Collective modes in superfluid helium when there is a relative velocity between the normal and superfluid components

Transcription

1 Fizika Nizkikh emperatur, 008, v 3, No /5, p Colletive mode i uperfluid helium he there i a relative veloity betee the ormal ad uperfluid ompoet IN Adameko, KE Nemheko, ad VA lipko VN Karazi Kharkov Natioal Uiverity, 61077, Ukraie valipko@mailru AFG Wyatt hool of hyi, Uiverity of Exeter, Exeter, EX QL, Uited Kigdom AFGWyatt@exeterauk Reeived November 6, 007 Colletive mode are tudied i uperfluid helium he the ormal ad uperfluid ompoet have a relative veloity I thi paper the geeral diperio relatio for firt ad eod oud mode i obtaied for arbitrary value of, ad e have foud the relatiohip betee the amplitude of the oillatig variable for firt oud It i ho i a firt oud ave, that both temperature ad preure a oillate, ad moreover, the ormal fluid veloity a exeed the uperfluid veloity i the ave I the geeral ae of firt oud, the ormal fluid ot oly ha a veloity ompoet parallel to the ave vetor, but alo a travere veloity ompoet It i ho that he there i oly a phoo ytem i the helium, the amplitude of the temperature oillatio i a firt oud ave i a aiotropi phoo ytem, a exeed that i a eod oud ave i a iotropi phoo ytem, for imilar value of the ormal fluid deity AC: 675D uperfluid phae; 675dt oud ad exitatio Keyord: ormal ad uperfluid ompoet, firt ad eod oud, phoo ytem i helium 1 Itrodutio I uperfluid helium ytem ith high value of the relative veloity of the ormal ad uperfluid ompoet are tudied experimetally i variou ay Oe of them ue pule of thermal exitatio i helium [1] uh pule are reated by a heater immered ito helium at very lo temperature he a hort eletrial urret i applied to the heater he heater the radiate exitatio ito the helium Aother ay to reate a large relative veloity i to make the helium move through arro hael I thee ytem, the veloitie of the ormal or uperfluid ompoet a be very loe to the ritial veloity uh ytem have attrated the attetio of both theoretiia [ ] ad experimetalit [5 9] he to-fluid hydrodyami equatio of uperfluid helium aue a fudametal differee betee helium ad laial liquid he proee that a take plae i pule of laial gae, or i movig liquid, ha i priiple the ame propertie a proee i a tatioary ga or liquid hi i due to the fat that the oordiate frame a be traformed ito a frame here the ga pule, or movig liquid, i tatioary I uperfluid helium the ituatio i very differet If the relative veloity v v,herev ad v are the ormal ad uperfluid veloitie, i ot equal to zero, the it i impoible to fid a frame here veloitie of both ompoet of helium are equal to zero hi fat mea that mot pheomea ill have differet propertie i uperfluid helium ad, moreover, e propertie ompared to tatioary helium he 0 he theoretial deriptio of thee problem a firt made by [10] here eod oud propagatio a aalyzed at very mall value of veloity Reult o eod oud, at arbitrary value of, ere preeted i [11] here the diperio equatio for eod oud a obtaied ith the approximatio that the oeffiiet of thermal expaio, ad mometum deity of helium, are equal to zero hee approximatio are ot valid i the geeral ae IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt, 008

