Electrode thermal processes leading to the initiation of pulsed vacuum breakdown
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A.
THERMAL OF
PROCESSES
I~ULSED
Litvinov
and
VACUUM A.
F.
LEADING
TO T H E
BREAKDOWN
Shubin
UDC 537.521.7
One p r o c e s s which m a y lead to the initiation of vacuum breakdown is electrode heating by a t h e r m a l f i e l d - e m i s s i o n current. The initial instability caused by the heating m a y occur at the cathode or anode, depending on the applied voltage, the electrode shape, the electrode material, etc. An attempt was made in [1, 2] to determine the conditions under which a given electrode is responsible for vacuum breakdown. A combined Joule-heating and e m i s s i o n - h e a t i n g model was adopted in [1, 2] in a calculation of the cathode t e m p e r a t u r e , while the anode t e m p e r a t u r e was calculated under the assumption that it was heated by electron bombardment. The following simplifying assumptions were used: 1) the cathode t e m p e r a t u r e was found f r o m the solution of the s t e a d y - s t a t e heat-conduction equation, which does not hold under pulsed conditions (~'pul -< 10-8 sec); 2) in a study of the events at the anode, no account was taken of the distribution of the e l e c t r o n - e n e r g y loss with r e s p e c t to thickness in the target. The transient p r o b l e m of the heating of a cathode s u r f a c e due to combined Joule heating and emission heating was solved in [3]. The basic theoretical r e s u l t is an equation giving the time tde which elapses before destruction of the cathode point in t e r m s of the c u r r e n t density and the p r o p e r t i e s of the cathode m a t e r i a l : o,55~%c~
Idc=
(1)
x0~]K2
where PK, c~, and • are the density, heat capacity, and t e m p e r a t u r e coefficient of the r e s i s t i v i t y • (X = • Time tdc is defined as the time at which the c u r r e n t exceeds i I by 10% (Fig. 1). The validity of Eq. (1) for tungsten was demonstrated experimentally in [4]. We a r e concerned here with the change in the anode t e m p e r a t u r e caused by the e l e c t r o n beam; in c o n t r a s t with [1, 2] we will take into account the distribution of the e l e c t r o n - e n e r g y loss with r e s p e c t to depth. F i g u r e 2 illustrates the model of the m i c r o s c o p i c p r o t u b e r a n c e at the cathode. The cathode is a s sumed a cylindrical point, in c o r r e s p o n d e n c e with the usual situation in vacuum-breakdown experiments. Assuming that the t e m p e r a t u r e field p e n e t r a t e s only a s h o r t distance into the m a t e r i a l during the experiment (pulsed conditions), i . e . , a distance small in c o m p a r i s o n with the radius R a of the anode spot, we can d e s c r i b e the t e m p e r a t u r e distribution in the anode by the following boundary-value p r o b l e m : OT
~ O~T
d~
O_T
p,~ca~- = a ~ + ~ d-~, r/t=~ = To, Ox ::~o= O.
(2)
The quantities Pa, Ca, and Xa are the density, heat capacity, and t h e r m a l conductivity of the anode material, and co is the density of the e n e r g y flux incident at the anode surface. The function
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