Numerische Tabellen für Beta-Zerfall und Elektronen-Einfang / Numerical Tables for Beta-Decay and Electron Capture

322 Pages·1969·30.225 MB·German

by  H. Behrens, J. Jänecke (auth.), H. Schopper (eds.)

LANDOLT -BÖRNSTEIN
NUMERICAL DATA AND FUNCTIONAL RELATIONSHIPS
IN SCIENCE AND TECHNOLOGY
NEWSERIES
EDITOR IN CHIEF
K.-H. HELLWEGE
GROUP I: NUCLEAR PHYSICS AND TECHNOLOGY
VOLUME4
NUMERICAL TABLES FOR BETA· DECAY
AND ELECTRON CAPTURE
BY
H. BEHRENS· J. JÄNECKE
EDITOR
H. SCHOPPER
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1969
LANDOLT - BÖRNSTEIN
ZAHLENWERTE UND FUNKTIONEN
AUS NATURWISSENSCHAFTEN UND TECHNIK
NEUE SERIE
GESAMTHERAUSGABE
K.-H. HELLWEGE
GRUPPE I: KERNPHYSIK UND KERNTECHNIK
BAND 4
NUMERISCHE TABELLEN FüR BETA· ZERFALL
UND ELEKTRONEN· EINFANG
VON
H. BEHRENS . J. JÄNECKE
HERAUSGEBER
H. SCHOPPER
SPRINGER-VERLAG BERLIN HEIDELBERG GMBH
1969
ALLE RECHTE VORBEHALTEN.
KEIN TEIL DIESES BUCHES DARF OHNE SCHRIFTLICHE GENEHMIGUNG
DES SPRINGER-VERLAGES üBERSETZT ODER IN IRGENDEINER FORM'
VERVIELFALTIGT WERDEN.
© SPRINGER-VERLAG BERLIN HEIDELBERG 1969
URSPRÜNGLICH ERSCHIENEN BEI SPRINGER-VERLAG, BERLINIHEIDELBERG 1969
SOFTCOVER REPRINT OF THE HARDCOVER 1ST EDITION 1969
LIBRARY OF CONGRESS CATALOG CARD NUMBER 62-53136
ISBN 978-3-662-37716-1 ISBN 978-3-662-38531-9 (eBook)
DOI 10.1007/978-3-662-38531-9
Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere
Kennzeichnung nicht Zu der Annahme, daß solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu
betrachten wären und daher von jedermann benutzt werden dürften
Titel-NUIllIIlC1" 6185
Vorwort
Die beim ß-Zerfall und beim Elektroneneinfang experimentell beobachtbaren Größen hängen
nicht nur vom Kern selbst ab, sondern sie werden auch durch die Vorgänge in der Atomhülle
bestimmt. Diese Effekte werden mit Hilfe von Funktionen berücksichtigt, für die eine Reihe
von Tabellen vorliegen. Diese weisen aber zum Teil erhebliche Widersprüche auf. Ein großer Teil
der früher publizierten Meßergebnisse wurde infolgedessen mit unzureichenden Tabellen aus
gewertet.
In den letzten Jahren gelang es, durch eine neue Formulierung der Theorie des ß-Zerfalls die
kernphysikalischen Effekte sauber von den Hülleneinflüssen abzutrennen. Auch wurde die Behand
lung verschiedener Korrekturen dadurch transparenter. Da außerdem der Einfluß verschiedener
Näherungsverfahren auf die Berechnung der Elektronenwellenfunktionen jetzt besser zu verstehen
ist, können verschiedene Rechenverfahren nunmehr zu übereinstimmenden Ergebnissen führen.
Aufgrund dieser neuen Erkenntnisse ergab sich ein dringender Bedarf an verbesserten Funktionen
für den ß-Zerfall und den Elektroneneinfang. Diese Lücke wird durch die hier publizierten Tabellen
geschlossen.
Da die Bezeichnungsweise in der Literatur sehr stark variiert, und in einigen Fällen die Formeln
nur unvollständig angegeben wurden, ist den Tabellen eine Zusammenstellung von Formeln voran
gestellt, die alle Korrekturen höherer Ordnung enthalten.
