Elastic scattering of 4 He atoms at the surface of liquid helium

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1 Indin Jounl of Pue & Applied Physics Vol. 48, Octobe, pp Elstic sctteing of 4 He toms t the sufce of liquid helium P K Toongey, K M Khnn, Y K Ayodo, W T Skw, F G Knyeki, R T Eki, R N Kimengichi & S Rotich Deptment of Physics, Moi Univesity, Box 5-Eldoet-Keny Deptment of Physics, Msinde Mulio Univesity, Box 9-Kkmeg-Keny E-mil: yudh@yhoo.com Received 3 Apil ; ccepted August Elstic sctteing of 4 He toms when they ppoch the sufce of 4 He liquid, hs been studied. The liquid sufce is ssumed to be unifom nd the density pofile is the sme long nd pependicul to the sufce. The incident 4 He tom will intect with lge numbe of 4 He toms in the liquid ne the sufce of liquid 4 He. Hence, the effective intection of the incident pticle will be due to mny-body foces. The mny-body foces e epesented by the t-mtix. In the eqution fo potentil enegy pe tom in the bulk liquid, the pi potentil ws eplced by the t-mtix. The Gussin potentil used in clculting the expecttion vlue of the t-mtix is equivlent to the Lennd-Jonnes potentil. The esults give quntittive geement with the coesponding expeimentl vlues fo 4 He fo the vious vlues of χ. The potentil enegy pe tom in the bulk liquid obtined in this clcultion is -.3 K. The expeimentl enegy of intection is -.8K, which implies tht the fomul deived by us fo the potentil enegy pe tom in the bulk liquid woks well. Keywods: Elstic sctteing, 4 He toms, t-mtix, Potentil enegy Intoduction The fee sufce of liquid 4 He is vey simple, clen nd menble to theoeticl nd expeimentl investigtions. The density pofile of the sufce hs been mesued with incesed ccucy nd it hs been found to extend unusully becuse of the quntum ntue of liquid helium. The sufce suppots quntized cpilly wves clled ipplons, which e themlly excited whees elementy excittions t solid sufce e the well undestood, eltively little is known bout the elementy excittions t fluid sufce (i.e. ipplons). In noml fluid, these excittions e ovedmped t lge wve vectos due to the effect of viscosity. In supefluid, the dmping is expected to be much lowe, hence, the obsevtion of well-defined excittion with wvelengths down to inte-tomic distnces should be possible. An obvious system fo this pupose is 4 He below the lmbd tempetue nd, in fct, the existence of lge wve-vecto ipplons both on bulk 4 He nd in films hs been pedicted by theoy nd hs indiectly been confimed expeimentlly 3. Theoists inteested in the popeties of inhomogenous quntum mny-body systems investigted on the subject 4,5 nd detiled clcultions fo the fee sufce of bulk 4 He nd films of tomic thickness bsobed on solid substte t covege hve yielded the dispesion eltion nd the dynmic stuctue fcto of the elementy excittions in the micoscopic egime, i.e., fo lge wve vectos. The theoies bove do not involve ipplons. Attempts t obtining eflectivity R(k z ) hve been mde by consideing the cetion of ipplons on the liquid sufce by the incoming tom 6,7. The mesued eflectivity could only be epoduced by such models in the limit of vey low enegy ipplons such tht ipplon poduction by the eflecting tom is not impotnt. Edwds nd co-wokes 8- hve mesued the eflectivity of 4 He fom the sufce t low tempetue (T<4 mk) whee theml ipplons e unimpotnt. Edwd nd Ftouos developed theoy fo the eflectivity t tnsltionlly invint liquid helium sufce (T= K), which shows tht eflectivity depends upon the density pofile, especilly in the low-density egion. They obtined density pofile fo the sufce by fitting this model to the mesuements. The extensive mesuements of Edwds hve shown tht the eflectivity R(k z ) is function of the incident ngle. These pmetes e independent nd so eflectivity R(k z ) cn be expessed s function of the wve vecto component k z tht is pependicul to the plne of the sufce. This ises becuse the sufce is tnsltionlly invint when thee e no ipplons o othe defects pesent, then, the Schödinge eqution fo the tomic wvefunctions cn be esolved into independent components pllel nd pependicul to the sufce. It is found tht R(k z )

