn 3He, 4He atoms respectively. 34 are the masses of For the halo at low temperature at c 0, with RH = 13 A, eq. (17) yields = 2.13 X lO10Hz, (18a) = 2.91 X 1O10Hz (18b) =
—
.
We believe these modes could be observed using microwave absorption * I should like to thank J.B. Ketterson for communicating his results before publication, M.W. Cole for useful correspondence, and A.J. Dahm and A.L. Fetter for useful discussions. During the course of this work I was the recipient of a N.A.T.O. post-doctoral fellowship Greatawarded Britain.by the Science Research Council of *2
We have followed the general method indicated here, assuming constant density within the halo, and boundary conditions which do not allow transport of material across the phase interface. The fluids are almost immiscible, so we believe this not to be a bad approximation. I am indebted to A.J. Dahm for this suggestion.
References [1] P.D. Roach, J.B. Ketterson and P.R. Roach, Phys. Lett. 63A (1977) 273. (21 A.C. Anderson, M. Kuchnir and J.C. Wheatley, Phys. Rev. 168 (1968) 261. 131 M. Kuchnir, J.B. Ketterson and P.R. Roach, J. Low. Temp. Phys. 19 (1975) 531.
[41C.N. Barber, P.V.E. McLintock, I.E. Miller and G.R. Pickett, Phys.Lett.54A(1975)241. [5] P.D. Roach, J.B. Ketterson and P.R. Roach, in: Proc. Internat. Conf. on Quantum Liquids, Sanibel Island (1977). [61 J.P. Laheurte and J.P. Romagnan, Phys. Lett. 50A (1974) 213, and references therein. 171 M.W. Cole and T.J. Sluckin, J. Chem. Phys. 67 (1977) 746.
213
Volume 64A, number 2
PHYSICS LETTERS
[8] K.R. Atkins, Phys. Rev. 116 (1959) 1139. [91 L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon, London, 1958) Section 85. [10] W.F. Saam and J.P. Laheurte, Phys. Rev. A4 (1971) 1170. [11] H.M. Guo, D.O. Edwards, R. Sarwinski and J.T. Tough, Phys. Rev. Lett. 27 (1971) 1259.
214
12 December 1977
[12] A.L. Fetter, in: The physics of liquid and solid helium, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1976) Vol. 1. [13] B.D. Josephson and J. Lekner, Phys. Rev. Lett. 23 (1969) Ill. [14] L.D. Landau and E.M. Lifshitz, Fluid mechanics, (Pergamon, London, 1959) Section 61.