Average ion density and fundamental frequency vs applied voltage for the p ϭ 0.2 Torr, d ϭ 20 cm case. The regimes where self-oscillation, period-doubled, and chaotic oscillations appear are marked 1P, 2P, and C, respectively. 

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... n g 3.25 10 p Torr m for room temperature ͑ 297 K ͒ . This equation does not include the effect of the beam current, since we have neglected the effect of the virtual cathode. However, if the injected current increases, the potential difference due to the virtual cathode affects the size of the AGR, and the boundary moves down to a lower voltage regime. The effect of a large cathode temperature plays a similar role to that of a large current density, which moves the boundary down to the lower voltage regime. In Fig. 2, the boundary lines from Eq. ͑ 13 ͒ compare well with our simulation results for two gas species, helium and argon, which have different ion masses. The mass ratio of an ion to an electron plays an important role in the transition as shown in Eq. ͑ 13 ͒ . The simulated current density of the beam is J b ϭ 1A/m 2 with cathode temperature kT ϭ 1 eV Ͻ ␧ iz , which satisfies the condition of Eq. ͑ 9 ͒ , where ␧ iz ϭ 24.5 and 15.76 eV for He and Ar, respectively. The typical temperature of hot filament is near or above 0.2 eV, and it plays a minor role if it is much smaller than the sheath potential and ionization energy and not so much smaller than the virtual cathode potential. We have varied p and d values with the same pd product. Simulated hysteresis curves are plotted in Fig. 5, which are obtained by increasing the applied voltage slowly and decreasing it after the mode transition. The rising time is 1 ms, which is much larger than the mode-transition time-scale. The mode transition happens at V ϭ 72 V for the pd 0.02 Torr cm case, and at V d 37 V for the pd ϭ 0.08 ͑ Torr cm ͒ case. The mode-transition voltage is affected only by the product of p and d , but the areas of the hysteresis curves are different for different d values. This means that even though the mode transition is determined by the value of pd , the dynamics of the system is strongly related with each value of p and d . The low frequency oscillations in the AGM before and after the mode transition are also shown. In the double-layer regime, self-oscillations are observed just above the transition boundary between the AGM and the DLM as shown in Fig. 6 ͑ a ͒ . The oscillation frequency is very low, approximately tens of kHz, which is much smaller than the ion plasma frequency of approximately an order of MHz. Since the elastic collision is dominant and the injected beam spreads out by the collision, there exists no longer beam– plasma or Pierce–Buneman instability. Thus the oscillation mechanism of this mode is different from those of the AGM and the TLM. As the external voltage increases, this oscillation becomes a period-doubled oscillation ͓ Fig. 6 ͑ b ͔͒ , and finally a chaotic oscillation ͓ Fig. 6 ͑ c ͔͒ . The fluctuation amplitude decreases as the external voltage increases, finally generating a steady-state DL. It was reported that the propagating DL in collisionless finite-length plasmas shows a low-frequency oscillation ͑ 2.5 kHz ͒ . 15 There are two stages of the oscillation; the first one is a strong propagating DL which is accompanied by a broad negative potential at its low potential side, and the second one by a fast increase of the potential in the entire plasma shortly after the DL reaches the anode. This is a potential- relaxation oscillation, 10 which is also shown in the plasma diode with injected electron and ion beams with equal temperatures. 18 The propagation velocity is associated with the ion transit-time of the DL. The oscillation mechanism in the collisional discharge plasma is similar to this, except that there is no negative potential at its low potential side, and that the DL formation and oscillation are related to the collision effect. The time evolutions of oscillating DL are shown in Fig. 7. If the potential difference of the DL is larger than the ionization energy ␧ iz , the densities of electrons and ions increase at the high potential side. When the densities increase sufficiently, the breaking of charge neutrality due to fast electron loss induces large electric field which drives the ions to the low potential side ͑ from t ϭ 0.73 to t ϭ 0.77 ms ͒ . After the ion loss of the high potential side, the increased potential difference causes fast ionization, and the densities increase again ͑ from t ϭ 0.77 to t ϭ 0.803 ms ͒ . Repetition of these processes is the mechanism of self-oscillation in the collisional DLM, which is the same with that of the reported experiment. 16 Experimentally, low frequency fluctuation of the ionization front in the TLM → AGM transition regime was observed. 7 . The fluctuating ionization front produces ion bunches that propagate in the direction of the electric field, which agrees very well with our simulation results shown in Fig. 7 b . The propagation speed of the ion bunches in Fig. 7 ͑ b ͒ is about 1.6 ϫ 10 3 m/s, which is ion thermal velocity rather than ion-acoustic wave velocity. For a large applied voltage ͑ V d տ 35 V ͒ , when the ion loss due to the large electric field exceeds the ion creation due to ionization, the oscillation disappears. The DL, how- ever, is still maintained because of the decelerated ions by a charge-exchange collision. For a larger applied voltage, the ions accelerated by the large electric field move to the cathode, and the mode transits to the TLM. Figure 8 shows the dependencies of the dominant oscillation frequency and the average density to the applied voltages for the p ϭ 0.2 Torr and d ϭ 20 cm case. The oscillation frequency suddenly increases where V d Ͼ 32 V, and satu- rates again where V d Ͼ 33 V. Between these two values at 32.5 V, a period-doubled ͑ 2P ͒ oscillation appears. It is very similar to the experimental results, 16 where a period- doubling route to chaos of the DL oscillations in the collisional discharge plasma was reported. The density decreases as the voltage increases, since ion loss exceeds ion creation. For a larger voltage, the density increases again, especially in the TLM. The plasma density is mainly affected by the neutral gas pressure, while the oscillation types—1P, 2P, or chaotic oscillations—are mainly affected by the applied voltage. For a fixed voltage case ( V d ϭ 30 V ͒ , the average density is 8.7 ϫ 10 13 m Ϫ 3 for p ϭ 0.15 Torr which linearly decreases to 2.4 ϫ 10 13 m Ϫ 3 for p ϭ 0.22 Torr, and the oscillation frequency is 4.74 kHz for p ϭ 0.15 Torr which linearly increases to 13.4 kHz for p ϭ 0.22 Torr. All of these are 1P oscillations with the same voltage. The mode transition and the nonlinear oscillations in the dc-driven beam-injected collisional discharge plasma are studied using a particle-in-cell simulation. For various products of pressure and system length ( pd ) and dc voltage ( V d ) values, three different modes are observed. The mechanism of mode transition between the AGM and the TLM is eluci- dated. The estimated boundaries of those two modes agree well with the simulation results. In addition to the previously reported AGM and TLM, the slow ion-acoustic double-layer mode is observed for large pd and intermediate V d values. The ion-collision effect plays an important role in the formation of double layers. In this regime, low frequency self- oscillation and the period-doubling route to chaos were observed, which is similar to the reported experimental results. The high frequency oscillations in the TLM are caused by a beam–plasma-type instability in the bounded system. A di- agnostic parameter, defined as the ratio of the interaction length to the slippage length, is a good measure for period- doubled and chaotic oscillations of this system as shown in other beam–wave interaction systems. Helpful discussions with Professor M. H. Cho, Professor M. Yoon, and Professor S. H. Kim are greatly acknowl- edged. The present studies were supported ͑ in part ͒ by the BSRI-Special Fund of the Pohang University of Science and Technology, the Basic Science Research Institute Program ͑ Ministry of Education 1997, Project No. 97-2439 ͒ , and the PDP ...