2 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt I Ref ad 1, a ell a i Ref 3 the temperature depedee of the ritial veloity of motio i helium ( )a tudied hi alloed u to determie the limit of thermodyami tability i helium I Ref 3 ad it a ho that quaipartile ytem, ith large value of, poee uique thermodyami propertie I thi paper, the geeral diperio relatio for firt ad eod oud, at arbitrary value of, i obtaied We have obtaied the relatiohip betee the amplitude of the hydrodyami variable for firt oud he reult obtaied i thi paper are igifiatly differet from the ell-ko reult hih are obtaied for iotropi phoo ytem ( 0) i uperfluid helium oud mode i uperfluid helium he hydrodyami equatio for uperfluid helium aordig to Ref 13, 1 a be ritte a follo: div ( v t v) 0 ; (1) div ( v ) 0 ; t () v ( v) v 0 ; t (3) Ai v k ( v ) Ai Ak t x i x i () Here i the deity of helium; ad are deitie of the ormal ad uperfluid ompoet, repetively; v ad v are veloitie of ormal ad uperfluid ompoet, repetively; i etropy of uit of volume; i the hemial potetial of uit of ma of helium, A / ; v v i the relative veloity of the ormal ad uperfluid ompoet After liearizatio of thi ytem of equatio, e oider the olutio, for mall deviatio, a plae ave i a ifiite medium: ( v ) u v v 0, (5) ( v u) v 0, (6) ( ) k v u v 0, (7) k ( ) v k u v 0 (8) k Here v vk / k, ad imilarly for v ad Vetor k i the ave vetor ad k it modulu, u i the modulu of the phae veloity ad u / k,hereitheagular frequey of the mode mall deviatio of the parameter are marked ith the ymbol «tilde» From Eq (7) it follo that the oillatio of the uperfluid veloity i alay logitudial if u v he reao that the olletive mode ith u v doe ot exit, i beaue the uperfluid y alay ha potetial flo At the ame time from Eq (8) it follo that the oillatio of the relative veloity ad the ormal fluid veloity v, have i geeral both logitudial ad travere ompoet relative to the ave vetor k We hooe the oordiate frame, ith axi x direted alog the vetor of the relative veloity, axiy lie i the plae determied by the vetor ad k ad k y 0 (ee Fig 1) Equatio (5) (8) are a ytem of five equatio for ay to alar variable, for example, preure ad temperature, ad the uperfluid veloity v (it projetio o vetor k) ad to ompoet of the vetor of relative veloity No e hould take ito aout the thermodyami equatio:,, ( / ),,, ( / ),, ( / ) ad the relatio for the hemial potetial, ;, (9) ; (10) ; (11) 1 d d d d, (1) hih give a oetio betee the mall deviatio 1, (13) ad determie the relatio betee the derivative of the thermodyami variable: 1 k Fig 1 he oordiate frame ith the x axi direted alog, ad ith the y axi lyig i the plae hih defied by vetor ad ave vetor k he agle i betee vetor k ad,, x ; (1) 358 Fizika Nizkikh emperatur, 008, v 3, No /5

3 Colletive mode i uperfluid helium he there i a relative veloity betee the ormal ad uperfluid ompoet ( / ) 1 ( / ),,,, ; (15) (16) he ytem (5) (8) ha a otrivial olutio, for ay value of, if it determiat i equal to zero hi oditio give a equatio of fifth order ith repet to the phae veloity u he diperio equatio for oe of the mode a be eaily foud diretly from the ytem of Eq (5) (8) hi i beaue the vetor Eq (8) beome a alar equatio he u v o, a otrivial olutio of Eq (5) (8) hould exit if u v vk / k We all thi mode the travere mode ad have aalyzed it i Ref 15 3 Diperio equatio for firt ad eod oud at arbitrary value of relative veloity o obtai the diperio relatio for firt ad eod oud, e a tart from the Eq (5) (8) ith u v, hih expliitly exlude the travere mode o e take the expreio for v from (6) ad for v from (7) ad put them ito Eq (5) Fially e get the equatio ( ) ( ) u v u v ( u v ) (17) he e get for the y ompoet of relative veloity from Eq (8) i y x (18) u v u v For the x ompoet of Eq (8) ad the amplitude of the temperature oillatio, e obtai folloig equatio: x o ( u v ) i o o ( u v ) 0, (19) o ( u v ) x o ( u v ) u v [ oi ] 0 (0) Uig relatio (13), for mall deviatio of the hemial potetial, i the Eq (17) e fid 1 x o ( u v ) ( u v )( u v ) ( u v ) 0 (1) he ytem of Eq (19) (1) together ith thermodyami relatio (9) (11) form a ytem of three equatio for the three idepedet variable,, x For a otrivial olutio of thi ytem of equatio to exit, the determiat from thee equatio mut be zero hi give a fourth order equatio for u, ie Eq (19) ad (0) are liear ad Eq (1) i quadrati ith repet to u Aareult, e get the diperio relatio a a algebrai expreio of the fourth poer hi i expliitly preeted i Appedix A he four root of thi diperio relatio are the to mode of firt oud ad the to mode of eod oud, at arbitrary value of the relative veloity I the limitig ae of 0, the geeral relatio (A1) (A6) give the reult for the veloitie of oud obtaied i Ref 13 I the ae of mall value of,ithe liear approximatio ad egletig thermal expaio, the geeral relatio (A1) (A6) give the reult i Ref 10, obtaied there for the ae ad 0 Firt oud at arbitrary value of relative veloity Coider the firt oud mode i the importat pratial ae he,ad at arbitrary value of hi ae, i partiular, our i experimet [1,5,8,9] We hooe the oordiate frame here j 0 hi a be alay obtaied by the appropriate Galilea traformatio Uig the iequality, i thi frame e get v, v 0 () At relatively lo temperature, he, the geeral diperio relatio (ee Appedix A) ha a olutio u (3) for firt oud o get the relatio betee the oillatig variable, e tart from the iitial ytem of Eq (19) (1), here e ubtitute the diperio relatio (3) ad take ito aout Eq () ad oditio A the reult, e get the folloig expreio for the mode u : x C D, () E D (5) Here e itrodue variable C, D, ade defied i Appedix B Fizika Nizkikh emperatur, 008, v 3, No /5 359

