Deep Inelastic Neutron Scattering Studies of Atomic Momentum Distributions in Condensed Argon and Neon*

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1 Deep nelastc Neutron Scatterng Studes of Atomc Momentum Dstrbutons n Condensed Argon and Neon* M. A. Fradkn, S.-X. Zeng, and R. O. Smmons Physcs Department and Materals Research Laboratory, Unversty of llnos at Urbana-Champagn, USA Z. Naturforsch. 48a, (993); receved December 23, 99 Usng deep nelastc neutron scatterng, drect measurements have been made, ) of the dynamc structure factor of a seres of condensed argon samples over the temperature range 8 to 8 K, near meltng (for wave-vector transfers 2 to 26 Ä - ) and ) of lqud neon, near 27 K (for wave-vector transfers.4 to 27.6 Ä - ). Neutron tme-of-flght chopper spectrometers were employed. Snglepartcle knetc energes, < k >, can be obtaned from the analyss of the Doppler-broadened recol spectrum of the target partcles. For argon the temperature dependence of < k > can be compared to expectatons from theory, from thermodynamc data, and from prevous neutron scatterng measurements on collectve vbratonal modes. For lqud neon the wave-vector-transfer dependence of J{y), the longtudnal momentum dstrbuton functon, s beng analyzed for fnal-state effects. Key words: nelastc neutron scatterng; Momentum dstrbutons. PACS Scheme 6.2.Gz, 6.2.B, 63.2.Pw.. Purpose Usng epthermal neutrons, drect measurements of sngle-partcle momentum dstrbutons n condensed matter are possble through deep nelastc scatterng. uanttes drectly related to the sngle-partcle momentum dstrbuton, such as the average snglepartcle knetc energy, < k >, a part of the total energy htherto naccessble to drect measurement, can be determned. The condensed noble gases provde a prototype famly that exhbts ranges of behavor, such as quantum to classcal, ncreasng mportance of multbody nteratomc nteractons wth ncreasng atomc number, and ncreasng lattce vbratonal anharmoncty wth decreasng mass. Determnaton of the snglepartcle knetc energy provdes nformaton as to the mportance of these behavors. 2. Method Pulsed neutron spallaton facltes, such as the Argonne Natonal Laboratory ntense Pulsed Neutron Source (PNS), are capable of producng the large flux of epthermal neutrons necessary to carry out deep nelastc neutron scatterng studes. Chopper spectrometers, n whch the ncdent neutron energy selecton s performed by a phased Ferm chopper and the fnal neutron energy s determned va tme of flght, were employed n these measurements (Fgure ). The nstruments LRMECS and HRMECS were used to take the argon data and neon data, respectvely [], n the mpulse approxmaton for neutron scatterng, where the recol of the target partcle s assumed to be free, the dynamc structure factor S A (, E) s related to the atomc momentum dstrbuton n(p) smply by S A (ß, E) = \n(p)ö(e-e r -h-p/m) dp, () + Supported by U.S. Department of Energy BES-Materals Scences under contract No. DE-FG2-9ER4439. * Presented at the Sagamore X Conference on Charge, Spn and Momentum Denstes, Konstanz, Fed. Rep. of Germany, September -7, 99. Reprnt requests to Prof. R. O. Smmons, Physcs Department, Unversty of llnos, W. Green St., Urbana, L 68, USA. where s the wave-vector transfer, E the energy transfer, E T the recol energy of the target partcle, and M the atomc mass. The argument of the delta functon expresses conservaton of energy and momentum for the scatterng process. For an sotropc Gaussan n(p), n(p) = (na)~ 32 e\p{ p 2 /A), (2) / 93 / -438 $.3/. - Please order a reprnt rather than makng your own copy. Unauthentcated Download Date //8 :3 PM

