- Email: [email protected]
The microscopic derivation of the quantum-hydrodynamical model for a dilute solution of impurities in superfluid helium
Volume 31A, number 9
PHYSICS
LETTERS
4 May 1970
THE MICROSCOPIC DERIVATION OF THE QUANTUM-HYDRODYNAMICAL MODEL FOR A DILUTE SOLUTION IMPURITIES IN SUPERFLUID HELIUM
Max-Planck-Institut
H. WINTER fdr Physik und Astrophysik
Miinchen,
OF
Germany
Received 26 March 1970 The quantumhydrodynamical model for dilute solutions of impurities in superfluid helium is derived microscopically starting with a particle Hamiltonian.
The most important diagrams contributing to the imaginary part of the impurity self energy at low temperatures are those with the least possible number of lines in a cut:
i
i
Fig. 1 Contributionsto Imi%@p) = F The spectral functions there are given by: A(p, w) = g(p, w + in)F(p, w)g(p, w - in)
(1)
Using unbroadened spectral functions diagr. 1 vanishes on account of Landau’s criterion. The lowest nonvanishing orders are diagr. 2 with unbroadened propagators and diagr. 1 with one propagator broadened. (According to form. (1) we insert for l? diagrams with cuts of two unbroadened lines.) These terms sum up to the scattering between an impurity and a phonon. The other ones contribute to higher order processes. Introducing one-particle sources (v , v_), variable chemical (p) and vector potentials tA) for each component we are able to represent the vertices in fig. 1 by the functional derivatives:
(2)
Where the G’s are the variables conjugate to
the v’s. The contributions giving rise to self energy insertions into the propagators are not taken into account. Using the conservation laws for the number of particles and the momentum these vertices are expressed by functional derivatives of the impurity mass operator, the densities and the currents with respect to A, I_(. In F they sum up to thermodynamic derivatives and we obtain as a result the phenomenological scattering probability [1,2]. So we have shown that no particle-like scattering remains and the impurities effectively interact with correlation functions of helium (which is a supposition in ref. [3]) via deformation potentials. In addition we calculate the linear response of the system to an external vector field acting on the impurities by renormalizing the impurity momentum (density)-momentum correlation function. Similar calculations were carried out in refs. [4-‘71 for other systems. Up to the first order in the impurity concentration the transport properties of the impurities are determined by a distribution function Q (k, w), which obeys the phenomenological 8 oltzmann equation, if we use the forementioned result for l? and approximate the imaginary part of the impurity-impurity scattering matrix consistently. The author is indebted to Prof. W. Brenig and Prof. W. Gijtze for discussions. References 1. I. M. Khalatnikovand V. N. Zharkov, Sov. Phys. JETP 5 (1957) 905. 2. G. Baym and C. Ebner, Phys. Rev. 164 (1967) 164. 3. W. F. Saam, Ann. of Phys. 53 (1969) 239. 489
Volume
31A. number 3
PHYSICS
LETTERS
4. A. B. Migdal. Sov. Phys. JETP 16 (1964) 1366. 6. G.M.Eliashberg. Sov. Phys. JETP 14 (1963) 886.
6. J. Gavoret 7. H. Winter,
4 May 1970 and P. Nozieres, Ann. of Phys. 28, 349. Z. Phys., to be published.
*****
EVIDENCE
FOR FOR
A SUPERFLUID-INDUCED ELECTRONS IN LIQUID
SURFACE HELIUM
BARRIER
W. SCHOEPE
Physik-Department
der Technischen
Hochschule
Miinchen.
Germany
and C. PROBST Zentralinstitut
fiir Tieftemperaturforschung der Bayerischen Akademie Garching bei Miinchen, Germany Received
24 March
der Wissenschaften
1970
The free surface of liquid helium acts as an energy barrier on electron currents across the surfac into the vapor, if the liquid is superfluid. In zero electric field, the barrier vanishes at Th. The measurements were performed in pure 4He and in a solution of 3He in 4He.
