THE TWO ROTON BOUND STATE

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THE TWO ROTON BOUND STATE

T. Greytak, R. Woerner

To cite this version:

T. Greytak, R. Woerner. THE TWO ROTON BOUND STATE. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-269-C1-274. �10.1051/jphyscol:1972146�. �jpa-00214936�

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JOURNAL DE PHYSIQUE Colloque C1, supplkment au no 2-3, Tome 33, Fkvrier-Mars 1972, page C1-269

THE TWO ROTON BOUND STATE (*)

T. J. GREYTAK and R. L. WOERNER

Physics Department and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U. S. A.

Resume. - De rtcents travaux sur la diffusion Raman indiquent I'existence d'un Btat B deux rotons lies. L'knergie necessaire pour creer deux rotons a 1,2 OK est infkrieure au double de l'knergie propre d'un seul roton pris B cette m&me temperature, resultat qui ne peut pas 6tre explique si I'on considkre que les rotons sont sans interaction mutuelle. La comparaison des spectres Raman obtenus et des calculs theoriques montre que l'interaction est attractive et donne lieu a un Btat D lie (moment angulaire L = 2), l'knergie de liaison &ant 0,37 f 0,10 ^{O K .}Ces r&sultats, ainsi que les mesures de largeur de raie des rotons peuvent &re utilises pour obtenir une description detaillee de l'interaction roton-roton.

Abstract. - Recent Raman scattering experiments have found evidence for the existence of a bound state of two rotons. The energy necessary to create two rotons at 1.2 OKis less than twice the single roton energy at the same temperature, a result which cannot be explained on the basis of non- interacting rotons. Comparison of the spectra with theoretical calculations shows that the interaction is attractive and gives rise to a bound D state (angular momentum L = 2) with binding energy 0.37 & 0.10 ^OK.These results, together with roton linewidth measurements, can be used to obtain a detailed picture of the roton-roton interaction.

Two-excitation Raman scattering has become a valuable probe of the elementary excitations in superfluid helium [I], [2]. The most recent experiments involve the discovery of a bound state of two rotons [3]. We would like to review here the evidence which points toward such a bound state, describe the measurements of its properties, and discuss our current understanding of its origin.

1. Raman scattering in liquid helium. - In these experiments the scattering of a photon is accompanied by the creation in the helium of two elementary excitations with nearly equal and opposite momenta, hk and hk'. By conservation of momentum the total momentum of the pair, hK ⁼hk + hk', is equal to the change in the momentum of the photon. However, since the momentum of the photon is small compared to the momentum of the excitations being studied (less than lod3 times the momentum of a roton for example), K can be taken to be zero and k can be considered to be equal t o - kt. Because the liquid is isotropic, the energy of an excitation depends only on the magnitude of its wavevector and, as shown schematically in figure 1, the energies ~ ( k ) and &(I?) of the excitations created in the scattering process are equal. Since there is no restriction on the magnitude of k, one can scatter from a pair formed from any region of the dispersion curve. The photon, by conservation of energy, suffers an energy loss E equal to the energy necessary to

(*) Supported by the Advanced Research Projects Agency under Contract No. DAHC 15-67-C-0222.

create that pair. More specifically Halley [4] shows that the intensity of the Raman spectrum at an energy loss E is the product of two terms : the density of two-excitation states evaluated at a total wavevector K = 0 and total energy E, p, (K = 0, E), and a factor which indicates the strength of the coupling between these excitations and the electric field of the light beam. The density of states can change very rapidly with E ; the coupling to the light, on the other hand, has a relatively smooth variation with E and can be taken as constant over the various sharp features of p , (K = 0, E).

Energy (OK)

~ O T

Momentum (he')

FIG. 1. - The dispersion curve for liquid He4 taken from neutron inelastic-scattering data. ^{E k}and EP are the energies of a possible pair of excitations, of nearly equal and opposite momenta k and k', created in the liquid during the scattering

of a photon.

