SHOCK-WAVE

PASSAGE

INHOMOGENEOUS V.

I.

THROUGH

AN

MEDIUM

R omanova

UDC 533o6.011.72

The p a s s a g e of a m e d i u m - p o w e r shock wave through a condensed m e d i u m with smoothly changing density is considered (initial speed of sound is v a r i e d ) . The solutions obtained d e s c r i b e the t h e r modynamic p a r a m e t e r s of the m e d i u m behind the s h o c k - w a v e front. P r o p a g a t i o n of shock waves in inhomogeneous media has been studied by a n u m b e r of authors [1-7]. In t h o s e studies the equation of state of the m e d i u m was chosen in the simplified f o r m of that for an ideal g a s . However, it is of physical and p r a c t i c a l i n t e r e s t to study the r u l e s of s h o c k - w a v e p r o p a g a t i o n in condensed media with v a r i a b l e c h a r a c t e r i s t i c s . 1.

Equation

of State

The p r e s e n t study will examine media with a L a n d a u - Stanyukovich-type equation of s t a t e [8], which allows v a r i a t i o n of both density and the speed of sound in the condensed m a t e r i a l through which the wave p r o p a g a t e s . At m o d e r a t e p r e s s u r e s in the shock wave (~10 ~ arm) the dependence of e n e r g y and p r e s s u r e on t e m p e r a t u r e and density may be w r i t t e n in the f o r m of two components, one of which is r e l a t e d solely to elastic f o r c e s , while the other is connected solely with t h e r m a l motion [9]. They a r e usually t e r m e d the "cold" (E c and Pc) and " t h e r m a l " (Et and Pt) c o m p o n e n t s . The t h e r m a l p r e s s u r e Pt is p r o p o r t i o n a l to t e m p e r a t u r e T (with T f a r exceeding the initial t e m p e r a t u r e of the m a t e r i a l before c o m p r e s s i o n ) and i n v e r s e l y proportional to specific vo]Lume V. The proportionality coefficient F (Gruneisen coefficient), g e n e r a l l y speaking,, is density dependent. However, this dependence may be neglected [9]. We take F = 2, which c o r r e s p o n d s to a number of metals at slight c o m p r e s s i o n . F o r the cold p r e s s u r e we use the function proposed in [10]:

w h e r e , as G a n d e l ' m a n indicated [11], a = (1/3) P0c~, Then the total e n e r g y E ; E c + E t of the m a t e r i a l has the form

E= 2.

Shock

Adiabat

and

c~

6VoV ~

(V o - - V) [V2o-k VVo ~ 2V~] -k 3RT.

(1,2)

Isentrope

The equation of state chosen gives v e r y s i m p l e e x p r e s s i o n s for the shock adiabat. obtain the function for the change in p r e s s u r e amplitude P~ = : c~

Vo

On the wave front we

vo-vf

2Vf - - Vo

(2.1)

and the t e m p e r a t u r e change

Co Tf = 18R-"

(Vo + vf ) (vo - - vf )' VoVf(2Vf--Vo)

(2.2)

Behind the wave front the motion of the m a t e r i a l is i s e n t r o p i c . Since TV 2 = TfV}, the equation of the i s e n t r o p e r e l a t i n g values on the front to values within the volume e n c o m p a s s e d by the shock wave t a k e s on the T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 30, No. 3, pp. 478-482, March, 1976o a r t i c l e submitted October 21, 1974.

Original

This material is protected" by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York, N. Y 1001 t. N o part ] ]of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying, . o f the p ubltsher " . A copy o f this . arttcle . is available . . from . the pubhsher f o r $ 7-->0. microfilming ' recording or otherwise ' w i t h o u t written permission

] 315

for m

C2 3.

Basic

V~

2

= co

2V0--Vf 2Vf - - Vo

VW o

(2.3)

Equations

Since a shock wave propagating through a m a t e r i a l with v a r i a b l e density c r e a t e s a v a r i a b l e entropy at each m a s s e l e m e n t , it will be convenient to solve the p r o b l e m in Lagrange v a r i a b l e s . With the above a s s u m p t i o n s the s y s t e m of equations for the p r o b l e m * will have the f o r m OU

3Vo

,

(3.1)

Om

OV Ol

p--c~

OP

.....

