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Spectroscopic quadrupole moment of holmium from pionic X-ray measurement
Volume 53B, number 1
PHYSICS LETTERS
11 November 1974
SPECTROSCOPIC Q U A D R U P O L E M O M E N T O F H O L M I U M F R O M PIONIC X-RAY M E A S U R E M E N T ~ P. EBERSOLD, B. AAS, W. DEY, R. EICHLER, J. HARTMANN*, H.J. LEISI and W.W. SAPP Laboratory for High Energy Physics, Swiss Federal Institute of Technology, Zurich, c/o SIN, Vtlligen, Switzerland Received 7 October 1974 From the observed quadrupole splitting of the 5g-4f X-ray transition in n-16SHo we determine the spectroscopic quadrupole moment of 16s Ho to be Q = 3.47 ± 0.11 b. The strong interaction shift and the width of the 4f level are found to be eo = 0.35 ± 0.08 keV and r o = 0.21 ± 0.04 keV, respectively.
It has been shown that the quadrupole splittings of X-rays from highly excited levels of muonic atoms are suitable for obtaining precise values of nuclear spectroscopic quadrupole moments [1,2]. The quadrupole interaction in pionic atoms, in contrast to the muonic case, is composed of two parts. The first part is the usual electromagnetic quadrupole coupling, and the second part is due to the strong interaction between the pion and the nucleons [3]. To the extent that the strong interaction quadrupole effect is known, quadrupole splittings in pionic atoms may be used to determine spectroscopic quadrupole moments of nuclei. In an experiment at SIN (Swiss Institute for Nuclear Research), using the 600 MeV isochronous cyclotron, we have observed the quadrupole splitting of the 5g-4f X-ray transition in pionic holmium (see fig. 1). Pions at the intermediate focus of the nM1 beam were stopped in a Ho-oxide target. The X-rays were observed with a 3 cm 3 Ge(Li) detector in coincidence with a stop signal from a conventional counter telescope. The electron contamination of the pion beam was suppressed by requiring a coincidence with the 50 MHz signal of the accelerator delayed by an amount appropriate to the time of flight of the pions. Both the initial and the final state of the 5g-4f transition are split by the quadrupole interaction into energy levels characterized by a total angular momentum F This work was partially supported by SIN. * On leave from the Teehnische Universit~lt,Munich, Germany.
48
EF = (A2-e2)R(F),
(1)
where A 2 is the electromagnetic quadrupole coupling constant, e 2 is the quadrupole coupling constant of the strong interaction [3] and R (F) is an angular momentum factor [1 ]. We define an effective quadrupole moment Qefr by writing eq. (1) in the form E F -- Qeff(A~2)/Q)'R (IV)
(2)
where A(2p) is the electromagnetic quadrupole coupling constant appropriate to a point nucleus and a relativistic wave function of the pion, and Q is the
I 2OO
tt, .~ ~
[
1;-165Ho 1,
5g - 4 f
lOO
a6o
a65
key
Energy
Fig. 1. The 5g-4f X-ray transition in pionic 165Ho. The ener83"resolution of the Ge (Li) detector was 1.2 keV FWMH at 412 keV.
Volume 53B, number 1
PHYSICS LETTERS
spectroscopic quadrupole moment of the nucleus in its ground state; Qeff may then be expressed as Qeff =
Qfl+81+82-e2/A(2P))"
(3)
In eq. (3) 51 describes the correction due to the finite size of the nucleus (effect of overlap between nucleus and pion) and 82 takes account of the distortion of the pionic wave function due to both the finite nuclear size and the strong interaction. The widths of the X-ray transitions between the various F states are determined by the strong interaction width of the final F state, which is given by
[3] PF ffi P0 + P2" R (F)
(4)
and by the radiation width l'rad of the initial state. The Lorentzian width of each X-ray transition is the sum of PF and l'rad. The instrumental width is described by a Gaussian distribution, modified by a low-energy tail. The width of the Gaussian and the parameters fixing the tail are taken from fits to the calibration spectrum which was recorded simultaneously with the X-ray spectrum and to the pionic Xrays from higher levels. The hyperfine structure components of the initial 5g level are assumed to be statistically populated. Hence the relative intensities of all 20 hyperfine transitions are known, and the observed structure in fig. 1 is determined by the three unknowns Qeff, I'0 and P2" We take the ratio P2/Vo = -0.31 from the theoretical work of Scheck [3] and obtain from a leastsquares fit to the data of fig. 1 Qeff = 3.65 + 0.07 b,
(5)
and
PO = 0.21 + 0.04 keV.
