^ l
DEEPLY BOUND ORBITS IN PIONIC ATOMS
AND THE OPTICAL POTENTIAL
r [ f m ]
Arie Taal
TR diss
1705
AND THE OPTICAL POTENTIAL
r [ f m ]
Arie Taal
TRdiss
1705
AND THE OPTICAL POTENTIAL
Proefschrift
ter verkrijging van de graad van
doctor aan de Technische Universiteit
van Delft, op gezag van de Rector Magnificus,
prof. drs. P.A. Schenck, in het openbaar te verdedigen
ten overstaan van een commissie door het College
van Dekanen daartoe aangewezen, op
21 maart 1989 te 16.00 uur
door
Arie Taal
geboren te Scheveningen,
natuurkundig ingenieur
TR diss
1705
prof. dr. A. H. Wapstra, prof. dr. G. van Middelkoop.
Dr. ir. J. Konijn heeft als begeleider in hoge mate bijgedragen aan het totstandkomen van dit proefschrift. Het College van Dekanen heeft hem als zodanig aangewezen.
This work is part of the research programme of NIKHEF at Amsterdam, made possible by financial support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands' Organization for the Advancement of Scientific Research (NWO). It was also supported in part by the Swiss National Foundation and by the Bundesministerium fur Forschung und Technologie of the Federal Republic of Germany and by the University of Warsaw.
1 Introduction 1
2 Theory of the pionic atom 5
2.0 Introduction 5 2.1 General formalism of multiple scattering 6
2.2 Pion absorption 12 2.3 Shifts and widths of pionic energy levels 15
2.3.1 Non strong interaction contributions to pionic energy levels 17
2.3.2 Vacuum polarization 17 2.3.3 Orbital electron screening 17 2.3.4 Electromagnetic polarization 18
2.3.5 Lamb shift 18 2.3.6 Electromagnetic form factor 18
2.3.7 Reduced mass effect 18 2.4 Hyperfine structure 19 2.5 Calculation of the strong interaction shift and width 20
3 Optical potential parameters 21
3.0 Introduction 21 3.1 Parameter sets 21 4 Experimental methods 27
4.0 Introduction 27 4.1 Experimental facilities and applied electronics 27
4.2 Pionic 24Mg and 27A1 measurements 33
4.3 Pionic Si measurement 39 4.4 Pionic 93Nb, "alRu, natAg and na,Cd measurements 42
5 Methods of data analysis 43
5.0 Introduction 43 5.1 Detector response function 43
5.2 Background and step function 45 5.3 Hyperfine complexes 46 5.4 Compton edges 47 5.5 Fit program 49
6.0 Introduction 6.1.1 Pionic 1 s-level in MMg
6.1.2 Pionic ls-level in 27A1
6.1.3 Pionic ls-level in ^Si 6.2.1 Pionic 2p and 3d-level in 93Nb
6.2.2 Pionic 2p and 3d-level in naIRu
6.3.1 Pionic 3d-level in ""Ag and natCd
6.4 Conclusions
7 Fits of the optical potential parameters 7.0 Introduction
7.1 Extension of the optical potential 7.2 Parameter fits
7.2.1 Fits of s-wave (and p-wave) strong interaction parameters 7.2.2 Best fit to all pionic atom data
7.2.3 Fits using scattering lengths and volumes 7.2.4 Fits using the constraint ReB0 = -ImB0
7.3 Conclusions 8 Summary and conclusions
References Appendix A Samenvatting Curriculum vitae
Chapter 1
Introduction
Pionic atoms offer an attractive possibility to investigate the strong pion-nucleus interaction in distinctly selected angular momentum states, which was recently reemphasized by a proposal to create and implant a negatively charged pion in a deeply bound pionic orbit at the COSY facility under construction at Jiilich.
A pionic atom is formed when a negative pion is stopped in a target and captured into an atomic orbit by the Coulomb field of a host nucleus. The initial state into which the pion is captured is not precisely known. From X-ray studies it follows that it is in a high-lying state with principal quantum number n > 20. Well within its life-time (26 ns) the pion cascades down to deeper-bound energy levels with the emission of Auger electrons and X-rays. This cascade takes about 10"15 seconds, depending on the nuclear charge of the host atom. The pion finally reaches
circular orbits close enough to the nucleus to feel the short-range (< 1 fm) strong pion-nucleus interaction. For these orbits the pion does not feel electron screening, hence to a good approximation a pionic atom can be considered as a hydrogen-like system. This simplifies the theoretical description.
The effect of the strong interaction on the pion increases very rapidly when the overlap of the pion wave function with the nucleus increases, and manifests itself as an energy shift of the pionic level relative to the point nucleus Coulomb value. Another consequence of the interaction is the broadening of the energy level due to pion absorption by the nucleus. These strong interaction level shifts and widths of pionic atom orbits can be calculated from a modified Klein-Gordon equation by considering the phenomenological optical potential as representing the strong interaction. A rather successful optical potential of the Kisslinger type is the one derived by Ericson and Ericson [Eri 66]. As the basic approach they used a multiple scattering formalism for the pion-nucleus interaction. By using the low energy pion-nucleon scattering amplitudes they derived an optical potential as a function of those parameters. The specific values for some of these optical potential parameters can be determined from the experimentally obtained pionic X-ray shifts and widths.
This thesis is concerned with the study of the pion-nucleus interaction by means of the measurement and the analysis of X-ray transitions in pionic atoms. In the thesis of De Laat [Laa 88] new experimental data of pionic 3d-levels in heavy elements like 181Ta, nalRe, nalPt, 197Au, 208Pb and 209Bi revealed values for shifts and widths, which could not simultaneously be
reproduced by standard optical potential calculations. For the 3d-levels of these heavy pionic atoms the data extended almost up to the end of the stable isotopes of the periodic system. The 3d-data have in the present work been measured to lower Z by measuring the 3d-level shifts and widths in 93Nb, nalRu, nalAg and natCd. In order to search for further discrepancies the pionic
orbits of other deeply bound orbits have been measured. Measurements were performed on pionic atoms with the nuclei 24Mg, 27A1 and 28Si (ls-levels), and 93Nb and natRu (2p-levels). The pionic
ls-level measurements were of special interest, since Olivier et al. [Oli 84] had shown that the earlier observed deviations for deeply bound 3d-states could be caused by an enhanced s-wave repulsion. It was found, however, that the pionic ls-level shifts and widths were not anomalous. The 2p-data from^Nb and "j^Ru were expected to be of great interest, since Krell and Ericson [Kre 69) predicted the strong attractive pionic p-wave interaction (resulting in a positive shift with respect to the pure electromagnetic value for the level energy) to be dominated by the repulsive s-wave (negative shift) in the region of Z=36. In the present measurements, the 2p-level shifts in
93Nb and "alRu were indeed found to be negative. They showed the first negative shifts ever
observed in the pionic 2p-level.
For the pionic 2p—»Is X-ray transition in the isotopes 4Mg, Aland Si, the yield is rather
low. Only a few percent of the captured pions manage to reach the ls-orbit due the large absorption probability in higher orbits. Therefore, these transitions occur as broad low intensity peaks in the recorded energy spectra. In order to measure these transitions in a way suitable for a reliable analysis, special modern spectroscopic techniques were applied. The employment of large-volume high-purity Ge-detectors in combination with Compton suppression BGO-shields provided energy spectra with improved peak-to-background ratios. Additional coincidence techniques, selecting proper pion stops in the target, made it possible to discriminate against certain pion absorption channels accompanied by disturbing nuclear y-rays in the region of interest. The influence of neutrons emitted after pion absorption, which produce broad structures due to inelastic scattering in the Ge-detectors, was suppressed by means of time-of-flight discrimination. These methods resulted in reliable measurements of shifts and widths of pionic atom data.
All experiments were performed by using the very intense, high-duty factor pion beam at the meson factory of the Paul Scherrer Institute (formerly Schweizerisches Institut fiir Nuklearforschung, SIN). This was especially necessary to obtain satisfactory counting statistics for these low intensity pionic transitions within a reasonable measuring period.
