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INSTANTANEOUS X-RAY EMISSION INDUCED BY ION-ATOM COLLISIONS
K. Hino, T. Watanabe
To cite this version:
K. Hino, T. Watanabe. INSTANTANEOUS X-RAY EMISSION INDUCED BY ION-ATOM COLLI- SIONS. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-271-C9-274. �10.1051/jphyscol:1987947�.
�jpa-00227365�
INSTANTANEOUS X-RAY EMISSION INDUCED BY ION-ATOM COLLISIONS
K. HINO and T. WATANABE
Atomic Processes Laboratory, The Institute of Physical and Chemical Research ( R I K E N J , Wako, Saitama, 351-01, Japan
~ 6 s u m e - ~'Amission des rayons X dans la coll ision ion-atorne est c o n s i d k r B e d u point d e v u e d e I 1 i n c o n s i s t e n c e theorique.
L'attention est faite particul ikrement b la capture radiative d'klectrons (REC). Nous avons m i s en relief le fait que le principe-guide pour la consistence du traitement theorique est la demande de Itinvariance de jange pour la formation de la matrice S. Nous avons du le resul tat que ltapproximat ion des impulsions et la forte approximation Born potentielle pour R E C satisfait approximatiotivement h cette demsnde. Un applic t'on est faite pour le processus R E C dans les chocs Xe'4+ et U'" sur la but Be.
Abstract
-
X - R a y emission in ion-atom collision is considered from the standpoint of theoretical consistency. Particular attention is payed for radiatvie electron capture (REC). I t is shown that a guide principle to obtain the formula is the gauge invariance requirement. W e have obtained a results that the impulse approximation and the strong-potential Born approximation for REC satisfy approximatgie this ~2quirement. An application is made forRECprocess i n X e a n d U + impacts o n B e target.Instanteneous X-ray emmi s ion processes induced by the interact ion of an energetic heavy ion with an target atom are considered from the standpoint of theoretical consistency.
I t is known that there are some inconsistencies in the theoretical
t r e a t m e n t on r a d i a t i v e p r o c e e s e s induced by i o n - a t o m collisions.
Particularly the calculated cross section for radiative electron capture (REC) process from a neutral target atom by a high velocity projetile ion does depend on the coordinate frame emplyed in the formula of non- relativistic quantum mechanicsllf. T h e cross section obtained in the projectile coordinate frame is different from that obtained in the target coordinate frame. I t was pointed out by Shakeshaft and Spruch in 1977 121 that this is due to the difference of the basic equations which satisfy for particle motions and for field interacbtion. T h e former follows the Schroedinger equation which is invariant in the Galilean transformation and*
the latter does the Maxwellian equation which is invariant in the Lorentz transformation. The photons which are relevant to the motion of the center- of-mass of the total system is derived and i t is named by Shakeshaft and Spruch as "spurious radiation" and i t has been told that if one take the center-of-mass coordinate system, this spurious radiation term can be eliminated.
I t w a s , however, shown by Hino in 1984 131 that the so called "spurious
radiation" can not be removed by the full quantum mechanical calcualtion (Born approximation) using the center-of-mass coordainte frame. This calculation predicted also the fact that the R E C cross section depends on th mass of the target atom. T h e cross section for REC in collision of a NePo' on a 4 ~ e is expected to be about t w o times as m u c h as the cross
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987947
C9-272 JOURNAL DE PHYSIQUE
section of the same ion impact on 3 ~ e in the Born calculation using center- of-mass frame but i t was denyed by an experiments. The both cross sections w e r e measured te be the same within 7% error by Kambara et. al. / 4 / . Furthermore, i t was found that the usual S matrix for REC process written by Fermi's golden rule have counted the same physical process (the Feynman diagramme) by two times in the lowest order. I t w a s found out that this double counting comes from the fact that w e do not know the exact solution of scattering state of three particles. There exists some ambiguity in choosing the sattering state wavefunctions of initial and final states.
Here the inital and the final states do m e a n those just before and after the X-ray emission and these are different from asymptotic formulas in scattering problems.
W e have come to the conclusion that the gauge invariance property should be examined throughout the whole formulation of the present problem. In the present research, w e are main1 y concerned with electron bremss trahlung (EB), atomic bremsstrahlung ( A B ) , radiative electron capture (REC) and molecular orbital X-ray emmission (MI X-ray). Particular attention is paid for REC process. The radiation process treated here classified into three, i. e. two-body collision process and the interaction with radiation fields, three-body non-rearrangement collision processes and the interaction with radiation fields, and three-body rearrangement collision process and the interaction w i t h radiation fields. For simplisity i t is assumed that the system consists of Coulomb-interact ing two or three particles, i.e. an electron (e) and a nucleus (A) or a parojecti le nucleus (A) and a hydrogen-
like target atom [(B,e)l.
