Diode laser absorption spectroscopy of lithium isotopes

Abstract

We study Doppler-limited laser intensity absorption, in a thermal lithium vapor containing ⁷Li and ⁶Li atoms in a 9 to 1 ratio, using a narrow-linewidth single-longitudinal-mode tunable external cavity diode laser at the wavelength of 670.8 nm. The lithium vapor was embedded in helium or argon buffer gas. The spectral lineshapes were rigorously predicted for $D_1$ and $D_2$ for the lithium 6 and 7 isotope lines using reduced optical Bloch equations, specifically derived, from a density matrix analysis. Here, a detailed comparison is provided of the predicted lineshapes with the measured ⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$ lines, in the case of high vapor density and with intensity above the saturation intensity. To our knowledge, this is the first time that such detailed comparison is reported in the open literature. The calculations were also extended to saturated absorption spectra and compared to measured Doppler-free ⁷Li-$D_2$ and ⁶Li-$D_2$ hyperfine lines.

1 Introduction

In the last few decades, several new spectroscopic techniques have been developed to study the absorption in neutral vapors utilizing tunable diode lasers. Those include saturated and Doppler-limited absorption spectroscopy, polarization spectroscopy, multiphoton spectroscopy, saturated interference spectroscopy and heterodyne polarization spectroscopy. Doppler-free and Doppler-limited spectroscopy demands knowledge of the hyperfine structure and the use of density matrix equations. In this regard, the absolute Doppler-broadened absorption by rubidium atoms has been studied theoretically and experimentally for atoms below the saturation intensity [1, 2].

Doppler-free and Doppler-limited lineshape calculations have been performed in detail using density matrix equations only for ⁶Li-$D_2$ in [3]. By using a two-level approximation, only the Doppler-limited spectra of the lines ⁶Li-$D_1$, ⁷Li-$D_2$ and ⁷Li-$D_1$ were able to be modeled, as the Doppler-free spectra cannot be obtained by the same approximation. In this work, we extended the analysis to these lithium lines in detail using rate equations that include all the multiplets.

Reduced optical Bloch equations for the Zeeman multiplets were obtained starting from Zeeman ⁶Li-$D_2$ magnetic sublevels [3] considering the effect of collisions with buffer gas atoms. The method was also extended to the case of ⁸⁵Rb-$D_2$ line and was used to calculate the lineshapes in a collisionless rubidium vapor with intensities above saturation [4]. In a previous work, the atom density in lithium beams was studied at low laser intensities [5]. The atom density determination of a given substance requires a carefully selection of the laser conditions such as laser intensity, beam size and shape, and center wavelength in a mode hop-free region. Well resolved spectra can be obtained at laser intensities higher than the saturation intensity, where the density determination has to consider the measurement of the laser intensity. At low laser intensities, the transmission remains constant and it is possible to obtain absolute measurements without considering laser intensity. In this case, the signal to noise ratio of the absorption spectra decreases.

Here, we provide a new set of detailed equations for the absorption of a narrow-linewidth tunable diode laser at the Doppler-broadened ⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$ isotopic transitions and verify the validity of these equations with spectral measurements obtained at relative high lithium density and laser intensity above the saturation intensity which is typically $10\,\hbox {W/m}^2$. To our knowledge, this is the first time that such detailed comparison between theory and experiment is reported in the open literature. The absorption equations were also extended to the Doppler-free case and compared to the measured Doppler-free ⁶Li-$D_2$ and ⁷Li-$D_2$ hyperfine lines with excellent results.

The technique described in this paper can also be applied to provide practical and useful equations needed for the description of various other atomic physics experiments such as laser-cooling experiments  [6], electromagnetically induced transparency [7], resonance ionization spectroscopy [8], and laser isotope separation [9–11] studies.

Absolute absorption spectroscopy can yield the number density of the sample being studied and has many applications in physics, laser and plasma spectroscopy, chemistry, metallurgy and industry [12–14]. The technique is independent of the absolute intensity measurement at low laser intensities, where the experimental effort should consider the noise suppression or phase sensitive method to increase the signal to noise ratio. At higher intensities, where signals are strong and the spectra more defined, it is necessary to consider the absolute measurement of the incident intensity as the transmission depends on the intensity, and the model should account for this intensity dependence. At high vapor pressure it is necessary to use high laser intensity to obtain the transmitted signal.

