A strong-coupling theory of superfluid ${^4}$He

a r X i v :c o n d -m a t /9910395v 1 25 O c t 1999

A strong-coupling theory of super?uid 4He.

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.html

?

and E.Akkermans ??

?

Department of Physics,Technion,32000Haifa,Israel

?

Physique des Solides et LPTMS,Universite‘Paris-Sud 91405,Orsay,Cedex,France.

We propose a new theoretical approach to the excitation spectrum of super?uid 4He .It is based on the assumption that,in addition to the usual Feynman density ?uctuations,there exist localized modes which describe the short range behaviour in the liquid associated with microscopic cores of quantized vortices.We describe in a phenomenological way the hybridization of those two kinds of excitations and we compare the resulting energy spectrum with experimental data,e.g.the structure factor and the cross section for single quasi-particle excitations.We also predict the existence of another type of excitation interpreted as a vortex loop.The energy of this mode agrees both with critical velocity experiments and high energy neutron scattering.In addition we derive a relation between the condensate fraction and the roton energy and we calculate the reduction of the ground-state energy due to the super?uid order.

PACS:67.40-w,67.40.Vs,63.20.Ls,62.60+v

I.INTRODUCTION

In this paper we present an alternative description of the energy spectrum of super?uid 4He.Among the stan-dard approaches to this problem,we mainly ?nd either the e?ective Hamiltonian of Bogoliubov 1,or the varia-tional approach proposed by Feynman 2.

The Bogoliubov description relies on the assumption of a weakly interacting Bose gas.Its main success was to obtain a linear dispersion for the long wavelength excita-tions starting from the quadratic spectrum of free bosons.This linear phonon branch cannot be obtained from sim-ple perturbation theory but corresponds to a RPA-like description of this problem.This linear behavior is in-deed one of the main features of the experimental spec-trum (?g.1).Nonetheless this success,neither the initial slope of the dispersion ε(k ),nor the rest of the spectrum beyond the linear part,is quantitatively obtained from this approach.

A alternative description was proposed by Feynman 2,based on a variational form of the wavefunction for the low-lying excited states which leads to an excitation spec-trum of the form

ε(k )=

ˉh

2k 2

glasses11where such a picture,based on two-level tun-neling states,has been successfully used to describe the low temperature excitations measured by speci?c heat or heat transport experiments10.

The TLS we consider are localized on atomic scales and may correspond to quantum coherent rings of a few atoms only,which are supposed to be responsible for the super?uid behaviour of the liquid.These short rings cor-respond to vortex cores of sizes R?1?A,and may be considered as local http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlrge ring structures arise from the long-range interaction between the TLS.

The e?ective Hamiltonian H loc we propose for these interacting local modes can be diagonalized by a Bogoli-ubov transformation.The resulting spectrum bears sim-ilarity with the weakly interacting Bose gas expression, although it does not assume a weak interaction.Indeed, in order to recover a gapless phonon branch at low mo-mentum(k?0),we need to consider strong interac-tions,namely interactions of the order of the bare energy E0.Unlike ordered structures of dipoles on a lattice for which it is possible to calculate the interaction term,for liquid4He(and for glasses),the interaction term is ob-tained from the experimental energy spectrum.Despite this shortcoming,this model gives closed analytical ex-pressions for quantities such as the scattering intensity of neutrons by single quasiparticle excitations,the relative variation of the ground-state energy and the condensate fraction in terms of a single quantity,namely the strength of the interaction,so that it becomes possible to relate them.

In order to compare this approach to the Feynman vari-ational scheme,we use an equivalent formulation of our problem based on a description of the interaction between the localized modes as due to a virtual exchange of Feyn-man phonons.This is very much in the way Hop?eld30 and Anderson12considered the exciton problem in dielec-tric media.This description is equivalent to the previous one based on the e?ective Hamiltonian H loc provided we use the dipolar approximation which corresponds to large momentum.For low momentum,both the density?uctu-ations and the large scale exchange cycles correspond to the same degrees of freedom and cannot be disentangled anymore.In the large momentum limit the energy spec-trum results from the hybridization of these two kinds of excitations,which represent independent degrees of freedom of the super?uid.This way we gain a better understanding of the roton part of the spectrum.In ad-dition we?nd a new localized excitation branch at energy E2=2E0,which we interpret as microscopic vortex-loop excitations of the super?uid.These type of excitations are usually thought as being completely independent of the phonon-roton excitations,while here we propose a model that describes them both together.

