Measurements tend to indicate ne w 0.6-0.9, which means that ionization tends to occur early in the channel. This is to be expected, because that is where the backstreaming electrons have had the most chance to gain energy by falling"up the potential The factor n, = is related to the ionization fraction. Putting mi=neCA, m=m+mn=(nc +ncn)A, (11) 1+(%) Since ci /cn is large(cn w neutral speed of sound, i.e., a few hundred m/sec, while C:= gIsp-20,000m/sec), nu can be high even with ne /nn no more than a few percent. Datall show n, ranging from 40% to 90% The factor ne = IB/I requires some discussion. Most of the ionization is due to the backstreaming electrons, so that we are not really free to drive IB towards Ia (Ies =I-I). What we need to strive for is (a) Conditions which favor creation of as many ions as possible per backstreaming electron, and (b) Minimization of ion-electron losses to the walls, once they are created This can be quantified as follows: Let b be the number of secondary electrons(and of ions) produced per backstreaming electron and let a be the fraction of these new e-i pairs which is lost by recombination on walls. Then, per backstreaming electron, (1-a)B ions make it to the beam, and an equal number of cathode electrons are used to neutralize them therefore I_(1-a)阝 I31+(1-a)阝 Clearly, we want B>>1 and a<<1. The first(B>>1)implies lengthening the electron path by means of the applied radial magnetic field, and also using accelerating potentials which are not too far from 5/2 times the range of energies where ionization is most efficient(typically 30-80 Volts). This last condition creates some difficulties with heavy ions, which require higher accelerating potentials for a given exit speed The condition a < 1 implies minimization of insulation surfaces on which the recombination can take place and arrangement of the electric fields such that ions 16.522, Space P pessan Lecture 18 Prof. Manuel martinez16.522, Space Propulsion Lecture 18 Prof. Manuel Martinez-Sanchez Page 4 of 20 Measurements[1] tend to indicate ηΕ ~ 0.6 - 0.9 , which means that ionization tends to occur early in the channel. This is to be expected, because that is where the backstreaming electrons have had the most chance to gain energy by “falling” up the potential. The factor u mi = m η i i is related to the ionization fraction. Putting e i m = n c A , i i n ( ) ei nn m=m+m = n c +n c A i i ii , e i n n u e i n n n c n c = n c 1+ n c ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ η ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (11) Since i n c c is large (cn ~ neutral speed of sound, i.e., a few hundred m/sec, while i sp c gI 20,000m/sec  ∼ ), ηu can be high even with n n e n no more than a few percent. Data[1] show ηu ranging from 40% to 90%. The factor ηE Ba =I I requires some discussion. Most of the ionization is due to the backstreaming electrons, so that we are not really free to drive BI towards I I =I -I a BS a B ( ) . What we need to strive for is (a) Conditions which favor creation of as many ions as possible per backstreaming electron, and (b) Minimization of ion-electron losses to the walls, once they are created. This can be quantified as follows: Let β be the number of secondary electrons (and of ions) produced per backstreaming electron, and let α be the fraction of these new e-ipairs which is lost by recombination on walls. Then, per backstreaming electron, ( ) 1 - α β ions make it to the beam, and an equal number of cathode electrons are used to neutralize them. Therefore ( ) ( ) B a a I 1 - = = I 1+ 1- α β η α β (12) Clearly, we want β >> 1and α << 1. The first (β >> 1) implies lengthening the electron path by means of the applied radial magnetic field, and also using accelerating potentials which are not too far from 5/2 times the range of energies where ionization is most efficient (typically 30-80 Volts). This last condition creates some difficulties with heavy ions, which require higher accelerating potentials for a given exit speed. The condition α << 1implies minimization of insulation surfaces on which the recombination can take place, and arrangement of the electric fields such that ions