(6)
The energy corresponding to the position of the unsplit 5g-4f X-ray line was found to be ~75g.4f = 381.04 + 0.08 keV.
(7)
From this value and calculated electromagnetic level positions [5] we find for the strong-interaction shift eo = 0.35 + 0.08 keV.
(8)
This value is significantly larger than the theoretical
11 November 1974
value, e(th) = 0.20 keV, as computed from the pionicatom program with Kisslinger potential, LorentzLorentz effect and standard parameters [4]. The value given here differs somewhat from the examples published in ref. [3], insofar as eo is obtained from the exact energy eigenvalue of the Klein-Gordon equation, whilst e 2 has been calculated perturbatively with distorted pion wave functions. In ref. [3], the quantities e o and e 2, for the case of pions, were calculated perturbatively and with hydrogenic wave functions. Relativistic wave functions appropriate to a point nucleus have been used, and first-order vacuum polarization corrections were applied. In order to obtain Q from Qeff, we need a value for e 2. Since the theoretical ratio e2/e o is nearly independent of the parameters of the strong-interaction optical potential, we use the relation e 2 = (e2/eo)(th)e(oexp).
(9)
Taking (e2/eo)(th) = --0.17 and e(exp) from eq. (8) we find e 2 = -0.060 + 0.014 keV.
(9')
The remaining correction terms in eq. (3) are found to be small, namely 81 = -0.001 and 82 = 0.007. From eqs. (3) and (9') we then obtain Q = 3.47 + 0.11 b.
(10)
This value depends on the theoretical ratio (e2/eo). In principle, it can be measured directly from an experiment on muonic 165Ho, similar to the one performed with 175Lu [2]. Small corrections due to the Ml-hts splittings and E2 vacuum-polarization are included. This value of the spectroscopic quadrupole moment is more precise than any value by any other method [5]. Two immediate consequences of this result are: first, the rotational model of nuclei may be checked by combining our result with the determination of B(E2) from Coulomb excitation. Within the rotational modelB(E2) is related to the intrinsic quadrupole moment Qo" According to ref. [6], Qo = 7.62 + 0.12 b; Qo, again assuming the validity of the rotational model, leads to a prediction for the spectroscopic quadmpole moment, namely Q = 3.55 + 0.06 b. This value is in good agreement with eq. (10). Second, the quadrupole coupling constant of the elec49
Volume 53B, number 1
PHYSICS LETTERS
tronic holmium atom, as measured, e.g., by atomic beam experiments, may be combined with our value of Q to derive the electric field gradient at the Ho nucleus, a quantity which is very difficult to calculate for a multi-electron system [5]. The precise knowledge of Q may, similarly, be used to determine field gradients in other electronic systems (crystals, molecules), for which quadrupole coupling constants have been measured. These systems may then in turn be used to measure spectroscopic quadrupole moments of other isotopes and excited nuclear states. It is a pleasure to thank Dr. F Scheck for most valuable discussions, and for permitting us to use his
50
11 November 1974
computer programs. We also wish to thank the staff of the SIN accelerator for their skillful and persevering efforts during the first beam period at SIN.
References [1] [2] [3] [4] [5]
H.J. Leisi et al., J. Phys. Soc. Japan Suppl. 34 (1973) 355. W. Dey et al., Heir. Phys. Acta 47 (1974) 93. F. Scheck, Nucl. Phys. B42 (1972) 573. M. Krell and T.E.O. Ericson, Nucl. Phys. B11 (1969) 521. G.H. Fuller and V.W. Cohen, Nuclear Data Tables A5 (1969) 433; in particular see comment on p. 437. [6] K.E.G. L6bner, M. Vetter and V. H6nig, Nuclear Data Tables A7 (1970) 495.