This paper is organized as follows.We begin by intro-ducing the e?ective Hamiltonian of the strong coupling description in section II.In section III,we obtain expres-sions for the scattering intensity of the single-excitation branch and compare it to the experimental data.In sec-tion IV,we evaluate the reduction in the ground-state energy of the super?uid and the condensate fraction.In section V,we consider the dipolar approximation and the resulting hybridization scheme.We compare the re-sulting spectrum and the static structure factor to the experimental data.Finally,in section VI,we discuss the vortex-loop branch and we conclude in section VII.

II.THE EFFECTIVE HAMILTONIAN

DESCRIPTION

We consider super?uid4He as a set of local states de-scribed as two-level systems of bare energy E0.Their Hamiltonian is then

H0loc= k E0b k?b k(2)

where translational invariance allows us to write opera-tors in k-space.The operators b k obey bosonic commuta-tion relations.This holds in the limit of a low density of localized modes12,i.e.for the case of no multiple occupa-tion of a site.This condition corresponds to vortex-cores or small exchange rings which cannot be multiply excited on the same site.The value of E0will be determined later on.

The interaction between these localized-modes can generally be written as12

H loc= k E0+X(k) b k?b k(3)

+ k X(k) b k?b??k+b k b?k

The diagonal part in this Hamiltonian corresponds to the hopping of a local mode between sites,and the o?-diagonal part describes the creation(or the anihilation) of pairs of local modes on distinct sites.The matrix el-ement X(k)depends on the microscopic details of the interaction.For a lattice structure like excitons in a crystal13or solid14bcc4He,it is possible to express X(k) as a summation of dipolar terms.This is not possible in the super?uid phase,for which we have no lattice struc-ture.

The Hamiltonian H loc is diagonalized by the Bogoli-ubov transformationβk=u(k)b k+v(k)b??k.The re-sulting spectrum is

E(k)=

2 E0+X(k)

2 E0+X(k)

The

corresponding

ground-state wavefunction

for

the lo-cal modes is 15

|Ψ0 = k exp

v k

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

2

=??(7)

Notice that for dipoles on a lattice 14,this condition is a self-consistent de?nition of E 0since the energy re-quired to excite locally a dipole from the ground-state corresponds to the one needed to ?ip it with respect to the coherent background.This excitation changes the sign of the interaction energy and costs 2|X (k =0)|=2?=E 0,which corresponds to the previous de?nition of the energy of the local mode.

For arbitrary k ,X (k )is obtained by comparing Eq.(4)with the measured energy spectrum (?g.2).This shape of X (k )is similar to the expression calculated for dipoles on a simple cubic lattice along various directions 13,14.This could be helpful towards the development of a mi-croscopic model for the super?uid phase.

It is useful at this stage to compare our approach with the weakly interacting Bose gas limit.There,the Hamil-tonian is given within the Bogoliubov approximation by

H W IBG =

k

(εk +N 0V k )a ?k a k

+

k N 0V k

a ?k a ?

?k

+a k a ?k

(8)

where N 0=| a 0 |2,εk =ˉh 2k 2/2m ,and V k is the e?ec-tive potential between the bosons at wave vector k .The

operators a ?k and a k correspond to the creation and ani-hilation of an atom and not of a localized-mode as in (3).

The Bogoliubov excitation spectrum which corresponds

to (8)is

E = N 0V 0/m .

The physics described by the two Hamiltonians H loc and H W IBG is very di?erent.The weakly interacting Bose gas has a non-interacting limit given by the ideal Bose gas which,at T=0,is known to be fully condensed in the state k =0and which has already a broken gauge symmetry.The interaction depletes the condensate and changes the spectrum to linear at low momentum.On the other hand,the Hamiltonian H loc corresponds to a set of independent localized modes with no broken symmetry in the non-interacting limit (X (k )=0).To restore in the linear momentum regime both the condensation and broken gauge symmetry,we need a non-zero interaction X (k ).Since this interaction is large (X (0)=?E 0/2),it is not a small perturbation to the non-interacting case.In that sense,it is a strong-coupling description of the super?uid.

