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Measurement of the Lamb shift in the n = 4 state of ionized helium
Volume 58A, number 4
PHYSICS LETTERS
6 September 1976
MEASUREMENT OF THE LAMB SHIFT IN THE n =4 STATE OF IONIZED HELIUM A. EIBOFNER Physikalirches Institut der Universitit Tübingen, Germany
Received 12 July 1976 A value of (1769.0 ±2.0) MHz has been found for the 4 ~j,2—~ P112 Lamb shift in ionized helium (theory: (1769.1 ±0.1) MHz). The precision of the measurement has been limited by cascading processes from P-S transitions in higher n states.
The S112—P112 Lamb shift in the n
=
4 state of
._~
,i~,/
‘I~~
ionized eral investigations helium hasusing been optical determined [1] and since radio 1966frequenin sevcy method [2, 3]. In the most accurate measurement so far [4], the precision of the result of (1768 ±5) MHz has been limited by an unexplained dependence of the positions of the resonance signals on r.f. power. The purpose of the r.f. experiment described in this paper was to determine the 4 S112 —4 P112-splitting with an accuracy comparable to that in a preceding 4P312-4~1/2 measurement [51. Each one of the four possible r.f. transitions between 4 S1,2 and 4P!/2 Zeeman states is influenced more or less by disturbing effects. The resonance signals will be superposed by cascading signals originating from S-P transitions in higher n states, by 4 D—4 F r.f. signals or by the cyclotron resonance. Additionally the signal centers wifi be displaced by the electric fields in the interaction region. In the experiment, the ae transition ~1/2(m = 1/2)—P112 (m = 1/2) has been used, which is nearly unaffected by electric fields. The experimental arrangement has been similar to that in previous investigations. The He~states have been populated by electrons (6 mA, 300 eV) moving parallel to a variable magnetic field in a helium atmosphere (He pressure: 10 mTorr). By an electric r.f. field parallel to the magnetic field electric dipole transitions between the S and P Zeeman states have been induced resulting in a change of the intensity of the 468.6 nm radiation (n 4 -+ n = 3). The light emitted at right angles to the magnetic field has been selected with a monochromator. For lock in detection of the resonance signal the electric r.f. field was square wave modulated. The magnetic field has been swept
~
~
‘~
~26
•“
,‘: i
~“
,‘
~
030
0.~2
$
H ITsila)
Fig. 1. Transition frequencies for the 4 ~e and 4 ~eresonances (solid lines) and for the cascading resonances of interest in the experiment (dashed lines). The horizontal lines indicate the magnetic field ranges used at the seven r.f. frequencies Dots refer to the experimental centers of the 4 c~eresonance signals and of the cascading signals.
through the resonance at seven fixed r.f. frequencies in the range from 4.4 to 4.8 GHz (fig. 1). The measured signals have been fitted with a Lorentzian function at 19 magnetic field strengths. At first sight the results seemed to be quite satisfactory and an accurate value for the Lamb shift with a statistical error of less than 0.2 MHz was expected. But in the course of a careful examination of the fitting parameters some deviations from the theoretical signal function have been found: — The extrapolation of the squared line widths to zero r.f. power provided values for the natural line width exceeding the theoretical width by about 10% (statistical error about 3%). 219
Volume 58A, number 4
PHYSICS LETTERS
— The baseline of the resonance signals had a small slope. The relation of the slope to the height of the resonance signal (typically about 5 X l0~/Tesla) depended on r.f. frequency. — The centers of the resonance signals showed a small dependence on r.f. power. — From panoramic measurements made in the large magnetic field range from 0.2 to 0.4 T, an abnormal dependence of the heights of the 4 ~e signal on r.f. power has been deduced. In order to fmd the causes for these effects, the experimental signal values at 65 equidistant magnetic field strengths have been compared with the corresponding values of the fitted Lorentzians. Using this method, nonstatistical signallike deviations have been found. Considering several properties of the disturbing signals (height and sign related to the main resonance, half width, dependence of the signal heights on r.f. power) it can be concluded that the deviations have been caused by a superposition of cascading effects from the ~letransitions ~l/2 (m = — 1/2) — P112 (m =
