The Influence of Narrowing Grivet-Lenz Model on the Projector Properties of Magnetic Lenses

: Present work consist of a computational investigation aims at reveal the influence of narrowing or extending Grivet-Lenz model on its own projector properties. For this reason the well known formula of Grivet-Lenz model is raised to a power named in this work by “shape factor”. Projector properties and their related variables were determined for several values of the shape factor. Results have shown that as the shape factor value increases the magnetic field become more narrowing and its projector properties efficiency get worse. Accordingly to enhance the projector properties of Grivet model it should be extended over a wide axial interval


Introduction
Conventionally to get rapid approximation for the properties of imaging magnetic field, one has to use analytical function to represent the axial magnetic field.One of the most useful functions is the Grivet-Lenz model.This model had been extensively studied for both objective and projector properties by several author, see for example [Szilagyi, 1988;Al-Batat, 1996;Warid, 2004].It can be say that up to now Grivet-Lenz model can be investigated through three optimization variables namely the half width, magnetic peak value and the lens length.
However present work propose a new optimization parameter for this model named 'shape factor'.Where the conventional mathematical form of Grivet-Lenz is raised to a variable power and called shape factor.This new variable is capable for extending or localizing the magnetic field distribution corresponds to Grivet-Lenz model along the optical axis.Precisely, this work aims at investigating the influence of varying the value of the shape factor on the projector properties of the modified Grivet-Lenz model and their related parameters.

Theory
The conventional form of Grivet-Lenz model usually given by the following expression [Hawkes, 1982]: Where B z is the axial magnetic flux density distribution along the optical axis z which in practice start at z=z 1 and extend to z=z 2 .B m and W are the peak value and the half width of the distribution respectively.
Equation ( 1) is modified in this work to be as shown in the following expression: The Influence of Narrowing Grivet-Lenz Model on the Projector Properties of Magnetic Lenses ……………………………… Muhssen S. E. Al-Amshani The exponent n is the shape factor.It is added to the Secant function in equation ( 1) in order to controlling the B z distribution extension along the optical axis-z.In fact this will increase the degree of freedom for assigning B z distribution.Now by a suitable choice for B z , W and n the magnetic field along the required interval z 1 ≤z≤z 2 can be determined.
In order to examine the projector properties of the distribution computed according to equation ( 2) by means of the following two equations that describe respectively the radial and spiral distortion coefficients [Hawkes, 1972]: The paraxial ray equation [Szilagyi, 1988]: have to be solved initially to determine the charge particle trajectory 3r(z), of a charge to mass along the interval z 1 ≤z≤z 2 for a certain accelerating potential V r .
In this work equation ( 5) was solved for the case of zero magnification condition.

Results and Discussions
Present work is mainly concern with investigating the influence of   to enhancing image quality n should be taken to be small as possible as can.
Further more this figure shows that the magnification decrease as n increases due to the behavior of (F p ) min .

Conclusions
According to the above results one may say that the extending of the magnetic field over a wide interval along the optical axis will enhance the projector properties.Thus the long lenses is better than the short one to use as a projector lens.
-Hawkes P.W., (1972),"Electron-Optics and Electron Microscopy", (Taylor and Francis Ltd., London). ) n varying the shape factor (n) on the projector properties of the distributions obtained from equation (2).Thus the remaining parameters B m and W are maintained constant at the following values respectively B m =1T and W=1mm.It The Influence of Narrowing Grivet-Lenz Model on the Projector Properties of Magnetic Lenses ……………………………… Muhssen S. E. Al-Amshani / ‫والخمسون‬ ‫الثامن‬ ‫العدد‬ ‫األساسية‬ ‫التربية‬ ‫كلية‬ ‫مجلة‬ 9002 34 is important to mention thha the parameter W does not necessarily means the half width B z .

Figure ( 1
Figure (1) shows the B z distributions corresponds different values of n.It can be seen that the increasing of n leads the magnetic field to be more localized along the optical axis.Consequently the refractive power of B z will decreases.

Figure ( 2
Figure (2) shows variation of D r and D s as a function of n at the excitation parameter (NI/V r 1/2 ) that gives minimum value of the projector focal length (F p ) min .It is clear that D r and D s getting further worse as n increase i.e. the image become more distorted as long as n takes higher values.In other word, in order

Figure ( 1
Figure (1): The axial magnetic field distribution for different n value.

Figure ( 2
Figure (2): The projector properties D r , D s and (F p ) min at the first magnification point.

Figure ( 3
Figure (3): The spiral distortion D s for various values of n at D r =0.
. "Inverse Design of Magnetic Lenses Using A defined Magnetic Field", M.Sc.Thesis, College of Education, Al-Mustansiriyah University, Baghdad, Iraq.-Warid H. H., (2004), "Inverse Design of Symmetrical Magnetic Lenses Using Analytical Function to Approximate Magnetic Field", M.Sc.Thesis, College of Education, Al-Mustansiriyah University, Baghdad, Iraq.