Relativistic calculations of secondary ions produced in plasma desorption mass spectrometry

I d Time of Flight (TOF) mass spectrometry, the ratio; mass/charge of ions of the sample under investigation is determined by a relationship between time of flight and spectrometer parameters. Usually this relationship is derived using classical (Newtonian) mechanics. Since high energies are involved in the flight of the particles a relativistic treatment is considered. It is investigated whether the relativistic effects are significant or not, for the motion of secondary ions produced in the Plasma Desorption Mass Spectrometer. Treating the motion of these ions relativistically, a relationship between the Time of Flight (TOF) and mass/charge of the ion was derived. Using advanced mathematical software package Maple 6, the derived relationship was compared with classically obtained results. Then it was found that relativistic effects are not significant and they can be ignored for the linear Plasma Desorption Mass Spectrometer.


INTRODUCTION
The objective of the study is to investigate whether there are any significant relativistic effects in the motion of secondary ions produced in the Plasma Desotption

Mass Spectrometer (PDMS).
Plasma desorption mass spectrometry is developed by combining electronic sputtering and time of flight method. Sputtering can be loosely described as the process of ejection of molecules from a surface when it is bombarded with some energy source. In Time of Flight mass spectrometry method, ions of molecules whose mass is to be determined, are needed to be produced. This can be achieved by bombarding the sample by some energy source so that sputtering occurs. Thus sputtered ions are termed as "secondary ions" produced in the plasma desorption mass spectrometry.
This new method of mass spectrometry was created by Macfarlane and his coworkers. This was following the discovery in 1974 of fission fragments from Cf 252 source can sputter and ionize intact organic molecules from a surface 6

RELATIVISTIC TREATMENT OF THE SECONDARY IONS
The sputtered ions from the sample, in the PDMS, undergo acceleration voltage V/, which is typically of the order 10 4 volts. Therefore the ions can reach relativistic velocities.
Since the time resolution of the PDMS at the Department of Physics is as high as 1 nanosecond 2 at high velocities, classical approximation of motion may be inadequate.
Thus it is worthwhile to see if the relativistic effects in the motion of secondary ions could be detected.
When treating the motion of secondary ions, the mass spectrometer (and the lab in which it is installed in) is taken as the rest frame. The motion of the ions is considered relative to this lab-frame. Kinetic energy and the momentum p, of the ions are taken as given by the following relativistic equations.
Since the horizontal acceleration is much greater than the gravitational acceleration and vertical initial velocity, it was assumed in the derivation that the vertical motion of an ions is negligible.

ERROR OF MASS
Consider the mass/charge of the secondary ion as a function as shown in the following

equation. m/q = f{S"S 2 ,S 3 ,V a ,V i ,V 2 ,T <) ,T TOF )
The error in m/q, can be found as;

37;
^0 +<5r 7Wf 0 / (18) By substituting appropriate values in to the above equation, the error of m/q can be found.

CALCULATIONS AND ANALYSIS
The relativistic equation (15) which is used throughout this project happens to be non-linear. This equation is basically used for calculating the Time of Flight for given values of spectrometer parameters, or for finding the mass/charge for a given Time of Flight. For calculations in both of these cases, basically two options were available. One was, writing a computer program by using some high level language and the second, using a standard mathematical software package.

Results of calculations of relativistic and classical masses for Times of Flight are tabulated below. The other mass spectrometer parameters have the usual values, given earlier.
* e is taken as charge of an electron. Throughout the report this notation will be used.   Table 2, it is seen that the difference in masses when calculated using relativistic and classical relations decrease as the time of flight increases. This difference shows it self at fifth decimal place in units of a.m.u/e. Also this difference is seen to be decreasing. It can be concluded that the significance of the difference rapidly fades as the measured time of flight increase (i.e. for heavy secondary ions produced in PDMS). From the results tabulated above, it can also be concluded that, in the TOF range of order of 10' 7 seconds to lO* seconds, relativistic effects are negligible.       The graph in figure 5 is plotted in the range of acceleration distance, Si from 0.4 cm to 2 cm, whereas the typical value for 5y is 1.23 centimeters. The rime difference is still seen to be of the order 10" 13 seconds.  The graph in figure 6 is plotted for acceleration voltage V/, from 9000 volts to 40000 volts. The time difference is still seen to be of the order 10" 13 seconds. Since the time resolution of the PDMS at the department of physics is 1 nanosecond, this difference is very much negligible.

CONCLUSIONS
Corresponding m/q for a given TOF and spectrometer parameters could be calculated by applying classical equation (1) or relativistic equation (14). But it was seen that the difference between the two results is of order 10" 5 a.m.u. per electron charge.
Further more, the instrumental errors suppress the relativistic corrections. Hence it could be concluded that the relativistic effects are negligible.
Also, it was seen that even when spectrometer parameters such as pre acceleration voltage (Vi), pre acceleration distance (5/) and free flight distance (S2) was increased to promote relativistic effects, still the difference between the two values of the TOF was of the order 10' 13 seconds. Therefore it can be concluded that for the current type of PDMS, relativistic effects are negligible and classical TOF equation (1) is sufficient for calculating m/q.