Ot

OU Om

(3.2)

'

[ ( - ~ ~ 3 2V~ 2Vf - - V o

1].

(3.3)

As boundary conditions on the shock wave we take the equations for velocity of the m a t e r i a l

1 Vo and for the front t r a j e c t o r y in L a g r a n g i a n f o r m

d/TLf

Co i2 Vf

dt

Vo

i)

1 2

(3.5)

Vo

We a s s u m e that the shock wave in the m e d i u m is excited by detonation of a v e r y thick layer of explosive substance attached to one side. The detonation wave striking the boundary of the medium m = 0 is r e flected and p r o p a g a t e s in the opposite d i r e c t i o n f r o m the explosion products. Because the explosion p r o ducts f o r m a thick layer, on the boundary with the c o m p r e s s e d m a t e r i a l a constant p r e s s u r e is maintained. T h u s , at m = 0, P = const, and, consequently, the specific volume is constant. F r o m continuity conditions for the value of this p r e s s u r e we have 0"578-- Pp-~V

V~ ' Co p --Vo l t-~0Vo + 2 -~o ) -----~ / ( - ~ o - - I ) [ I - -

(-~)~-1"

(3.6,

which was obtained by use of the s i m p l e s t f o r m of the equation of state for the detonation products [12]. 4.

Asymptotic

Solution

for

Initial

Stage

of Process

T o e s t i m a t e the c h a r a c t e r of• change in the shock wave upon propagation through the m e d i u m of inhomogeneous density we use an expansion of the dependent v a r i a b l e s for s m a l l m and t i n t h e f o r m of a power s e r i e s in m and t. We a s s u m e that V0 = V00(1 + am + ~rn2),

(4.1)

co = Cuo(1 + xm -+- &n2),

(4.2)

w h e r e V00, ~, /3, Coo, v~, 6 a r e s o m e a r b i t r a r y c o n s t a n t s .

We s e e k t h e expansions n e a r m = 0 i n t h e f o r m

V = V (I + Vim -4- Vgn ~ -i-' V6nt),

(4.3)

U = U (I + U~t + Up_m"+ U3t2 + U~mt),

(4.4)

if--

mf D

(1 + Dlmf).

(4.5)

T o d e t e r m i n e the coefficients of the expansion we substitute Eqs. (4.3) and (4.4) in Eqs. (3.1)-(3.3) and use boundary conditions (3.4)- (3.6).

/

*The b r i e f p a p e r by Kompaneets, Romanova, and Y a m p o l ' s k i i in Zh. Eksp. T e o r . F i z . , P i s ' m a Red., [16, No. 4, 259 (1972)] unfortunately p r e s e n t s t h e s e equations in an e r r o n e o u s f o r m , as was indicated to the authors by V. N. Svidinskii. 316

The e x p r e s s i o n s for p r e s s u r e , m a s s velocity, and s h o c k - f r o n t velocity at m = 0 a r e written in t e r m s of the constant initial c o m p r e s s i o n value 0 = (V00/V): 5

~ 0--1 ----c~oPoo - 2--0

D=C0o

,

(4.6) (4.7)

0 -- l ~(2--o)

'

o0 /P ~- CooPo0 V / 2 --

(4.8)

The coefficients of the linear t e r m s , written in t e r m s of the dimensionless p a r a m e t e r ~ = (0 -- 1)/(1 -- 20), have the f o r m

U,D ----D1 = a Vl = ( z

n+l

1

3n+2

3n~+4ar+2

(~ + I) (3g ~- 2)

• - 3n+2

+z

,

3n

(4.9) (4.10)

(n + I) (33 ~- 2)