Bruschi et al. [l] have reported that for temperatures less than 1.7OK electrons encountered increasing difficulty to pass through the free surface of liquid 4He into the vapor phase, if the temperature is lowered. Below 1.5oK the electron current decreased rapidly according to exp (- @//kBT), with @B = 25 f 1oK. This behavior was mainly attributed to the image-force potential which acts on a charge near the interface of two media having different dielectric constants. The effect of the image-force potential outside the liquid was recently discussed in detail by Cole and Cohen [2]. Using the assumption of the image-force, however, it can be readily shown, that the work function $ must depend on the extracting field E, in contradiction to the results of ref. [l]. In addition, the saturation of the currents at 1.7oK remained unexplained since this temperature is much smaller than the barrier. Therefore, we made a special effort to determine $ with a higher accuracy. Bergides pye 4He we also investigated a solution of He in He. Our results demonstrate that the barrier depends not only on the applied field but also on the temperature, giving clear evidence that the barrier is present only in the superfluid state. The measuring cell was similar to that shown in ref. [l], consisting of three electrodes as indicated in fig. 1. We measured the currents when 490
the liquid level was above the collector and compared them with the corresponding currents when the liquid surface was below the collector. The currents with the collector being immersed were nearly temperature independent, in agreement with ref. [l]. On the other hand, when the liquid level was lowered below the collector the current showed a striking dependence on the temperature as plotted in fig. 1 for different extracting fields. For example for a field of 154 V/cm the current remains temperature independent from 4.2oK down to 1.7oK. Only below 1.7oK the current decreases rapidly with temperature as exp (-$//kgT), indicating an effective energy barrier @&?B = 30.4oK, which the emitted electrons have to overcome. It is evident that this positive energy barrier only exists well below l.‘7’K and apparently disappears when this temperature is exceeded. It is essential to point out that for lower extracting fields the energy barrier disappears at higher temperatures, which approach the X-point for zero extracting fields. The same behavior is observed in a solution of 32% 3He in 4He below the shifted X-point of 1.66oK. Thus a finite energy barrier for the emission of electrons from liquid helium seems to be closely related to the superfluid state. Besides this temperature dependence of the barrier it is important to emphasize that @ also
PHYSICS
LETTERS
4 May 1970
THE MICROSCOPIC DERIVATION OF THE QUANTUM-HYDRODYNAMICAL MODEL FOR A DILUTE SOLUTION IMPURITIES IN SUPERFLUID HELIUM
Max-Planck-Institut
H. WINTER fdr Physik und Astrophysik
Miinchen,
OF
Germany
Received 26 March 1970 The quantumhydrodynamical model for dilute solutions of impurities in superfluid helium is derived microscopically starting with a particle Hamiltonian.
The most important diagrams contributing to the imaginary part of the impurity self energy at low temperatures are those with the least possible number of lines in a cut:
i
i
Fig. 1 Contributionsto Imi%@p) = F The spectral functions there are given by: A(p, w) = g(p, w + in)F(p, w)g(p, w - in)
(1)
Using unbroadened spectral functions diagr. 1 vanishes on account of Landau’s criterion. The lowest nonvanishing orders are diagr. 2 with unbroadened propagators and diagr. 1 with one propagator broadened. (According to form. (1) we insert for l? diagrams with cuts of two unbroadened lines.) These terms sum up to the scattering between an impurity and a phonon. The other ones contribute to higher order processes. Introducing one-particle sources (v , v_), variable chemical (p) and vector potentials tA) for each component we are able to represent the vertices in fig. 1 by the functional derivatives:
(2)
Where the G’s are the variables conjugate to
the v’s. The contributions giving rise to self energy insertions into the propagators are not taken into account. Using the conservation laws for the number of particles and the momentum these vertices are expressed by functional derivatives of the impurity mass operator, the densities and the currents with respect to A, I_(. In F they sum up to thermodynamic derivatives and we obtain as a result the phenomenological scattering probability [1,2]. So we have shown that no particle-like scattering remains and the impurities effectively interact with correlation functions of helium (which is a supposition in ref. [3]) via deformation potentials. In addition we calculate the linear response of the system to an external vector field acting on the impurities by renormalizing the impurity momentum (density)-momentum correlation function. Similar calculations were carried out in refs. [4-‘71 for other systems. Up to the first order in the impurity concentration the transport properties of the impurities are determined by a distribution function Q (k, w), which obeys the phenomenological 8 oltzmann equation, if we use the forementioned result for l? and approximate the imaginary part of the impurity-impurity scattering matrix consistently. The author is indebted to Prof. W. Brenig and Prof. W. Gijtze for discussions. References 1. I. M. Khalatnikovand V. N. Zharkov, Sov. Phys. JETP 5 (1957) 905. 2. G. Baym and C. Ebner, Phys. Rev. 164 (1967) 164. 3. W. F. Saam, Ann. of Phys. 53 (1969) 239. 489