Die Ausgabe-Daten der Rechenmaschine wurden unmittelbar als Druckvorlage benutzt. Dieses
Verfahren, das sich bereits früher bewährt hat, wurde auch in diesem Fall vom Verlag mit Sachkunde
und Verständnis gehandhabt.
Karlsruhe, Dezember 1968
Der Herausgeber
Preface
The measurable quantities in ß-decay and electron capture do not depend on the nucleus alone,
they are also determined by processes within the atomic shell. These effects have been described by
functions presented in a number of tables which, however, contain serious discrepancies. Conse
quently, many of the experimental results published in the past have been evaluated on the basis of
inadequate tables.
Thanks to a new formulation of the theory of ß-decay, it has in recent years become possible
to distinguish the physical effects in the nucleus from the influence exerted by the atomic shell.
This also made the treatment of various corrections much easier to follow. Moreover, since the
influence of various approximation procedures for calc.ulating the electron wave functions is now
better understood, coincident results can be obtained by different methods of computation.
This changed situation has generated an urgent need for improved ß-decay and electron-capture
functions, a gap which is closed by the tables published here.
Since there is considerable variation in the nomenclature used in the literature and in some cases
the formulae have not been published completely, the tables are preceded by a collection of formulae
which contain all higher-order corrections.
The book was reproduced directly from the computer output. This procedure has already proved
its worth and was handled here by the publishers with technical competence and good judgement.
Karlsruhe, December 1968
The Editor
Inhaltsverzeichnis
Numerische Tabellen für Beta-Zerfall und Elektronen-Einfang
Von H. BEHRENs, Institut für experimentelle Kernphysik der Universität und des Kemforschungszentrums Karlsruhe,
und ]. ]ÄNECKE, The University of Michigan, Department of Physics, Ann Arbor, Michigan/USA
1 Einleitung . . . . . . . . . . . . 1
1.1 Einfluß des Coulomb-Feldes 1
1.2 .Ältere Tabellen für den tJ-Zerfall ....... 1
1.3 Die hier abgedruckten Tabellen für den tJ-Zerfall . 2
1.4 Ältere Tabellen für den Elektronen-Einfang. . . . . . . . 2
1.5 Die hier abgedruckten Tabellen für den Elektronen-Einfang. . . . . . . . . 2
1.6 Neuere zusammenfassende Darstellungen des tJ-Zerfalls und Elektronen-Einfangs 3
2 Formeln für den tJ-Zerfall und Elektronen-Einfang 3
2.1 Klassifizierung der verschiedenen tJ-Zerfälle 3
2.2 Allgemeine Formeln für den tJ-Zerfall 3
2.2.1 Form der fJ-Spektren . . . . . . . 3
2.2.2 Elektronen-Polarisation . . . . . . . . . . . . . . 7
2.2.3 tJ-y-Richtungs-und Zirkularpolarisations-Korrelation 8
2.2.4 Elektronen-Emission von ausgerichteten Kernen . . . 8
2.3 Spezielle Formeln für erlaubte, einfach verbotene, L-fach nicht unique und (L-l)-fach unique verbotene
tJ-Zerfälle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Teilchen-Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9
2.3.2 Linear-Kombinationen ML(ke, k.) und mL(ke, k.) von Formfaktor-Koeffizienten . . . . . . . . .. 11
2.3.3 Spektrums-Form, Elektronen-Polarisation, tJ-y-Richtungs- und Zirkularpolarisations-Korrelation und
Verteilung der von ausgerichteten Kernen ausgesandten Elektronen. 13
2.3.3.1 Erlaubte Zerfalle .......... 13
2.3.3.2 Einfach nicht unique verbotene Zerfälle . 14
2.3.3.3 Einfach unique verbotene Zerfalle . . 14
2.3.3.4 L-fach nicht unique verbotene Zerfälle 15
2.3.3.5 (L-1)-fach unique verbotene Zerfälle 15
2.4 JI-Werte und integrierte Fermi-Funktion . . . . . 16
2.5 Allgemeine Formeln für den Elektronen-Einfang . ........ 17
2.6 Spezielle Formeln für erlaubte und einfach verbotene Elektronen-Einfänge 20
2.6.1 Erlaubte übergänge ......... 20
2.6.2 Einfach nicht unique verbotene übergänge ...... 20
2.6.3 Einfach unique verbotene übergänge . . . . . . . . . 21
2.6.4 Verhältnis von Elektronen-Einfang zu Positronen-Zerfall 21
3 Beschreibung der Tabellen einschließlich der verwendeten Rechenmethoden 21
