Neutron-electron scattering length extraction from the neutron diffraction data measured on noble gases

Transcription

1 Neutron-electron cattering length extraction from the neutron diffraction data meaured on noble gae L.V.Mityna, V.G.Nikolenko, S.S.Parzhitki, A.B.Popov, G.S.Samovat Frank Laboratory of Neutron Phyic, Joint Intitute for Nuclear Reearch, 14198, Dubna, Ruia Abtract The propoed new method for extraction of the neutron-electron cattering length b ne, which i connected directly with the fundamental phyical value of neutron mean quared charge radiu, from the low neutron diffraction on noble gae wa developed and performed by analyi of literature data on tructure factor for gae Kr, 36 Ar and liquid Kr. For the mot part, different variant of analyi of thee experimental data allow u to obtain b ne value with the tatitical accuracy 1 %, but in ome of them the real poibility for the acceible accuracy no wore than 3% wa hown. In order to remove ome ytematical error and to reach uch good accuracy the execution of the comparative diffraction meaurement i propoed with a pair of gae having cloe atomic propertie and different n,e-cattering contribution, uch a Ar 36 Ar or Xe Kr. PACS. 8..-v, q, 14..Dh,13.4.-f 1. Introduction Hitory of the n,e-interaction invetigation began nearly 7 year ago, the firt mention of thi effect wa made by D.Ivanenko in 1941 [1]. However, in pite of exitence of a dozen of the mot precie experimental b ne value with error <.5 1 fm obtained by different method (ee, for example []) there were doubt about the validity of their declared accuracy, a they differed by ~5 tandard error. In all method there are principal trouble connected with the neceity of introducing large correction, whoe ize of order of the invetigated effect. Thu, further invetigation of thi problem with the aim to adjut bne value are deired and required a before. The intrigue i that in the interval about ±1% all b ne value were cattered around the o-called Foldy cattering length μe = = fm, Mc b F which, according to the Dirac generalized equation for neutron, i connected with the interaction between the neutron anomalou magnetic momentum and electric field. One may note that the term attached to dive r in the Dirac generalized equation for neutron and in the Dirac equation for electron have the ame tructure h eh r [ ε + ( + μ)] dive 4Mc Mc and depend on the particle magnetic momentum μ only, and parameter ε appear jut in the equation for neutron. In cae of b ne and b F value the coincidence parameter ε mut be equal to zero. Poible equality of b and b value would be urpriing, a it ignifie that ne F 1

2 the charge ditribution of a neutron cattered on the outide charge become diplayed through magnetic momentum only. So cloene of b ne and b F value will confirm thi nontrivial phenomenon. From the point of view of comprehenion of all correction introduced into the experimental data for making effect the implet experiment to obtain b ne value i the low neutron cattering by noble ga. Neverthele, in thee experiment it i impoible to ignore the diffraction on nuclei of neighboring atom even at low ga denitie. So, weak dependence of b ne value obtained from the data of [3] on the ga preure wa noticed and taken into account in [4], whereupon the extracted b ne value were decremented by 1% for Kr and by 5% for Xe. Seemingly, the negative role of neutron diffraction at the obervation of n,e-interaction effect wa noticed firt by A.I.Akhieer and I.J.Pomeranchuk [5] oon after appearance of the claical paper by E.Fermi and L.Marhall [6], which wa one of the pioneering attempt to dicover n,e-cattering. The warning [5] wa evidently known to the author of one of the mot accurate reult for the n,e-cattering length obtained in [3]. The author of [3] ued gae at low preure atm jut to detect poible diffraction ditorting the reult. The new method to obtain b ne value propoed in Dubna [7] permit to ue much more dene gae, inamuch a the diffraction can be taken into account with atifactory accuracy when extracting the n,e-effect. Our method i baed on two fact: 1) diffraction and n,e-interaction contribution to cattering cro ection have very different dependence on the momentum tranfer ћq (ocillating and monotonou); ) diffraction contribution to the cattering i proportional to a ga denity n (or even ha a term with n ), but the nuclear and n,e-cattering do not depend on n. Invetigation of the neutron diffraction in monatomic gae in quet of interatomic interaction potential are progreing in the lat decade. A for ga denitie, the preure up to hundred atmophere are ued in thee experiment. So, the effect of n,e-cattering in them i eentially more than in the reearche like [3].. Baic relation According to conception accepted in literature we decribe the neutron cattering intenity per one target atom and unity neutron flux with the accuracy enough for our purpoe by the following expreion di( σ ( ) nc q σ coh = { F ( V, q, A)(1 + B) + F ( V, q,a), 4 1 ( ) + B (1) dω π nc q σ 8π acohbnezf ( where B =, q i the wave number of momentum tranfer, σ and σ total and coherent nuclear cro ection, the number of electron in atom, q a coh σ coh are the i the length of coherent nuclear cattering, Z i / [ 1+ 3( q / ] f ( = q ) i the electron form factor of an atom, i the Hartree-Fock contant, which characterize atomic propertie (ee [8]). For taking into account the influence of the atom thermal motion on the neutron cattering ditribution in expreion (1) we did not ue the Placzek correction, which are generally applied in the work for diffraction invetigation, but more correct (ee [9]) kinematic decription. It wa ued in [3] and et out in detail in V.Turchin book [1]. The thermal motion effect i decribed by function F