We would like to point out that the entire excitation spectrum given by (4)is unique to the super?uid phase,as only there we can de?ne the function X (k )and local mode energy E 0.In particular,the phonon mode is dif-ferent from the zero-sound mode that appears in the nor-mal ?uid 20.We shall now use this approach to obtain an-alytical results that we shall compare subsequently with experimental measurements.

III.SCATTERING INTENSITY

The neutron scattering intensity is a direct probe of the

density ?uctuations in the liquid and may be described using the dynamic structure factor S (k,ω).It is usually accepted since the work of Miller,Pines and Nozieres 21that we can split the total contribution to S (k,ω)into two parts,

S (k,ω)=NZ (k )δ(ˉh ω?ε(k ))+S (1)(k,ω)

(10)

where the ?rst term accounts for single quasi-particle

excitations of weight Z (k ),while the second describes multiparticle excitations.This separation is justi?ed at low temperature and low momentum,typically for

k ≤0.5?A ?1

.In this regime,integrating (10)over the energy and noticing that S (1)(k,ω)vanishes in the low momentum limit we obtain Z (k )=S (k ),which results also from the Feynman theory.The comparison with the experimental data 3shows that it works indeed in the low momentum regime mentionned above,but fails to describe the non linear part of the spectrum,except perhaps for the position of the maximum.

Since the excitations of the local modes from the ground state (6)involve breaking pairs of coherent local

modes,the probability of a neutron to scatter inelasti-cally on such an excitation is proportional to the occupa-tion density of these pairs at each momentum given by b k ?b ??k =u k v k ,so that

Z (k )=4πk 2I 0u k v k

(11)

where I 0is an arbitrary normalization constant and where the factor 4πk 2comes from the three dimensional phase http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmling the expression (5)for

u

k

and v

k

,

we obtain u k v k =1E (k )which together with (4)gives

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

Z (k )=πk 2I 0

E 0

E 0

2

?1

(12)

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

Using the experimental E (k )in expression (12)we ob-tain a di?erential cross-section which agrees well with the experimental results 3)obtained at saturated vapor pressure (S.V.P.)as shown in ?g.3.In the low momen-tum limit,we recover the proportionality between S (k )and Z (k )and therefore Z (k )is linear with k .We empha-size that as a result of both (12)and the saturation of E (k )to 2?at large k ,the scattering intensity Z (k )van-ishes identically at this point.This indeed corresponds to the experimental measurement of Z (k ),and could not be obtained neither from weakly interacting Bose gas de-scription nor by variational approaches.

IV.DESCRIPTION OF THE CONDENSATE

A strong-coupling theory of superfluid ${^4}$He

A.Condensate fraction

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

A strong-coupling theory of superfluid ${^4}$He

In the microscopic description of a Bose liquid,the condensate is usually characterized by the condensate fraction,n 0/n ,which measures the relative occupation by the atoms of the lowest energy state.At zero tem-perature it is equal to one for an ideal Bose gas,while a ?nite interaction depletes the lowest energy state with a corresponding decrease of n 0/n .It is possible to cal-culate this fraction in terms of the interaction potential only within the Bogoliubov description of a weakly inter-acting Bose gas.More generally,it is known 22that this ratio n 0/n is related to the zero momentum limit of the non-condensate particle distribution in the ground state,n k ,namely

n k →0=

n 0

2ˉh k

(13)

where c is the sound velocity and m is the mass of a 4He atom.For the real liquid,we do not known how to ob-tain n 0/n in terms of other measurable quantities.This ratio has been obtained indirectly 23from a measurement of the distribution of the non-condensed atoms.Numer-ically,using path integral Monte-Carlo 8similar values have been obtained.An analytical expression has been

proposed which is based on the assumption that the de-pletion of the condensate is related to the thermally ex-cited rotons 24.

In the present model,it is also possible to obtain the divergent part of the ground state occupation number n k =v (k )2of the local http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmling (5),it is given by

n k →0=

?

n

=

?