Volume
31A. number 3
PHYSICS
LETTERS
4. A. B. Migdal. Sov. Phys. JETP 16 (1964) 1366. 6. G.M.Eliashberg. Sov. Phys. JETP 14 (1963) 886.
6. J. Gavoret 7. H. Winter,
4 May 1970 and P. Nozieres, Ann. of Phys. 28, 349. Z. Phys., to be published.
*****
EVIDENCE
FOR FOR
A SUPERFLUID-INDUCED ELECTRONS IN LIQUID
SURFACE HELIUM
BARRIER
W. SCHOEPE
Physik-Department
der Technischen
Hochschule
Miinchen.
Germany
and C. PROBST Zentralinstitut
fiir Tieftemperaturforschung der Bayerischen Akademie Garching bei Miinchen, Germany Received
24 March
der Wissenschaften
1970
The free surface of liquid helium acts as an energy barrier on electron currents across the surfac into the vapor, if the liquid is superfluid. In zero electric field, the barrier vanishes at Th. The measurements were performed in pure 4He and in a solution of 3He in 4He.
Bruschi et al. [l] have reported that for temperatures less than 1.7OK electrons encountered increasing difficulty to pass through the free surface of liquid 4He into the vapor phase, if the temperature is lowered. Below 1.5oK the electron current decreased rapidly according to exp (- @//kBT), with @B = 25 f 1oK. This behavior was mainly attributed to the image-force potential which acts on a charge near the interface of two media having different dielectric constants. The effect of the image-force potential outside the liquid was recently discussed in detail by Cole and Cohen [2]. Using the assumption of the image-force, however, it can be readily shown, that the work function $ must depend on the extracting field E, in contradiction to the results of ref. [l]. In addition, the saturation of the currents at 1.7oK remained unexplained since this temperature is much smaller than the barrier. Therefore, we made a special effort to determine $ with a higher accuracy. Bergides pye 4He we also investigated a solution of He in He. Our results demonstrate that the barrier depends not only on the applied field but also on the temperature, giving clear evidence that the barrier is present only in the superfluid state. The measuring cell was similar to that shown in ref. [l], consisting of three electrodes as indicated in fig. 1. We measured the currents when 490
the liquid level was above the collector and compared them with the corresponding currents when the liquid surface was below the collector. The currents with the collector being immersed were nearly temperature independent, in agreement with ref. [l]. On the other hand, when the liquid level was lowered below the collector the current showed a striking dependence on the temperature as plotted in fig. 1 for different extracting fields. For example for a field of 154 V/cm the current remains temperature independent from 4.2oK down to 1.7oK. Only below 1.7oK the current decreases rapidly with temperature as exp (-$//kgT), indicating an effective energy barrier @&?B = 30.4oK, which the emitted electrons have to overcome. It is evident that this positive energy barrier only exists well below l.‘7’K and apparently disappears when this temperature is exceeded. It is essential to point out that for lower extracting fields the energy barrier disappears at higher temperatures, which approach the X-point for zero extracting fields. The same behavior is observed in a solution of 32% 3He in 4He below the shifted X-point of 1.66oK. Thus a finite energy barrier for the emission of electrons from liquid helium seems to be closely related to the superfluid state. Besides this temperature dependence of the barrier it is important to emphasize that @ also