4 For the mode u, the relatio a be obtaied by ubtitutig, i Eq () ad (5), the agle for, hih orrepod to ubtitutig u for u By projetig the oillatio of uperfluid veloity oto the vetor k, ad ith Eq (7) ad relatio (), e get for the firt oud mode u : v u v 1 u, (6) u here e have ued, i aordae ith Eq (13) or (), the approximate equality, (7) hih applie for firt oud beaue the otributio of thermal exitatio to a be egleted a ubtitutig expreio (), (5), ad (7) ito Eq (18) give the projetio of the oillatio of the relative veloity oto the axi that i perpediular to vetor, y, for the firt oud mode ith u : i o D ( E C ) y ( o ) D (8) For the mode u, e ubtitute for i Eq (8) ad multiply the right had ide by 1 he relatio () (8) determie the oillatio of the mai thermodyami variable, the uperfluid ad the relative veloity ompoet, for the firt oud mode u hi ae i very importat i pratie a it i the ae here the otributio of the thermal exitatio i egligible ( ) ad the relative veloity a take ay value I the limit 0, from Eq (B1) (B3) e get o C E o,, l, 1 o (9) D l, (30) here e ue the equality (1) for the traformatio of the thermodyami derivative I the limit 0, the oillatio of all vetor value are logitudial (ie, alog k), o, e a take 0i (9) ubtitutig (9) ad (30) IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt ito () ad (5) e get, uig (6), the relatio betee the amplitude of the oillatig variable hee oiide ith the reult of Ref 13 At 0 there i o differee betee mode u For firt oud, at 0, preure ad the ormal ad uperfluid veloitie oillate, ad the temperature oillatio are determied by the mall value of the oeffiiet of thermal expaio i helium A ill be ho belo, a uuual ituatio appear he i ot mall 5 Colletive mode of firt oud i a phoo ytem at arbitrary value of Let u oider a phoo ytem he e oly have phoo i the uperfluid helium For example, helium at 06 K for the iotropi ae, ad a phoo pule i helium at 005 K i the aiotropi ae he geeral relatio a be ritte expliitly for the phoo ytem ith liear diperio: p, (31) here / i the oud veloity of liquid helium I thi ae [13] ad, (3) (33) ( 1 / ) Calulatig the repetive derivative for the phoo ytem, e get l 3 ( )o [ 3 ( u 1) ( 3u )]o x G G G G G G [ ( u 1) ( u 1) ( 3u 1)]o [( u 1) l ( ) 3,, 5 3 G G u, (3) l u, (35), here ug 8 (36) i the Grueie otat Uig thee expreio for the thermodyami derivative e get from Eq (19) (1) the folloig expreio for the amplitude of the firt oud mode u i a phoo ytem: ( u 1)] 1 ( o ) G ; (37) 360 Fizika Nizkikh emperatur, 008, v 3, No /5