2

E (k )?(E 0+X (k ))

(16)

The second term in (16)is the shift in the ground state energy ?E G .Using Eq.(4)we obtain

?E G =?

1

32π2?

k max

k 2(E (k )?2?)2

dk

(18)

where k max is the largest value of the momentum and corresponds to the termination point of the quasi-particle spectrum.V sp is the speci?c volume per http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmling the experimental3,9,32(?g.1)energy spectrum for E(k), we obtain

?E G??5.0±1.0K(SV P)

?E G??3.0±1.0K(P=24atm)(19) for saturated vapor pressure and P=24atm respectively. In?g.4we plot the integrand k2(E(k)?2?)2in(18). It shows that the main contribution to the reduction in the ground state energy comes from the roton part of the spectrum due to the larger phase space volume,compared with the phonon region(k→0).

Both from experimental data23and path integral Monte-Carlo calculations8,it is possible to deduce the values for the change in the kinetic zero point energy of the atoms between the normal phase at Tλand the super?uid phase in the limit T→0

?E kin?2.5±3.0K(SV P)

?E kin?1.3±3.0K(P=24atm)(20) for saturated vapor pressure and P=24atm respectively. First,we notice that the large experimental uncertainty makes any quantitative comparison di?cult.Moreover, the two quantities?E G and?E kin do not measure exactly the same property.While?E G measures the change in the ground state energy of the local modes between the non-interacting and interacting limits,both at T=0,?E kin measures the change in kinetic energy per atom between two di?erent temperatures.Since the variation of the volume of the super?uid between Tλand T=0is small27,it is usually accepted to atribute the change in the kinetic energy mainly to the condensation phenomenon.In our model this condensation energy is calculated per local mode,nevertheless we?nd that both quantities decrease for an increasing pressure.

C.E?ective mass

Although the relations(15)and(18)for the conden-sate fraction and the condensation energy give both the right order of magnitude and the expected bahaviour as a function of pressure,they are consistently larger than the experimental values by a factor of about2.This fac-tor may stem from the fact that our expressions are given as a function of the local modes while the experimental data is obtained per4He atom,and there is no direct one-to-one correspondence between these two.There-fore,the mass term which appears in(15)instead of be-ing the bare mass m of a4He atom is the e?ective mass m eff of the bare local mode.From the above compari-son between n0/n and?E G with experiments we obtain that m eff?2m.Although such a relation can be ob-tained only from a microscopic calculation beyond our phenomenological model,it is interesting to notice that

a similar result was obtained numerically using path inte-gral Monte-Carlo calculations8.There,the e?ective mass

m?of a tagged4He atom which does not participate in the Bose permutations was found to be m??2m.Since such a tagged atom is a local node or defect in the su-per?uid,it corresponds to the bare local mode we con-

sider in the Hamiltonian H loc.As we saw in(5)and(6), the bare local mode of energy E0appears for the non-interacting case(X(k)=0and v(k)=0).Then,it does not contribute to the coherent(super?uid)ground state, i.e.behaves as a local node of the order parameter.This correspondence may justify taking m eff?2m bringing both(15)and(18)into agreement with the experimental data represented in?g.5,where the experimental values23 for n0/n,?and c are obtained independently27.

V.HYBRIDIZATION OF LOCALIZED AND

DENSITY MODES

In the previous sections we described the super?uid as resulting from the condensation of a set of localized and interacting modes in the strong coupling limit.This de-scription,as we emphasized,is very di?erent from the Bogoliubov weak coupling limit.Since it is aimed to de-scribe the full energy spectrum,we would like to relate it to the Feynman variational scheme.In this section we shall show that in addition to the phonon-roton branch of excitations around the roton momentum,there is another branch of excitations.We shall discuss the properties of this excitation in the light of experiments on thermal nu-cleation of vortices28and on large momentum neutron scattering29.To that purpose,we propose in this sec-tion a description equivalent to the previous interacting local mode problem,where the interaction between these modes results from the exchange of virtual phonons.This is very much in the same spirit of Hop?eld30and An-derson considering the exciton problem in dielectric me-dia.There localized excitons,taken as dipoles,interact through the exchange of photons.For the super?uid,the local modes play the role of the excitons,and interact through the exchange of Feynman phonons.This picture is only valid in the range of large enough momentum be-yond the linear part around k→0,which corresponds to density excitations(i.e.Feynman phonons).It is only in this regime that the two types of excitations repre-sent independent degrees of freedom of the system.The Hamiltonian describing Feynman density excitations is