3 F ( V ( A + 1), θ, A) = A π V U V + V V V V coθ A V V coθ ( V V ) exp A 1 A dv. A () 4( ) U ( V + V V V coθ ) A + 1 In expreion () θ i the cattering angle of neutron in the laboratory reference frame, V and V are the initial neutron velocity and it velocity after cattering, А i the atom ma kt T number, U = = [m/], T i the ga temperature in K. ma A Function C ( i connected with the tructure factor S( by formula nc( S( 1 =. (3) 1 nc( The peculiarity of expreion (1) applied by u i the ue of the function F attached to diffraction term with doubled atom ma, a the diffraction, firt of all, i connected with the interaction between the neutron and pair of atom having coincident velocity vector. F 1,4 1, 1, 1,98,96 B -,5 -,1 -, o q, A -1 Fig.1. Contribution of thermal motion correction to the neutron cattering intenity for Kr calculated for neutron diffraction cattering on ingle atom (curve 1) and on two atom (curve ) and calculation of n,e-contribution. The value of eparate effect for gaeou krypton are hown in Fig.1 and Fig.. It i een, that n,e-contribution to nuclear cattering i ~1% or le and doe not depend on the 3

4 ga denity. And the diffraction amplitude rie trongly with the ga denity increae at preerving poition of their maximum and minimum. di nucl 1,15 1 1,1 1,5 di dif,5 4, 3 -, o q, A -1 Fig.. Calculation for Kr of the nuclear and n,e-cattering term contribution to the neutron intenity, which doe not depend on the ga denity curve 1 and diffraction term dependent on the ga denity: curve at the preure 1 atm., curve 3 at 1 atm., curve 4 at atm. The problem of b ne value determination come to accurate eparating the n,econtribution from the diffraction wave with taking into account the thermal motion of atom. Unfortunately, for diffraction decription there i no trict theory yet, o in our analyi we ued different expreion decribing relaxing ocillation of the correlation function C(, uch a πq C ( = ( A1 An)exp( A3in( + A5 ), (4) A4 where one or two out of five parameter A1 A5 are ometime fixed. Teting the different phenomenological formula for C( decription including hardcore model approach (a in []) adjudicated that all conidered variant with the exception of the lat one fit to the experimental data atifactorily. 3. Reult The experimental data of [11 13] were ued, where the S( value were obtained for gaeou Kr, iotope 36 Ar and for liquid Kr, in order to tet the propoed method of the b ne value obtaining. Moderate accuracy (~5 1-3 ) of thee experimental S(, which wa ufficient to obtain information about interatomic interaction potential, neverthele, allowed u to extract b ne value with the accuracy 1 %. To achieve better accuracy it i deirable to meaure the cattered neutron intenity for noble gae with the relative accuracy

5 3.1. Gaeou Kr In [14] we ued experimental data S(q,n) meaured for gaeou Kr and publihed in [11]. The tructure factor were obtained for 17 ga denitie n = ( ) 1 1 cm -3 and 78 value of q up to 4 Å -1. To normalize the neutron cattering intenity the author of [11] ued vanadium, auming that it cattering cro ection i iotropic in the interval of q from up to 4 Å -1, and the calculated S(q,n) value uing the expreion di( ϑ) σ = [ 1+ B f ( ] { [ S( q, n) ] γ + 1+ P1 ( }, (5) dω 4π where BB i the correction for n,e-cattering with b ne =.34 1 fm, P 1 ( i the well known Placzek correction, which i uually applied to decribe thermal motion of atom in olid tate experiment. We ued all S(q,n) from [11] and decribed them by fitting formula, which enued after equating the right part of equation (1) and (5) and finding a olution of combined equation relative to the experimental S(q,n) from (5). We kept the normalizing multiplierα before the fitting formula to take into account inaccurate knowledge of krypton and vanadium cro ection. In our expreion (1) term with n appeared in diffraction dependency on ga preure due to linear in n functionc( = C ( nc1 (. Thu, in the fitting formula for decribing the experimental S( n, value for all n for each q there were three varied parameter: C,C 1and B. b ne,1-3 Фм -1 - C 1,,1, C,1, -, o q, A -1 Fig.3. Fitted parameter b ne, C and C 1 with F-correction and α =.975 (light point) and with the Placzek correction and α =1 (black point). 5