2 M. A. Fracjkn et al. Neutron Studes of Atomc Momentum Dstrbutons Sample Poston 439 Hydrogenous Sheldng Beam Montor 2 Detectors (4) Sheld Tanks Fg.. Schematc horzontal secton of the LRMECS spectrometer at PNS-ANL [], Scattered neutrons can be detected over the broad angular range to 6. where A = 2MT* a n d T* s an effectve t e m p e r a t u r e, the d y n a m c structure factor assumes the G a u s s a n form, S A (, E) = C exp { - (E - E f M2/(A h2 2)}. (3) < k > can be o b t a n e d by ntegraton over all a t o m c m o m e n t a, weghted by the m o m e n t u m dstrbuton. < k > s related to S A (, E) by < k> = (3/4)(X/M). (4) Especally at large wave-vector transfers, nelastc neutron d a t a are c o m m o n l y presented n t e r m s of J(y), the longtudnal m o m e n t u m dstrbuton f u n c t o n, nstead of S((2, E). F o r an sotropc system, J(y) s related to S(,E) by J(y) = (h/m)s(,e), () where the scalng varable s y = (M/h2)(E-ET). (6) Varous agents, such as n s t r u m e n t resoluton, multple scatterng, a n d fnal-state effects (owng to the falure of the mpulse a p p r o x m a t o n ) can cause the observed S(, E) spectrum to be n o n - G a u s s a n. T h e effects of these agents m u s t be ether reduced to a neglgble level or accounted for n order to obtan accurate m o m e n t u m dstrbutons. F o r the d a t a t a k e n on condensed argon, multple scatterng a n d nstrument resoluton were accounted for by c o n v o l u t n g a G a u s s a n J ( y ) wth a fttng functon. T h e fttng functon was determned by fttng d a t a f r o m lqud a r g o n samples, for whch t s assumed t h a t < k > s gven by Unauthentcated Download Date //8 :3 PM

3 M. A. Fracjkn et al. Neutron Studes of Atomc Momentum Dstrbutons the expanson [2] < k > = (3/2) kbt { + (/2)(@/T) 2 - ( / 2 4 ) ( / 7 T +...}, (7) where &2 = (h2/3mkb) ( A Y }. (8) F r o m molecular-dynamcs calculatons for a LennardJ o n e s lqud one can obtan, usng the L e n n a r d - J o n e s parameters of Ross and D a n o n [3], a value of 6 K/Ä 2 for the average value of the Laplacan of the potental energy for lqud a r g o n at s a t u r a t e d v a p o r pressure at a t e m p e r a t u r e of.6 K [4], We have assumed n o u r present analyss that o u r expermental condtons have satsfed the mpulse a p p r o x m a t o n a n d hence t h a t fnal-state effects are neglgble. Justfcaton for neglectng fnal-state effects n the case of lqud a r g o n can be obtaned f r o m the theoretcal w o r k of Sears []. n Sears' treatment the m e a s u r e d J(y) of an sotropc system s related to J A (y) n the mpulse a p p r o x m a t o n by J{y) = y (nverse angstroms) Fg. 2. Calculated fnal-state effects n lqud argon at = 24 Ä~. The sold curve s J A (y) for lqud argon at.6 K, taken to be a Gaussan functon havng a wdth correspondng to a sngle-partcle knetc energy of 2.7 K. The dashed curve s the frst antsymmetrc term n the Sears expanson, enlarged by a factor of for vsblty. Owng to the small sze of the antsymmetrc term the expected measured J(y) s essentally the same as J, A (y). JM-A3[d3JlA{y)/dy3] + X4[d4JA(>/d/]-..., (9) where J(y) s the measured longtudnal m o m e n t u m dstrbuton functon, J A (y) s the longtudnal m o m e n t u m dstrbuton functon n the mpulse a p p r o x m a ton, a n d A3, A4,... are wave-vector-transfer dependent coeffcents. All odd coeffcents are antsymmetrc n a n d hence antsymmetrc n y\ all even coeffcents are symmetrc n a n d symmetrc n y. n partcular, A3 = M (AY} / 36 t2, where V s the par-potental energy a n d M s the mass of the scatterer. At <2 = 2 Ä - we calculate for lqud a r g o n at.6 K the frst ant-symmetrc term at ts m a x m u m to be a p p r o x m a t e l y.% of J A (y) at that value of y. A correcton of ths sze leaves the wdth, a n d hence < k >, essentally u n c h a n g e d f r o m that of JlA(y). Ths can be seen n Fg. 2, n whch we have plotted two curves, ) JlA(y), a G a u s s a n functon havng a wdth c o r r e s p o n d n g to an energy gven by (7), a n d ) the frst antsymmetrc term n the Sears expanson enlarged by a factor of to ad vsblty. O w n g to the small sze of the frst antsymmetrc term the expected m e a s u r e d J(y), consstng of A ( y ) a n d the antsymmetrc term, would not be vsbly dfferent from J A (y) s h o w n n Fgure 2. n solds, the m a g n t u d e of fnal-state effects s stll uncertan, n part owng to the absence of realstc par dstrbuton functons a n d a c c o m p a n y n g quanttatve o u c 2 u 2 7 Temperature (K) Fg. 3. Sngle-partcle knetc energy of condensed argon versus temperature. The sold lne represents the value of < k > predcted by equpartton, (3/2) kbt. Fnte nstrument resoluton and multple scatterng have been accounted for usng a fttng functon derved from data on lqud argon. Lqud data taken at.6k were used to determne the fttng functon for the three low-temperature sold-argon data sets (open crcles), lqud data taken at 87.4 K were used for the three hgh-temperature data sets (flled crcles). calculatons of such effects. F o r present p u r p o s e s of analyss we have a s s u m e d that fnal-state effects are neglgble n o u r d a t a on sold argon, whch were taken at smlar wave-vector transfers as o u r lqudargon data. Unauthentcated Download Date //8 :3 PM