5 Colletive mode i uperfluid helium he there i a relative veloity betee the ormal ad uperfluid ompoet 3 3 o ( ug ) o [ ug ( 7uG 1) ] o ( ) ug ( 3 ) 1 (38) ( o ) 3 i y o [( 3uG) ( 1 ug) ]o ( 3uG 1) ( ug 1) ( o ) (39) I order to get the relatio for the mode u, oe eed to ubtitute for, ad multiply the right had ide of Eq (39) by ( 1) Coider the limitig ae 0, he the ave vetor k i direted alog the vetor of the relative veloity From Eq (37) (39) ad Eq (6) for the firt oud mode u e obtai: 1 v 1 ( u ) ( u ) ( u ) v, (0) x G G G v, (1) 1 ( u ) u ( u ) 3 G 1 G G 3 ( ) v () For the mode u e get 1 v 1 ( u ) ( u ) ( u ) v, (3) x G G G v/v u= u= / Fig he ratio of the amplitude v / v for the firt oud mode u ad u, a a futio of the relative veloity, for phoo ytem at 0, alulated from Eq (0) ad (3) v, () 1 ( u ) u ( u ) 3 G 1 G G 3 ( ) v (5) I Fig e ho the ratio of the amplitude v / v for the firt oud mode u ad u, a a futio of the relative veloity, for a phoo ytem at 0, alulated from Eq (0) ad (3) We ee that, i the mode u,he/ 067, the ormal ompoet doe ot oillate, ad he / 067 the ormal ad uperfluid ompoet oillate i atiphae I Fig 3 e ho the ratio of the relative amplitude /( v ) for the firt oud mode u ad u a a futio of the relative veloity, for a phoo ytem at 0, alulated from Eq () ad (5) We ee that i the mode u, he / 063the temperature doe ot oillate, ad for / 063the temperature ad v oillate i atiphae We ote, that for a phoo ytem i uperfluid helium, the veloity of the ormal fluid, i firt oud, i may time that of the uperfluid For the ae 0 i Eq (0) (/)(/v ) u= 0 u = / Fig 3 he ratio of the relative amplitude /( v ) for the firt oud mode u ad u a a futio of the relative veloity, for phoo ytem at 0, alulated from Eq () ad (5) Fizika Nizkikh emperatur, 008, v 3, No /5 361

6 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt 1 v [ ( u )] v x 1 G v (6) o, i firt oud v v he ame reult a be obtaied diretly from the relatio i Ref 13 It i iteretig to ote that i pite of the mall value of the oeffiiet of thermal expaio, the relatio (), (5) ivolve the ratio betee the mall value of the thermal expaio oeffiiet ad the etropy of phoo ga hi ratio i temperature idepedet i the phoo regio I detail: 1 1 0,, 0, 0 (7) From thi geeral equatio for the phoo ytem e get v/v v /v y v /v / x ph 3uG 1 3 (8) It follo from Eq () (or (5)) that the relative oillatio of temperature are ot mall ompared to the relative oillatio of v /,eveat 0 here are to reao hy, i the firt oud mode u at, the relative value of the temperature oillatio a reah large value i ompario ith relative oillatio v / i pite of mall value of thermal expaio oeffiiet Firtly, the thermal expaio oeffiiet ad etropy our i the ratio /, ad, eodly, i the mode u, he the relative veloity beome, the temperature oillatio beome very large he relatio betee the amplitude i firt oud i phoo ytem, he i ot very mall, trogly deped o the agle betee the ave vetor k ad the relative veloity vetor hi follo from Eq (37) (39) I the ae /e get for both mode u : 1 v 1 ( u ) ( u ) y G G v, (9) v ( u ) ( u ) x x G 1 G 1 v, (50) v, (51) 1 ( u ) ( u ) 3 G 1 G 1 v (5) I Fig e ho the ratio of the amplitude, v / y v ad v / x v, for the firt oud mode u, a a futio of the relative veloity, for a phoo ytem at /, alulated from Eq (9) ad (50) We ee that the ormal fluid doe ot oly oillate i the logitudial diretio, but that there are alo travere oillatio I Fig 5 e Fig he ratio of the amplitude, v y / v ad v x / v,forthe firt oud mode u, a a futio of the relative veloity, for phoo ytem at /, alulated from Eq (9) ad (50) ho the ratio of the relative amplitude /( v ) for the firt oud mode u a a futio of the relative veloity, for a phoo ytem at /, alulated from Eq (5) We ee that a / ireae, the amplitude of the temperature oillatio ireae At 0, he the phoo ytem ha o defiite diretio, e fid, a expeted, that the oillatio of the vetor variable are logitudial, v x x 0, ad Eq (9), (51), ad (5) oiide ith Eq (0), (1), ad (), repetively, at 0 I thi paper, e have foud the relatiohip betee the amplitude of the oillatig variable of firt oud (/)(/v ) / Fig 5 he ratio of the relative amplitude /( v ) for the firt oud mode u a a futio of the relative veloity, for phoo ytem at /, alulated from Eq (5) 36 Fizika Nizkikh emperatur, 008, v 3, No /5