2 Background

The angular momentum for different lithium lines are given in Table 1 [15, 16], I is the nuclear spin quantum number, J is the quantum number for the electronic angular momentum, which is the sum of the orbital angular momentum L and the electron spin S.

Table 1 Angular momentum for different lithium lines [15, 16]

The transition probabilities $N_{ifq}=\left| \left\langle i\right| D_q\left| f\right\rangle \right| ^2/\left\| D\right\| ^2$ were calculated in this work for the ⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$ hyperfine lines. Here, $\left| \left\langle i\right| D_q\left| f\right\rangle \right| ^2$ is the matrix element for the dipole moment [3, 4, 17] between the ground states i and the excited states f, $\left\| D\right\| ^2$ is the reduced dipole matrix element  [3], $q=-1 , 0 ,+1$, corresponds to transitions with left circular, linear and right circular polarization, respectively, when $m'-m=-1, 0, +1 , m$ and $\,m'$ are the magnetic moment of the Zeeman sublevels of the ground and excited states, respectively. An example for transition probabilities between the ground states i = 1–3 corresponding to the ground-state multiplet labeled j = 1 with total angular momentum F = 1, and the excited states f = 13–17 corresponding to the excited-state multiplet labeled k = 5 with $F' = 2$ of the hyperfine lines is shown in Table 2.

Table 2 Transition probabilities for ⁷Li-$D_2$ $F' = 2$, k = 5 , f = 13 to 17, F = 1, j = 1 for i = 1 to 3 between ground states i and excited states f, the ground and excited-state multiplets are labeled with i and f, respectively. m file and $m'$ column correspond to ground and excited magnetic moment of the Zeeman sublevels, respectively (this work)

The coherences in the density matrix between ground states or between excited states are negligible if the energetic separation between states is large compared with the natural linewidth and if the saturation parameter $S_0 \,{<<}\, 1$ or if the homogeneous linewidth is large compared with the natural linewidth [18]. The other coherences are approximated by stationary states (Wilcox-Lamb approximation) [19].

Next, the reduced transition rate for absorption or stimulated emission among complete ground (j) and excited (k) multiplets is found to be

$$\begin{aligned} W_{j-k}=\frac{\frac{1}{2} \left( \frac{8\pi K_e}{c \hbar ^2} \left\| D\right\| ^2 \right) \cdot {\Gamma }}{\left( \left( \omega -\omega _0 \right) -\left( \omega _j-\omega _k\right) -\mathbf {k} \cdot \mathbf {\upsilon } \right) ^2+{\Gamma } ^2} \cdot N_{j-k}\cdot I \end{aligned}$$

(1)

where ${\Gamma }= \gamma /2 + \mathrm {\Gamma }_L +\mathrm {\Gamma } _c +\mathrm {\Gamma } _T$ is the homogeneous broadening, $\gamma /2$ is the natural linewidth, $\mathrm {\Gamma }_L$ is the laser linewidth, $\mathrm {\Gamma } _c$ is the collision broadening, $\mathrm {\Gamma } _T$ is the transit-time broadening, $\omega -\left( \omega _j-\omega _k\right)$ is the laser detuning from resonance, and $\left( \omega _j-\omega _k\right)$ is the laser angular resonance frequency, $\omega _j$ and $\omega _k$ are the angular frequency shifts from the center of gravity of the states, $\omega _0$ is the center absorption angular frequency of the atoms at rest, $\mathbf {\upsilon }$ is the mean velocity of the atoms, $\mathbf {k}$ the wavenumber, $K_e=8.99 \times 10^9$ Vm/C is Coulomb’s constant, $N_{j-k}$ is the sum of the $N_{if0}$ values for transitions among the sublevels of the multiplets labeled j and k. Note that previously [3] we defined $N_{j-k}$ as the average of the values for transitions among the sublevels of the multiplets labeled jand k. Here, the new definition simplifies the calculation of the reduced transition probabilities as it eliminates the necessity to count the number of transitions. For example in the ⁷Li-$D_2$ transition $N_{1-5}=N_{1,14,0}+N_{2,15,0}+N_{3,16,0}=1/12+1/9+1/12=5/18$. The reduced transition probabilities $a_{j-k}$ for the different hyperfine lines are shown in Table 2. The reduced coefficient $a_{j-k}$ for spontaneous emission among the j = 1 ground state and the k = 5 excited state with degeneracy $g_5=5$ is

