Novel scaling laws for the Langmuir-Blodgett solutions in cylindrical and spherical diode

It is found that the Langmuir-Blodgett (LB) solutions for the space charge limited current density, for both cylindrical and spherical diodes, may be approximated by J_app =(4/9)ε₀(2e/m)^1/2E_c^3/2/D^1/2 over a wide range of parameters, where E_c is the surface electric field on the cathode of the vacuum diode and D is the anode-cathode spacing. This dependence is valid whether Ra/R_c is greater than or less than unity, where R_a and R_c is respectively the anode and cathode radius. Minor empirical corrections to the above scaling yield fitting formulas that are accurate to within 2.5 percent for 0.002 <; R_a/R_c <; 30000. An explanation of this scaling based on the transit time or capacitor model is given.