7 Colletive mode i uperfluid helium he there i a relative veloity betee the ormal ad uperfluid ompoet relimiary reult for eod oud ere preeted at QF oferee [16] ad ha bee publihed i Ref 17 A detailed tudy of eod oud ill be preeted i our ext paper We ompare the reult obtaied i thi paper to thoe for the iotropi ae I the ae of eod oud i the iotropi phoo ytem (ie, 0) (eeref13)therelatio betee the hydrodyami parameter i v v 3u 1 G 3uG 1 v 33 ( u 1) G, (53), (5) v (55) 33 ( ug 1) I the relatio (53) (55) ad belo, e hooe the idepedet variable v beaue the uperfluid veloity v pratially doe ot oillate, ee Eq (53) he relatio (53) ad (55) hould be ompared ith the repetive relatio for firt oud i a aiotropi phoo ytem For typial experimetal value from Ref 1, 5, 8, 9, for trogly aiotropi phoo ytem / 097 ad 005 K For eod oud i the iotropi phoo ytem, for the ame deity of ormal ompoet a i aiotropi ae (ie, at 0 K), it follo from Eq (53) (55) that: v 6 6 / , 9 10, 0 15 (56) v v v / For the mode u of firt oud i a phoo ytem ith / 097, it follo from Eq (0) () at 0 that: v / 0 053, 6, (57) v v v / ad for the firt oud mode u, i a aiotropi ytem ith / 097 at 0, it follo from Eq (3) (5) that: v / 059, 5 (58) v v v / For the firt oud mode i the iotropi phoo ytem it follo from Eq (3) (5) that: v / 017, 083 (59) v v v / We ee, eod oud, i the iotropi phoo ytem, i aordae ith (56) i maily a temperature ave ad a ave of v, ad preure ad the uperfluid ompoet pratially do ot oillate he ormal ad uperfluid urret are ot equal i a phoo ytem, ad a have the ame diretio For firt oud i the iotropi ae, it follo from (59) that the uperfluid ad ormal veloitie, preure ad temperature all oillate, ad the temperature oillatio i ot mall A for the mode of firt oud, i the aiotropi ae, for the mode u, maily the ormal veloity ad temperature oillate, ad preure ad uperfluid veloity pratially do ot oillate (ee (57)) At the ame time, i the firt oud mode u, i a phoo ytem ith / 097, i aordae ith (58), all the variable oillate ith imilar value 6 Coluio A otable feature of a trogly aiotropi phoo ytem i uperfluid helium, reated i experimet [1,5,8,9], i the large value of the relative veloity, of the uperfluid ad ormal ompoet he oud mode i tatioary ( 0) heliumadfortheaeofmall, have bee tudied for may year, but the aalyi of oud propagatio at arbitrary ha ot bee doe util o I thi paper the geeral diperio equatio for firt ad eod oud i uperfluid helium i foud, at a arbitrary thermodyamially table value of Ita ho, that i the limit of mall otributio of thermal exitatio, / 1, at arbitrary value of i uperfluid helium, the firt oud mode a be exited ith the diperio la k he geeral relatio betee the amplitude of the oillatig variable i the firt oud mode, i derived I pite of «iotropi type» of diperio la for the firt oud mode, the relatio betee the amplitude of the oillatig variable, trogly deped o the kid of mode (either k or k) ad o the agle betee the veloity ad the ave vetor k I the limitig ae 0, the geeral relatio for the amplitude of the oillatig variable i preeted i Ref 13 he relatio betee the amplitude of oillatig variable i firt oud are tudied i detail for the ae of a aiotropi phoo ytem ith arbitrary hi oditio i very importat i pratie beaue high value of are realized i phoo pule propagatig i uperfluid helium [1,5,8,9] It i ho, that for trogly aiotropi phoo ytem, a reated i experimet, the amplitude of uperfluid veloity, preure, ad temperature at a give oillatio amplitude of the ormal veloity a be the ame order of magitude a the orrepodig relatio i a ave of eod oud i the iotropi phoo ytem, ad eve exeed them It hould be oted, that eve i the iotropi ( 0) phoo ytem, the oillatio of veloitie Fizika Nizkikh emperatur, 008, v 3, No /5 363

