Electric acceleration
All thrust is based on action/reaction. That means you must push on something. A rocket accelerates propellant out the back. If it accelerates M mass of propellant at once and A acceleration that will require M*A force. That will produce an equal but opposite force on the engine, and presumably the engine is attached to the spacecraft. If F is the force applied by the engine and the spacecraft has mass Ms (Mass Spacecraft) then the acceleration applied to the spacecraft will be F/Ms. The longer the propellant is in the engine, the greater the total propellant mass to be accelerated. The greater the propellant mass the greater the propulsive force, but the greater the mass of propellant you must carry. The real key is to increase the exit velocity of the propellant. That can be accomplished by either keeping propellant in the engine longer or increasing the acceleration applied to the propellant. Propellant for a chemical rocket is the spent fuel, and the chemical reaction only creates enough energy to accelerate the exit velocity so far. Different fuel mixtures produce different energy, cryogenic hydrogen/oxygen seams to have the greatest.
With an ion engine the propellant does not undergo chemical reaction. Instead it is ionised and pulled with an electric field. A grid is charged with static electricity and the ionised gas is pulled toward it. Some of the gas will impact the grid, causing electric discharge and no propulsive force. Some of the gas will pass through the holes of the grid causing it to exit the engine; that gas provides propulsion. Xenon or Krypton gas is used since they have a high mass to electric charge ratio. That increases propellant mass per unit of electric charge. Notice, increasing propellant mass increases thrust force but also increases mass of propellant to be carried. In terms of reducing propellant mass to get to your destination, that cancels out. The key with Xenon and Krypton is they reduce electricity required. Also, if charged gas density in the engine is too high the positive charge of the gas at the acceleration grid will negate the force of negative charge on the acceleration grid. In other words, if gas density is too high the charges will "jam up" the engine causing reduced efficiency. Heavy gasses like Xenon and Krypton are a way to increase gas mass without increasing the number of ions.
The numbers to measure fuel efficiency for an engine in space is Specific Impulse, designated by the letter "I" with a subscript "SP", such as (Isp). It is measured in seconds; that is one pound of fuel can produce one pound of thrust for how many seconds. Or in metric, one kilogram mass of fuel can produce one kilogram force of thrust for how many seconds. Here kilogram force is defined as one kilogram mass multiplied by one standard gravity of acceleration. One standard gravity is the acceleration on Earth at sea level, or about 9.78 metres per second squared.
As a comparison, the solid rocket boosters of the space shuttle have an Isp of about 269 seconds. The main engines of the space shuttle are the most efficient chemical rockets every built, they produce 455 seconds in vacuum; their efficiency isn't quit as high at sea level. The ion engines used on Deep Space One had a nominal efficiency of 3,100 seconds, but the exact Isp depended on throttle setting. This demonstrates that ion engines are much more fuel efficient. However, electric engines are weak. The thrust produced by the engine on DS1 was about equal to the force of a single sheet of paper resting on your hand in normal Earth gravity. That is obviously not enough to lift the spacecraft off the ground. In space you keep going at whatever speed you have, so a low thrust can gradually increase speed over months. A low-thrust high-efficiency engine is great once you are in space, but it won't get you off the ground.
Today the Glenn Research Center where the engine for DS1 was developed is working on a larger version. They have produced an Isp of 5,600 seconds and are working on a still larger one that they hope will produce 8,300 seconds. A researcher at the Johnson Space Center is working on a Variable Specific Impulse Magnetoplasma Rocket (VASIMR) that will use hydrogen gas instead of xenon or krypton and permit either high-efficiency and low-thrust, or low-efficiency and relatively high-thrust. It still won't produce enough thrust to launch off the ground, but should exit Earth orbit quickly using high-thrust then cruise toward Mars using continuous thrust with high-efficiency and low thrust.
The problem with electric engines is electric power. The more thrust you want from them the more electric power they need. Solar power provides continuous power during the trip but it really isn't much. The high-power electric engines that some people hope to use to leave Earth orbit for Mars will require 10 megawatts or more. That will either require a giant solar array 700 metres wide or a nuclear reactor.