A significantly better reproduction of both the deviating 3d-level shifts and widths could be obtained by adding to the optical potential s and p-wave rtNN-isovector terms. In the region of Z > 67 the influence of the pionic s-wave interaction term in the optical potential is growing and begins to compensate the p-wave contribution in an analogous way as in the 2p-levels for high Z.
A new fit of the optical potential parameters has resulted in an improved matching of the calculated shifts and widths with the experimental values for all pionic atom levels.
Chapter 2
Theory of the pionic atom
2.0 Introduction
The pion-nucleus interaction in a mesic atom is generally described by a phenomenological optical potential, by which the energy and width of a pionic bound state can be calculated. A phenomenological approach is followed because the nucleus is a complex dynamical system and it is not so clear how the pion behaves inside the nucleus. One assumes that the interaction of the pion with the nucleus is that of elastic scattering. The advantage of this basic assumption is the connection which can then be made with low energy pion-nucleon scattering. From these experiments pion-nucleon scattering lengths can be deduced and incorporated into the optical potential. Pion absorption is accounted for by adding complex parameters yielding complex eigenvalues of the Klein-Gordon equation for the pionic orbits. The imaginary part of these eigenvalues is related to the width of the bound state. Several approaches lead to an optical potential for the pion-nucleus interaction. The distinction between them is mainly due to the nature of the approximations along the way to a calculable result. Most derivations make use of the multiple scattering formalism developed by Watson [Wat 53, Fra 53, Wat 57]. A clear outline of such a derivation starting from Watson's multiple scattering formalism together with the approximations made is given by Stemheim & Silbar [Ste 74]. We will use the multiple scattering approach employed by Ericson & Ericson [Eri 66]. Although the latter one is not fundamentally different from the former, it incorporates more clearly the so called Lorentz-Lorenz effect describing the polarization of the nuclear medium caused by the pion wave.
Once an optical potential is constructed the bound state eigenvalues of the pion can be obtained by a numerical solution of the Klein-Gordon equation, which describes the pion-nucleus interaction. The theoretical contributions of the strong interaction to the energy and the width of pionic atom levels can then be compared to experimentally derived values resulting in a specification of the optical potential parameters.
2.1 General formalism of multiple scattering
Assume that the pion-nucleus system is described by a Hamiltonian of the form
A
H = KJt + HN + 2 v . , (2.1)
where K^ represents the kinetic energy operator of the free pion, HN is the nuclear Hamiltonian
and Vj describes the interaction of the pion with the ith nucleon in the nucleus. Starting from the Schrodinger equation we can write die following representation-free relation for the scattered wave
IM>> = |cD> + G Tn AI O > , (2.2)
where l<t>> represents the incoming pion wave and G the Green's function, which is given by G=(E-K1t-HN+ie)"1. The pion-nucleus scattering matrix TnA satisfies the Lippmann-Schwinger
equation
T„A = V + V G Tr t . (2.3)
We now further assume that the pion-nucleus interaction may be treated as if it were composed of undisturbed individual JtN-scatterings. Therefore, in analogy with eqs.(2.2) and (2.3), the multiple scattering of the pion in the nucleus is defined by the equations
A I ¥ > = I < J > > + G E tnl 4 'n> , (2.4a) A i yn> = l < D > + G £ tml 4 'm> . (2.4b) m=l tn = vn+ vnG T „ . (2.4c)
where I *Pn > is the fractional wave incident on nucleon number n and t,, the pion-bound nucleon
scattering matrix. If we limit ourselves to coherent scattering, which means that the nucleus remains in its initial (ground) state, we can derive a one-body Schrodinger equation for the pion wave function, which wave function is defined as
\|/(r) = < r n0l ¥ > , (2.5)
many-particle Green's function by the free-pion Green's function
G = ( E - Kn- HA+ i e ) " ' = G0= ( E - Kn+ i e ) "1 . (2.6)
This approximation assumes that the excitation energies and kinetic energies of the nucleons can be ignored. Under certain restrictive conditions [Beg 61, Aga 73, Hiif 73] the in-medium TtN-scattering matrix T, may be replaced by the free space nN-matrix t on the energy shell, so that the pion-nucleus scattering matrix TnA is viewed as a sequence of elementary free space pion-nucleon
interactions wherein off-shell effects, which cannot be extracted from free space 7tN-scattering, are absent.
These conditions are:
a) the nucleons can be considered as frozen to their positions,
b) the range of the icN-interactions is such that the potentials generated by two adjacent nucleons never overlap.
In the case of eq.(2.4c) this is known as the impulse approximation and asserts that free and bound nucleons scatter in the same way. Under such conditions the free pion-nucleon scattering amplitude f can be introduced by means of the relation
< r ' - xnl tnl r - xn> = - ^rf ( r ' - xn; r - xn) , (2.7)
2m
where rn is the reduced pion-nucleon mass and xn the coordinate of the n1*1 nucleon. Because of
the frozen-nucleus approximation, the coordinate of the scattering nucleon in eq.(2.7) remains unchanged, which is justified since mN» mn .
In projecting out the pion wave function as defined by eq.(2.5) we obtain from eq.(2.4a)
\|/(r)=<))(r)+Jdr'g0(r,r')Jdxp(x)Jdr"f(r'-x,r"-x)v(r";x) . (2.8)
in which the nuclear density p(r) is introduced by means of the definition
A
p(x)V(r;x) = <Qol 2 6(xn-x)\|/n(r)IQo> , (2.9)
n = l
and the Green's function go(r,r') given by
Hence the pion-wave function in the nuclear medium as expressed in eq.(2.8) is given by the incident pion wave <|>(r) and the sum of all the scattered waves from the nucleons in the nucleus. The coherent part of eq.(2.4b) yields an expression for \|/(r;x)
\|/(r;x)=<t>(r)+[dr'g0(r,r')Jdyp(y) [ l + C ( x , y ) ] J d r " f ( r - y , r ' - y ) \|/(r"; x ;y ) , (2.11)
where the nuclear two-particle correlation function C(x,y) is introduced, defined by A A
p(x)p(y)[ l + C ( x , y ) ] = < f i0l E o ( xn- x ) X 5 ( xm- v ) I Q0> . (2.12)
n = l m*n
With eq.(2.11) we have expressed \|/(r;x), the wave incident on a nucleon at position x, in terms of the correlated wave \|/(r;x;y) incident on another nucleon at the position y. This procedure can be continued to find an equation for \|/(r;x;y) in terms of \|/(r;x;y;z) which will contain a three-nucleon correlation function, etc. In applying this method further we are left with an infinite system of coupled integral equations. To find in practice a solution for the pion wave function we must truncate the iteration at some stage. Simplest is the interruption at the earliest possible stage, neglecting all nucleon correlations.
The Schrodinger equation for the scattering of a pion in first order is obtained by applying the operator (A+k2) to eq.(2.8)
(A+k2)\)/(r)=-47tJdxp(x)fdr'f(r-x,r'-x)\)/(r') . (2.13)
To find an explicit expression for the optical potential, the pion-nucleon scattering amplitude has to be specified. Since the pion and the nucleon carry isospin, the scattering amplitude is an operator in the combined spin-isospin space. The scattering amplitude may be written in a partial wave expansion in the pion-nucleon center-of-mass system
f cm( k cm. kc m) = l QTS (2/ + l ) Ai ja ^T 2. ( k ) ^ ( kc n; kc m) , (2.14)
with k cm, kcm the final and initial momentum of the pion in the c.m.frame, respectively, Qr being
the projection operator onto isospin channels T = 1/2 and T = 3/2 and A,j the projection operator onto the states with total angular momentum j = / + 1/2. In the case of a bound pion, the interaction with a nucleon involves energies close to threshold (low momenta) which justifies the restriction to s and p-waves. This yields the following parametrization [Kol 69]
with t the isospin operator for the pion, x the isospin operator for the nucleon, a is the Pauli-spin operator and n=k'cm x k „„ the normal to the scattering plane.