8 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt of uperfluid v ad ormal v ompoet do ot equal oe to aother: v v,ad v [ 1( 3u 1)/ ] v 6 v G At 0 the iequality betee v ad v beome troger, ee Fig We hould ote here, that i the phoo regio, the amplitude of the relative temperature (38) i firt oud mode tur out ot to be mall i ompario ith the relative oillatio amplitude of the ormal (or uperfluid) ompoet ad trogly gro ith he i loe to for the mode k, ee Fig 3 o, the oud mode i uperfluid helium, at ozero value of the relative motio, poe uuual propertie, hih are mot apparet at large value of heauthor hope that the relatio betee the amplitude of the oillatig variable i the oud mode of uperfluid helium he there i relative motio of ormal fluid ad uperfluid ompoet, obtaied i thi paper, ill timulate e experimet to tudy oud mode i aiotropi quaipartile ytem of uperfluid helium Appedix A he geeral diperio equatio for firt ad eod oud i uperfluid helium at arbitrary value a be preeted a follo a u a u a u a u a (A1) Here e itrodue otatio a, (A) here ( / ) ( / ), ad / ; a3 o ( ) ( ) 3 ; (A3) a o ( ) a a, (A) 0 here 3 3 a ( 6 6 ) ( / ) , ad a 0 3 ( / ) ( / ) 36 Fizika Nizkikh emperatur, 008, v 3, No /5

9 Colletive mode i uperfluid helium he there i a relative veloity betee the ormal ad uperfluid ompoet ( / ) ; a o ( ) a o ( ) a, (A5) here 3 3 a ( )( ) ( / ) , ad a 11 ( ) ( / ) here ( / ) ( / ) 3 ( ); a o ( ) a o ( ) a ( ) a, (A6) a0 ( ) 3 ( / ) 3 3, a ( ) ( ) ( / ) 3 ( / ) 3 3 a 00 ( / ) ( / ) 3 3 ( / ) 3 ( / ) ( / ) ( ) ( )( 3 ), ( / ) Fizika Nizkikh emperatur, 008, v 3, No /5 365

10 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt l C l 3 o Appedix B 1 l l l o ( 1 )o 1 l / l /, (B1) hereedeote 1 l l l l l, 1 ; D l / l 3 l l o 3 o ( / ) l l l 3 l ( / ), (B) here e ue the folloig hortut 3 l l l ( / ) ; E l l 3 3 l o ( / ) l l o l l 1 o ( ) l / l l ( / ), (B3) here l l l l ( / ) Akoledgemet We expre our gratitude to ERC of the UK (grat E/F /1) for upport of thi ork 1 AFG Wyatt, NA Lokerbie, ad RA herlok, hy Rev Lett 33, 15 (197) AF Adreev ad LA Melikovky, JE Lett 78, 57 (003) [i ma Zh Ekp eor Fiz 78, 1063 (003)] 3 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt, hy Rev Lett 96, (006) IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt, J hy: Code Matter 18, 805 (006) 5 R Vovk, CDH William, ad AFG Wyatt, hy Rev Lett 91, 3530 (003) 6 RE Grieti ad J oeie, hy Rev Lett 90, 3501 (003) 7 JA Flate, CA Lidemith, ad W Zimmerma, J Lo emp hy 101, 73 (1995) 8 DH mith, RV Vovk, CDH William, ad AFG Wyatt, Ne J hy 8, 18 (006) 9 DH mith ad AFG Wyatt, aepted for publiatio i hy Rev B 10 IM Khalatikov, i'ma Zh Ekp eor Fiz 30, 617 (1956) 11 RG Arkhipov, Z Ekp eor hy 59, 055 (1970) 1 AF Adreev ad LA Melikovky, J Lo emp hy 135, 11 (00) 13 IM Khalatikov, A Itrodutio to the heory of uperfluidity, WA Bejami (ed), Ne York Amerdam (1965) 1 J utterma, uperfluid Hydrodyami, Horth Hollad, Amterdam (197) 15 to be publihed 16 Iteratioal ympoium o Quatum Fluid ad olid QF006, Kyoto (006) 17 IN Adameko, KE Nemheko, VA lipko, ad AFG Wyatt, J Lo emp hy 18, 57 (007) 366 Fizika Nizkikh emperatur, 008, v 3, No /5