He+ iyonunun ince yapı aralığındaki kesişmeyen sinyalleri

Abstract

İlk olarak Eck vd. (1963) tarafından gözlenen kesişmeyen sinyaller için Wieder ve Eck (1967), Maujean ve Descoubes (1978) sadece kesişen iki altduruma dayanan birbirine denk teori geliştirdiler. Ancak yüksek mertebeli sinyallerin diğer durumları da içerebileceğini hesaba katmadılar. Beyer ve Kleinpoppen (1978) 'in de hesaplamaları zamandan bağımsız idi. İlk olarak zamana bağlı, diğer durumları da göz önüne alan çalışmayı Tepehan (1990) yapmış ve deneylerle Tepehan vd. (1982a, 1982b) karşılaştırılmıştır. Ayrıca, Beyer (1973) tarafından geliştirilen teorinin asimetrik sinyalleri tanımlamakta yetersiz olmasına rağmen Tepehan (1990) da bu problem ortadan kalkmıştır. Bu çalışmada He+ iyonunun n=4 durumunda yapılan deneyler ile hesaplaması yapılmayan sinyaller hesaplanarak karşılaştırması yapılmıştır. Karşılaştırma için seçilen durumlar ise S-D ince yapı aralığındaki cJ ve S-F ince yapı aralığındaki $H durumlarıdır. Zamana bağlı hesap tekniğiyle bilgisayarda işlemler yapılmıştır. Simetrik sinyaller için iyi sonuç veren Lorentz tipi en uygun grafiğin (curve fitting) asimetrik sinyallerin (cJ gibi) genlik (amplitude) ve genişlikleri (FWHM) göz önüne alındığında göreli olarak iyi sonuç vermediği görülmüş, bunun için yedinci dereceden Lorentz tipine göre en uygun grafik çizimi yapılmıştır. Kesişme merkezi veya kesişmeme olarak bahsedilen noktada esasen kesişme olmamakta, fakat bu hale en yalan durumdan bahsedilmektedir. Kesişmeme oJ ve S-F ince yapı aralığındaki (3IT sinyalleri ve S-D ince yapı aralığındaki oJ sinyalleri, Zeeman ve Stark etkisinin bileşimi olarak zamana bağlı formda incelenmiş, etkileşme şekli olarak L.S eşlenmesinin (coupling) manyetik alanın uygulanmasıyla bozulacağım öngören Paschen-Back etkileşmesi seçilmiştir. Enerji seviyelerinin değişimi ve şiddet (intensity) manyetik alanın fonksiyonu olarak sabit elektrik alan altoda gerekli pertürbasyon terimleri yerleştirildikten sonra matris köşegenleştirme metoduyla elde edilmiştir. Elektrik alan herbir durum için belli bir aralıkta (od için 7-14 kV/m, (3H için 10-21 kV/m) değiştirilerek işlem tekrarlanmıştır. Elde edilen veriyle manyetik alan -şiddet grafikleri çizilmiş ve en uygun çizim metoduyla grafik çizilerek Lorentz fonksiyonundaki tanımlar yardımıyla kesişme merkezi, şiddet, genlik ve genişlik elde edilerek manyetik alana karşı grafikleri çizilmek suretiyle davranışları gösterilmiştir. Kesişmeme merkezi olarak sıfir elektrik alandaki değer alınmış, grafiğin eğimi ise Stark sabiti olarak elde edilmiştir. Bu işlemler sonucunda kesişmeme merkezi S-D ince yapı aralığında cJ durumu için 580.595 mWeber/m2, S-F ince yapı aralığında PH durumu için 740.875 mWeber/m2 bulunmuştur. Bu değerler daha önce yapılan deneylerle (deney hata payı dahil edildiğinde birebir uyuşmakta) en az % 0.07, en fazla % 0.08 fark etmekte; önceki yapılan hesap ve teorilerle ise en az % 0.0002, en fazla % 0.005 fark etmekte, Stark sabiti ise deneylerle en az % 0.9, en fazla % 12 fark etmektedir.