Because of considerable discussion of collision- induced Raman scattering at this conference we should point out that the coupling between the light and the

18'

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972146

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C1-270 T. J. GREYTAK AN ID R. L. WOERNER medium [5] which is assumed in the helium Raman

experiments is thought to be the same as that which gives rise to collision-induced scattering from normal monatomic liquids [6]. In such liquids the spectral distribution of the Raman scattered light is governed by the motion of individual atoms and the dynamics of their collisions with their neighbors. No theory or model of the dynamics of dense gases exists which can be used to predict the resulting spectrum ; however, experimentally it has been found that the spectrum can be roughly described by an exponential decrease of the scattered intensity with increasing energy shift : I(E) cc exp(- E/E0). In fact the spectrum for super- fluid helium at large energy shifts can also be put in this form with Eo = 33 OK (23 cm-I). This is similar to the behavior found in liquid argon by McTague, Fleury, and DuPre 161. In the case of normal fluids, however, this smooth dependence of the spectrum on energy persists as the energy shift E goes to zero ; whereas, the spectrum for superfluid helium exhibits definite structural features for energies E less than 36 ^O^K(25 cm-l). This structure arises from the elementary excitations which are the object of these studies. It can be said, then, that in this region of energies the dynamics of the liquid helium are determined by well defined collective motions of the atoms rather than by individual atomic motions. It appears that only the quantum liquids, He4 and He3, exist as liquids at temperatures sufficiently low that a major portion of their macroscopic properties and dynamical structure can be understood on the basis of a collection of elementary excitations.

The simplest model used to interpret the Raman scattering in helium assumes that the excitations making up the pair do not interact with each other.

However, the elementary excitations are allowed to possess an intrinsic temperature dependent line width 6~(k) (68 is taken to be the full width at half height of the spread in the energy associated with an elementary excitation). In this case

where p l ( ~ ) is the single excitation density of states. It can be seen from figure 1 that three peaks should occur in the Raman spectrum, corresponding to the three regions in which the dispersion curve has zero slope and, therefore, a high density of states. The excitations near the local minimum of the dispersion curve, e,, are the rotons and, together with the phonons near

E = 0, determine the thermodynamic properties of the helium below the A temperature. A peak should occur in the Raman spectrum near E = 2 eo due to the creation of pairs of rotons. If el and E, indicate the energies a t the local maximum and the plateau of the dispersion curve, then the Raman spectrum was also expected to show peaks at E = 2 el and E = 2 E,.

Experimentally, only two of these peaks are observed.

The peak due to scattering from the rotons is well

defined and the measurements of the temperature dependence of its width have allowed a determination of the roton line width [2] in the temperature region where it is dominated by roton-roton collisions :

A broad peak is also observed near 2 8, = 36OK and is attributed to the creation of excitations at the plateau, or end point, of the dispersion curve.

The absence of a peak in the spectra near

was very puzzling. One possible explanation was that the coupling of the light to these excitations was much weaker than the coupling to the rotons. Stephen [5]

has made a detailed theory of the light scattering process in liquid helium, and his results for the intensity and polarization ratios of the Raman scattering from the rotons are in good agreement with the experimental measurements. However, his theory gives a scattering from the excitations at the maximum of the dis- persion curve which is only a factor of 3 smaller than the scattering from the rotons [7]. Another possibility was that the linewidth of the excitations near the maximum is so large that they do not give a sharp peak in our spectra. In order for this to be the case, SE for these excitations would have to be large compared to the instrumental width of the spectrometer, 2.1 ^OK.

However, neutron scattering results [8] have placed an upper limit on 68 in this region of 1 OK for a temperature of 1.1 OK. These two effects were insufficient to explain the observed absence of a peak in the spectra at 2

2. Evidence for a two-roton bound state. - Ruvalds and Zawadowski 191 and Iwamoto [lo] have pointed out that in the presence of even a weak interaction between the excitations, $p1(E/2) is not a good approximation to the true density of two-exci- tation states, p, (K = 0, E), and that in certain cases the two functions may be qualitatively different. As a simple example of this phenomena consider a hydrogen atom. Its energy levels are well known ; but, how can one estimate the energy levels of two hydrogen atoms in an otherwise empty box ? The non-interacting approximation equivalent to eq. (I) says that the new energy levels will simply be all possible combina- tions of the single atom levels taken two at a time, with allowances made for symmetry. But it is known that the two hydrogen atorns will form a molecule which has energy levels, particularly the low lying ones, which cannot be related to such a scheme.