4 Tabellen und Diagramme 29
Tabelle I. Beziehung zwischen Impulsen, Gesamtenergien, kinetische Energien und Be-Werten der Elektronen für
diejenigen Impulswerte, die in den folgenden Tabellen auftreten. . . . . . . . . . . . . . . . . . 30
Tabelle 11. Unabgeschirmte Coulomb-Funktionen für den tJ-Zerfall . . . . . . . . . . . . . . . . 31
Tabelle 111. Verhältnis von abgeschirmten zu unabgeschirmten Coulomb-Funktionen für den tJ-Zerfall . 235
Tabelle IV. Integrierter statistischer Faktor . . . . . . . . . . . . . . . . . . . . . . . . . " . 293
Diagramm I und II. Integrierte Fermi-Funktion dividiert durch den integrierten statistischen Faktor für tJ--und
tJ+-Zerfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Diagramm III und IV. Abschirmkorrekturen zu der integrierten Fermi-Funktion für tJ--und tJ+-Zerfall . 298
Tabelle V. Coulomb-Funktionen der gebundenen Atom-Elektronen für den Elektronen-Einfang 301
5 Literatur . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . 315
Table of Contents
Numerical Tables for Beta-Decay and Electron Capture
By H. BEHRENS, Institut für experimentelle Kernphysik der Universität und des Kernforschungszentrums Karlsruhe,
and J. JÄNECKE, The University of Michigan, Department of Physics, Ann Arbor, MichiganjUSA
1 Introduction . . . . . . . . . .
1.1 Effect of the Coulomb field . . 1
1.2 Previous tabulations for ß-decay 1
1.3 The present tables for ß-decay . 2
1.4 Previous tabulations for electron capture 2
1.5 The present tables for electron capture . 2
1.6 Recent reviews of ß-decay and electron capture 3
2 Formulae for ß-decay and electron capture. 3
2.1 Classification of ß-decays . . 3
2.2 General formulae for ß-decay 3
2.2.1 Shapes of beta spectra . 3
2.2.2 Electron polarization . 7
2.2.3 ß-y directional and circular polarization correlation 8
2.2.4 Electron emission from oriented nuclei . . . . . . 8
2.3 Special formulae for allowed, first forbidden, Lth non unique forbidden, and (L-l)th-unique forbidden
ß-decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Particle parameters . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . .. 9
2.3.2 Linear combinations ML (ke, k.) and mL (ke, k.) of nuclear form factor coefficients . . . . . . . . . 11
2.3.3 Spectrum shape, electron polarization, ß-y angular and circular polarization correlation, and distribution
of electrons emitted from oriented nuclei. 13
2.3.3.1 Allowed decays . . . . . . . . 13
2.3.3.2 First non unique forbidden decays 14
2.3.3.3 First unique forbidden decays . . 14
2.3.3.4 Vh non unique forbidden decays 15
2.3.3.5 (L-l)th unique forbidden decays 15
2.4 ft-value and integrated Fermi function . . . . 16
2.5 General formulae for electron capture 17
2.6 Special formulae for allowed and first forbidden electron captures 20
2.6.1 Allowed transitions. . . . . . . . . 20
2.6.2 First non unique forbidden transitions. . . 20
2.6.3 First unique forbidden transitions. . . . . 21
2.6.4 Ratio of electron capture to positron decay . 21
3 Description of the tables, including the calculation methods 21
4 Tables and diagrams . . . . . . . . . . . . . . . . . . 29
Table I. Relations between electron momenta, total energies, kinetic energies and Be-values used in the following
tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Table H. Unscreened ß-decay Coulomb functions . . . . . . . . . 31
Table IH. Ratio of screened to unscreened ß-decay Coulomb functions 235
Table IV. Integrated statistical factor. . . . . . . . . . . . . . . 293
Diagrams land H. Integrated Fermi-function divided by the integrated statistical factor for ß-- and ß+-decay,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 295
Diagrams In and IV. Screening corrections to the integrated Fermi-function for ß--and ß+-decay, respectively 298
Table V. Coulomb functions of the bound orbital electron for electron capture 301
5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Ref. p. 315] 1 Introduction
1 Introduction
1.1 Effect of the Coulomb field
The main object of experiments in the field of ß-decay is to obtain information about form factors.