He+ ION WITH n = 4 Line width problem could not be solved until 1966 that Lea et al. (1966) reported fully successful measurements of intervals 4S - 4P, followed by Hatfield and Hudges (1967), Beyer and Kleinpoppen, Jacobs et al. (1971) measured the 4D - 4F intervals with an accuracy limited by the weakness and the large width of the signals. All these measurements use AL = 1 electric dipole transitions which can easily be induced by RF fields. Anticrossing signals were first observed by Eck et al. (1963) during a level crossing investigation of the fine and hyperfine structure of the 2^P term of lithium. Since then the technique has been used to study the structure of excited states with high accuracy. A theory was developed by Wieder and Eck (1967) describing level and anticrossing signals between two states in a combined equation. Glass-Maujean and Descoubes (1978) derived an equivalent equation using the density matrix formalism. The signals in both calculations are based on the two crossing substates only although coupling scheme may involve other states for higher order signals. Beyer (1973) working on anticrossing signals in He+, calculated the crossing position independently of the theory Wieder and Eck (1967) two different methods. Zeeman and the Stark effects were treated independently in the first method, based on the assumption that a small electric field has little influence on the energy of the magnetic substates. This approach represented a good approximation for electric fields below the 15 kV/m. By increasing the electric field to have Stark energies comparable with that of Zeeman, this method was accepted to be less accurate. Second calculation was made with a combined treatment of the Zeeman and Stark effects. Using the time-independent approach, the full fine-structure system was diagonalised applying electric field and magnetic field simultaneously: The interaction element, V^, derived from this calculation can be used together with the theory by Wieder and Eck (1967) to calculate the width and the degree of saturation of the signals. The signal shape is symmetric. It does not take into account the influence of other states which may cause some degree of asymmetry in the experimental anticrossing signals. This may affect the crossing positions and Stark shifts of the substates, and does not give the exact energy eigenvalues of the intersecting Zeeman substates near their closest approach, especially when the interaction energy is less than or equal to one quarter of the difference between the line widths (i.e. V^ _ i/4|ya - yj, | where of the corresponding substates ya, yj, are the widths of the corresponding states (Lamb 1952). Although the crossing - and anticrossing-signal was used as a term in realty there exist not such thing. It is just the intensity changes in the line emitted from the corresponding substates are observable as a result of the state mixing at and near the crossing position (Beyer and Kleinpgppen 1978). The substates still cross in the time-dependent but not in the time-independent approach below this critical energy. Crossing center and the Stark shift can be derived by finding the closest distance of the two anticrossing states via the time independent matrix diagonalisation. In the present work, the energy matrix of the full fine structure system was used in a time dependent calculation (Tepehan 1990). Since the imaginary part of the eigenvalues represents the lifetime of the states at the corresponding signals if the crossection of the substates are known. The n = 4 state of the He+ has been chosen for this detailed line shape analysis of anticrossing signals since experimental data (Tepehan et al 1982a) and last theoretical calculation (Tepehan 1990) are available for comparison. Moreover, chosen substates limited to aG', aJ, (3IT (look at Tablo A.1 in Ek A section for substates) because of the lack of the theoretic calculation for last two, and the first one to explain some of the criteria on it. Theoretical background: The Anticrossing signals depend on the population of states, and in the case of mixed level-crossing-anticrossing signals, also on the coherence induced by the static perturbation between the crossing states. The shape of the signal can be obtained if the intensity of the observed spectral transition is calculated as a function of the magnetic field. The light propagates with right-hand (transition Am= 1) or left-hand (transition Am= -1) circular polarization i.e.a- polarized light. Intensity is proportional to I J| and |b> the most common pure anticrossing signal. This Lorentzian curve centered at Av =0: vin 35.229 - r r I 35.229 E- t t 35.229 E- r t c S 35.229 E- S ^ S E.a t !