$$\begin{aligned} a_{1-5}= & {} \left( 2L_f+1\right) \gamma \left( N_{1,13,q}+\sum _{i=1}^{2}N_{i,14,q}+\sum _{i=1}^{3}N_{i,15,q}\right. \nonumber \\&\left. +\sum _{i=2}^{3}N_{i,16,q}+N_{1,17,q} \right) /g_5 \end{aligned}$$

(2)

with $L_f=1$ is the orbital angular momentum of the excited state. Values for the normalized reduced coefficient for spontaneous emission $a_{j-k}/\gamma$ , calculated here, are listed in Table 3.

Table 3 Reduced transition probabilities $N_{j-k}$ and normalized reduced coefficient $a_{j-k}/\gamma$ for spontaneous emission

Using the procedure described above, we obtain a new set of reduced optical Bloch equations valid for magnetic field B = 0 for all complete multiplets of ⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$ hyperfine lines

$$\begin{aligned} \dot{\rho }_{jj}= & {} \sum _{k,|\Delta F|\le 1}\left( W_{j-k}\left( \rho _{kk}/g_k-\rho _{jj}/g_j\right) +a_{j-k}\rho _{kk}\right) +\gamma _B\left( \frac{g_j}{g_k}-\rho _{jj}\right) \end{aligned}$$

(3)

$$\begin{aligned} \dot{\rho }_{kk}= & {} \sum _{j,|\Delta F|\le 1} W_{j-k}\left( \rho _{jj}/g_j-\rho _{kk}/g_k\right) -\left( \gamma +\gamma _B\right) \rho _{kk} \end{aligned}$$

(4)

$$\begin{aligned} 1= & {} \sum _{jj} \rho _{jj}+\sum _{kk} \rho _{kk} \end{aligned}$$

(5)

and $\gamma _B=\gamma _T+\gamma _{\upsilon c}$ is the sum of the transit-time relaxation rate and the velocity changing collisions rate due to collisions between the Li and the buffer gas atoms which was described previously [3], $\gamma _T=\upsilon /d$ is the transit-time relaxation rate where $\upsilon$ is the mean velocity of the atoms and d is the diameter of the laser beam, $\rho _{jj}$ and $\rho _{kk}$ are the sum of the populations of the multiplets of the ground and excited states, respectively. The procedure to calculate the absorption (for Doppler-limited and Doppler-free spectra) of a vapor illuminated by a laser beam was described elsewhere [3]. The method considers the populations of the involved Zeeman multiplets obtained solving the reduced optical Bloch equations in the case of steady state as the laser used is a CW laser. In this work, we calculated the absorption of all the lithium lines (⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$) using Eqs. (3–5). The 0.1 m path length of laser light inside the lithium cell was divided in 20 parts. We considered the resonance wavelengths [15, 16] of the different transitions.

3 Experiment

The narrow-linewidth emission is provided by a diffraction grating-tuned external cavity diode laser utilizing a Littman Metcalf configuration [12, 13] and yielding a linewidth of  100 kHz. The single-transverse-mode, and single-longitudinal-mode, laser emission was directed through a heat-pipe cell (Comstock, Model HP-802) containing a lithium vapor with a total path length inside the vapor of 0.1 m. The temperature was measured by a thermocouple. The heat-pipe cell was evacuated to approximately 13 mPa and filled with a buffer helium gas to avoid lithium vapor condensation on the heat-pipe windows. Alternatively, we used argon as a buffer gas. In the Doppler-limited experiment, the laser beam diameter was reduced to 2.4 mm prior to the entrance to the heat pipe. The incident laser power measured with an optical power meter was 2.4 mW, and the calculated intensity was 530 W/m$^2$. After passing the heat-pipe cell the beam passed an iris diaphragm with 2.4 mm diameter. We adjusted the beam direction to ensure maximal signal into the photodiode. The transmitted light was detected with a high speed Si detector. The active area of the detector was $0.8 \,\hbox {mm}^2$, with 1.0 mm diameter.