Form factors can be expressed in terms of matrix elements if it is assumed that the weak interaction
causes the nucleons inside the nucleus to decay like free particles. Precise values for the above quantities,
however, can be extracted only if all effects due to the electromagnetic interaction between the ß-particles
and the nucleus are properly taken into account. Of major importance is the distortion of the wave
function of the emitted ß-particle by the electrostatic Coulomb field of the nucleus. The most obvious
effect of this distortion of the wave function is that it strongly affects the shape of all ß-decay spectra
and makes it necessary to introduce the so-called Fermi-function. To obtain the distortions of the wave
functions, one has to solve the Dirac equation for an unbound electron in the field of a nucleus. For
the exact solution of this problem one has to consider the finite size of the nucleus as well as the screen
ing of the nuclear electrostatic field by the orbital electrons. Such a solution can only be obtained
numerically.
1.2 Previous tabulations for ß-decay
There are a number of published tables of the Fermi-function and of electron radial wave functions
along with the phase shifts. However, in most cases screening corrections were not considered or
were applied only in an approximate way. The Fermi-function of a point nucleus without screening
corrections is tabulated in the NBS-tables [1]. RosE, PERRY and DISMUKE [2] introduced a finite de
Broglie wave length correction by including higher terms in the confluent hypergeometric function.
These authors [2] give tables of Coulomb functions for allowed and forbidden ß-transitions. DZHELEPOV
and ZYRIANOVA [3] published tables for the Fermi-function, for the integrated Fermi-function, and
for forbidden ß-decay Coulomb functions which included corrections for finite size and for screening.
These tables, however, were not obtained from an exact solution of the problem of a nucleus of finite
size screened by orbital electrons but by adding corrections to the functions which were obtained for
a point nucleus. Exact solutions of the Dirac equation for a nucleus with finite size but without screening
were calculated by SLIV and VOLCHOK [4] and more extensively by BHALLA and RosE [5]. The former
tabulated Coulomb phase shifts and amplitudes. The latter tabulated Coulomb phase shifts and radial
wave functions evaluated at the nuclear radius. It has been pointed out [6, 7] that the tables of BHALLA
and RosE [5] are not entirely correct for positrons of higher momentum.
Until recently it was difficult to treat the screening effect of the orbital electrons properly because
the numerical integration of the Dirac equation is particularly complicated and lengthy for this case.
RosE [8] showed that for higher electron momenta, the calculation of screening corrections for the
Fermi-function is simplified by using the WKB method (see also ref. [9]). A compilation of the para
meters needed for the WKB screening correction, i. e., the potential at the nuclear radius caused by
the orbital electrons, is given by GARRET and BHALLA [10]. A certain discrepancy exists between the
values given by these authors [10] and the values given by MATESE and ]OHNSON [11].
The first detailed calculations of screening corrections to the Fermi-function were carried out by
REITZ [12] who numerically integrated the Dirac equation. These corrections have been used in the
above mentioned tables of DZHELEPOV and ZYRIANOVA [3]. Severalauthors [11, 13, 14] have noted that
the corrections of Reitz are not quite correct for higher values of the electron momentum. An exact
numerical integration of the Dirac equation which takes into consideration the finite size of the
nucleus as well as screening was carried out by BÜHRING [16]. For several values of Z, his tables
give certain combinations of the radial wave functions and phase shifts for allowed and first forbidden
ß-transitions. These are needed in the formulation of the ß-decay theoryas discussed below. For a few
values of Z, MATE SE and ]OHNSON [11] calculated Fermi-function screening corrections by integrating
the Dirac equation numerically with a self-consistent atomic potential. These numerical values agree
with the values given by BÜHRING [16]. Fermi-function screening corrections can also be obtained
by solving the Klein-Gordon equation analytically using a Hulthen model for the atomic field [13].