4 44 M. A. Fracjkn et al. Neutron Studes of Atomc Momentum Dstrbutons.2 r.8.4 " : -.8 ** -.2 = 3. A" h «. >. } V «, - / «V = 2.9 A.4 - y (nverse angstroms) - y (nverse angstroms) «.8 Ml - / l -,' - r = 6.2 A" r l - y (nverse angstroms) : $. A" = y (nverse angstroms) Fg. 4. Longtudnal m o m e n t u m dstrbuton functon, J(y), for lqud neon at approxmately 27 K for four wave-vector transfers. These data were collected at a mean scatterng angle of 87, at ncdent neutron energes of 2, 9, 3, and mev, respectvely. They span a range of wave-vector transfers not prevously explored, and t can been seen that the shape of J(y) progressvely changes. The explct study of fnal-state effects n the scatterng process s now possble. 3. Results for Condensed Argon 4. Data on Lqud Neon The temperature dependence of < k > for condensed argon shows a devaton from equpartton behavor at low temperatures (Fgure 3). Several models, ncludng the self-consstent-average-phonon model (SCAP) [6], are currently beng nvestgated n order to account for the observed temperature dependence. ualtatvely the temperature dependence s that expected for a system of q u a n t u m oscllators. Wth an approprate model t should be possble to deduce a value for the ground-state knetc energy of sold argon. A Debye model employs the zero-temperature lmt of the heat capacty, yeldng < k ( T = )> = (9/6) C(T = ) = 2 K, from a heat capacty <9C value of Fnegold and Phllps [7]. t s of nterest to nvestgate more explctly the doman of applcablty of the mpulse approxmaton. We have therefore undertaken to do ths on flud neon, an sotropc scatterng system for whch the Sears analyss [] should apply. Pror to our study there was a wde gap n the avalable data. The measurements of Buyers et al. [8] had a maxmum wave-vector transfer of 2. Ä " whle those of Peek et al. [9] are for wavevector transfer greater than 2 Ä ~. We studed a wde range of wave-vector transfers,.4 to 27.6 Ä ~, usng the avalable range of scatterng angles (87 to 36 ) on the ANL spectrometer H R M E C S [] and dfferent ncdent neutron energes. Unauthentcated Download Date //8 :3 PM

5 442 M. A. Fracjkn et al. Neutron Studes of Atomc Momentum Dstrbutons 442 Neon brdges the gap between the quantum behavor of condensed helum and the classcal behavor of the heaver noble gas solds. t s therefore a good system for the study of fnal-state effects n the neutron-atom scatterng process. Shown n Fg. 4 s a subset of our data on flud neon. From estmaton of the value of A 3 (7) at near Ä - one expects to see a sgnfcant dfference between J(y) and J la {y). ndeed, by comparson wth the J(y) data at larger n Fg. 4, one sees, qualtatvely, the expected change of shape, ncreased ntensty at negatve y and reduced ntensty at postve y. The data shown nclude the effects of fnte nstrument resoluton and of multple scatterng, so further analyss s underway to refne ths qualtatve result. Acknowledgements Ths research has benefted from the use of the ntense Pulsed Neutron Source (PNS) at Argonne Laboratory. Ths faclty s funded by the U.S. Department of Energy, BES-Materals Scences under contract No. W-3-9-ENG-38. [] C. K. Loong, S. keda, and J. M. Carpenter, Nucl. nstrum. Methods A 26, 38 (987). [2] V. F. Sears, Can. J. Phys. 63, 68 (98). [3] J. C. Ross and F. Danon, Dscuss Faraday Soc. 4, 97 (96). [4] L. Verlet, Phys. Rev. 9, 98 (967). [] V. F. Sears, Phys. Rev. B 3, 44 (984). [6] K. Shukla, A. Paskn, D. O. Welch, and G. J. Denes, Phys. Rev. B 24, 724 (98). [7] L. Fnegold and N. E. Phllps, Phys. Rev. 77, 383 (969). [8] W. J. L. Buyers, V. F. Sears, P. A. Lonng, and D. A. Lonng, Phys. Rev. A, 697 (97). [9] D. A. Peek, M. C. Schmdt,. Fujta, and R. O. Smmons, Phys. Rev. B 4, 967 (992). Unauthentcated Download Date //8 :3 PM