6 The reult of fit are hown in Fig.3. The upper picture how a contancy of b ne parameter within the error bar, which mean good quality of the experimental data decribed with the accuracy 3 1. The performed fit with replacement of function F by Placzek P ( q 1 ) howed that parameter C and C 1 (ee Fig.3) are practically inditinguihable, but b ne ytematically differ by fm. Thi circumtance denote once more the importance of correct calculation of atomic thermal motion influence on the extracted b ne etimation. Furthermore, the experimental data for all q and all n imultaneouly were fitted by ten parameter: α, b ne and eight parameter Ai for decribing function C and C 1 by formula (4) (auming that A =). Example of experimental data decribing for four choen ga denitie demontrate the atifactory quality of thi fitting at the lower part of Fig.4. The reult obtained with free parameter α i placed in the firt line of Table. C,1, -,1 C 1,, S-1,5, n=.58 -,5 n=1.96 n=4.1 n=6.19 -, o q, A -1 Fig.4. The reult of imultaneou fitting to the data for all q and all n. Thereby, in the propoed method the diffraction on neighboring atom doe not diturb obtaining the b ne value, and that differentiate our method profitably from the reearche with liquid metal [], where correction for diffraction, which i more than n,eeffect, i calculated analytically. 3.. Gaeou iotope 36 Ar A 36 Ar ha an anomalouly large cattering cro ectionσ 78 b, and n,e-effect for it mut be 1 time le than for natural Ar, o extracting b ne value from the 36 Ar diffraction data we made a check of our method enitivity. 6

7 In [15] we ued S( data obtained in the Intitute Laue-Langevin in Grenoble [1] for q = (.4.1) Å at four denitie of gaeou iotope 36 Ar with the accuracy from.6% up to.%. Small number and narrow range of denitie n did not allow u to get b ne value for each given q eparately (a in the cae of gaeou Kr, ee 3.1). We fitted thee data for each denity of 36 Ar in the wide interval of q value uing formulae (1) and (4). The author of [1] normalized experimental S( value on the data at large q neglecting the diffraction there and not taking into account n,e-cattering. Their thermal motion correction differed from F -correction () by a contant multiplier cloe to 1 only, o their intenity we decribed by expreion (1) with F 1 = F = 1 and with varied multiplier α taking into account the difference of normalization. During data proceing a trong correlation took place between b ne value and α ( Δ α ~ 1-3 changed b ne by 1%), and inufficient preciion of normalization did not allow u to get b ne value with the accuracy better than 1 %. We could decribe experimental data of S( atifactorily at q 3 Å only. Moreover, all the data for the lowet denity of argon were rejected becaue of large χ criterion. The weighted average reult for the ret three denitie i hown in Table (line ) with the tatitical error. However, the ytematic error of normalization evaluated over a wide interval q = ( 8 ) Å -1 turned out two time more than the tatitical one Liquid Kr Although b ne value extracting from the neutron cattering on liquid i not o clear in comparion with the cattering on gae becaue of the problem of adequate conideration of thermal motion for atom of liquid, the b ne value wa alo obtained from the diffraction data meaured in Grenoble [13] for even cloe denitie of liquid Kr in the wide range q ~ (,4 7) Å. We ued thee experimental data to verify different verion of bne value extracting. They were decribed in detail in [16,17]. Out of ~5 experimental data preented in [13] we ued S(n, in the interval q 4 6 Å, becaue we could not get fit with χ 1 analyzing all the data (with q < 4 Å ). Simultaneou fitting of all even denitie with the ame parameter A i of function C( gave the reult = (1.39 ±.4) 1 fm. The reult of fitting i hown in Fig.5. b ne However, the correlation between bne value and normalizing contant α led to b ne value pread up to ± 3%. Uing individual diffraction parameter fitted for each of even ample and fixing them afterward in the following fitting with free b ne value and with even different normalizing parameter α i we obtained b ne = (1.63 ±.) 1 fm. Thi error i 1 15 time le than for gaeou Kr and 36 Ar, where it i caued by correlation between bne and α. We ucceeded in the break off correlation here due to wider region of q reached q > 16 Å, where n,e-contribution i negligible. Thi example demontrate future propect of obtaining bne value with the accuracy ~ % by mean of performing imilar meaurement with gaeou ample. 7

