Extensions of the longitudinal envelope equation Page: 2 of 8
This report is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided to Digital Library by the UNT Libraries Government Documents Department.
The following text was automatically extracted from the image on this page using optical character recognition software:
In addition, there may be a bunching force due to a ramp in a longitudinal accelerating field.
This force may be written as:
FZ =-q E/(s) z (2)
where -EO'(s) is the local ramp in the longitudinal field and we have chosen to use a linear,
position-dependent bunching force. The coordinate z is position with respect to the bunch center.
In reference 4, a self-consistent longitudinal distribution with an envelope equation was
developed, consistent with these force equations.
In ref. 4, the self-consistent longitudinal distribution for a particle bunch is written as:
f(z,z',s) = 3N 1 _ - - -(z- z)2 (3)
27E z02 E2 zo
within the region in z, z' that the argument of the square root is positive (f = 0 outside). In
reference 4, position along the accelerator s is used as the independent variable (z' = dz/ds). z,
z' are particle coordinates, N is the total number of particles in the bunch. zo is the bunch half-
length (full length = 2zo) or beam envelope amplitude. EL is the longitudinal emittance
(unnormalized) in z - z' coordinates. (Lawson' includes an excellent discussion of this distribution
but unfortunately his equation 4.49 (in the current edition) contains a typo: the square root is
missing. His symbols are also somewhat different: his definition for N differs and his y" refers
to the bunching force parameter and not the second derivative of the central beam energy.) The
bunch density profile corresponding to eq. 2 is parabolic:
3N z2' (4)
X(z,s) = 1 - -
The equation for the beam envelope amplitude zo(s) is:
Here’s what’s next.
This report can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Report.
Neuffer, David. Extensions of the longitudinal envelope equation, report, April 30, 1997; Batavia, Illinois. (digital.library.unt.edu/ark:/67531/metadc624405/m1/2/: accessed May 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.