2 744 INDIAN J PURE & APPL PHYS, VOL 48, OCTOBER tends to unity s k This is due to the sufce potentil pofile ppeing moe bupt nd moe eflecting fo the incident toms with longe wvelengths. Edwds nd Ftouos obtined the following eqution elting the sttic effective potentil U(z) to the supefluid density. This cn be obtined s follows: V (z) m z η ψ + ψ = EB ψ () whee U(z) is the sttic effective potentil, m is the mss of the 4 He tom nd E B is the binding enegy of the tom in the bulk. The z-dependence of the potentil U(z) is defined by the locl cuvtue of the supefluid density. A functionl fom fo ψ ws chosen, which gve the symptotic behviou U(z) = Cz 3 fo lge z, i.e., consistent with the long nge Vn de Wls ttction. The mgnitude of C ws detemined fom n intetomic potentil. The Schödinge eqution fo the sctteed tom ws numeiclly integted though the sufce to obtin R(k z ) so s to compe with the mesued eflectivity. In vition upon this model, Edwd nd Ftouos lso used the diffeent ppoch of integting ove ll the intetomic potentils to get U(z). Howeve, they found tht the density n(z) could not be detemined uniquely. In the pesent study, we hve not clculted R(k z ). We hve clculted the enegy of intection V B nd the binding enegy E B. Pevious studies hve been cied out unde the ssumption tht when 4 He tom ppoches the sufce of 4 He liquid, it will intect with one tom in the liquid. Hence, the effective intection used in the pevious studies is two-body intection. When clculting the potentil bove the liquid sufce, the coection to the sum of thei pi potentils due to the mny-body coeltions becomes impotnt becuse of the close poximity of the condensed toms to ech othe 3. In liquid helium, the highe ode coeltions incese the binding enegy ove tht clculted one only using pi potentils 4,5. Consequently, this would give n ppent incese in the stength of the 6 tem in the pi potentil ove the dilute gs vlue. When 4 He tom ppoches the sufce of 4 He liquid, the incident tom is likely to intect with lge numbe of 4 He toms in the liquid ne the sufce of 4 He liquid. In moe elistic sense, the effective intection of the incident pticle will be due to mny-body foces. Hence, thee is need to tke into ccount multi-pticle intections. In this study, since the density pofile cnnot be uniquely detemined, the liquid sufce will be ssumed to be unifom nd the density pofile is the sme long nd pependicul to the sufce. The incident 4 He tom intects with lge numbe of 4 He toms in the liquid ne the sufce of the liquid 4 He. Theefoe, the effective intection tht the incident pticle will hve is due to the mny-body foces. Hence, we intend to pusue the following objectives in this study to deive the expession fo potentil enegy pe tom in the bulk liquid, nd to clculte its numeicl vlue. Theoy of 4 He Atom Sctteing t the Liquid 4 He Sufce When 4 He tom ppoches the sufce of 4 He liquid, it will be sctteed. The liquid sufce will be ssumed to be unifom nd the density pofile is the sme long nd pependicul to the sufce. The incident 4 He tom intects with lge numbe of 4 He toms in the liquid ne the sufce of liquid 4 He. Hence, the effective intection of the incident pticle will be due to the mny-body foces. The mny-body intection is epesented by the ective mtix t k. In the equtions fo the potentil enegy pe tom in the bulk liquid, the pi potentil is eplced by the ection mtix: t = V ( ) + V ( ) G V ( ) + V ( ) G V ( ) G V ( ) +... () k whee V() is the two-body intection potentil nd G the Geen s function. The Gussin fom of the potentil 5 ws used insted of the Lennd-Jones potentil since integtions cnnot be done using Lennd-Jonnes potentil. To deive the expession fo the potentil enegy pe tom in the bulk liquid 4 He using the t-mtix fomlism, the expecttion vlue of the t-mtix cn be witten s: = τ (3) ik ik tk e t ke d whee t k is given by Eq. () nd dτ is expessed in spheicl co-odintes s: dτ= sin θdθdφ d (4) Using Eqs ( nd ), t k becomes: ik ik tk = e V e sin θdθdφd ik ik + e VG Ve sin θdθ d +... (5)