In fact Ruvalds and Zawadowski showed that the absence of a peak in our c;pectra near 2 E, can be explained by a depletion of the density of pair states in this region caused by an interaction between the excitations. They also showed that the same interaction:

if assumed to exist between the rotons, would enhance

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THE TWO ROTON i BOUND STATE C1-271

the density of pair states in the vicinity of 2 E,, a condition which they referred to as a two-roton reso- nance. It was later pointed out by Stephen [ I l l that the equations of Ruvalds and Zawadowski actually describe a bound state of two rotons [12].

In Stephen's theory [5] of Raman scattering from liquid helium the light can couple only to the D wave portion of the wave function for the pair of excitations.

If evidence for a bound state is found by Raman scattering, that bound state must then be a D state, that is it must have an angular momentum L = 2. Stephen therefore suggested that the coupling between the excitations be studied on the basis of the simplest model interaction which can give rise to a D wave bound state. For such a model the interaction in k space between two excitations has the separable form U(k, k') = 5 gP2(cos Qkk,) (3) where R(k - k') is the momentum transfer, P, is a Legendre polynomial, Q,,, is the angle between k and k', and g is a coupling constant. Although this differs from the interaction U(k, k') = g used initially by Ruvalds and Zawadowski (which can have only a S wave bound state), the analytical form for

based on either interaction is identical [lo], [12].

Consider first the behavior of p2 near E = 2 ^E, and for zerq temperature where 6~ -+ 0. If the dispersion curve near the maximum is represented by

the unperturbed density is given by

where Q(x) is a unit step function which is one for x > 0 and zero for x < 0, and A , = (k1/2 a)' p? fi-I.

This density diverges at E = 2 el as shown in figure 2.

FIG. 2. - Schematic representation of the modification of the density of pair states due to an attractive interaction. The dashed

curves correspond to the non-interacting case.

To obtain a depletion of the density of states it is necessary that the coupling constant in this region be negative, g, < 0. This corresponds to a repulsive interaction between these excitations in the medium.

The resulting zero temperature density is given by

where El = A: g:. This function approaches the unperturbed density for 2 E, - E > E,, reaches a maximum at 2 - E = El, and goes to zero at 2 ^E, = E. This behavior is shown schematically in figure 2. The area which has been lost under p, due to the presence of the interaction is aA1 E?, an amount proportional to gl.

In the theory of Ruvalds and Zawadowski the spectrum is temperature dependent only through the temperature dependence of the line widths 66 of the individual excitations. Assuming that 64k) is a constant in the interval of energies being considered, it can be shown that the expression for pz at finite temperatures is simply the convolution of eq. (5) with a Lorentzian lineshape of unit area and width 2 68,. Ruvalds and Zawadowski have used the measured Raman spectra in this region to estimate gl to be about

The roton region of the dispersion curve can be represented by ~ ( k ) = E, + R2(k - k0)'/2 p,. In this region one then has at zero temperature

where A, = (k0/2 a)' ,u% h- I. This expression diverges at E = 2 E, and is zero for E < 2 E,. It is not unrea- sonable to assume that the coupling constant of eq. (3) has the same sign for the rotons as it has for excitations near the maximum, go < 0. However in the roton region of the curve this choice of sign leads to an attractive interaction between the excitations ! This behavior can be attributed to the fact that the effective mass of the excitations changes sign in going from one region to the other. A force which would cause repulsion between excitations at the maximum (negative effective mass) would cause an attraction between the rotons (positive effective mass). For go < 0 the expressions of reference [9] lead to the following form for p2 at zero temperature :

This density is compared with the non-interacting result in figure 2. The 6 function represents a single bound D state with binding energy Eo = A; gg. ^The

remaining states are unbound and it can be seen from figure 2 that for E - 2 E, > E, their density approaches that found for non-interacting rotons. The area

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C1-272 T. J, GREYTAK AND R. L. WOERNER

gained in the bound state is twice that lost in the unbound states giving a net increase in the area of p, with respect to its unperturbed value of zAO E?.