The experimental setup of the Doppler-free absorption experiment is depicted in Fig. 1. The diode laser was slowly scanned close to the resonance using a function generator. A strong pump beam (red line) is driven in the opposite direction to the weak diagnostic beam (blue line). Both beams are derived from the same laser and therefore have the same frequency. The pump beam amplitude was modulated with an optical chopper with typical chopping frequencies of 2079 Hz for the ⁷Li-$D_2$-line and 1046 Hz for the ⁶Li-$D_2$-line. We used a 60 slot disk. A lock-in amplifier was used to remove the background and obtain a Doppler-free absorption signal. The atoms in the heat pipe have two ground states (separated by the hyperfine interaction) and a set of closely separated excited states. Our Doppler-free absorption signal consists of two ordinary saturated peaks at the two resonance frequencies of the ground states separated by less than the Doppler width, and a middle peak (cross-over), which is at the average frequency value of the two resonance frequencies. In the case of Doppler-limited absorption experiment, the pump beam was blocked and we did not use the lock-in amplifier.

Fig. 1

Experimental setup of the absorption (Doppler-free and Doppler-limited) measurements. M mirror, BS beamsplitter, ID iris diaphragm and PD photodiode, FPI Fabry Perot interferometer. DSO digital storage oscilloscope. The red line corresponds to the intense pump beam and the blue line to the probe beam

To obtain the absorption spectrum, we measured first the photodiode signal with the heat-pipe cell temperature at 450 °C. Simultaneously, we measured the voltage ramp of the laser piezo and the Fabry Perot signal. We let the heat pipe cool down to 268 °C and measured the transmission with negligible lithium vapor inside the cell when no absorption was observed. This signal was considered as the baseline of the absorption spectrum and was measured simultaneously with the voltage piezo ramp and the interferometer signal. The shape of the baseline is produced by the interference due to the heat-pipe sapphire windows (MDC-Vacuum, Mod. 450002) with 2.7 mm thickness and refraction index approximately n = 1.76 and corresponds to the transmission profile of an etalon with nearly 37 GHz free spectral range. The baseline and the absorption curve have a small slope due to the increasing laser power.

As the laser was not frequency locked, we recorded each absorption spectrum, scanning ramp (not shown here) and the interferometer signal simultaneously. A small drift of the baseline frequency regarding the full spectrum was observed in some cases. We corrected this small drift using the Fabry Perot spectra as references. The absorption line and baseline measurements at 450 and 268 °C can be seen for example in Fig. 2. In this example, the Fabry Perot signal corresponding to the baseline and absorption spectrum are overlapped. The transmittance is obtained dividing the absorption spectrum by the baseline.

Fig. 2

Photodiode signal in blue for absorption and in black for baseline recorded simultaneously with Fabry Perot transmissions. The Fabry Perot interferograms corresponding to the absorption signal and the baseline are overlapped

The lithium gas density and the absorption increase with temperature and depend on the buffer gas pressure. At very low buffer gas pressure, the lithium gas density and the absorption diminish as seen in Fig. 3. In this case, the pump was extracting more gas from the heat-pipe.

Fig. 3

Comparison of transmittance at same temperature (500 °C) and different He buffer gas pressure. Red line 573.3 Pa and blue line 13.3 Pa

4 Results

Figure 4 shows the measured transmission of the Doppler-broadened absorption spectrum of lithium isotopes and the theoretical spectrum of the complete line calculated with our new set of density matrix equations. The relative frequency scale was obtained using the values of the line center of the ⁷Li-$D_2$ and ⁶Li-$D_1$ spectra. The incident laser intensity was $530\, \hbox {W/m}^2$, the heat-pipe cell temperature was 450 °C, and the heat pipe was filled with 533 Pa of helium as buffer gas.

The lithium density n, the homogeneous broadening $\mathrm {\Gamma }$ and the relaxation rate due to velocity changing collisions $\gamma _{\upsilon c}$ were obtained from a fit. The density obtained was $n\left( ^7\mathrm {Li}\right) =4.6\times 10^{17}\,\mathrm m^{-3}$ and $n\left( ^6\mathrm {Li}\right) =5.1\times 10^{16}\,\mathrm m^{-3}$ comparing our experimental spectra with the theoretical and considering abundances of the ⁷Li and ⁶Li isotopes in our sample as $10.0 \%$ and $90.0 \%$, respectively.