The constants can be expressed in terms of pion-nucleon scattering lengths and scattering volumes, b0= i ( a1 +2 a3) , b ^ I ^ - a , ) , (2.16) C0 = - j f <ai l+ 2 a3 l ) + 2 ( ai 3+ 2 a3 3) ) • Cl=j { (a3 l 'ai l) + 2 ( a3 3 -an)} ' d0 = I { ( au+ 2 a3 1) - ( a1 3 + 2a33)} , d, = ± {(a31 - a„) - ( a „ - a13)} , with 0 0 1 2 1 3
a™ = lim a1 T 1 = Um51T / k , a™,,, = l i m a ,T,I/ k = l i m 8 , _ , , / k , (2.17) 21 k^o lul k ^ o2 1 , 1 UM k-»o ■"••" k->o li-i>
where 82T,i. 82721 a r e t n e Pu r e strong interaction s and p-wave phase shifts, respectively. As is
obvious the nucleon scattering amplitude can be written as a function of individual pion-nucleon scattering lengths and volumes which are measurable quantities.
Ignoring for the moment spin and isospin dependence in the pion nucleon scattering amplitude, only the terms bg and Cgof eq.(2.15) are retained, the Schrodinger equation as expressed in eq.(2.13) can be written as
( A + k V ( r ) = 2 m U(" ( r ) T ( r ) , (2.18)
with IjC(r) the first order optical potential found by Kisslinger [Kis 55]
U< 1 )( r ) = - ^ l ( l + ^ ) b0p ( r ) -(l+^ ) -,V . cop ( r ) V } . (2.19)
Since the expression for the scattering amplitude is given in the pion-nucleon center-of-mass system, therefore, we need to transform the scattering amplitude from the pion-nucleon cm. system to the pion-nucleus cm. system. As a consequence from the transformation kinematical factors arise
(1 + J22L) and ( 1 + ^ ) - ' , (2.20)
mN mN
In order to account for correlations we must proceed one step further in the iteration. By closing the iteration at the second step with the approximation \|/(r;x;y)=\|/(r;x) a direct connection between \y(r;x) and \|/(r) can be obtained from eq.(2.8) and (2.11)
V ( r ; x ) = \)/(r)+jdrg0(r)r)Jdy p ( y ) C ( x , y ) | d r " f ( r - y , r " - y ) \ | / ( r " ; y ) . (2.21)
After substitution in eq.(2.8) we find besides the first order potential given by eq.(2.19) an expression for the potential in second order, which equation can only be solved by defining a simple correlation function.
A simple correlation function C(r,r') which depends only on the relative distance between the nucleons is
r-1 r < r
C(lr-r1) = C(r)= ( o r>r° • <2-22)
c
with the following properties common to all correlation functions
lim—>~ C(r,r') = 0 , no correlation at long distance (2.23)
Ir-r'l
lim—» 0 C(r,r') = - 1 , two nucleons cannot occupy the same place. (2.24)
lr-r'l
This requirement on the correlation function means that the nucleons in the nucleus are supposed to move independently as long as they are further apart than the distance rc but can never come
closer to each other than this distance. For this reason the correlation function defined in eq.(2.22) is called the hard-core correlation function and rc the hard-core radius. With this property for the
nucleon-nucleon correlation function and assuming low energy pions ( k rc« l ) the optical potential
in second order, often referred to as the Ericson-Ericson potential [Eri 66], is found to be
U(r) = - j g - ( q ( r ) - V - ° *r ) V) , (2.25) 2m 1 + 4/37t^a(r) with q(r) = ( l + ^ ) b0p ( r ) , (2.26) mN u and a(r) = ( l + $L) ' ' c0p ( r ) . (2.27)
Compared to the first order potential in eq.(2.19) the p-wave part is renormalized due to the pair correlation. This phenomenon reminds one of the Lorentz-Lorenz effect in electrodynamics where the local field E]oc is larger than the averaged field E in a polarized medium
given by
with T| the mean polarizability of the medium (see for instance Born & Wolf [Bor 75]). From eqs.(2.8) and (2.11) a similar relation follows
E < 1 ) ( r ) = 7T^^p- W i t h E(1) = [ V ,V( r ' ; r ) ] ,= r . (2.29)
The parameter \ in eq.(2.25) determines the strength of the Lorentz-Lorenz effect; it depends sensitively upon the range of the 7tN-interaction, rnN. For zero range TiN-forces L, is related to the
two body NN-correlation function by £=-C(0). Thus for hard core NN-interactions C(0)=-1, one has £=+1. The range of the pion-nucleon forces, r ^ , was examined by Hiifner and Iachello [Hiif 75] and concluded to be 0.25 < r,jN < 0.50 fm. As a consequence of the finite range r„N they
showed that the Lorentz-Lorenz effect is largely quenched which makes the nucleon-nucleon correlations hard to detect by mesic atom experiments.
Up to now, the isospin and spin dependence of the pion-nucleon scattering amplitude was ignored. Defining the nuclear isospin density x(r) by
A
x(r)= "cOlSXjSCr-rjJIO , (2.30) i = 1
extra terms bjt.xtr) and C]t.x(r) occur in the optical potential. The effect of these terms may be included in eq.(2.25) by the replacements
b„ p(r) _» b0p(r) + b, { pp(r) - pn(r) } , (2.31)
c0p(r)-> c0p(r)+C ]{pp(r)-pn(r) } ,
with pp(r) and pn(r) the proton and neutron density respectively, and p(r)=pp(r)+pn(r). Single
In analogy with the isospin density the spin density o(r) is defined by
o(r) = < O l E a 5 ( r - r,) I O > = 2A-,Sp(r) = 2 A -14 7 ^ p ( r ) , (2.32) i = 1 J(J + 1 )
with S the total spin of the scattering system and J the total angular momentum. Since the o\n operator in eq.(2.15) is an anti-diagonal matrix in spin space, the do-term represents spin-flip of the scattering nucleon. For elastic scattering Pauli blocking prohibits the spin-flip of a nucleon in a closed shell or sub-shell and, therefore, the do-term equals zero for an even-even nucleus like
24Mg and 28Si. In 27A1 there is only one proton in the ld5/2 shell which spin can be changed.
Hence the spin-flip term is of order A"1 compared to the isospin terms.
In the impulse approximation the in-medium scattering matrix x was replaced by the free space TtN t-matrix, and one neglects nuclear binding and the Pauli principle. The significance of the Pauli principle was stressed for instance by Delorme and Ericson [Del 76]. But the complexity of the problem prohibits an incorporation of these effects in the optical potential parameters, stressing the phenomenological character of these parameters. That the impulse approximation is possibly not justified for pion scattering at zero energy may be concluded from the binding corrections and the corrections due to the Pauli principle for very low energy scattering far below the 3,3 resonance as calculated by De Kam [Kam 81]. Like the Lorentz-Lorenz effect, Pauli corrections and binding corrections are still under study.
2.2 Pion absorption
To make the optical potential suitable to describe pion absorption, parameters must be added which reflect the absorption mechanism in the nucleus. One expects that absorption on a single nucleon is highly suppressed due to energy and momentum conservation. If a stopped pion is to be absorbed on a single nucleon, the nucleon must have an initial momentum of roughly 500 MeV/c. The Fermi momentum-distribution makes this rather improbable. Hence the dominant absorption process involves at least two nucleons. Experiments on multi-nucleon removal after pion absorption reveal that the main process is pion absorption on a correlated nucleon pair, although pion absorption on a cluster of more than two nucleons is not much less important. Measurements to reveal the absorption mechanism are strongly hindered by the multi-nucleon knock-out due to final state interactions. An important quantity is the ratio R of np to pp pairs that can absorb the pion. Experimentally, the value of R is not determined unambiguously. This is illustrated for instance by the different values of R for 12C. Nordberg et al. [Nor 68] obtained R =
2.5±1.0 whereas Ozaki et al. [Oza 60] found a value of R = 5.011.5 and Lee et al. [Lee 72] R = 8.8±1.3. Statistically speaking the value for R should be given by the ratio 2N/(Z-1), for a nucleus with Z protons and N neutrons. Since the experimental values for R are significantly
larger than the statistical expectation, the pion absorption must occur preferentially on a proton in a strongly correlated np-pair, called quasi-deuteron absorption(QDA).
