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beam plasma instabilities are ubiquitous in physics and have been investigated for many decades @xcite .
the topic currently undergoes a renewed interest through the fast ignition scenario for inertial fusion @xcite or some scenarios of gamma ray bursts production in astrophysics @xcite .
indeed , astrophysics offers a very wide range of unstable systems which can be magnetized , relativistic , homogenous or not .
counter - streams instabilities are usually studied assuming parallel streams . but why should real systems systematically fit this scheme ? admittedly , when an electron beam enters a plasma , a return current is prompted in opposite direction to neutralize it @xcite .
but two already current and charge neutralized plasma shells could perfectly collide over a background plasma at an arbitrary angle .
it seems the problem of non - parallel streams has not be addressed so far , and the goal of this letter is to show that non negligible growth rates can arise from non - parallel streams interactions as well .
let us consider two non - relativistic beams , both charge and current neutralized . for simplicity
, we consider here two electron - proton beams where both species have equal densities so that charge and current neutrality are guarantied regardless of the beams relative motion .
the problem of the beams respective orientation is interesting only if there is a background plasma .
otherwise , one just needs to consider the reference frame of one of the beams to cancel any orientation parameter .
we eventually come up with the setting pictured of figure [ fig:1 ] .
calculations are conducted in the reference frame where the background plasma of electronic and protonic density @xmath4 is at rest . in order to focus on the parameter @xmath0 , the two beams have equal protonic and electronic densities @xmath5 , and both flow at the same velocity @xmath6 along their respective direction .
we now study the stability of harmonic perturbations @xmath7 where @xmath8 and @xmath9 .
note that the all @xmath10 unstable spectrum is evaluated in order to be able to spot the fastest growing modes for any given configuration .
finally , we neglect proton velocities perturbations in view of their much larger inertia .
as a first approximation , we implement a cold three fluids model for the two beams and the plasma . linearizing the conservation and euler equations for the three species ,
the dispersion equation is found evaluating the dispersion tensor in a standard way in terms of the dimensionless variables , @xmath11 where @xmath12 is the background plasma frequency , @xmath13 the electron charge and @xmath14 the electron mass . due to the arbitrary orientation of the wave vector and of the two beams ,
the full dispersion tensor is too large to be reported here .
still , the dispersion equation remains polynomial and can easily be solved numerically . for @xmath15 , the @xmath16 component of the tensor
give the growth rate of a two - stream like instability , as least for @xmath1 and @xmath17 .
the corresponding dispersion equation reads , @xmath18    the growth rate map in terms of @xmath19 for @xmath1 , @xmath20 and @xmath17 on figure [ fig:2 ] . the first and the last cases pertain to well - known systems as @xmath1 eventually comes down to one single beam of density @xmath21 interacting with the plasma , while @xmath2 corresponds to two counter - streams crossing over a background plasma .
these situations have been well studied , and it is known that within the present non - relativistic regime @xcite , they are governed by the two - stream instability which dispersion equation is precisely given by eq .
( [ eq : disperts ] ) .
the maximum growth rate @xmath22 is in the diluted beam regime @xmath23 , @xmath24 for @xmath1 , the two beams act as one , and a resonant unstable mode is feed by the free energy of both at the same time . for @xmath2
, resonant unstable modes can only travel with one single beam , yielding a smaller growth rate .     for @xmath25 , @xmath26 and @xmath1 , @xmath20 and @xmath17.,scaledwidth=90.0% ]
besides these extreme orientations of the two beams , fig . [ fig:2 ] clearly displays some interesting features for the case @xmath3 .
one the one hand , the fastest growing mode is here found for @xmath27 so that it can stay is phase and exchange energy with both beams at the same time .
such non - trivial orientation of the fast growing mode has been so far related to relativistic effects @xcite .
we find here that some unusual system geometry can produce the same effect as the only way for a mode to move in phase with two non - parallel beams is to follow an oblique direction . on the other hand
, the maximum growth rate seems very close to the one reached for @xmath1 , and some finer numerical evaluation shows that it is almost the same .
such equality can be demonstrated setting @xmath3 from the very beginning , and considering the growth rate for wave vectors fulfilling @xmath28 .
the dispersion equation for these modes reads , @xmath29p(x)=0,\ ] ] where @xmath30    since the first factor in eq .
( [ eq : pisur2 ] ) is strictly equal to the dispersion equation ( [ eq : disperts ] ) for @xmath1 , we here prove that the maximum growth rate for this branch is rigourously the same , and that it is reached for the very same @xmath31 component
. a closer look at @xmath32 shows that it yields another unstable mode . by setting @xmath33 in the expression of @xmath32 and developing the result near @xmath34
, we find the growth rate of this second oblique unstable mode , @xmath35 for @xmath36 and @xmath37 lower than unity , this secondary growth rate remains smaller than @xmath38 .
units ) for @xmath39 , in terms of the angle @xmath0 and for various beam to plasma density ratios @xmath36.,scaledwidth=45.0% ]
the overall system is thus found exactly as unstable for @xmath1 as it is for @xmath3 . beyond this value of @xmath0 , the maximum growth rate must decrease to its final value @xmath40 , as given by eq .
( [ eq : gr - ts ] ) .
figure [ fig:3 ] shows how @xmath22 evolves between @xmath1 and @xmath17 , for @xmath39 and various beam to plasma density ratios @xmath36 . as expected
, @xmath22 remains almost constant between 0 and @xmath20 while it falls down to its final value @xmath41 for @xmath42 . because @xmath22 is independent of @xmath37 for @xmath1 , @xmath20 and @xmath17
, we can expect an overall weak dependance for other angles .
this is confirmed through numerical calculation as the three curved displayed on fig .
[ fig:3 ] are indistinguishable from their @xmath43 counterparts .
ton conclude , it is important to emphasize that there is no such thing as a stable configuration . on the contrary ,
the system remains unstable regardless of the beams orientation , and the evolution of the maximum growth rate is limited .
the most unstable wave vector is two - stream like for parallel and anti - parallel orientations , but turns oblique in the intermediate case to stay in phase with both beams at the same time .
noteworthy , relativistic and/or kinetic effects should force an oblique regime regardless of the beam orientation so that an evaluation of the whole unstable spectrum becomes mandatory .
this work has been achieved under projects fis 2006 - 05389 of the spanish ministerio de educacin y ciencia and pai-05 - 045 of the consejera de educacin y ciencia de la junta de comunidades de castilla - la mancha .
unstable modes growing when two plasma shells cross over a background plasma at arbitrary angle @xmath0 , are investigated using a non - relativistic three cold fluids model . 
 parallel flows with @xmath1 are slightly more unstable than anti - parallel ones with @xmath2 . 
 the case @xmath3 is as unstable as the @xmath1 one , but the fastest growing modes are oblique . 
 while the most unstable wave vector vary with orientation , its growth rate slightly evolves and there is no such thing as a stable configuration . 
 a number of exact results can be derived , especially for the @xmath3 case .