PHYSICS LETTERS
11 November 1974
SPECTROSCOPIC Q U A D R U P O L E M O M E N T O F H O L M I U M F R O M PIONIC X-RAY M E A S U R E M E N T ~ P. EBERSOLD, B. AAS, W. DEY, R. EICHLER, J. HARTMANN*, H.J. LEISI and W.W. SAPP Laboratory for High Energy Physics, Swiss Federal Institute of Technology, Zurich, c/o SIN, Vtlligen, Switzerland Received 7 October 1974 From the observed quadrupole splitting of the 5g-4f X-ray transition in n-16SHo we determine the spectroscopic quadrupole moment of 16s Ho to be Q = 3.47 ± 0.11 b. The strong interaction shift and the width of the 4f level are found to be eo = 0.35 ± 0.08 keV and r o = 0.21 ± 0.04 keV, respectively.
It has been shown that the quadrupole splittings of X-rays from highly excited levels of muonic atoms are suitable for obtaining precise values of nuclear spectroscopic quadrupole moments [1,2]. The quadrupole interaction in pionic atoms, in contrast to the muonic case, is composed of two parts. The first part is the usual electromagnetic quadrupole coupling, and the second part is due to the strong interaction between the pion and the nucleons [3]. To the extent that the strong interaction quadrupole effect is known, quadrupole splittings in pionic atoms may be used to determine spectroscopic quadrupole moments of nuclei. In an experiment at SIN (Swiss Institute for Nuclear Research), using the 600 MeV isochronous cyclotron, we have observed the quadrupole splitting of the 5g-4f X-ray transition in pionic holmium (see fig. 1). Pions at the intermediate focus of the nM1 beam were stopped in a Ho-oxide target. The X-rays were observed with a 3 cm 3 Ge(Li) detector in coincidence with a stop signal from a conventional counter telescope. The electron contamination of the pion beam was suppressed by requiring a coincidence with the 50 MHz signal of the accelerator delayed by an amount appropriate to the time of flight of the pions. Both the initial and the final state of the 5g-4f transition are split by the quadrupole interaction into energy levels characterized by a total angular momentum F This work was partially supported by SIN. * On leave from the Teehnische Universit~lt,Munich, Germany.
48
EF = (A2-e2)R(F),
(1)
where A 2 is the electromagnetic quadrupole coupling constant, e 2 is the quadrupole coupling constant of the strong interaction [3] and R (F) is an angular momentum factor [1 ]. We define an effective quadrupole moment Qefr by writing eq. (1) in the form E F -- Qeff(A~2)/Q)'R (IV)
(2)
where A(2p) is the electromagnetic quadrupole coupling constant appropriate to a point nucleus and a relativistic wave function of the pion, and Q is the
I 2OO
tt, .~ ~
[
1;-165Ho 1,
5g - 4 f
lOO
a6o
a65
key
Energy
Fig. 1. The 5g-4f X-ray transition in pionic 165Ho. The ener83"resolution of the Ge (Li) detector was 1.2 keV FWMH at 412 keV.
Volume 53B, number 1
PHYSICS LETTERS
spectroscopic quadrupole moment of the nucleus in its ground state; Qeff may then be expressed as Qeff =
Qfl+81+82-e2/A(2P))"
(3)
In eq. (3) 51 describes the correction due to the finite size of the nucleus (effect of overlap between nucleus and pion) and 82 takes account of the distortion of the pionic wave function due to both the finite nuclear size and the strong interaction. The widths of the X-ray transitions between the various F states are determined by the strong interaction width of the final F state, which is given by
[3] PF ffi P0 + P2" R (F)
(4)
and by the radiation width l'rad of the initial state. The Lorentzian width of each X-ray transition is the sum of PF and l'rad. The instrumental width is described by a Gaussian distribution, modified by a low-energy tail. The width of the Gaussian and the parameters fixing the tail are taken from fits to the calibration spectrum which was recorded simultaneously with the X-ray spectrum and to the pionic Xrays from higher levels. The hyperfine structure components of the initial 5g level are assumed to be statistically populated. Hence the relative intensities of all 20 hyperfine transitions are known, and the observed structure in fig. 1 is determined by the three unknowns Qeff, I'0 and P2" We take the ratio P2/Vo = -0.31 from the theoretical work of Scheck [3] and obtain from a leastsquares fit to the data of fig. 1 Qeff = 3.65 + 0.07 b,
(5)
and
PO = 0.21 + 0.04 keV.