In an analogous way as was done for the TcN-scattering amplitude, a low energy JI2N-amplitude in the zero range approximation can be constructed [Eri 66], with the following parametrization
f. = B0 + B^x.-x.) +B2(o.-cj) + B3[l - (o.-oj)][(xi+ x.)-t ]+ B / a o ^ t j - x p +
B5[l-(a..oj)][(t.x.)(t.xj) + (t.xj)(t.x.)] + k'.k {C0 + C1(x.-Tj)+... ) , (2.33)
where the indices i,j refer to different nucleons. This parametrization is incomplete since it is assumed that the two nucleons involved are in a relative s-state and, therefore, certain possible terms are absent. Furthermore, spin-flip contributions are neglected. It is in fact a phenomenological two-nucleon amplitude wherein the constants simulate all the information on the short ranged absorption process and even short ranged nucleon pair correlations. The parameters are complex numbers with the imaginary part accounting for the width of the bound pion level.
The isoscalar s-wave part of the potential is represented by
( l + ^ ) B0p2( r ) , (2.34)
z mN
with (1+m /2mN) arising from the transformation of the Jt2N cm. system to the lab. system. To
discriminate between the two main absorption channels, one may make the replacement
B0 P V ) "» { B0(pp, P>> + B0(pn) PpW P „ « ) • <2-35>
with B0(pp) representing the absorption on a proton-proton pair and B0(pn) for the absorption on a
proton-neutron pair. The isoscalar term in the p-wave part is represented by the complex number C0 where one can make the same replacement in C0(pp) and C0(pn). Neglecting p-wave
non-isoscalar terms, the influence caused by the spin and isospin terms B, to B5 can be seen as a
correction AB to B0, since this correction is also proportional to p (r). Retaining the isovector pan
of the Jt2N-amplitude explicitly with the distinction between the main absorption channels, the s-wave and p-s-wave part of the potential are extended respectively by
( l + ^
2- ) f B
1, V ( r ) - B . , ,p(r)p(r)} ,
2mN l K p p )Kpw Kpn)Kpv " nv "
( i +_n2n_)-1{r V ( r ) - C . , , p ( r ) p ( r ) } 2 mN ( p p ) P Kpn)KPv / Knv '>
As already mentioned, the absorption on a cluster of nucleons is also of importance. The process of pion absorption is still not fully understood. A study of ten absorption channels with stopped pions on Li by coincidence detection of two outgoing nucleons was performed by Dorr et al. [Dor 85]. The quasi-deuteron process was found to be responsible for 72.5% of the absorption strength while Isaak et al. [Isa 82] found a percentage of 51%. Three other channels of somewhat less importance are characterized by the outgoing particles n-p, n-d and n-t, each having a strength of about 10%. The six channels involving the coincidence of two charged particles leaving the nucleus after pion-absorption are weak(< 1%). For heavy nuclei, the QDA may be the dominant process again. This is confirmed for instance by Isaak et al. [Isa 83] and Shinohara et al. [Shi 86] who could explain the experimental data after stopped pion absorption in Ni-isotopes and
209Bi, respectively, to a considerable extent, using quasi-deuteron absorption in an extremely
peripheral region of the nucleus. The cluster absorption channels cannot be assumed to be governed by the parameters mentioned so far since these reactions should be represented in the optical potential by terms proportional to p3(r) for 3N-absorption and to p4(r) for a-cluster
absorption and so on.
Resuming the above mentioned parameters, an optical potential results of the following form as used by different authors in fits to pionic atom data
U(r) = --fe ( q(r) - V- "( r ) V ) , (2.37) 2m 1 + 4/37t^a(r) with q ( r ) = ( l + ^ ) { bop ( r ) + b1[ pn( r ) - pp( r ) ] } + (l + ^ ) { B0p2( r ) + B1p ( r ) 6 p ( r ) } , and a ( r ) = (l + ^ ) "1{ c0p ( r ) + c, [ pn(r) - Pp(r) ] } + ( 1 + J ^ . )1^ p2(r) + C] P(r)5p(r)} ,
where 5p(r)=pn(r)-pp(r). Several authors do not take the renormalization due to the
Lorentz-Lorenz effect into account for the p-wave part of the 2N-interaction, since it is not clear how this phenomenon should occur in the potential. In those fits the C0 and Cj terms are not incorporated
into the Lorentz-Lorenz term. Isovector terms in the absorption, represented by B, and Cj, are mostly neglected in the fits to pionic atom data.
2.3 Shifts and widths of pionic energy levels
Until now the pion-nucleon scattering formalism has been discussed. In the case of a pionic atom, however, the pion is in a bound state with an energy below threshold. The energy levels of a pionic atom are obtained by the bound-state eigenvalues of the Klein-Gordon equation of the form (h=c=l)
{A + [(E-VC(r))2-m2]}V(r)=2rnU(r)V(r) , (2.38)
where Vc(r) is the Coulomb potential, E the pion binding energy including the pion rest mass m,
m=m[l+(m/M)]"1 the reduced pion mass with M the nuclear mass and U(r) the potential,
describing the 7t-nuclear interaction as given e.g. by eq.(2.37). This form of the Klein-Gordon equation entails that the 7t-nuclear interaction can be treated in a non-relativistic way. Whether such an approach is justified depends on the values of the optical potential (see for some comments and further references Giovanetti et al. [Gio 87]).
For a nucleus of finite size the Coulomb potential is given by
C fPp( r , )
VV) = - a J
1^
Tdr' , (2.39)
with a the fine structure constant and pp(r) the proton density. The proton density function is
represented by a two-parameter Fermi-distribution
pp(r) = N (1 + exp[ 4 In3 (r-c )/t])"' , (2.40)
where N is the normalization constant chosen so that the volume integral over pp(r) equals the
total number of protons, and c represents the radius of the distribution and t the skin thickness. Since the optical potential U(r) contains complex parameters, the eigenvalues of the Klein-Gordon equation (2.38) for a given pionic orbit (n, / ) are also complex
En; = Re(En /) + i I m ( En /) = R e ( En /) - i rn ; , (2.41)
where the imaginary part of the eigenvalue is equated to the width rn / for the given orbit. The level
energy shift caused by the strong interaction, e„j, is defined as the difference between the energy in the field of a point charge Ze and the energy in the combined Coulomb field of the
extended nuclear charge distribution pp(r) and the optical potential
Using this convention a negative value for en/ corresponds to a repulsive shift. The shift so
defined contains strong interaction effects as well as the finite size effect. In simplifying the Klein-Gordon equation by neglecting the finite size of the nucleus and the strong interaction the solution is exact and is given by
2 -1/2
E^, = rn { 1 + ( ^ _ ) } ; (2.43)
"' n - / - l / 2 + ((/ +l/2)2-(Za)2)1 / 2
see for instance [Sch 83]. In order to compare the theoretically calculated shift e„, (for a given set of optical potential parameters) with the value deduced from experiments, we must subtract the electromagnetic contribution from the measured energy value Eexp for the pionic transition
between the levels with quantum numbers (n+1,/ +1) and (n,/). Since the effect of the strong interaction on an upper atomic level is usually down by at least two orders of magnitude with respect to the lower level of the transition, the shift en/ is related to the measured energy of the
corresponding transition by the following relation
en / =Ec x p(n+l,/+l->n,/ ) - ( EC+ E°" ) , (2.44)
where Ec is the transition energy according to the point nucleus approach and E the other
contributions to the pionic transition energy. These other contributions due to effects like vacuum polarization, orbital electron screening, nuclear polarization etc. are discussed in subsections 2.3.1-8.
Two processes in pionic atoms determine the width rn ( of a given orbit: the electromagnetic
transition probability (including Auger-transitions) to a lower level and the pion absorption from this level. For the strong interaction width of the ls-level one can write
rS re x p rad S A exp
Is 2p-»ls 2P 2P 2P 2p->ls
with rrad and TA the radiative and Auger-widths, respectively. The radiative and the Auger-width
can be calculated by the computer code STARKEF [Tau 78]. As can be read in chapter 6 the contributions of the upper level are negligible compared to the lower level which justifies the approximation made in eq.(2.45). By means of an intensity balance for the 2p-level, the strong interaction width rfp can be obtained. From the measured X-ray spectrum one can deduce the ratio of the yield of the 2p—>ls X-ray transition, Yrad(»ls), versus the population of the
2p-level, P(2p), according to
Yra"(2p->ls) Yrad(2p->ls)
P(2P) I Yrad(n',/ '->2p) + Z YA(n',/ ' -»2p)
In this expression Yrad equals the intensity of the observed X-ray transition whereas the
Auger-yield can be calculated. For this ratio the following expression also holds
. rr a d
Yr a a(2p->ls) = X2P- > 1 . ( 2 4 ? )
P(2
P)
= r-
+r
2 s p +r
2 A p from which two relations rfp follows.2.3.1 Non strong interaction contributions to pionic energy levels
All effects which contribute to the pionic energy level and which are not governed by the Klein-Gordon equation as given by eq.(2.38) are to be calculated separately. The experimental value of a pionic transition is then corrected according to eq.(2.44) to abstract the shift which can be compared to the prediction of a particular set of optical potential parameters as defined in eq.(2.42). A brief description of these corrections is given below whereas a list of the estimated values can be found in chapter 6.