Firstly w e consider the radiative process on the occasion of two-body collisions as a preliminary discussion on the problems which concerns three bodies and radiation fields. The S matrix formulas for bremsstrahlung and radiative recombinat ion, i. e.
and
are known to maintain the gauge invar iance property if w e use the exact initatial and final wavefunctions. W e examine h o w the validity of gauge invariance property holds in the perturbat ion (Born) expansion series on scattering states. W e have found that the gauge invariance holds in the respecitve order of Born expansion series in the case of bremsstrahlung, but the gauge invariance does not neccessarily hold in the respective order of the series in the case of radiative recombination processes.
Secondly, the discussion is made on the three-body collision processes and the reconsideration is m a d e from the viewpoint of gauge invariance requirement. When w e make a similar consideration on three-body problems by using perturbation expansion method. W e take the case of radiative electron capture (REC) as a typical example;
the initial and the final state wanvefunctions can only be given in terms of a perturbation exansion series, because w e do not have the exact forms of three-body scattering states; A + (B,e) and (A,e) + B. I t can be shown that all these unreasonable points such a s frame dependent results and double conunt ing of Faynman's di agramme etc. are removed by taking gauge invariance requirement into consideration for the expression of the initial and the final-state wavefunctions.
-- bJe have been investigated h o w the gauge invar iance property maintain in
interaction, Vm and explands the S matrix in terms of the interaction between A and e, VAe, or B and e, VBe, with no-poteni tal Green's function) and in the Born approximation (that expands the S matrix in terms of VAB +
" ~ e ~i th no-potent ial Green's function) the gauge invar iance does
tVgl; '~ph e ~ m p u l s e ' approximation (IA) and the strong-potetial Born approximation (SPB) 151 (that expands the S matrix in terms of V with Green's f u n c t i o n under VAe w i t h o u t ( I A ) and w i t h ( S P B ) of P-eshel 1 contributions) holds the gauge ~nvariancefor the respective order of power expansion. W e a l s o h a v e a c o n c l u s i o n that w h e n w e u s e the B o r n approximation for the calculation of R E C cross sect ion, the projectile frame is most proper among three coordinate reference frames (projectile, target, and center-of-mass) from the view point of the gauge invariance. In this sense, the concept that "REC is regarded as a reverse process of photoionization" is valid within the velocity region of "not-extremely relativistic".
In the case of radiative deexcitation and radiative excitation:
and
the gauge inveriance holds even in the Born approximation (that expands the S matrix in terms of VA, with no-potential green,^ function).
A remark is due about the X-ray angular distribution for REC processes in a relativistic range. The angular distibution of phtons from moving particles in the moving projectile frame deviates from the sin% -rule. By the Lorentz transformat ion from the moving frame into the laboratory frame, however, this deviation is almost cancele$out. W e obtain an angular distribution approximately propertional to sin 0 where O L is the photoemission angle w i t h respect to the projetile direction in the laboratory frame. T h e present formalism warns, however, that this kind of treatment requires a careful study of the r e f e r e n c e f r a m e f r o m the v i e w p o i n t of g a u g e invar inace. I t has been proved that a proper angular distribution can be obtained only by calculations in the moving projetile frame followed by the transformation into the laboratory frame,
if
w e employ an approximation that destroys the gauge invariance. According to the argument given above, a proper angular distribution can be obtained using the laboratory frame from the beginning. However, the use of the S P B approximation that satisfies the gauge invariance avoids such a difficulty completely.Finaly w e have calcu ted the total cross sections for the radiative lectron capture by XekBjf and LJg2+ projectile ions on B e targets and the angular distributions of emitted X-rays using the relativistic strong- potential Born (RSPB) approximation /5,6/ and also using other methods. W e have compared the results w i t h experimental data 171 and with the calculations using other theories 161. In Figs. 1 and 2 the total cross section for REC by RSPB approximation for are shown together with experimental data and the results of calculation by other theoris.
REFERENCES
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/2/ R. Shakeshaft and L. Spruch, Phys. Rev. Lett. 38 (1977) 175.
131 K. Hino, P.XS Thesis, Department of Applied Physics, The University of
C9-274 JOURNAL DE PHYSIQUE Tokyo (1984)
/4/ T. Karnbara, Y. Awaya, M. Kase, H. Kurnagai, H. Shibata and T. Tonuma, J.
Phys. Soc. Jpn 56 (1987) 1907.
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Fig. 1. Fig. 2.
Figs. 1 and 2. Tot54+cross %$ctions for the radiative electron capture processes in t h e X e
-
a n d U +-Be collisions versus the incident velocity V/C and the Coulomb parameter between an active electron and the projectleion / 6 / . Solid line (- ) : the relativistic strong potential Born
(RSPB) apprximation, chain line (--- -): the RSPR calculation without the Lorentz contraction factor based on lthe motion of a target nucleus in the moving frame (RSPB-11), dotted chain line ( - 0 - ) : the relaivistic Born c a l c u l a t i o n (RB), break line ( - - - ) : the n o n - r e l a t i v i s t i c S P B c a l c u l a t i o n , d o t t e d line ( * - * * a s - - + * ) : the r e l a t i v i s t i c i m p u l s e approximation calculation (RIA) and Experimetnal results are quated from the paper by Meyerhof et a1 171.