(6)
The energy corresponding to the position of the unsplit 5g-4f X-ray line was found to be ~75g.4f = 381.04 + 0.08 keV.
(7)
From this value and calculated electromagnetic level positions [5] we find for the strong-interaction shift eo = 0.35 + 0.08 keV.
(8)
This value is significantly larger than the theoretical
11 November 1974
value, e(th) = 0.20 keV, as computed from the pionicatom program with Kisslinger potential, LorentzLorentz effect and standard parameters [4]. The value given here differs somewhat from the examples published in ref. [3], insofar as eo is obtained from the exact energy eigenvalue of the Klein-Gordon equation, whilst e 2 has been calculated perturbatively with distorted pion wave functions. In ref. [3], the quantities e o and e 2, for the case of pions, were calculated perturbatively and with hydrogenic wave functions. Relativistic wave functions appropriate to a point nucleus have been used, and first-order vacuum polarization corrections were applied. In order to obtain Q from Qeff, we need a value for e 2. Since the theoretical ratio e2/e o is nearly independent of the parameters of the strong-interaction optical potential, we use the relation e 2 = (e2/eo)(th)e(oexp).
(9)
Taking (e2/eo)(th) = --0.17 and e(exp) from eq. (8) we find e 2 = -0.060 + 0.014 keV.
(9')
The remaining correction terms in eq. (3) are found to be small, namely 81 = -0.001 and 82 = 0.007. From eqs. (3) and (9') we then obtain Q = 3.47 + 0.11 b.
(10)
This value depends on the theoretical ratio (e2/eo). In principle, it can be measured directly from an experiment on muonic 165Ho, similar to the one performed with 175Lu [2]. Small corrections due to the Ml-hts splittings and E2 vacuum-polarization are included. This value of the spectroscopic quadrupole moment is more precise than any value by any other method [5]. Two immediate consequences of this result are: first, the rotational model of nuclei may be checked by combining our result with the determination of B(E2) from Coulomb excitation. Within the rotational modelB(E2) is related to the intrinsic quadrupole moment Qo" According to ref. [6], Qo = 7.62 + 0.12 b; Qo, again assuming the validity of the rotational model, leads to a prediction for the spectroscopic quadmpole moment, namely Q = 3.55 + 0.06 b. This value is in good agreement with eq. (10). Second, the quadrupole coupling constant of the elec49
Volume 53B, number 1
PHYSICS LETTERS
tronic holmium atom, as measured, e.g., by atomic beam experiments, may be combined with our value of Q to derive the electric field gradient at the Ho nucleus, a quantity which is very difficult to calculate for a multi-electron system [5]. The precise knowledge of Q may, similarly, be used to determine field gradients in other electronic systems (crystals, molecules), for which quadrupole coupling constants have been measured. These systems may then in turn be used to measure spectroscopic quadrupole moments of other isotopes and excited nuclear states. It is a pleasure to thank Dr. F Scheck for most valuable discussions, and for permitting us to use his
50
11 November 1974
computer programs. We also wish to thank the staff of the SIN accelerator for their skillful and persevering efforts during the first beam period at SIN.
References [1] [2] [3] [4] [5]
H.J. Leisi et al., J. Phys. Soc. Japan Suppl. 34 (1973) 355. W. Dey et al., Heir. Phys. Acta 47 (1974) 93. F. Scheck, Nucl. Phys. B42 (1972) 573. M. Krell and T.E.O. Ericson, Nucl. Phys. B11 (1969) 521. G.H. Fuller and V.W. Cohen, Nuclear Data Tables A5 (1969) 433; in particular see comment on p. 437. [6] K.E.G. L6bner, M. Vetter and V. H6nig, Nuclear Data Tables A7 (1970) 495.