H0= kε(k)a?k a k(21)

where a k are Bose operators andε(k)is the Feynman spectrum2given by the relation(1),expressed in terms of the static structure factor S(k).The Hamiltonian de-scribing the coupling between these phonons and the lo-calized modes is30

H c= k(λ(k,E0)b k+μ(k,E0)a k)(a k?+a?k)(22)

+h.c.

and the total Hamiltonian is

H=H0+H0loc+H c(23) Since H and H loc provide two equivalent descriptions of the same problem,the functionsλandμare related to X(k).If we now assume that the localized modes are local dipolar excitations,then there is a speci?c relation30,31between these functions,given byλ(k)= iE0 ?3X(k)2andμ(k)=?E03X(k)

A(k)

a?k a??k exp D(k)

B(k)

a?k a??k exp C(k)

2ε(k)=

ˉh2k2

A(k) and

D(k)

A(k)

=

ε(k)?E(k)

B(k)

=

E0?E(k)

physical picture of this mode is therefore of an excita-tion made out of two vortex-core excitations.Such an excitation in three dimensions may be viewed as a mi-croscopic vortex-loop,or a localized defect.The radius R of a vortex-loop can be estimated using a Feynman-type formula for the energy of the associated circulating current37

E vortex=2π2ρˉh2R

a (30)

where at T=0,we take the density of the super?uid ρs to be the total densityρand a is the core size equal to the atomic radius namely a?1.4?A38.The radius R obtained from(30)and which corresponds to E vortex= 4?=34.4K is R?5.1?A.

Experimental support for the interpretation of the hy-bridized localized state at E=4?as an intrinsic exci-tation of the?uid is provided by critical velocity exper-iments28.In phase-slippage studies of the critical veloc-ity through an ori?ce,the critical velocity is driven by the thermal nucleation of vortex-loops.The correspond-ing activation energy E v is determined by the nucleation rateΓgiven by the Arrhenius lawΓ=Γ0exp ?E v

2mR ln R

2m

(31)

where n is the number of emitted vortex-loops and k c,n

is given by the equality of the free-recoil energy and the

n th vortex-loop energy,so that k c,n?2.3√

not yet been observed directly shows up indirectly and allows to interpret phase slippage exper iments and to get a hint for the puzzling broadening observed at high en-ergy and momentum transfer neutron scattering.Since we interpret this excitation as the smallest vortex-loop (or local defect)of the super?uid,our model o?ers a way of describing both the phonon-roton and the localized vortex-loops within one scheme45.

Acknowledgement It is our pleasure to acknowledge the very kind support of both the LPTMS(Bat.100)and the department of Solid state physics(Bat.510)at the university of Paris Sud(Orsay),and G.Montambaux and http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmltet for their interest.

A strong-coupling theory of superfluid ${^4}$He

Wavevector (A ?1

)

10

2?

2030E n e r g y (K )

FIG.1.The experimental energy spectrum at saturation vapor pressure [3,9,32]compared with the Feynman phonon spectrum (Eq.1)based on the experimental static structure factor S (k )[33-35].Also marked is the roton energy ?and the termination of the quasi-particle branch at energy 2?(circled point).0

1

23

Wavevector (A ?1

)

?8

???6

?4

?2

X (k ) (K )

???8?6

?4

?2

X (k ) (K )

(a)

(b)

FIG.2.The interaction matrix X (k )from Eq.(4),based on the experimental energy spectrum at (a)saturation vapor pressure [3,9,32]and (b)P=24atm [3].The bare energy of the local mode is taken to be E 0=2?.Notice that X (k )tends to zero at the termination point of the spectrum at high momentum.