2.3.2 Vacuum polarization
The electron of a virtual electron-positron pair creation in the Coulomb field of the nucleus tends to be attracted to the nucleus. This phenomenon is referred to as vacuum polarization. The vacuum polarization changes the electrostatic potential of the nucleus over a distance of the order of the Compton wavelength of the electron (~10"ncm) and since the average pionic orbit size is
comparable to or smaller than this wavelength, the orbiting pion is inside the vacuum-polarization charge cloud of the nucleus. As a consequence of this phenomenon, the energy levels in a pionic atom obtain a negative shift (stronger binding): the pion senses a charge larger than the nominal charge Ze of the nucleus. Using the expressions for the vacuum polarization found in the references [Bio 72, Bor 76, Rin 75] the shifts can be calculated to the orders of a(aZ), a2(aZ),
a(aZ)3, a(aZ)5 and a(aZ)7.
2.3.3 Orbital electron screening
The electron cloud in a pionic atom decreases the effective charge of the nucleus experienced by the pion and results in an energy shift of the pionic atom level. For pionic orbits with principal quantum numbers n < 10 most of the atomic electrons are far outside the region spanned by such orbits. Screening depends almost only on the K and L-electrons. This effect can be calculated
using a relativistic Hartree-Fock procedure as given by Vogel [Vog 73]. The main uncertainties in the calculations result from the unknown numbers of K and L-electrons taking part in the screening. Owing to the Auger effect during the cascade of the pion, the electronic shells are partly emptied. The calculations take into account the Auger effect and the refilling of the electronic K and L-shells during the cascade to obtain the mean population of these shells.
2.3.4 Electromagnetic polarization
Nuclear polarization is die effect that the orbiting pion influences the nucleus by separating the protons and neutrons with respect to the nuclear center-of-mass. In return, the nucleus can polarize the orbiting pion due the large potential difference over the pion radius (=1 fm). The additional energy shifts produced by the polarization of the nucleus by the electric field from the pion (and vice versa) are e.g. given by Ericson and Hiifner [Eri 72].
2.3.5 Lamb shift
The 'bare' mass of the pion is essentially the mass which appears in the Klein-Gordon equation. But the observed mass has a component associated with the electromagnetic self-interaction of the pion. This process consists of the emission and the subsequent re-absorption of a virtual photon by the pion, which results in an energy shift of the pionic level. Calculations of this effect are performed according to the method of Klarsfeld and Maquet [Kla 73].
2.3.6 Electromagnetic form factor
The pion is represented as a point charge in the Klein-Gordon equation which is not fully correct since the pion has a charge structure (=1 fm), i.e. a form factor. The perturbing potential introduced by the pion form factor can be found in reference [lac 71]. It is found that the energy shift due to this effect is strongly n-dependent and is observable only in the ls-levels.
2.3.7 Reduced mass effect
In the Klein-Gordon equation the reduced mass was used instead of the relativistic pion mass and, therefore, does not take all relativistic effects into account. Relativistic reduced mass
corrections are discussed by Barrett et al. [Bar 73] for the Dirac atom case. The result is assumed to hold also for the Klein-Gordon situation.
2.4 Hyperfine structure
The non-spherical part of the strong and electromagnetic interaction gives rise to a hyperfine splitting of each mesic level for nuclei with spin I > 1. Under the assumption that the deformation of the nucleus is of quadrupole shape only, the energy shift e(F) and level width T(F) of any given member (/ ,I,F) of the hyperfine multiplet, relative to the point-nucleus value, are given [Sen 72] by
e(F) = A j X ( / , I , F ) + e0+(E2-ReA2)C(/,I,F) , (2.48)
r(F) = r0+ ( r2- 2 I m A2) C ( / , I , F ) .
In this expression n,/ are the quantum numbers of the unperturbed mesic orbit, and 1 is the nuclear spin. The shift e0 and width F0 stem from the monopole part in the Coulomb potential Vc and the
strong interaction potential U. Due to the quadrupole part of the strong interaction potential, an additional shift e2 and width T2 occur, which are calculated by Koch and Scheck [Koc 80]. The
magnetic dipole moment of the nucleus and the magnetic field created by the pion's orbital motion causes a hyperfine interaction indicated by Aj [Dey 75]. Furthermore, A2 is the electric hyperfine
constant due to the electric quadrupole part of V~ and C(/, I,F) is the angular momentum factor of quadrupole hyperfine structure
<»■■■»-
3
™i';;r" .*-<«>-•<-»«->.
We assume that the initial hyperfine levels are statistically populated. The relative intensity Ircl
of an El hyperfine transition between levels of the total angular momentum quantum numbers F; and Ff is then given by
2
The parametrized natural line shape obtained in this way is folded with the detector response function and compared with the experimental results. We will show in chapter 6 that excellent fits can be obtained, which we consider to be a confirmation of the correctness of the procedure described above.
2.5 Calculation of the strong interaction shift and width
For the calculation of the strong interaction shift and width we make use of two different computer codes, PIATOM written by Tauscher [Tau 78] and MESON written by Koch [Koc 73, Koc 80]. These codes use the Numerov method [Bla 67] for integrating the Klein-Gordon equation with the optical and Coulomb potential. The input for these codes consists of relevant parameters for the optical potential. Neither code calculates all the higher order corrections and deformation which are listed above. The code MESON calculates the monopole shift and width, e0
and r0, with respect to the point Coulomb value and includes finite size effects. This code also
calculates nuclear deformation effects yielding the parameters e2, T2 and A2, see eq.(2.48). All the
higher order coiTections like vacuum polarization, electron screening, electromagnetic polarization, Lamb shift, electromagnetic form factor and reduced mass effect are calculated by the code PIATOM. Together these computer codes give an adequate picture of the mesic atom calculations.
Chapter 3
Optical potential parameters
3.0 Introduction
Experiments in the late sixties, using solid state detectors as high-resolution y-ray detectors, yielded new accurate data for the strong interaction shifts and widths in pionic atoms. These data have been reviewed by Backenstoss [Bac 70]. Data of pionic 2p—>ls transitions up to Z=12, 3d->2p transitions for 13 < Z < 30, 4f-»3d transitions for 20 < Z < 30 and 5g—>4f transitions for 73 < Z < 92 allowed to perform the first empirical fits of the optical potential parameters throughout the periodic system. One of the first fits was performed by Krell and Ericson [Kre 69]. Better experimental techniques, like improved background suppression, gradually yielded data from deeper bound pionic atom levels. Predictions from known parameter sets could be compared to the new data, resulting in new, adjusted parameter sets. Measurements of pionic atoms are still in progress and 2p—»ls transitions have presently been measured up to Z=14 and 3d-»2p transitions up to Z=44. Recently measured 3d-levels in 181Ta, natRe, natPt, 197Au, 208Pb and 2 Bi indicate the beginning of a saturation effect, giving smaller experimental widths than are
predicted by standard optical potential calculations, [Laa 88].
3.1 Parameter sets
Out of the many published parameter sets, four have been chosen for comparison with the new pionic data presented in this thesis. The values of these parameter sets are listed in table 3.1. The set by Batty et al. [Bat 78] corresponds to an optical potential that slightly deviates form the one given in eq.(2.37). In order to account for the dominance of quasi-deuteron absorption in the optical potential the p2 terms were replaced by 4pppn. By writing 4pppn instead of (pp + pn )2,
the absorption part of the potential (ImB0, .ImC0) is more appropriately described. Yet, for the
dispersive part (ReB0, ReC0) this substitution is rather doubtful. There is no experimental
evidence that scattering on a pair of neutrons is suppressed (which corresponds to ignoring the terms pn 2).