3 TORONGEY et l.: ELASTIC SCATTERING OF 4 He ATOMS 745 π π tk V d sin d d = θ θ φ π π V GV d sin d d + θ θ φ+ othe tems π π k = V d[ cosθ] [ φ] t π [ cos ] [ ] π + VG V d θ φ + othe tems k t = 4 V d + V G V d + othe tems (6) (7) The two body intection potentil V() cn be expessed s : < V ( ) = 4 e e (8) ξ whee is the hd-coe dimete, is the epulsive nge of the Gussin potentil, the ttctive nge of the Gussin potentil nd ξ is the enegy in eg. The vlues of G (, ), G (, ) cn be expessed s 5,6 : 3 mρ G (, ) = G (, ) = (9) χ η N m whee χ = nd η G (, ) (, ) m G = 4 π Since the est of the quntities in Eq. (9) e constnts, χ depends on the vible (hd-coe dius). The potentil enegy pe tom in the bulk liquid cn be clculted using the t-mtix fomlism expessed in Eq. (7). Substituting Eq. (9) into (7), we get: k t = 6πξ e d e d 3 ρ 8πξ m e d χ η 3 ρ 8πξ m e d χ η πξ mρ e d χ η () Upon integtion of Eq. () nd using the substitution =x nd theefoe, d=dx, the following equtions e obtined: 3 e d = π + + π () 4 3 e d = π + + π () 4 π e d = + (3) 4 π e d = + (4) e 4 π d = ( ) (5) Substituting Eqs (-5) into Eq. (), we get the expession fo the potentil enegy pe tom in the bulk liquid s: = πξ π + + π 4 3 k 6 t 3 6 πξ π + + π πξ mρ π π χ η πξ mρ π + + χ η ( + ) + (6) The genel expession fo the zeo-point kinetic enegy is:

4 746 INDIAN J PURE & APPL PHYS, VOL 48, OCTOBER η K = m ( q) (7) whee q is the distnce between two helium toms t the time of elstic sctteing of the incident helium tom fom the sufce of the liquid helium. 3 Results Since the sctteing of 4 He pticles fom the sufce of 4 He liquid is ssumed to be elstic, the kinetic enegy is conseved. The potentil enegy pe tom in the bulk liquid ws found by integting ove the expecttion vlue of ection mtix fom the hd-coe dius to infinity. The fomul fo t k tkes into ccount the mny body intection since the sctteing of 4 He tom t the sufce of 4 He liquid is mny-body intection nd not two-body intection. When 4 He tom ppoches the sufce of 4 He liquid, it will intect with lge numbe of 4 He toms ne the sufce of the liquid 4 He. The effective intection of the incident pticle will be due to the mny-body foces. The mny-body intection is epesented by the ection mtix t k. The Gussin potentil ws used nd not the Lennd-Jones potentil. Khnn nd Ds 5 fitted the expeimentl cuve with the potentil, which ws used in this study. The Gussin potentil used is equivlent of the Lennd-Jones potentil. The potentil enegy pe tom in the bulk liquid ws obtined. The vlue of χ ws vied between nd 58 fo ech vlue of the ection mtix. Tucke nd Wytt obtined potentil enegy pe tom in the bulk liquid 4 He s V B =.8 K whees the mesued binding enegy 7 hs vlue E B = 7.6 K. The contibution to the kinetic enegy fom the supefluid density gdients is zeo in the bulk liquid. This fixes the zeo-point kinetic enegy in the bulk liquid t K =3.65 K, which is close to the vlue obtined using neuton sctteing mesuements 8 of K==3.3±.337 K. Figue shows the gph of expecttion vlue of the ection mtix ginst chi-sque fo vious vlues of the hd-coe dius of the toms. Incesing the vlues of hd-coe dii nd t the chi-sque vlues of χ =3, 4, 3 nd 38 give Vn de Wls potentils of V B = k =.337,.47,.3 nd.576 K, espectively. It cn be seen in Fig. tht s χ ppoches 58, expected vlues of K nd K e obtined, which e close to the mesued binding enegy 7 of E B = 7.6 K. This vlue coesponds to the lge vlues of the hd-coe Fig. Reltion between the expecttion vlue of the ection mtix t k nd the Chi-sque. Seies : =.5Å, Seies : =.Å, Seies 3: =.556 Å, Seies 4: =.84Å dius. We cn similly locte vlues of t k, which e close to E B fo some definite vlues of χ. The zeo-point kinetic enegy somehow does not ppe in the finl vlue of t k. The vlues of t k fo the vious vlues of χ wee computed nd tbulted fo ech potentil width. The vlues of the coe dii used include.5,.,.56 nd.84å. Fo coe dius of.5å, the gdient of the gph chnged shply when the enegy of intection ws educed to bout K. As the vlue of χ ws incesed, the enegy of intection ws educed to bout K nd, hence, the chnges in the gdient become insignificntly smll. With coe dius of.