Energy Distribution of Free Space Field Ionization : Application to Field Strength Calibration

We discuss the method of field strength calibration from energy distributions measurements of free space field ionization. An alternative approach is considered and an improved calibration is suggested. The method of electric field strength calibration from considerations of energy distribution of field ionization in free space above a field ion emitter tip is due to Sakurai and Muller [1,2]. Until their proposal the calibration of the field strength at a metal surface, Fo, was made by comparison with the Fowler Nordheim equation for field emission as reported by MUller and Young [ 3 ] . The accuracy of such calibration is thought to be about 2 15%. Sakurai and Mullerrs method could improve the calibration up to 5% precision. Their basic assumption is that the peak of the ionization distribution moves away from the tip with increasing tip voltage, so that the field strength at the maximum of the distribution is constant. They performed measurements using a He-D2 gas mixture, from which the D + 2 ions energy deficit was measured. The apex tip radius was determined from the ring counting method[4]. A theoretical determination of the field strength at the points where the ionization distribution peaks is proposed in Appendix B of ref. 2 taking the frequency factor, V, with which an electron strikes the potential barrier, as an adjustable parameter. In the present work we use both a numerical JWKB calculation and an analytical expression [51 for the ionization rate constant in order to study the effects of the basic assumption of Sakurai and MUller. A comparison between the results obtained by the two different approaches is made. Furthermore we present an alternative Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984914 C9-78 JOURNAL DE PHYSIQUE determination of the geometrical field factor which is independent of the behaviour of the field strength at the points where the distribution function peaks. These considerations and proposals could lead to a determination of the best image field (BIF) for different imaging gases and different metal tips. Considering a hyperboloid approximation for the tip shape the electric field strength is given by F(V,x) = V /[k Rt (1 + 2x/Rt)] (1) where V is the tip voltage, R is the tip apex radius, k is the geometrical field t factor and x is measured from the tip surface along the tip symmetry axis. This corresponds to a surface field strength equal to Fo(V) = V / k Rt (2) An ion formed at point x above the surface has an energy deficit, compared with the full acceleration case for an hypothetical ion originating at the surface, equal to X AE(V,x) = 1 F(V,xt) dx' = V [In (1 + 2x/Rt)] / 2k (3) for a singly charged ion. Calling the ionization rate constant I and the velocity of the gas atom v, we have where p is the ionization probability per unit of distance. Neglecting the thermal velocity, v is given by v = [ a/m ] 112 ( 5 ) where a is the atomic polarizability and m the atomic mass, leading to p = [ m/a l1l2 I/F The ionization distribution, D (x), is given by X m Dx(x) = P(X) exp [I ~(x') dx' 1 X It should be emphasized here that there is a distinction between the ionization distribution as a function of position, Dx, and as a function of the energy deficit, De, the latter being the one that is usually determined experimentally. The relation between the two is De(AE) = Dx(x)/F(x) (8) The diagrams of Dx(x) and De( AE ) for Argon and values of tip fields between 5.0 and 3.0 V/A can be seen in Figs. 1 and 2. The x coordinate at which D (x) peaks, xm, corresponds, by use of eq. 7, to the point where d p(x)/dx = 2 P (x) (9) The value of De(AE(xm)), where x is the point of maximum D (x), is different m X from the maximum value of D (AE), though the difference is small. iONiZATiON DISTRIBUTION IONiZATION DISTRIBUTION DISTANCE FROM TIP ENERGY DEFICIT Fig. 1 Ionization Distribution as Fig. 2 Ionization Distribution as function of position (A). Argon on function of energy deficit (eV). Argon Tungsten with a 600 A tip apex radius. on Tungsten with a 600 apex radius. We performed a three-dimensional JWKB calculation of the energy deficit distribution for different values of the field strength Fo for all noble gases from Helium to Xenon. He and Ne were included for completeness even though the fields considered are in excess of normal evaporation field strengths. The corresponding values of the field strength at points where the energy distribution peaks, F were P' determined and it was noticed that the value of F decreases with increasing values P of the tip field, in contrast to the constant value assumption of Sakurai and Mfiller. From the numerical calculations of F for the different values of F we P could obtain as a fitting relation F 'IF = exp[B (Fol/ Fo 1 ) ] P P (10) From eq. 10 we obtain which is eq.7 of Sakurai and Mfiller [2], with the addition of the final correction term. Using eqs. 3 and 11 we have which differs from eq.8 of Sakurai and Mtiller [2] again only by the addition of term B(V1/V 1). For a variation of 10% in the voltage we have a change of about 5% C9-80 JOURNAL DE PHYSIQUE between the values of k determined in this way, compared with Sakurai and Mtiller's determination. We studied the variation of the field strength where the energy distribution peaks using another approach, an analytical one. Haydock and Kingham151 have derived a formula for the field ionization rate constant, which for the case of free-space ionization gives: ~ ( 2 1 ~ ) ~ I(F) = 37r v F [16 B ~ / Z F] exp[ (21/2~/3B112) (2512~312/3~) (13) (2B) 312 which is expressed in atomic units and was derived from eq. 9 of ref. 5, where B is the ionization potential, Z is the effective atomic charge and v is the frequency factor as before. The A~ factor is 2 213 -1 A = ( 1 2 n v h e ) which is the corrected form of eq.10 in ref. 5 and the electron frequency in hydrogen ,vh, can be related to v through v = 2 B v / m h 0 which is eq. 3.35 of ref. 6. By the use of equations 6, 7 and 9 we obtain Fo Rt A1 Fpa exp(C/F ) = 2 [ a F + C ] P P (16) where Al, C and a depend only on the imaging gas. An iterative solution of eq.16, which is rapidily convergent, gives us a variation of F with F (or V) with the P same general trend as obtained by the numerical JWKB calculation. The behaviour of the field strength at the points where D (x) peaks, determined both numerically and analytically, for Argon with tip fields between 5.0 and 3.0 V/A is shown in figs. 3 and 4. The same general trend can be seen in both cases and eq.10 constitutes only an approximate fitting equation. We will not assume the eq.10 behaviour, nor any other as a basis for the determination of k. FIELD STRENGTH A T MAXIMUM D(X) FIELD STRENGTH hr Nihti"li!Y a:<! DISrANCE FROM TIP 2 -8 2 ) s a W 2 2 8 2 , s ; Fig. 3 Numerical determination of the Fig. 4 Analytical determination of the Field strength (VIA) at points where field strength (VIA) at the points where D(x) peaks. Tip radius equal to 600 A. D(x) peaks. Tip radius equal to 600A. I , . , I , , , l , , , , l , , , , l , , , , ~ ,