This procedure was applied by EMAN [15]. His screening corrections, given for several values of
Z, are also in satisfactory agreement with the results of BÜHRING [16].
Behrensl Jänecke 1
1 Landolt-Börnstein, Neue Serie 1/4
1 Einleitung [Lit. S. 315
1.3 The present tables for (3-decay
The following ß-decay tables represent a tabulation of certain combinations of the electron radial
wave functions and phase shifts for the cases with and without screening. These combinations are
needed for the formulation ofthe ß-decay theory as given by BÜHRING, STECH and SCHÜLKE [17"·21].
In arecent book by SCHOPPER [22], this formulation is discussed in detail for allowed and forbidden
ß-transitions. The tabulated solutions of the Dirac equation including the finite size of the nucleus
were obtained numerically, but otherwise exactly. For the calculation of unscreened and screened
ß-decay functions, a uniform charge distribution potential was taken inside the nucleus. For the screened
functions, Hartree-Fock potentials (for parent nuclei with Z ~ 36) or Thomas-Fermi-Dirac potentials
(for Z > 36) were used outside the nucleus.
The functions without screening are tabulated for all values of Z, and over a wide range of electron
momenta. The functions with screening are not given directly. Instead, ratios of the function with
screening to the function without screening are listed. These ratios 'fJ are tabulated for all values of Z,
but for a more limited number of electron momenta for which 11 - 'fJ 1 ~ 0.02. These restrictions
became necessary because of the computer time required to calculate the screened functions. The text
(chapter 2) for the present tables (chapter 4) includes a compilation of the formulae relevant to the
various measurable quantities in ß-decay and electron capture [17 .. ·22]. In addition to the tables,
a graphical representation of the integrated Fermi-function as a function of the maximum kinetic
energy of the electrons or positrons is included for several values of Z.
1.4 Previous tabulations for electron capture
For the case of electron capture, the electron radial wave functions of the bound orbital electrons
at the nuclear radius are needed for theoretical calculations as well as for the analysis of experimental data.
Nomograms for these functions and for the Fermi-function were first obtained by using hydrogen like
Coulomb functions with screening constants according to SLATER [23,24]. Alternatively, one may obtain
the electron radial wave function by numerically integrating the Dirac equation for the Thomas-Fermi
Dirac atom [12] or by employing the Hartree-Fock self-consistent field method [25]. A compilation of
all previous calculations and nomograms is given by DZHELEPOV and ZYRIANOVA [3]. BRYSK and
RosE [26] were the first to derive detailed graphical representations of the K and L shell electron radial
wave functions at the nuclear radius which took into account both screening and finite size of thc nucleus
(see also ref. [24]). Extensive tables of functions needed for determining the probability of allowed
and forbidden K and L capture were published by BAND et al. [27,28] for a large number of Z-values.
The electron radial wave functions were obtained by integrating the Dirac equation with a uniform
charge distribution potential inside the nucleus and with a Thomas-Fermi-Dirac potential outside the
nucleus. Using the same potential, BREWER, HARMER and HAY [29] calculated, for several values of
Z, the Mshell electron radial wave functions at the nuclear radius. Electron capture ratios LI/K for the
region 3 ~ Z ~ 42 were calculated by WINTER [30] using analytical Hartree-Fock wave functions.
His results agree with ours and those of BAND et al. [27,28] as well as with other new calculations [31]
but are not in as good agreement with those of BRYSK and RosE [26].