å, the gdient of the gph chnged shply when the enegy of intection ws bout 5 K. As the vlue of c is incesed, the enegy of intection educed to bout K, befoe the gdient becomes insignificntly smll. Fo the dii of.56å nd.84å, the gphs chnged shply when the enegy of intection ws bout K nd 3 K, espectively. Once gin s the vlue of χ ws incesed, the enegy of intection educed to bout K. At the point whee thee is insignificnt chnge in the gdient of the gph s the vlue of χ ws incesed, ll the toms e ssumed to hve equl potentil enegy in the bulk liquid. It ws noted tht this point ( K) ws the sme fo ll potentil widths used in this clcultion nd, theefoe, gives the potentil enegy pe tom in the bulk liquid. This vlue is iespective of the potentil used. Fom the

5 TORONGEY et l.: ELASTIC SCATTERING OF 4 He ATOMS 747 Tble Reltion between the expecttion vlue of the ection mtix t k nd χ t k fo Hd-coe dimete χ =.5Å =.Å =.556Å =.84Å Tble Summy of impotnt esults obtined fom the clcultions Coe dius Å Chi-sque χ Expecttion vlue, t k Potentil V B Binding enegy, E B wok done, it ws noted tht the following pis of the χ nd hd-coe dius, espectively, gve the vlue of the expecttion vlues tht gee with the mesued dt s pesented in Tble. Fom the dt, the vege vlue of the Vn de Wls V B = t k =.3 K. This is with geement with the vlue obtined by Tucke nd Wytt, who obtined V B =.8 K. 4 Discussion nd Conclusions The question of intection between pticles is mtte of pobbility, whose function follows chisque distibution with vious vlues of χ. Unlike pevious studies tht nlyzed 4 He sctteing t the liquid helium in tems of the two-body intection, this study hs tken moe elistic ppoch of the mny-body intections due to the close poximity of the condensed toms to ech othe. The incident 4 He tom intects with lge numbe of 4 He toms in the liquid ne the sufce of liquid 4 He. To sum up the epeted sctteing of pticles, the t-mtix hs been used. Using the Gussin potentil with the Geen function, the expecttion vlue of the ection mtix t k is clculted. Results show genel incese in the expecttion vlue of the ection mtix t k s the hd-coe dius of the tom deceses. As the vlue of χ nd the coe dii incese, the enegy of intection educed to bout K, befoe the gdients becme insignificntly smll. At the point whee thee is insignificnt chnge in the gdient of the gph s the vlue of χ inceses, ll the toms e ssumed to hve equl potentil enegy in the bulk liquid. The point of K shed by ll the potentil widths used in this study gives the potentil enegy pe tom in the bulk liquid. Results show the vege vlue of Vn de Wls V B = t k =.3 K which is in geement with the vlue obtined by Tucke nd Wytt, thei vlue being V B =.8 K. Now, the mesued vlue of binding enegy is E B = 7.6 K; hence, the zeo point kinetic enegy in the bulk liquid ccoding to ou clcultions is K =.97 K, which is close to the vlue obtined by neuton sctteing expeiment 8, K =3.3±.3 K. The distnce between the helium toms t the time of elstic sctteing of the incident helium tom fom the sufce of liquid helium is q=.7 m. Acknowledgement The uthos e vey gteful to Moi nd Msinde Mulio Univesities fo viling the necessy texts nd jounls tht wee useful in the peption of this mnuscipt. Refeences Tucke M A H & Wytt A F G, J Low Temp Phys, (995) 5. Luio L B, et l, Phys Rev Lett, 68 (99) Mnninen M T & Peliol J P, J Low Temp Phys, 5 (983) Sm F & Cole M W, Phys Rev, B (975) 86 5 Lute H J, Godfin H, Fnk V L P, & Leidee P, Phys Rev,68(99) Echenique P M & Rendy J B, J Physics C: Solid Stte Phys, 9 (976) Swnson D R & Edwds D O, Phys Rev, B37 (988) Dupont R J, Himbet M, Pvloff N & Teine J, J Low Temp Phys, 8 (99) 3.

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