The area added to p, (K = 0, E) by the interaction is of importance when comparing experimental Raman scattering cross-sections to values computed from various theories of the coupling between the light and the liquid helium which are based on non-interacting rotons. Assume a spectrometer with a rectangular instrumental profile of energy width A& 9 E, is centered at E = 2 .so. A simple integration shows that the ratio of the increase in the measured power due to the presence of the interaction to the power that would be measured if go = 0 is given by 'Jl - ( ~ , / d e ) % . Intensity measurements

2

described in reference [l] used BE

-

3.6 OK and in reference [3] it was found that Eo ^N0.37 OK. Under these conditions the enhancement due to the interaction is about 50 %.

Several authors [lo], [12] have emphasized the fact that the Raman spectrum in the presence of an attractive interaction between the rotons will exhibit two maxima, one above and one below E = 2 E,, instead of a single peak near E = 2 8,. This is of course true mathematically, but in figure 3 it is shown that this is of little consequence in interpreting an actual spectrum at finite temperatures. p , ( K = 0, E ) for the rotons at finite temperatures can be shown to be the convolution of eq. (7) with a Lorentzian of unit area and width 2 68, where 68, is the linewidth of a single roton. This result is shown in figure 3 for a binding

FIG. 3. - Calculated density of two-roton states due to an attractive interaction as a function of temperature. The dashed curve on the right corresponds to the non-interacting case. Note the well-defined separation of the bound state even at 1.2 ^OK.

energy E, = 0.37 OK and roton linewidths given by eq. (2) at several temperatures around 1 OK. It can be seen that the ((peak )) near E = 2 8, + E, is too broad and low to be a significant feature of the spectrum in the presence of the strong narrow peak representing the bound state. In fact the experimentally important feature of the attractive interaction is that a measura-

ble portion of the scattering occurs at energies less than 2 ^{6 , .}An experimental trace of the roton region of the spectrum showing just such an effect is presented in figure 4.

FIG. 4. -The Raman spectrum of liquid He4 at 1.2 OK. The strong peak at zero energy shift is caused by Brillouin scattering and indicates the instrumental profile. The dotted curve is a theoretical fit to the data based on a two-roton bound state. The dashed curve would correspond to non-interacting rotons. The

dot-dashed line is the background level.

The maximum in the Raman spectrum shown in figure 4 occurs at an energy shift of 17.022 + ^{0.027 OK.}

This shift is 0.32 OK less than twice the energy of a single roton, 2 8, = 17.34 + .08 OK, determined by neutron scattering [I31 and confirmed by measurements of the temperature dependence of positive ion mobilities in superfluid helium 1141. The fact that the maximum occurs at an energy shift of less than 2 8,

cannot be due to the lifetime broadening of non- interacting rotons, or to the convolution of {he true spectrum with the instrumental profile ; both of these effects would cause the maximluk of the Raman spectrum to occur above 2 .&, The dotted line in figure 4 is a theoretical fit to the data based on a two-roton bound state. It was obtained1 by using eq. (7) with E, = 0.37 OK, a single roton width 6.5 = 0.1 5 OK, and the measured instrumental profile. The heights of the theoretical and experimental curves are matched at the top of the peak. Setting Eo = 0 in this procedure gives the dashed curve corresponding to non-interacting but lifetime broadened rotons. These values for Eo and BE give the best fit to the data for the choice 2 E,, = 17.34 OK. However, as the value of 2 8, is varied throughout the neutron range of uncertainty, other values of E, and 68 optimize the fit to the data. In fact for the smaller values of 2 E , the fit to the spectra is improved in the region above 17.5 OK, and the opti- mum values of Eo and BE decrease in magnitude. This suggests the possibility that higher resolution experiments may be able to determine both Eo and ^EOself consistently. At present the neutron uncertainly in 2 8, is the source of most of thr: experimental uncertainty of + 0.10 OK in the values of both E, and BE. The present value of Eo corresponcls to

very close to the value of g , estimated by Ruvalds and Zawadowski. The value of BE is about twice that

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THE TWO ROTON BOUND STATE C1-273 expected from eq. (2) due to roton-roton collisions or about 4 % of k,. From this figure one can estimate alone, but shows considerable experimental uncer- the mean radius of the bound state R

-

l/Ak. This

tainty. gives R

-

¹³^Awhich is to be compared to the mean

atomic spacing in liquid helium of 3.5 A.