ÜDIOT and DAUDEL [32] showed that for the exact treatment of electron capture one has to take
into consideration not only the radial wave function at the nuclear radius of the captured electron,
but also the corrections due to an exchange interaction between this particular electron and the elec
trons in the other shells. There are cases where the latter eifect changes the transition probability by
almost 20%. BAHCALL [33] calculated such exchange corrections, and also corrections for imperfect
atomic overlap, for a variety of Z-values. Recently V ATAI [34J has pointed out, that these corrections
may be not entirely correct.
1.5 The present tables for electron capture
The present tables for electron capture give the amplitudes and the values of the electron radial
wave function at the nuclear radius for the K, Land M shell for all values of Z (L and M shell ex
cluded for the very light nuclei). These tabulated quantities were calculated by integrating the Dirac
equation with the same potentials as were used for the ß-decay functions, but modified to include
exchange terms in the potential. As before, abrief compilation of all relevant formulae is given. The
notation follows closely the one introduced by STECH, SCHÜLKE and BÜHRING for ß-decay [17"'21].
As in ß-decay, only the amplitudes of the bound electron radial wave functions are needed and not
2 Behrensl Jänecke
Ref. p. 315] 2 Formulae for ß-decay and electron capture
the electron radial wave functions at the nuelear radius. Nevertheless, both quantltles are listed in
the present tables to facilitate the discussion and the comparison with other representations [26,35]
and existing electron capture tables. Possibly both functions will be useful in connection with other
applications. The exchange corrections given in chapter 2.5 were taken from BAHCALL [33].
1.6 Recent reviews of ß-decay and electron capture
To elose this introduction, we refer the reader to several recent reviews of ß-decay and electron
capture. Books by KONOPINSKI [36], Wu and MOSZKOWSKI [37], PRESTON [38] and DE BENEDETTI [39]
treat both topics as does the previously mentioned monograph by SCHOPPER [22]. Review articles have
been written by KONOPINSKI and RosE [40], WEIDENMÜLLER [41], TOLHOEK [42], and BLIN STOYLE and
NAIR [43].
For electron capture, the most detailed reviews are the articles of BRYSK and RosE [26], BOUCHEZ
and DEPOMMIER [35] and BERENYI [44]. Adescription of more special aspects of ß-decay is also given
in a number of articles in ref. [45].
Arrangement, Acknowledgments
In chapter 2, abrief compilation of all the relevant formulae used in the treatment of ß-decay
and electron captute is given.
In chapter 3, we explain the details of the computations and describe the tables presented in chapter 4.
The references are listed in chapter 5.
Sincere thanks are due to Professor H. SCHOPPER for initiating this work and for his continued
interest. The authors are particularly grateful to Dr. W. BÜHRING for making available to them his com
puter programms which provided the starting point for our calculations, for numerous valuable dis
cussions, and for the careful reading of the manuscript. Furthermore, we are indebted to Dr. H. ApPEL
for constructive comments and for reading the manuscript. The elose and excellent co operation with
the editors and the Springer-Verlag is greatly appreciated.
2 Formulae for ß-decay and e1ectron capture
2.1 Classification of ß-decays
The various types of ß-transitions are defined by the following selection rules:
+
L = 0, 1 ni . nf = 1 allowed transitions
L = 0, 1 ni • nf = - 1 first non unique forbidden transitions
L > 1 ni • nf = (_l)L Lth non unique forbidden transitions
ni . nf = (-1) L-l (L -1) th unique forbidden transitions
with L = t1J= IJ - J'j.
Here (j, ni) and (j', nf) denote the spins and parities of the initial and final nuelear states, respectively.
2.2 General formulae for ß-decay
2.2.1 Shapes of beta spectra
+
The number of ß-partieles with momentum p in the interval between p and p dp emitted per
unit time is:
(1)
where
g = weak interaction coupling constant,
Pe = electron momentum in units of moc,
We = VP: + 1 = total electron energy in units of moc2,
Wo = maximum value* of We,
Pv = Wo- We = energy or momentum of the neutrino in units of moc2 or moc, respectively,
Z = atomic number of the daughter nueleus.
*
For ß--decay the maximum kinetic energy Wo-I is equal to the difference of the atomic masses, for ß+-decay
2moc2 must be substracted from this difference in order to obtain Wo - 1.
Behrensl Jänecke 3