A strong-coupling theory of superfluid ${^4}$He

Wavevector (A ?1

)

0.2

0.4

0.6

0.8

1

I n t e n s i t y (a r b . u n i t s )

o

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlparison between the experimental scatter-ing cross-section [3]Z (k )of single quasi-particle excitations (points)at 1.1K and the theoretical curve (solid line),ob-tained using the experimental data for the energy spectrum at saturated vapor pressure E (k )[3,9,32]in Eq.(12).

1

2

Wavevector (A ?1

)

100200k 2

(E (k )??)

2

100

200300

k 2

(E (k )??)

2

(a)

(b)

FIG.4.The integrand k 2(E (k )?2?)2appearing in the calculation of the reduction in the ground state energy ?E G in Eq.(18).(a),(b)are at saturation vapor pressure and P=24atm respectively.Notice the large contributions around the roton momentum.

0.14

0.15

0.16

0.17

0.18

Density (gr/cc)

02

4

68

1012

14C o n d e n s a t e f r a c t i o n (%

A strong-coupling theory of superfluid ${^4}$He

)

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlparison between the experimental condensate fraction at di?erent densities (solid circles)[23]and Eq.(15),using an e?ective mass for the local modes of m ef f =2.3m (solid line).

A strong-coupling theory of superfluid ${^4}$He

Wavevector (A ?1

)

010

20

30

E n e r g y (K )

10

20

30

40

E n e r g y (K )o

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlparison between the experimental energy spectrum [3,9,32](points)and the energy E 1(k )in Eq.(28)(solid line),where the structure factor S (k )is obtained from independent measurements [33-35].(a)and (b)correspond re-spectively to the saturation vapor pressure and to P=24atm.The dashed line at energy E 2=4?indicates the position of the localized branch of excitations (vortex-loop).

A strong-coupling theory of superfluid ${^4}$He

Wavevector A

?1

0.00.5

1.0S (k )

0.0

0.5

1.0

1.5

2.0

S (k )

o

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlparison between the experimental structure factor S (k )[33-35](solid circles)and expression Eq.(28)(solid line)for the same two pressures as in Fig.6,where the energy E (k )is obtained from independent measurements [3,9,32].

01

23

45

Wavevector (A ?1

)

4?

8?

E n e r g y (m e V )

n=0 (Free recoil)

n=1

n=2

FIG.8.The dispersion relations of the free recoil (n =0)and the lowest two levels of recoiling atoms that correspond to vortex-loops emission (n =1,2)(Eq.31).

A strong-coupling theory of superfluid ${^4}$He

E (mev)

0.00

0.020.040.060.08S (k ,E )

0.000.050.100.150.200.25S (k ,E )

0.000.100.200.30S (k ,E )

00.10.20.30.4S (k ,E

A strong-coupling theory of superfluid ${^4}$He

)

http://m.wendangku.net/doc/301886d96f1aff00bed51e09.htmlparison between the experimental scatter-ing pro?les [29](solid circles)and the free-recoil Lorentzian (heavy dashed-line)at di?erent momenta:(a)k =2.8?A ?1,(b)k =3.0?A ?1,(c)k =3.2?A ?1,(d)k =4.0?A ?1.The remaining extra scattering is the heavy solid-line.The vor-tex-loop energy (4?)is indicated by the vertical dotted-line,the free-recoil energy by the vertical long-dashed-line,and the n =1splitted energy Eq.(31)by the vertical dashed-dot line.At relatively low momenta (a,b)there is still a noticable phonon-roton peak around the 2?energy.In (d)the heavy vertical lines indicate the n =2vortex-loop emission by the recoiling atom.

Energy/4?

2

4

6

8

10

W a v e v e c t o r A

?1

00.5

1

1.52

S c a t t e r i n g i n t e n s i t y

o

FIG.10.(a)Experimental scattering pro?le [44]at k =10.0?A ?1(solid circles).The free-recoil Lorentzian (heavy dashes-line)and extra scattering (heavy solid-line)are plot-ted.The free-recoil energy is indicated by the long-dashed vertical line.(b)The spectrum of the free-recoil (dash-dot)and the 17splittings due to vortex-loop excitations (Eq.31)(alternating solid and dashed lines).

相关推荐
相关主题
热门推荐