In order to evaluate the different parameter sets qualitatively, the generated shifts and widths are compared to the experimental compilation given in appendix A. For the calculation of the shifts and widths the program MESON (see section 2.3.9) was used. The experimental data of table A.l through table A.8 differ from the data sets used in [Tau 71], [Bat 78] and [Sek 83]; we also
Table 3.1 Four parameter sets describing the optical potential given in eq.(2.37), obtained from fits to then available experimental pionic level shifts and widths. Also given are the results from a phase shift analysis of pion-nucleon scattering data [Row 78].
bo b. ReB0 ImB0 Co C] ReC0 ImC0 [Tau71] $=0 -0.0296+ 0.0005 -0.077 + 0.007 0 0.0436+ 0.0015 0.172 + 0.008 0.22 + 0.03 0 0.036 +0.013 [Tau71]
<H
-0.0293 ± 0.0005 -0.078 ±0.007 0 0.0428 ±0.0015 0.227 ±0.008 0.18 ± 0.03 0 0.076 ±0.013 [Batty 78]*'S=l
-0.017 -0.13 ±0.02 -0.0475 0.0475 0.255 ± 0.003 0.17 0 0.090 ± 0.005 [Sek 83] %=l 0.003 ±0.008 -0.143 ±0.006 -0.15 ±0.04 0.046+0.003 0.21 0.18 0.11 ±0.01 0.09 ±0.01 [Row 78] -0.004 ±0.003 -0.094 ±0.003 0.23 ±0.07 0.17 ±0.03 IT.-* m- lml
mn <m'l
m6nmi
*) different expression for the optical potential
included the new experimental results presented in this thesis. In order to make the comparison of the calculated shifts and widths with the experimental values meaningful, we have in all cases taken the uncertainties in the shifts to be >5% of the measured widths. The resulting errors are never smaller than the ones given in the corresponding original publications. The reason for doing this lies in the fact that experience has taught us that one is never certain that the broadened lines do not contain contaminant lines that affect the actually measured energy to some extent. One should also note that Tauscher [Tau 71], Batty [Bat 78] and Seki [Sek 83] have used different experimental inputs, simply due to the fact that these fits cover a time span of 12 years.
In appendix A, table A.l, the experimental values of the pionic ls-level shifts and widths are listed including the presently measured values for Mg, Al and Si. The ls-level shifts are all negative, all higher level shifts positive except for the presently measured shifts of deeply bound pionic 2p-orbits in 93Nb and natRu (see chapter 6). This means that the pion experiences a
repulsive interaction in the ls-orbit but an attractive one in higher orbits. Therefore, the parameters b0, b, and ReB0 are generally negative, as these parameters mainly determine the ls-level shifts in
the calculation. Furthermore, the shift values increase steadily with the atomic mass A and additional isotopic effects seem to be hardly present. One can expect that the addition of neutrons increases the probability for scattering on nucleons, causing an increase in the shift. The values for the width are more sensitive to isotopic effects. The widths for 13C, 1 80 and 22Ne are
significantly lower than those for the isotopes 12C, I 60 and 20Ne, respectively. Here the statistical
ratio of pn to pp pairs (see remarks section 2.2) increases when the number of neutrons increases which would lead to the expectation of a corresponding increase of the level width; this is in
contrast to the experimentally observed decrease. In Appendix A are also listed the calculated shifts and widths for the 140 data points using the four parameter sets as input for the computer code MESON. To illustrate the deviations between experimental and calculated values, the quadratic differences, weighted by the experimental uncertainties, £22=(yexp-ycal)2/0"exp2, f°r t n e
shifts and the widths of each level are calculated according to each parameter set. From table A.l and A.2 it can be noticed that for the ls-levels the predictions of the sets [Tau71] (4=0,1) are on the average the best, the total £22-values for the 28 data points are 142, 147, 163 and 239 for the
parameter sets [Tau 71] (4=0,1), [Bat 78] and [Sek 83], respectively. Only for the set [Bat 78] about 75% of the total £)2-value is on account of the shifts. Common for all sets are the poor
values given for the ls-level shift in 1 80 and for the ls-level width in 20Ne. The 'wrong'
predictions for the ls-level shift in 1 80 claim each about 20% or more of the total Q2(En) for the ls-shifts of each set, whereas the predictions for the ls-level width in 20Ne contribute for over
30% to the total 22(r] s) for the ls-widths of each set.
Considering the pionic 2p-data in table A.3 and A.4 the shifts for the Ca-isotopes show a clear isotopic effect as do the widths of the Cr-isotopes. As already predicted by Krell and Ericson [Kre 69] the effective potential predicts a change of the attractive p-wave interaction to a repulsion for deeply bound pionic 2p-orbits in the region of Z=36. This is illustrated by the recently measured 2p-shifts in 9 3Nb and "^Ru (see chapter 6) for which values were found of e2p= -11 ± 3 keV and
-48 ± 7 keV, respectively. By adding the Q2-values for all 2p-cases one finds that the total Q2 -value varies considerably for the different parameter sets, ranging about 300-1750 for the 58 2p-data points. The 2p-level shift in O is by all sets mispredicted resulting in a percentage of 20% or more of the total Q (^p) for the 2p-shifts of each set, the same holds for the 2p-level width in
26Mg with respect to Q2(r2p).
The parameter set [Tau 71] (4=0) gives the worst predictions for the 3d-shifts and widths (see tables A.5 and A.6), in which the presently measured 3d-shifts for 93Nb, nalRu, nalAg and nalCd
are included. The 3d-shifts of the heavier elements like I 8 iTa, nalRe, nalPt, 197Au, 2(,8Pb and
Bi are underestimated by the set [Sek 83] by more than 60%. For this set about 80% of the total Q2-value results from the shifts, whereas for the other sets about 80% of the total g2-value is
on account of the widths. Considering the 3d-widths of these heavier elements, they are clearly overestimated by at least 35% for the sets [Tau 71] (4=0,1) and [Bat 78]. The total Q2-value for
the 34 3d-data points are about 1132, 652, 408 and 661 for the sets [Tau 71] (4=0,1), [Bat 78| and [Sek 83], respectively.
All sets with 4=1 predict the 4f-shifts worse than the 4f-widths, only about 25% or less of the total (22-value is attributed to the widths (table A.7 and A.8). The total Q2-values for 20 4f-data
points are 47, 40, 71 and 296 for the sets [Tau 71] (4=0,1), [Bat 78] and [Sek 83], respectively. The 4f-level shift in 237Np claims over 40% of the total 22(e4f) for the 4f-shifts of each set with
4=1, while 40% or more of the total Q (T4t) for the 4f-widths of each set with 4=1 is on account of 197Au.
Since the parameter sets of table 3.1 are not fit to our data compilation of appendix A, we can not say that the distribution of Q2 should follow a ^-distribution. If we nevertheless assume a %2 -distribution, the probability that the total Q2-values have the values as presented is negligibly
small. Thus statistically speaking, the optical potential with the given parameter sets yields a poor prediction of the shifts and widths of all pionic atom levels. The largest discrepancies between theory and experiment occur in the 2p and 3d-data, where the predictions of the 3d-level widths are the poorest. The ls-level shifts are the best predicted, namely all deviations between theory and experiment are less than 10%. In order to produce a better parameter fit the main attention has to be paid to a better match of the 2p and 3d-data.