8 Unfortunately, thi reult can not be accepted a ignificant one on account of eential change of the extracted b ne value at varying the limit of the working q-interval. It can be evidently explained by inufficiently atifactory decription of neutron diffraction in liquid tate of Kr. S 1,1 χ =3748 /568 Kr 1,, , 1,1 1,,99, o q, A -1 Fig.5. The imultaneou fitting for S( n, of liquid Kr for even denitie. The data for three denitie n = 11.86, and 17.1 nm -3 are hown only. Point are experimental data, curve are fitting reult. Table Target Fitting <b ne > 1 3, fm Kr for all q and all n imultaneouly 1,53±,4 36 Ar averaged value of imultaneou fit for each n 1,33±,8±,57 liquid Kr average value obtained by interval dividing 1,38±,7 average value of fit for three ample with imilar n 1,4±,1 To ecape afely a correlation between α and b ne we eparated n,e-part in our fitting function (detail ee in [16]): exp fit p( = α[1 + Bf ( ] = S ( /{1 + γ[ S ( ]}, where S exp ( are the experimental and S fit ( are the fitted tructure factor, and after dividing the working q interval into ix part, which correponded to ix viible period of 8

9 the diffraction, obtained three b ne value from the um ratio p / for i / j = 1/ 4, /5 and 3/6 (indexe i, j belong to different part of the interval). Thu, we excludeα. The average of three obtained b ne value i placed in the third line of Table. The averaged reult of fitting to the experimental data for three group of ample with imilar denitie, which were done in [17], i preented in the 4-th line of Table. 4. Concluion Poibilitie of the new method to extract the n,e-cattering length from the meaured tructure factor were entirely hown with the data of experiment [11 13]. The data analyi for gaeou Kr demontrated two variant of the data proceing for b ne value extracting: with the ue of n- and q-dependence of experimental tructure factor. Fundamental capability to olve the problem of b ne obtaining by new method wa alo hown by the model experiment performed by the Monte Carlo procedure [18], where angle ditribution of neutron with the energy.143 ev cattered by gaeou Kr were conidered. Calculation allowed u to come to a concluion that having the data for or more angle at 3 a few ga denitie in the range from cm to cm with the relative accuracy of experimental point 3 1, we could extract the bne value with the accuracy no wore than 5%. High enitivity of the new method to the n,e-cattering length extracting wa alo hown in the work with 36 Ar, where n,e-effect i 1 time le than for natural argon. A problem of the extracted neutron-electron cattering length preciion in the propoed method depend not only on the tatitical accuracy of the meaured angle ditribution of neutron cattered by ga but alo on an extent of diffractometer iotropy the contancy of the product of detector efficiency and the olid angle at all angle. The excluion of the dependency of neutron probability regitration by the detector on a cattering angle i fundamentally important tak. In [11] thi problem wa olved by normalizing the angle ditribution of neutron cattered by gaeou krypton on uch ditribution of neutron cattered by vanadium. However, the order of vanadium iotropy i not known with the accuracy, which would be ufficient for our purpoe (ee [19,]). To obtain b ne value with the accuracy ~ % the facility aniotropy mut be le than 4 ~ 3 1, that i practically impoible condition. The only poibility of thi trouble avoidance i to realize relative meaurement of two noble gae with trongly different n,econtribution to cattering cro ection, for example Ar and 36 Ar, which have n,econtribution 1.7% and.%, or Xe and Kr, whoe one are.1% and 1.%, correpondingly. The author are grateful to Dr. Renato Magli for valuable dicuion and for hi kind ending to u the numerical data for 36 Ar and liquid Kr, and alo to Dr.Yu.A.Alexandrov for ueful dicuion. The work i upported by RFBR, grant i p j Reference 1. D.Ivanenko, ZhETF, 11, 197(1941) (in Ruian).. S.Kopecky, J.A. Harvey, N.W. Hill, M. Krenn, M.Pernicka, P. Rieh, and S. Steiner, Phy. Rev., C56, 9(1997). 3. V.E. Krohn, G.R. Ringo, Phy. Rev., 148, 133(1966); Phy. Rev., D8, 35(1973). 4. L.V.Mityna, V.G.Nikolenko, A.B.Popov and G.S.Samovat, in X International Seminar оn Interaction of Neutron with Nuclei, Dubna, May -5,, JINR, 9

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