The space-charge-limited current density (SCLCD) for a one-dimensional vacuum diode for zero initial electron velocity, $u_0=0$, in vacuum is given by the Child-Langmuir law, $J_{\text{CL}}$. For electrons with $u_0\ne 0$ in vacuum, Jaffé modified $J_{\text{CL}}$ to obtain the SCLCD as $J_{\mathrm{Jaff}\acute{\mathrm{e}}}$. For current densities $J>J_{\mathrm{Jaff}\acute{\mathrm{e}}}$, electron reflections occur along with virtual cathode (VC) oscillations. Past studies have discussed hysteresis in diodes by sustaining these oscillations until $J=J_{\text{hys}}$, often claiming $J_{\text{hys}}\approx J_{\text{LD}}<J_{\mathrm{Jaff}\acute{\mathrm{e}}}$, where $J_{\text{LD}}$, also called the bifurcation solution, represents the steady-state current density where the electron velocity is zero at the VC. In this study, we demonstrate that $J_{\text{LD}}$ is not a valid steady-state solution since it represents a higher charge density than $J_{\mathrm{Jaff}\acute{\mathrm{e}}}$, the true SCLCD. We further demonstrate that $J_{\mathrm{Jaff}\acute{\mathrm{e}}}$ mathematically corresponds to the static bifurcation solution. Using particle-in-cell simulations across various gap distances and voltages, we demonstrate that $J_{\text{hys}}$ consistently differs from $J_{\text{LD}}$ and can be represented by a simple semiempirical relationship as a function of $J_{\mathrm{Jaff}\acute{\mathrm{e}}}/J_{\text{CL}}$.

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• Electron emission
• Space charge in beams
• First-principles calculations in plasma physics
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