Also given in table 3.1 are the values for b0, b1; c0 and C] which are calculated using the
formula (2.16) and making use of the free pion-nucleon scattering lengths from a phase shift analysis performed by Row et al. [Row 78]. As can be noticed the in-medium values for b0 and b,
obtained from pionic atom fits deviate more from the free pion-nucleon values than the parameters c0 and c, do. This can be understood since the non-local part (p-wave part) of the interaction is
confined to the surface region of the nucleus, where the free-pion-nucleon interaction could be more probable. The values ReB0 = 0.008, ImB0 = 0.027, ReC0 = 0.036 and ImC0 = 0.043 as
calculated by Chai and Riska [Cha 79] are in less good agreement with the pionic atom values. Most theoretical predictions yield ReB0 > ImB0 which is in contrast with the empirical fits ReB0 =
counter
Fig.4.1 Experimental setup for the measurement ofpionic X-rays. Incident pions are detected
by a beam telescope consisting of four scintillation plastics S]-S4 (NEW2A). The plastics S, and S2 both measure 150 mm x 130 mm x 5 mm, S3 [100 mm x 100 mm x 3 mm] defines the time of a pion stop in the target and the stop counter S4 [250 mm x 250 mm x 7 mm] determines whether a particle has passed the target. A pion stop in the target is denoted by 5/ x
S2 x S3 x S4. Four Ge-detectors are arranged at a distance of about 40 cm from the target. Two are placed in an asymmetric BGO Compton suppression shield and two in a symmetric BGO-crystal.
Chapter 4
Experimental methods
4.0 Introduction
The pionic atom experiments described in this thesis were performed at NIKHEF in Amsterdam and at the Paul Scherrer Institute (formerly SIN) at Villigen in Switzerland. At NIKHEF, measurements of pionic MMg and 27A1 were started to determine the properties of the
deeply bound pionic 2p—>ls X-ray transition. Our experimental method, though including modern spectroscopic techniques such as the use of Compton suppression BGO-crystals and time-of-flight discrimination to reduce neutron induced background, was found to be insufficient to measure these low-intensity transitions which have appreciable widths. Moreover, in pionic 24Mg and 27A1, and also in 28Si, the pionic 2p—»ls X-ray transition is obscured by strong nuclear y-ray
transitions. To reduce the effect of this v-interference additional coincidence requirements had to be applied. Therefore, these experiments were repeated at the Paul Scherrer Institute with much better beam properties (100% duty factor versus 1% at NIKHEF and a higher beam intensity). The result was an order of magnitude improvement of both the statistics and the peak-to-background ratio in these X-ray spectra. For the measurement of the pionic 3d-»2p X-ray transition in 93Nb and nalRu and of the 4f->3d transition in 93Nb, na,Ru, na,Ag and nalCd a
conventional setup without the application of additional coincidence techniques was found to be adequate at the Swiss facility.
4.1 Experimental facilities and applied electronics
The pion facility at NIKHEF in Amsterdam at the 500 MeV linear electron accelerator MEA had a pion production target consisting of a 1 mm thick tungsten radiator in combination with a 5 cm thick water-cooled pyrolytic graphite photon to pion converter. The secondary beam channel was designed to transport low energy pions produced by a pulsed beam of typically 400 MeV electrons, in bunches of 35 u.s at a repetition rate of 300 Hz and a 10 mA maximum peak current. The pion channel was built at an angle of 120° with respect to the incident electron beam. It consisted of a series of quadrupole and dipole magnets, where the intermediate part of the transport line had a 45° upward slope. From this channel a pion beam enters the experimental area with a focus of approximately 3 cm (height) x 5 cm (width).
100 [keV]
Fig.4.2a Time spectrum of stopped pions versus Ge-signals. The prompt part designates y and X-rays followed by a delayed part due to the longer time-of-flight of the neutrons.
J i i i i I i i i i_J i i i i i ■ i i i I
~-i—i—I—i—i—i—r
800 850 1000
energy , [keV]
Fig.4.2b Prompt and delayed (neutron) Ge-spectra ofpionic 93Nb. Due to the time-of-flight discrimination the prompt part of the region of interest around 900 keV is free from the neutron induced background, e.g. the reaction 72Ge(n,n'y) with a y-ray of 834.14 keV.
At the Paul Scherrer Institute a ring cyclotron provides a fixed energy 590 MeV proton beam from which secondary pion, muon, neutron and polarized proton beams are generated. The high-intensity high duty factor 50 MHz proton beam has a maximum high-intensity of about 250 mA at extraction. Several experimental areas suitable for muon and pion experiments are situated around a Be-target station in which pions and muons are produced. Of these experimental sites the p.E4 area was chosen, because of its low background of neutrons and y-rays. A superconducting solenoid (field strength 4 T) with three bending magnets along with the necessary quadrupoles produces a pion beam with a focus of 2 cm x 2 cm. With the optimum beam setting, corresponding to a pion momentum of 100 MeV/c, a stop rate of approximately 106 negatively
charged pions per second was achieved versus a stop rate of 2.10 at the NIKHEF facility. The conventional experimental setup comprised a conventional beam telescope consisting of four scintillation counters as shown in fig.4.1. The first scintillation plastic was chosen to have a size totally covering the incident beam. By operating the threshold of the corresponding discriminator at the lowest possible setting, every incoming charged particle that was able to reach the target area could be detected. A second plastic scintillator of comparable size to the first one was installed to trigger mainly on pions by adjusting the threshold of its discriminator in such a way that it ignored particles with a lower energy loss, like electrons and muons. The time of a pion stop was defined by a third plastic counter, which also mainly triggered on pions. This counter mounted close to the target was of the same size as the target. In this way background lines caused by muonic X-rays and muon capture are suppressed. A veto signal from stop counter S4 completed the telescope trigger with S[XS2xS3xS4 defining a pion stop. In order to stop the
pion beam efficiently in the target a Be-degrader was inserted between the first two plastics. The optimum amount of degrader material was determined by adjusting the amount of Be in such a way that the number of pion stops was maximized. On the coincidence signal S,xS2 a pile-up time
of about 150 ns was imposed to prevent ambiguity in the pion-stop signal. Only incident pions within a minimum-time interval of 150 ns isolated from a previous and a later pion were allowed to trigger the electronics.
High purity n-type Ge-detectors with a relative efficiency (with respect to a 7.62 cm diameter x 7.62 cm long Nal(Tl) scintillation detector) of about 30% and a resolution of 1.8 keV at 1.33 MeV were arranged around the target for y and X-ray detection. In a conventional arrangement the Ge-diodes were placed at a distance of about 40 cm from the target to obtain a good off-line time-of-flight discrimination probability. A typical time spectrum, designating the time between a pion stop and a Ge-signal, is shown in fig.4.2a. One can see that signals from neutrons emitted after pion absorption in the target nuclei are well separated in time from the prompt y-rays. By imposing a prompt time window on the recorded events an energy spectrum essentially free of neutron-induced y-rays can be created, see fig.4.2b. This is important for experiments where y-ray transitions from (n.n'y) reactions in the Ge-crystal are situated near X-ray peaks of interest, which would otherwise make the analysis cumbersome.
Fig.4.3a View of a symmetric Compton detector composed of BGO and a Nal(Tl)
scintillator at the front. All measures are in mm.
■ ' ■ '
a ioi
3 O I ■ I ' I ' I ' I ■ I ' I ■ I ■ I ' I ' I—■ I ' I 200 400 600 800 1000 1200energy
y[keV]
Fig.4.3b Energy spectrum of60 Co y-rays recorded by a Ge-detector placed in a symmetric BGO-shield as displayed above. The upper curve is the ungated y-ray spectrum and the lower one results after ignoring those events accompanied by a signal in the BGO(NaI)-shield.
Fig.4.4a View of an asymmetric Compton detector composed of BGO and a Nal(Tl)
scintillator at the front. The entrance hole for the Ge-detector is perpendicular to the axis of the crystal. All measures are in mm.
I . I . I . i
200 400 600
e60r
800 1000 1200 energy > [keV]
Fig.4.4b The Ge-energy spectrum of Co frays measured with an asymmetric BGO-shield
as displayed above. The ungated y-ray spectrum is given by the upper curve and the lower one results after ignoring those events accompanied by a signal in the suppression shield.
S ^ J D i s c U ^ -1 0 1 8 0 S,_|Disc S Disc _T *-y Computer Busy 71 stop /T-i BGO _ TFA Ge, ' JCFhrJ~l-S GateADC , Stop TDC! Spec Amp Pile Up ADC,
Fig.4.5 Simplified electronic scheme for the measurement ofpionic X-rays with Germanium detectors fitted in anti-Compton shields.