§ 35.229 E- o a: o / 35.223 E- 35 223 *- V ^ o-o-^ -t i.5716 0.57-6 5.5717 0.5717 5.5713.0.5713 5 Magnetic Field (Weber/m2) 11.642 r 1.64-2 11.642 - _p-°-a. /^ d 9 \> 11.641.57160.5716 5.5717 0.5717 5.5713 0.6718 5 Magnetic Field (Weber/m2) Figure 2. Calculated lifetimes of aG' signal of He+ ion n=4 states. Care on the near sides of anticrossing point 0.6717 Weber/m2. IX 5(Al/) = dl + A,2/fl2 (S.8) where the amplitude is A, the fUll width at half maximum is 2B, and d is the instrumental parameter. Equating Av = X - Z with X magnetic field axis and Z is the crossing center, and adding dispersion parts one gets (Beyer 1973): S = A[l + CAv] / [1 + (Av)2 / B2] + DAv + E (S.9) with now, A and B as they were but of absorption and dispersion, C is the parameter for the dispersion signal amplitude, D is the slope of the baseline, E is the background at X = Z. Observation and Comparison with Calculation: After finding the intensities in certain electric field with the change of magnetic field in computer, intensities versus magnetic field graphed by applying best curve fitting method with the use of Lorentz curve type. In choosing function for best curve fitting there were no problem of using equation (S.9) for aJ, but it was needed to use seventh degree Lorentz function -stated at eqn. (S.10)- for (3FT signal (figure 3). S' = S + F(Av)2 + G(Av)3 + H(Av)4 + I(Av)5 + J(Av)6 + K(Av)7 (S. 1 0) where F, G, H, I, J, K are any constants. However, in the case of experimental data there is no doubt to use equation (S.9) since the sinusoidal actions in small intervals [look at Tepehan et al. (1982a) figure 7]..5650.5700.5750.5800.5850.5900.3950.SOPO Magnetic Field (Weber/m2) c '.5S50.5700.5730.5800.5352.5900.5950. Magnetic Field (Weber/m") Figure 3. Best fit of calculated values of He+ n=4 pH' states w.r.t. Lorentz graphed at the left, and w.r.t. 7th degree Lorentz type at the right. type After finding the constants in Lorentz curve fits, anticrossing points versus squared electrical field plotted to calculate Stark constant from the slope of the graph, besides anticrossing center taken to be at zero electrical field. Yet there is only one Stark constant for each state via calculation -calculation for PH first done at this work- it differs experimentally depending on the pressure of helium (figure 4). This experimental difference is removable with extrapolation to zero pressure (Tepehan et al. 1982b). Action of the states also tried to understand by graphing amplitudes w.r.t. electric field, and the full width at half medium (FWHM) versus squared electric field (figure 5). As a result; when looked through the crossing centers in the fine structure interval by changing electric fields, observed graphic was said to be linear in the accuracy interval of the experiment (±0.9 - ±30 nVWeber/m2) (figure 4). So it has best fitted to linear equation, and the intersection point with the crossing centers axis at zero electrical field gave anticrossing center (table 1), with curve's slope to be Stark constant. Thereafter, it should have been possible to compare time dependent matrix method with the time independent matrix method in addition to experimental results (table 2). Table 1. Anticrossing position in He+, n = 4 Anticrossing This work Theories Experiments signal (mWeber/m2) (mWeber/m2) (mWeber/m2) 580.6253b S-D oJ 580.5949 S-F (3H 740.8753 740.8927b 740.8735e 580.67±0.25a 581.05±0.5C 580.587±0.072d 740.82±0.25a 740.794±0.042d a Beyer and Kleinpoppen (1971, 1972) b Beyer (1973) c Billy etal (1977) d Tepehan et al (1982a) e Tepehan (1990) Table 2. Quadratic Stark constant of the S - F anticrossing signals in He+, n = 4 Anticrossing signal Nearly magnetic field Electric field interval (mWeber/m2) (kV/m) (xlO-^Weber/V2) (xlO-^Weber/V2) (xlO-13Weber/V2) (x 10-13 Weber/y2) This work Beyer etal (1973) Tepehan et al (1982a) Tepehan et al (1982b) Comparing this work with the old experiments for anticrossing points, least difference was 0.07 % and the most difference was 0.08 %, and at least 0.0002 % at most 0.005 % difference with the old theories. However, when experimental XI 07U - a O 0.7«- - 0740 100 400 Squared Electric Field (x 106 V2/m2).7450 r C4 c O 'ö0 e '35 C/5 a.7445 -.7440.7 405 1 ı ı ı ı ı ı ? I ı ı ı ı ı ı ı ı I I I I I I I I I I I I I I ''''I''''''''' o 100 200 300 400 Squared Electric Field (x 106 V2/m2) 500 Figure 4. Crossing centers anticrossings S - F interval ŞET in n = 4 of He+ as a function of the squared electric field; for the observed data (cutted lines) and the theoretic calculation (solid line) (Tepehan 1990) at the top graph, and this work of calculation at the bottom. Xll ?§ 4> > 1 r 0 5000 (a) (b) 10000 15000 20000 Electric Field (V/m) 5000 * f 5 h.- v- 5000 10C00 15000 20000 Electric Field (V/m) 25000 Figure 5. Action of the J3H states of He+ for amplitudes of unpolarized and ct- polarized light (a) for 7t-polarized (b) w.r.t. electric field at the top graph, and that of width w.r.t. squared electric field at the bottom graph circles represents unpolarized and a-polarized light with triangle represents 7t-polarized light. xui errors taken into considerations these differences fall. Taking Stark effect into consideration for comparison; there exist difference at least 0.9 % at most 12 % with experiments. Since the experimental accuracy increased by the time these results may be used to identify signals, although this seems not possible by the experiment of Beyer (1973) because of intermingling between states (Tepehan 1982b table 1). Conclusion; differences in between this work (and other theoretical works) with experiments may be said like; resulting from experiments) 1. magnetic field uncertainties, 2. electric field uncertainties, 3. asymmetry uncertainties in the signals, 4. statistical uncertainties near the crossing points (Tepehan 1982b), resulting from theories; 5. Paschen - Back effect's usage boundary is not certain for applied magnetic field (Sakurai 1985), 6. so then, L.S coupling ions' population does not known whether to be taken as perturbation or not, 7. instead of using ellipsoidal/paraboloidal Schrödinger equation which may give good result (look at the sect. 51(3 of Bethe and Salpeter (1957) for the Stark effect example), spherical Schrödinger equation used.