Fig. 4

Doppler-broadened Li absorption spectrum in red and full theoretical spectrum in black. $\mathrm \Gamma = 1.2\times 10^8\mathrm s^{-1}, \gamma _{ \upsilon c} = 2.4\times 10^7\,\mathrm s^{-1}, \gamma _{T} = 3.9\times 10^5\,\mathrm s^{-1}, n\left( ^7\mathrm {Li}\right) =4.6\times 10^{17}\mathrm m^{-3}$ and $n\left( ^6\mathrm {Li}\right) =5.1\times 10^{16} \mathrm m^{-3}$ corresponding to the abundances of the $^7\hbox {Li}$ and $^6\hbox {Li}$ isotopes 0.90 and 0.10, respectively. The incident laser intensity was $530 \,\hbox {W/m}^2$, the heat-pipe cell temperature was 450 °C, and the heat-pipe was filled with 533 Pa of helium as buffer gas

In our density matrix approach, we obtained from a fit the homogeneous broadening $\mathrm \Gamma = 1.2\times 10^8\,\mathrm s^{-1}$ and the relaxation rate due to velocity changing collisions $\gamma _{\upsilon c} = 2.4\times 10^7\,\mathrm s^{-1}$ and $\gamma = 1/ \tau$ with $\tau =27.29$ ns the lifetime of the lithium transition, the transit-time relaxation rate was $\gamma _{T} = 3.9\times 10^5$ $\,\mathrm s^{-1}$ which is practically negligible compared to $\gamma _{\upsilon c}$. Thus $\gamma _B \approx 2.4\times 10^7$ . In this experiment, the saturation parameter [3, 12]

$$\begin{aligned} S=\frac{1}{2}\left( 1+\frac{g_e}{g_g}\right) \frac{1}{\gamma _B \mathrm {\Gamma }}\frac{\lambda ^3 I_0}{2 \pi c h \tau } \end{aligned}$$

(6)

was 1.65, where $g_e = 1$ and $g_g=1$ are the degeneracies of the excited and ground states, respectively, of the lithium 6 isotope.

The effect of the velocity changing collisions is best observed with saturated absorption spectroscopy. Nevertheless, it is possible to observe the effect of the velocity changing collisions in our Doppler-limited spectra. Inspection of Fig. 4 indicates that the theory for transmission at resonance, given in this paper (solid black line), agrees very well with the experimental data (red dots). Figure 5 shows a saturation curve for the ⁶Li-$D_1$ calculated with our theoretical model with $\mathrm \Gamma = 1.2\times 10^8\,\mathrm s^{-1}, \gamma _{\upsilon c} = 2.4\times 10^7\,\mathrm s^{-1}, \gamma _{T} = 3.9\times 10^5\,\mathrm s^{-1}$ and $n\left( ^6\mathrm {Li}\right) =5.1\times 10^{16} \mathrm m^{-3}$. The heat-pipe cell temperature was 450 °C and the helium buffer gas pressure was 533 Pa. The saturation curve consists of the transmittance of the ⁶Li-$D_1$ line at resonance as function of laser intensity, and it is shown in Fig. 5. The red asterisk shows the measured transmission which is in perfect agreement with the theoretical curve for the above given experimental conditions.

Fig. 5

Saturation curve with $\mathrm \Gamma = 1.2\times 10^8\,\mathrm s^{-1}, \gamma _{\upsilon c} = 2.4\times 10^7\,\mathrm s^{-1}, \gamma _{T} = 3.9\times 10^5\,\mathrm s^{-1}$ and $n\left( ^6\mathrm {Li}\right) =5.1\times 10^{16} \mathrm m^{-3}$, heat-pipe cell temperature 450 °C, and helium buffer gas pressure 533 Pa. The red asterisk shows the measured transmission