Furthermore, the Ge-detectors were placed in Compton suppression shields of mainly BGO (see figs.4.3a and 4.4a for the two types used). The front parts of the suppression shields are composed of Nal(Tl). The scintillation efficiency of Nal is a factor of eight higher than that of BGO, which makes Nal more suitable to detect low energy y-rays produced by backward Compton scattering. Compton-scattered y-rays from the Ge-counter seen by the suppression shield are used to produce a veto signal for the corresponding Ge-energy signal. The performance of this Compton suppression system is illustrated in figs.4.3b and 4.4b.
A combination of spectroscopy amplifiers (Canberra, model 2021) and pile-up rejectors (model 1468A) provided signals, corresponding to y-ray energies measured with the Ge-detectors, for 8192 channel Laben ADC's (model 8215) as is depicted in the electronic scheme of fig.4.5. For simplicity we limit the discussion to the case of no additional requirements, such as signals from extra Nal-detectors and the like as discussed later. Pile-up rejectors were necessary to prevent pile-up of subsequent y-ray signals from a Ge-detector. The pile-up rejectors are capable of discriminating between two events having a minimum separation of 500 ns. Outgoing count rate (OC) pulses provided by the pile-up rejectors were incorporated in the event trigger such that only single-energy events during the processing time of the amplifiers were taken into account. Ge-energy signals were split and fed into pulse-shaping filter amplifiers (TFA 474, Ortec) and constant fraction discriminators (CFD 934, Ortec) for amplification and timing of the Ge-signals, respectively. A typical CFD-delay used for the Ge-signals was about 24 ns providing amplitude-and rise-time compensation. The chosen delay along with the zero-crossing adjustment of the CFD's were mainly responsible for the time resolution which was about 10 ns for the prompt time peak as shown in fig.4.2a. By applying an anti-coincidence of the CFD-output with the BGO-crystal signal, a trigger indicating a "Compton-free" Ge-energy was obtained. The ultimate event trigger occurred after a coincidence with the pion-stop from the telescope. Every Compton free Ge-energy was stored in 16k, 24 bits Camac histogram modules (LeCroy model 3588), regardless of the presence of an event trigger. Secondly only those energies accompanied by an event trigger were registered into 64-words(16 bits) deep first-in first-out (FIFO) modules. In this way a general data acquisition program [Laa 85] allowed to write coincident data in list mode on magnetic tape as well to accumulate on-line spectra which could be displayed on a terminal screen during the measurement.
4.2 Pionic 24Mg and 27AI measurements
Some years ago the maximum Z-values for which strong interaction shifts and widths of pionic levels had been investigated were Z=10 and Z=33 for the Is and 2p-orbits, respectively. Beyond these Z-values, the relevant X-ray transitions become increasingly wider and weaker and
c 3 o energy [keV] t 1 0 -energy . [keV]
Fig.4.6 Spectra of pionic 27Al measured with a target of 3.97 glcm2thickness. The upper spectrum was recorded during a test run at NIKHEF. The lower one was measured at PSI with a thin target and a well tuned pion telescope in order to suppress the influence of the muonic 2p—>ls X-ray transition.
therefore more difficult to separate from the background, especially if these transitions are also obscured by nuclear y-ray transitions. In the case of 27A1 the pionic 2p—>ls X-ray transition at
about 363 keV is obscured by the muonic 2p—>ls transition at 345 keV and by a nuclear y-ray at 350 keV, as is illustrated in fig.4.6. This nuclear y-ray at 350 keV is mainly produced by the reaction 27Al(7t\2p4ny)21Ne which occurs after the pion ends its atomic cascade in the nucleus.
The spectra of fig.4.6 were obtained with a Compton-suppressed Ge-detector, in which case the suppression shield consisted of Nal scintillation material only, at the NIKHEF facility with a rather thick Al-target of 3.97 g/cm . To reduce the strong muon contamination, which is seen in the upper spectrum taken during a test run, a beam telescope was installed to provide a clean pion trigger. As is shown in the lower spectrum of fig.4.6 a significant improvement was obtained; the u.X-transition strength is reduced by a factor of about 3.5. This, however, is still not good enough for a meaningful analysis of the pionic 2p—»ls X-ray transition.
Besides the use of Compton suppression shields, necessary for a good peak-to-background ratio, an additional method has been applied to reduce the interference of possible nuclear y-rays. In the 27A1 measurement the aim was to reduce the occurrence of the 350 keV nuclear transition in
Ne in the energy spectra of the Ge-detectors. There are several possible strong nuclear y-rays besides the 350 keV transition in Ne, e.g. the 440 keV in Na and at higher transition energies the 1274 keV in 22Na and 1368 keV in MMg, which occur in different exclusive reactions after the
pion is absorbed from its atomic orbit. By measuring the energy spectra of the Ge-detectors in coincidence with an event in an array of Nal-crystals viewing the target, it may be possible to recognize these nuclear transitions in the spectra of the Nal-crystals. Off-line compilation of an energy spectrum of one of the Ge-counters in coincidence with windows, imposed on the spectra of the Nal-crystals covering the different nuclear y-ray energies seen in these spectra, would yield a Ge-energy spectrum "free" of the 350 keV photo peak.
01 0\ TX
The pion-absorption reactions in Al which end with a Ne nucleus and a Na nucleus in an excited state, producing the 350 keV and 440 keV y-rays, respectively, are of the same probability and by far the most important ones. An inconvenient side-effect of these two absorption reactions,
27Al(7t",2p4ny)21Ne and 27Al(;i",p3ny)23Na, is the emission of neutrons (of roughly 25 MeV).
Off-line discrimination against the (n,n') reactions of these prompt neutrons in the Nal-crystals by means of the time-of-flight method, requires measuring with a rather large solid angle.
The status of the data acquisition system at that time did not allow the recording without too many problems of several Ge-channels (both in energy and time) and Nal-channels. Furthermore, in view of the short allocated beam time we decided to arrange a total of eight 12.7 cm (diameter) x 12.7 cm (long) NaI(Tl)-crystals around the target at a rather short distance of about 12.5 cm, (see fig.4.7) and to measure without applying the time-of-flight method to the Nal-counters.
Analysis of the Nal-specrra showed that the nuclear y-rays were overshadowed by gamma radiation abundantly produced by (n,n') reactions in Nal and that, due to the lack of time spectra
Be-degrader
I
i Al target Beam.45°T
S, S , 20 cm 8 cm Beam ♦ S1W
lHjl^-5=g^a%7l
NalJ
counter VetoFig.4.7 Schematic view of the setup for the measurement of pionic 27Al. An array of eight 12.7 cm (diameter) x 12.7 cm (long) Nal(Tl)-crystals was used to detect nuclear y-rays after pion absorption. Plastic veto counters were placed in front of the NaI(Tl)-crystals to discriminate against charged particles. Compton-suppressed Ge-detectors were placed in the horizontale plane; the Na(Tl)-detectors viewed the target from above and below.
-us c ir>3 3 10 o u
J
102 , 1 . . i . . . . | . . . , 2 1 Mg 19 F T 127, i I <^ X luX 2 ,I * 25 >27, . i . . . . i . . . . i Na ' -511 ■ Mg ' i2
P
-* y w y j y * y w ^
200 250 300 350 400 450 500 550 energy , [keV] 200 250 300 350 400 450 500 energy > [keV]Fig.4.8 Pionic X and y-ray spectra of27Alfor one of the Ge-detectors. The upper spectrum is a singles y-ray spectrum of events only coincident with a valid pion stop in the target. The^ lower one results from the additional requirement of a coincidence signal in any of eight Nal(Tl)-crystals viewing the target.
S 500 550 ♦ [keV]
o
4_
-ol
-21 Mg , 9F
L
r
i
^ ^ ^ s s ^
21 2 ,Na 16III
j "
mk^
y
Ne1,1.
n X2p-» l s - H i j i l i . . . 1 . . . . 1 . 23, 12l J
7IiWftt
■fa 5-ii
I
ww
1 "If*1 200 250 300 350 400 450 500 550 energy , [keV]Fig.4.9 Pionic X and y-ray spectra of 24 Mgfor the same Ge-detector. The upper spectrum was measured in coincidence only with a valid pion stop in the target. The lower one was recorded in coincidence with an additional signal from an array of eight Naf(Tl)-crystals.