1/2) in the n = 6, n = 7 and n = 8 states. The three cascading signals are close together in the magnetic field range used in the experiment (fig. I) and cannot be separated. The height of the cascading signals has been about 3% of that of the main signal at small r.f. powers and about 1% at high r.f. powers. Some additional signals (smaller than 1% of the main signal) can be attributed to the cascading effect from the n = 5 f3e and ac transitions. The cascading signals result from a small component of the r.f. field perpendicular to the magnetic field. It can be shown that the cascading transitions nS—nP and the resonance transition 4S—4P are not independent of each other: the cascading effect will be much stronger if the resonance conditions are simultaneously satisfied both for the cascading and for the n = 4 transition. Theoretical estimations for the heights of the cascading signals are not possible because the excitation rates for the high n states are not known. The influence of the cascading transitions has been examinedby comparing the results of a single-Lorentzian fit with a fit including the cascading effects. The
220
6 September 1976
sum of the squares of the differences between the experimental and the fitted signals was reduced by about 50% and the resonance centers of the main signals idsplaced by about 0.1 to 0.2 mT. Within the precision of the experiment all the above-mentioned deviations from the theoretical signal function can be explained by the influence of the cascading signals from higher n states. The precision of the experiment has been limited mainly by the existence of the cascading signals. Additionally the possibility of smaller disturbing signals must be considered, which cannot be found by the method described above: an upper limit for the influence of such signals has been estimated. The contribution of other effects (statistical error, inhomogeneity of the magnetic field, influence of static electric fields [5]) to the uncertainty of 0.2 mT caused by cascading and disturbing signals can almost be neglected. From the experimental signal centers a value of (1769.0 ±2.0) MHz for the .4 ~1/2 —4 P112 Lamb shift has been deduced. There is a good agreement with the theoretical value of (1769.097±0.0 76) MHz [6], but the precision of the experimental value is somewhat frustrating. I wish to thank Professor M. Baumann for many useful discussions and Professor H. KrUger for the continuous support of this work. The generous financial assistance from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References [1] E.G. Kessler and F.L. Roessler, J. Opt. Soc. Am. 62 (1972) 440.
~ 102. [4] R.R. Jacobs, K.R. Lea and W.E. Lamb, Phys. Rev. A3
(1971) 884. [5] A. Eibofner, Z. Phys. A277 (1976) 225. [6] A. Peterman, Phys. Lett. 38B (1972) 330.
PHYSICS LETTERS
6 September 1976
MEASUREMENT OF THE LAMB SHIFT IN THE n =4 STATE OF IONIZED HELIUM A. EIBOFNER Physikalirches Institut der Universitit Tübingen, Germany
Received 12 July 1976 A value of (1769.0 ±2.0) MHz has been found for the 4 ~j,2—~ P112 Lamb shift in ionized helium (theory: (1769.1 ±0.1) MHz). The precision of the measurement has been limited by cascading processes from P-S transitions in higher n states.
The S112—P112 Lamb shift in the n
=
4 state of
._~
,i~,/
‘I~~
ionized eral investigations helium hasusing been optical determined [1] and since radio 1966frequenin sevcy method [2, 3]. In the most accurate measurement so far [4], the precision of the result of (1768 ±5) MHz has been limited by an unexplained dependence of the positions of the resonance signals on r.f. power. The purpose of the r.f. experiment described in this paper was to determine the 4 S112 —4 P112-splitting with an accuracy comparable to that in a preceding 4P312-4~1/2 measurement [51. Each one of the four possible r.f. transitions between 4 S1,2 and 4P!/2 Zeeman states is influenced more or less by disturbing effects. The resonance signals will be superposed by cascading signals originating from S-P transitions in higher n states, by 4 D—4 F r.f. signals or by the cyclotron resonance. Additionally the signal centers wifi be displaced by the electric fields in the interaction region. In the experiment, the ae transition ~1/2(m = 1/2)—P112 (m = 1/2) has been used, which is nearly unaffected by electric fields. The experimental arrangement has been similar to that in previous investigations. The He~states have been populated by electrons (6 mA, 300 eV) moving parallel to a variable magnetic field in a helium atmosphere (He pressure: 10 mTorr). By an electric r.f. field parallel to the magnetic field electric dipole transitions between the S and P Zeeman states have been induced resulting in a change of the intensity of the 468.6 nm radiation (n 4 -+ n = 3). The light emitted at right angles to the magnetic field has been selected with a monochromator. For lock in detection of the resonance signal the electric r.f. field was square wave modulated. The magnetic field has been swept
~
~
‘~
~26
•“
,‘: i
~“
,‘
~
030
0.~2
$
H ITsila)
Fig. 1. Transition frequencies for the 4 ~e and 4 ~eresonances (solid lines) and for the cascading resonances of interest in the experiment (dashed lines). The horizontal lines indicate the magnetic field ranges used at the seven r.f. frequencies Dots refer to the experimental centers of the 4 c~eresonance signals and of the cascading signals.