Figure 6 shows a typical Doppler-free spectrum for the ⁷Li-$D_2$ line. The probe laser intensity was $I_{0} = 30 \,\hbox {W/m}^2$ , the pump laser intensity $I_{p}=1000 \,\hbox {W/m}^2$, the temperature 408 °C, the argon buffer gas pressure $P_\mathrm{{Ar}}= 33$ Pa and the cell length $L=0.1\, {\rm m}$. The laser scanning frequency was 551 MHz $\mathrm {s}^{-1}$. The pump laser intensity was modulated with a chopper at 2079 Hz. Comparing the theoretical spectrum with the experimental, we obtained from the fit of the Doppler-limited spectrum with same experimental conditions the lithium density $n\left( ^7\mathrm {Li}\right) =2.7\times 10^{16}\,\mathrm m^{-3}$, and from the fit of the Doppler-free spectrum the homogeneous broadening $\mathrm \Gamma = 5.9\times 10^7\,\mathrm s^{-1}$, and the relaxation broadening due to transit time and velocity changing collisions $\gamma _B=\gamma _{\upsilon c} +\gamma _{T}= 9.0\times 10^6\,\mathrm s^{-1}$.

Fig. 6

Typical Doppler-free spectra for the ⁷Li-$D_2$ Line. Theory in black and experiment in red. $I_{0} = 30 \,\hbox {W/m}^2, I_{p}=1000\, \hbox {W/m}^2$, temperature 408 °C, argon buffer gas pressure $P_\mathrm{{Ar}}= 33$ Pa and cell length $L=0.1\,\hbox {m}, n\left( ^7\mathrm {Li}\right) =2.7\times 10^{16} \mathrm m^{-3}, \mathrm \Gamma = 5.9\times 10^7\,\mathrm s^{-1}$, and $\gamma _B=\gamma _{\upsilon c} +\gamma _{T}= 9.0\times 10^6\,\mathrm s^{-1}$

Fig. 7

Typical Doppler-free spectrum for the ${^6}\hbox {Li}-{\it{{D}}}_2$ line, $P_\mathrm{{Ar}}= 2.4$ Pa. Experiment [3] in red, and fit in black with equations derived in this work

Figure 7 shows a typical Doppler-free spectrum for the ${^6}\hbox {Li}-{\it{{D}}}_2$ line using argon as buffer gas. In this case, the $^6\hbox {Li}$ isotope concentration was 94.5 %. The probe laser intensity was $I_{0} = 0.8$ W/m$^2$ , the pump laser intensity $I_{p}=27 \,\hbox {W/m}^2$ , the temperature 375 °C, the argon buffer gas pressure $P_\mathrm{{Ar}}= 2.4$ Pa and the cell length $L=0.1\, \hbox {m}$. The pump laser was modulated with a chopper at 1046 Hz. The scanning frequency was 0.5 Hz. Comparing the theoretical spectrum with the experimental, we obtained from the fit of the Doppler-limited spectrum with same experimental conditions the lithium density $n\left( ^6\mathrm {Li}\right) =5.0\times 10^{15} \mathrm m^{-3}$ and from the fit of the Doppler-free spectrum $\mathrm \Gamma = 5.9\times 10^7\,\mathrm s^{-1}, \gamma _B=\gamma _{\upsilon c} +\gamma _{T}= 3.5\times 10^6\,\mathrm s^{-1}$.

5 Discussions and conclusions

A new set of reduced optical Bloch equations for the ⁷Li-$D_2$, ⁷Li-$D_1$, ⁶Li-$D_2$ and ⁶Li-$D_1$ hyperfine lines were obtained and rigorously compared with experimental measurements with excellent results. The derivation starts from the optical Bloch equations applicable to the relative populations of the ground and excited sublevels of the Zeeman multiplets for the case of a single laser beam emission transmitted across a lithium cell. We used the resulting reduced optical Bloch equations to calculate the absorption of a narrow-linewidth tunable diode laser emission in the Doppler-limited case. We found good to excellent agreement between experiment and theory. Furthermore, our fitting parameter were the density, the homogeneous broadening and the velocity changing collision relaxation rates, other parameters such as intensity and temperature were obtained by measurement. We also fitted the theoretical Doppler-free spectra for ⁶Li-$D_2$ and ⁷Li-$D_2$ hyperfine lines and found excellent agreement with the experimental data.