PPPL-3790 is available in <a href="PPPL-3790.pdf">pdf</a> format (1.8 MB).

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<b>Truncated Thermal Equilibrium Distribution for Intense Beam Propagation
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<b>Authors:</b> Ronald C. Davidson, Hong Qin, and Steven M. Lund
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<b>Date of PPPL Report:</b> February 2003
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<b>Presented at:</b> the Forty-fourth Annual Meeting of the APS Division of Plasma Physics, November 11-15, 2002, Orlando, Florida. Submitted to Physics of Plasmas.</p>

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An intense charged-particle beam with directed kinetic energy
 (<FONT FACE = "SYMBOL">g</FONT FACE = "SYMBOL">
<SUB>b</SUB>-1)m<SUB>b</SUB>c<SUP>2</SUP>
 
propagates in the z-direction through an applied focusing field with transverse focusing force modeled by
 
<B>F</B><SUB>foc</SUB> = -<FONT FACE = "SYMBOL">g</FONT FACE = "SYMBOL"><SUB>b</SUB>m<SUB>b</SUB><FONT FACE = "SYMBOL">w</FONT FACE = "SYMBOL"><SUP>2</SUP><FONT FACE = "SYMBOL"><SUB>b^</SUB></FONT FACE = "SYMBOL"><B>x</B><FONT FACE = "SYMBOL"><SUB>^ </SUB></FONT FACE = "SYMBOL">
 
in the smooth focusing approximation. This paper examines properties of the axisymmetric, truncated thermal equilibrium distribution F<SUB>b</SUB>(r,<B>p</B><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">) = A exp (-H<FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">/<SUB><IMG SRC = "./tee_cap.jpg" WIDTH = "9" HEIGHT = "20" BORDER = "0" ALT = "Tee Cap"></SUB><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL"><SUB>b</SUB>) <SUB><IMG SRC = "./oplus.jpg" WIDTH = "14" HEIGHT = "14" BORDER = "0" ALT = "OPLUS"></SUB>(H<FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">-E<SUB>b</SUB>), 

where A, 
<SUB><IMG SRC = "./tee_cap.jpg" WIDTH = "9" HEIGHT = "20" BORDER = "0" ALT = "Tee Cap"></SUB><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL"><SUB>b</SUB>, and E<SUB>b</SUB>

are positive constants, and H<FONT FACE = "SYMBOL"><SUB>^ </SUB></FONT FACE = "SYMBOL"> 

is the Hamiltonian for transverse particle motion. The equilibrium profiles for beam number density, n<SUB>b</SUB>(r) =<SUB><IMG SRC = "./integral.jpg" WIDTH = "15" HEIGHT = "15" BORDER = "0" ALT = "Integral"></SUB>d<SUP>2</SUP>pF<SUB>b</SUB>(r,<B>p</B><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">), 

and transverse temperature, T<FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL"><SUB>b</SUB>(r) =<SUB><IMG SRC = "./integral.jpg" WIDTH = "15" HEIGHT = "15" BORDER = "0" ALT = "Integral"></SUB> d<SUP>2</SUP>p(p<SUP>2</SUP><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">/2<FONT FACE = "SYMBOL">g</FONT FACE = "SYMBOL"><SUB>b</SUB>m<SUB>b</SUB>)F<SUB>b</SUB>(r,<B>p</B><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">), 

are calculated self-consistently including space-charge effects. Several properties of the equilibrium profiles are noteworthy. For example, the beam has a sharp outer edge radius r<SUB>b</SUB> with n<SUB>b</SUB> 

(r greater than or equal to r<SUB>b</SUB>) = 0, 

where r<SUB>b</SUB> depends on the value of E<SUB>b</SUB>/<SUB><IMG SRC = "./tee_cap.jpg" WIDTH = "9" HEIGHT = "20" BORDER = "0" ALT = "Tee Cap"></SUB><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL"><SUB>b</SUB>. 

In addition, unlike the choice of a semi-Gaussian distribution, F

<SUB>b</SUB><SUP>SG</SUP> = A exp (-p<SUP>2</SUP><FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL">/2<FONT FACE = "SYMBOL">g</FONT FACE = "SYMBOL"><SUB>b</SUB>m<SUB>b</SUB><SUB><IMG SRC = "./tee_cap.jpg" WIDTH = "9" HEIGHT = "20" BORDER = "0" ALT = "Tee Cap"></SUB><SUB><FONT FACE = "SYMBOL">^</FONT FACE = "SYMBOL">b</SUB><SUB><IMG SRC = "./oplus.jpg" WIDTH = "14" HEIGHT = "14" BORDER = "0" ALT = "OPLUS"></SUB>(r-r<SUB>b</SUB>), 

the truncated thermal equilibrium distribution F<SUB>b</SUB>(r,<B>p</B>) 

depends on (r,<B>p</B>) 

only through the single-particle constant of the motion H<FONT FACE = "SYMBOL"><SUB>^</SUB></FONT FACE = "SYMBOL"> 

and is therefore a true steady-state solution (<SUB><IMG SRC = "./partial.jpg" WIDTH = "14" HEIGHT = "14" BORDER = "0" ALT = "Partial"></B></SUB>/<SUB><IMG SRC = "./partial.jpg" WIDTH = "15" HEIGHT = "15" BORDER = "0" ALT = "Partial"></SUB>t = 0) 

of the nonlinear Vlasov-Maxwell equations.




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