through the resonance at seven fixed r.f. frequencies in the range from 4.4 to 4.8 GHz (fig. 1). The measured signals have been fitted with a Lorentzian function at 19 magnetic field strengths. At first sight the results seemed to be quite satisfactory and an accurate value for the Lamb shift with a statistical error of less than 0.2 MHz was expected. But in the course of a careful examination of the fitting parameters some deviations from the theoretical signal function have been found: — The extrapolation of the squared line widths to zero r.f. power provided values for the natural line width exceeding the theoretical width by about 10% (statistical error about 3%). 219
Volume 58A, number 4
PHYSICS LETTERS
— The baseline of the resonance signals had a small slope. The relation of the slope to the height of the resonance signal (typically about 5 X l0~/Tesla) depended on r.f. frequency. — The centers of the resonance signals showed a small dependence on r.f. power. — From panoramic measurements made in the large magnetic field range from 0.2 to 0.4 T, an abnormal dependence of the heights of the 4 ~e signal on r.f. power has been deduced. In order to fmd the causes for these effects, the experimental signal values at 65 equidistant magnetic field strengths have been compared with the corresponding values of the fitted Lorentzians. Using this method, nonstatistical signallike deviations have been found. Considering several properties of the disturbing signals (height and sign related to the main resonance, half width, dependence of the signal heights on r.f. power) it can be concluded that the deviations have been caused by a superposition of cascading effects from the ~letransitions ~l/2 (m = — 1/2) — P112 (m =
1/2) in the n = 6, n = 7 and n = 8 states. The three cascading signals are close together in the magnetic field range used in the experiment (fig. I) and cannot be separated. The height of the cascading signals has been about 3% of that of the main signal at small r.f. powers and about 1% at high r.f. powers. Some additional signals (smaller than 1% of the main signal) can be attributed to the cascading effect from the n = 5 f3e and ac transitions. The cascading signals result from a small component of the r.f. field perpendicular to the magnetic field. It can be shown that the cascading transitions nS—nP and the resonance transition 4S—4P are not independent of each other: the cascading effect will be much stronger if the resonance conditions are simultaneously satisfied both for the cascading and for the n = 4 transition. Theoretical estimations for the heights of the cascading signals are not possible because the excitation rates for the high n states are not known. The influence of the cascading transitions has been examinedby comparing the results of a single-Lorentzian fit with a fit including the cascading effects. The
220
6 September 1976
sum of the squares of the differences between the experimental and the fitted signals was reduced by about 50% and the resonance centers of the main signals idsplaced by about 0.1 to 0.2 mT. Within the precision of the experiment all the above-mentioned deviations from the theoretical signal function can be explained by the influence of the cascading signals from higher n states. The precision of the experiment has been limited mainly by the existence of the cascading signals. Additionally the possibility of smaller disturbing signals must be considered, which cannot be found by the method described above: an upper limit for the influence of such signals has been estimated. The contribution of other effects (statistical error, inhomogeneity of the magnetic field, influence of static electric fields [5]) to the uncertainty of 0.2 mT caused by cascading and disturbing signals can almost be neglected. From the experimental signal centers a value of (1769.0 ±2.0) MHz for the .4 ~1/2 —4 P112 Lamb shift has been deduced. There is a good agreement with the theoretical value of (1769.097±0.0 76) MHz [6], but the precision of the experimental value is somewhat frustrating. I wish to thank Professor M. Baumann for many useful discussions and Professor H. KrUger for the continuous support of this work. The generous financial assistance from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.
References [1] E.G. Kessler and F.L. Roessler, J. Opt. Soc. Am. 62 (1972) 440.
~ 102. [4] R.R. Jacobs, K.R. Lea and W.E. Lamb, Phys. Rev. A3
(1971) 884. [5] A. Eibofner, Z. Phys. A277 (1976) 225. [6] A. Peterman, Phys. Lett. 38B (1972) 330.