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The process of multiple scattering in rare earths
P. Takhar
To cite this version:
P. Takhar. The process of multiple scattering in rare earths. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-84-C5-85. �10.1051/jphyscol:1979532�. �jpa-00218949�
JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 5, Tome 40, Mai 1979, page C5-84
The process of multiple scattering in rare earths
P. S. Takhar
Royal Military College, Kingston, Canada (*)
Résumé. — Nous calculons les portées théoriques des électrons et positrons dans l'approximation d'un ralentis- sement continu. Le rapport théorique des portées R£ jR~ est calculé pour Se, Y, La, Nd, Eu, Ho et Yb et comparé à d'autres métaux. Nous citons quelques résultats pour des monocristaux.
Abstract. — Theoretical ranges for electrons and positrons were calculated based on continous slowing down approximation. The theoretical ratio of ranges Rp/Rp was calculated for Scandium, Yttrium, Lanthanum, Neodymium, Europium, Holmium and Ytterbium. The calculated values of the ratio of ranges for the rare earth metals is compared with the other metals. Also some results on single crystals are reported.
1. Introduction. — A number of workers [1-2]
have reported on theory of multiple scattering of electrons and positrons. Recently some experiments were performed to estimate electron-positron diffe- rences [3]. Berger and Seltzer [1] reported theoretical ranges. In this paper this theory fl] is extended to include other effects such as radiation losses.
2. Theory. — According to continous slowing down approximation Berger and Seltzer [1] calculated positron range R +, electron range JR ~ and their ratio using following equations :
(*) Present Address : A. M. University, Aligarh, India.
where
P \te)tal P \d XA o l l + P \d xA a d '
The + subscript stands for positrons and — super- script stands for electrons and p is density of the absorber.
Authors have modified ranges given by equation (1) to include the effects of multiple scattering and radiation losses. The new range is called projected range Rp and is given by the following equation :
R?(T) = RLaJT) - RL*.,(Tr±). (3) Where T is the energy of the particle and Tr corres- ponds to the energy where diffusion phenomenon sets. Thus straggling free range can be estimated by using equation (3).
Table I. — A) Comparison of theoretical projected ranges in mg/cm2 and ratio or ranges for 1.77 MeV electrons and 1.88 MeV positrons for rare earths.
Electron Positron Theore- Theore- Theore- Experi-
T~ tical Tr+ tical tical mental Element Z MeV r - ,d.a. R^JTT) K MeV K+5.d.a. R ^ t f ) K KIRV V+/ V
Sc 21 0.712 1025 386 639 0.680 1 163 375 788 1.23 — Y 39 0.950 1235 618 617 0.945 1 360 625 735 1.19 1.44 La 57 1.133 1313 803 510 1.126 1453 820 633 1.24 — Nd 60 1.150 1325 818 507 1.155 1460 846 614 1.19 1.67 Eu 63 1.183 1373 857 516 1.110 1470 843 627 1.22 — Ho 67 1.200 1350 887 463 1.190 1480 904 576 1.23 1.85 Yb 70 1.220 1 355 913 442 1.215 1490 937 553 1.25 1.47
B) Physical properties of rare earths
Element Sc Y La Nd Eu Ho Yb Atomic number 21 39 57 60 63 67 70 4f electrons — 0 3 6 10 10 14 Unpaired electrons — 0 3 6 4 4 1
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979532
THE PROCESS OF MULTIPLE SCATTERING I N RARE EARTHS C5-85
3. Calculations and results. - The values of R,f,.,,.(T) (see table I) were obtained from the tables compiled by Berger and Seltzer [I]. The values diffusion energy T: were calculated on the basis of new theory [4] and the corresponding values of R,f,,.,.(T,) are also listed in table I. Theoretical values of projected ranges (See table I) in a number of rare earths, such as Sc, Y, La, Nd, Eu, Ho and Yb were also calculated. The c.s.d.a. ranges and diffusion energy for rare earths listed in table I were obtained by interpolation, from the data provided by the other elements such as Al, Cu, Sn and Pb. The cal- culations were performed for 1.88 MeV positrons and 1.77 MeV electrons for comparison with the experiments [3] (See table I). Theoretical calculations
were also performed for NaCl, KCl, KBr and KI.
These are listed in table I1 for comparison with the rare earths (See table I).
A comparison of theory and experiment [3] indi- cates, that there is a fairly good agreement [4] for Al, Cu, Sn and Pb, however there are significant differences for Y, Nd, Ho and Yb (See table I). The experimental values of ratio of rages for Al, Cu, Sn and Pb ranging from 1.12 to 1.35, were reported elsewhere [4, 51.
The large discrepancy between experiment and theory, for rare earths, may be due to the fact, that rare earths have lots of electrons (See table I) in 4f shell.
Table 11. - Comparison of theoretical projected ranges in mg/cm2 and ratio of ranges for 1.77 MeV electrons and 1.88 MeV positrons in crystals.
Electron -
T; Theoretical
Element Z MeV R&A.~. R i s . d . a. (Ti) - - - - - RP
-
NaCl 28 0.805 1 140 501 639 KC1 36 0.923 1220 57 1 649 KBr 54 0.1108 1 300 763 537 KI 72 0.1330 1350 994 356
Positron -
T: Theoretical
MeV . . R: R;IR,
- - - -
0.840 1 280 511 769 1.20 0.930 1 350 595 755 1.16 0.1113 1440 772 668 1.24 0.1215 1467 895 572 1.61
References
[l] BERGER, M. J. and SELTZER, S. W., NRCPublication No. 1133, (1964) 305.
[2] BATRA, R. K. and SEHGAL, M. L., Nucl. Phys. A 156 (1970) 314.
[3] TAKHAR, P. S., Phys. Lett. 28A (1968) 423.
[4] TAKHAR, P. S. and GILL, T. S. et al., Proceedings 12th Rare Earth Research Conf., V-1 (1976) 855.
[5] TAKHAR, P. S. and GILL. T. S.. The Rare Earths in Modem Science and Technology (Plenum Press, N . Y . ) 1977 p. 355.