3. The bound state and roton-roton interactions. - For other cases of bound elementary excitations, for Eq. (3) is certainly not suggested as a complete example optical phonons in diamond [I71 and the representation of the actual roton-roton interaction. Cooper problem of two interacting electrons [18]

Yau and Stephen [I51 show that a potential of this it is possible to obtain a rather detailed description of form, in which scattering takes place in only one the microscopic disturbance of the medium represented angular momentum channel, is inadequate to explain by the bound pair. This is only possible, however, the roton-roton collision times measured from roton because the wave function of the non-interacting linewidths or from the roton contributions to the excitations is well understood. Unfortunately, this is normal fluid viscosity. However, eq. (3) does serve to not the case for rotons in superfluid helium. The p h ~ s i - give a simple expression for the D-wave part of cal picture of the roton which is most widely accepted

p.2 ( K = 0, E ) today [19] is the one developed by Feynman and Cohen [20]. It consists of two distinct regions. The and allows one to relate the position of the maximum core of a roton is thought to be a small localized in the spectra to the energy of the bound state. It is disturbance, a few interatomic spacings in extent, possible that the temperature dependent version of drifting through the medium with a speed given by eq. (7) is not a c o m ~ l e t e l ~ accurate descri~tion of even

the group velocity ofthe ~h~ exact structure the D-wave part of ~2 for the rotons. Yet is be

in the core is not well understood since it depends on 'lear from 'gure that in time a high the multipa*icle correlation functions for the helium.

resolution Raman study will be able to measure the Surrounding this disturbance is a backflow of atoms form Of P, giving Of which, at large distances, has a dipolar pattern. The the binding energy and width of the bound state and magnitude of the dipolar back flow, determined the effect of the interaction on the unbound states. The variationally, is about the same as would be caused best spectra at present have been taken with an ins-

by a He3 impurity atom moving through the liquid.

trumental width of 0.68 OK, almost twice the binding

For many purposes the backflow region about a roton

enera. These 'pectra have shown the presence Of a can be treated hydrodynamically and, to this extent, bound state and measured its binding energy to about rotons are sometimes thought of as microscopic vortex

25 %, but are not able to distinguish the fine fea-

rings.

Of the 'peCtrum and choose between Yau and Stephen have recently studied the interac- various models for the D-wave interaction. tion between rotons in superfluid helium 1151. By

The study Of the linewidth Of the two-roton bound comparing their theoretical results with experimental state promises to be Of particular interest* At Raman measurements they are able to determine some ciently low temperature the lifetime of the bound state

of the parameters involved in such an interaction.

will not be determined solely by the free lifetimes of its The interaction which they consider consists of two constituents, as is the case in the present theory. For parts. At separations greater than some distance example the decay of a single roton into two excitations the rotons interact by way of the velocity asso- is kinematically prohibited by the shape of the disper-

ciated with their backflow For separations sion curve, but the bound roton pair may into a less than there is a Strong repulsive interaction pair Of phonons of and 'pposite characterized by a hard core radius r,. They are able Thus one would expect that in the limit of very low to shown that the long range interaction will give rise temperatures the width of the bound state will be

to a bound state. The binding energy is insensitive greater than twice the width of a single roton. The

to the repulsive pad of the potential but is very sensi- effect of the ~ h o n o n s on the roton pairs is presently tive to ~h~ binding energy measured from the being studied by Baeriswyl and Jackle 1161. Raman experiments corresponds to a, = 5 A. On the The spectrum given in eq' (7) shows that the bound other hand, they find that the scattering of two rotons state is created at the expense of unbound pair states (the process which determines the linewidth of a roton whose energies to about E~ above '0. This

at temperatures above 1 OK) is dominated by the indicates that the wavefunction for the bound state

repulsive part of the potential. The measured values will be a linear combination of those free roton states of the roton linewidth are shown to be consistent whose energy is within A' = EoI' 0.2 OK on the

with a hard sphere radius of ⁼2.5 A ^fiis interest- minimum in the dispersion curve at &,. This band of

ing to note that the point contact typ of potential, states extends about % ^Of the distance to the v(r) ⁾⁼^yo which has been generally used [ZI] to maximum Of the curve at .'1 the treat roton-roton interactions, is shown by Yau and momentum these states are

*

^Ak k~ Stephen to be completely inadequate to describe the where Ak = (p, E0/fi2)'/". This corresponds to

observed values of roton linewidths.

Ak

-

^0.08^A-l We have tried, in this review, to point out that

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C1-274 T. J. GREYTAK AND R. L. WOERNER

Raman experiments can play a central role in esta- line. Also, Raman scattering can be extended to iso- blishing the detailed nature of the roton-roton inter- topic mixtures of He3 and He4 and should be able to action. Additional information will be gained by provide some information on .the interaction between extending measurements of this sort to lower tempe- the ~e~ quasi-particles and the elementary excitations ratures and to pressures as high as the solidification of the superfluid.

References

GREYTAK (T. J.) and YAN (J.), Phys. Rev. Letters, 1969, 22,987.

YAN (J.) and GREYTAK (T. J.), Proceedings of the Twelfth International Conference on Low Tempe- rature Physics, Kyoto, 1970.

GREYTAK (T. J.), WOERNER (R.), YAN (J.) and BEN-

JAMIN (R.), Phys. Rev. Letters, 1970, 25, 1547.

HALLEY (J. W.), Phys. Rev., 1969,181,338.

STEPHEN ( M . J.), Phys. Rev., 1969,187,279.

MCTAGUE (J. P.), FLEURY (P. A.), and DUPRE @. B.), Phys. Rev., 1969,188, 303.

In this regard it should be noted that a factor of Zz(kl)/Zz(ko) has been accidentally omitted from eq. (5.11) of reference [5].

YARNELL (J. L.), ARNOLD (G. P.), BENDT (P. J.), and KERR (E. C.), Phys. Rev., 1959, 113, 1379.

RUVALDS (J.) and ZAWADOWSKI (A.), Phys. Rev.

Letters, 1970, 25, 333.

IWAMOTO (F.), Prog. of Theor. Phys. (Japan), 1970,44, 1135.

[ll] STEPHEN (M. J.), Private Communication.

[12] ZAWADOWSKI (A.), RUVALIIS (J.), and SOLANA (J.), to be published.

[13] COWLEY (R. A.) and WOODS (A. D. B.), Canadian J. of Phys., 1971, 49, 177.

[14] SCHWARZ (K. W.) and S T ~ ~ R K (R. W.), Phys. Rev.

Letters, 1969, 22, 1278.

[15] YAU(J.) and STEPHEN (M. J.), Phys. Rev. Letters, 1971, 27. 482.

[16] BAEGWYL (D.) and J~;CKLE (J.), Helv. Phys. Acta, in press.

[17] COHEN (M. H.) and RUVALDS (J.), Phys. Rev. Letters, 1969.23, 1378.

[18] COOPER (L. N.), Phys. Rev., 1956,104,1189.

[19] PINES (D.), 1965, in 1965 Tokyo Summer Lectures in Theoretical Physics, Ed. by R. Kubo (Syokabo, Tokvo and Beniamin. Inc.. New York).

[20] F E Y N M A ~ (R. P.) ahd COHIN (M.), p h y s . ' ~ e v . , 1956, 102.1189. ₇_{- - - -}

[21] LAND*~ (L. D.) and KHALATNIKOV (I. M.), Zh. Ekspe- rim. i Teor. Fiz., 1949, 19, 637.