The e x p r e s s i o n s for coefficients of the quadratic t e r m s have a more c u m b e r s o m e f o r m and t h e r e f o r e will not be presented. Results of calculations p e r f o r m e d with the relationships obtained here permit the following general conclusions. Upon shock-wave propagation in a medium with d e c r e a s i n g density the p r e s s u r e at the wave front d r o p s , and the change in velocity of sound may r e a c h the energy output on the constant asymptote. Also, in the case of d e c r e a s i n g density, c o m p r e s s i o n in the wave i n c r e a s e s slightly, and with i n c r e a s e in the initial speed of sound the opposite o c c u r s ; analogous r e s u l t s a r e obtained for the t e m p e r a t u r e Tf and the r a t i o P t / P c at the front. The m a s s velocity Uf for the case of falling density i n c r e a s e s at any c 0. When the shock-wave propagates through a m a t e r i a l with increasing initial density o0 the p r e s s u r e at the front Pf i n c r e a s e s if c o is constant or i n c r e a s e s . For falling velocity of sound, although the dependence on c o is weaker than that on P0, the p r e s s u r e Pf may fall. Moreover, for i n c r e a s i n g P0 the :mass velocity Uf, c o m p r e s s i o n in the wave, t e m p e r a t u r e Tf, and P t / P c on the front all fall with wave p r o p a g a tion. A drop in initial velocity of sound affects the r e s u l t s negatively. Most interesting f r o m the viewpoint of change in t h e r m o d y n a m i c and mechanical p a r a m e t e r s in a m a t e r i a l disturbed by a shock wave a r e media with constant acoustic impedance (P0C0 = const). C o m p a r i s o n of the r e s u l t s with data of n u m e r i c a l solution by computer indicated that for some values the solutions obtained by use of the expansion a r e quite a c c u r a t e and, on the whole, do give a p r o p e r indication of the ~endencies of p a r a m e t e r v a r i a t i o n in condensed materials disturbed by a shock wave.

NOTATION R, gas constant, referred to 1 g of substance; F, Gruneisen coefficient; m, mass coordinate; t, time coordinate; tf, time at which shock wave reaches mass mf; U, mass velocity; Uf, mass velocity at front; D, shock-front velocity; V, specific volume; V0, initial specific volume; Vf, specific volume of compressed material at front; P, pressure; Pt, thermal pressure; Pc, cold pressure; Pf, pressure at front; Et, thermal energy; Ec, cold energy; T, temperature; Tf, temperature of material at front; c, velocity of sound; co, velocity of sound in material before compression; ef, velocity of sound at front; P0, density of uncompressed material. LITERATURE I~ 2~ 3. 4e 5. 6.

CITED

G. M. Gandel'man and D. A~ Frank-Kamenetskii, Dokl. Akad. Nauk SSSR, 107, 811 (1956). A. S. Kompaneets, Dokl. Akad. Nauk SSSR, 130, No. 5 (1960). ~~ I. Andriankin, A. M. Kogan, A. S. Kompaneets, and V. P. Krainov, Zh. Prikl. Mekh. Tekh. Fiz., No. 6 (1962). Yu. P, Raizer, Dokl. Akad. Nauk SSSR, 153, 551 (1963). Yu. P. Raizer, Zh. Prikl. Mekh. Tekh. Fiz., No. 4 (1964). W. Chester and R. Collins, Isr. J. Technol., 8, 345 (1970).

317

7. 8. 9.

R. Collins and Chen H s i a n g - t e h , Lect. Notes Phys., 8, 264 (1971). L . D . Landau and K. P. Stanyukovich, Dokl. Akad. Nauk SSSR, 46, 399 (1945). Ya. B. Z e i ' d o v i c h and Yu. P. R a i z e r , Physics of Shock Waves and H i g h - T e m p e r a t u r e Hydrodynamic Phenomena, Academic P r e s s (1966-1967). K . P . Stanyukovich, T r a n s i e n t Motions of a Solid Medium [in Russian], Gostekhizdat, Moscow (1971). G . M . G a n d e l ' m a n , A u t h o r ' s A b s t r a c t of D o c t o r a l D i s s e r t a t i o n , Moscow (1966). Ya. B. Z e l ' d o v i c h and A. S. K o m p a n e e t s , Detonation T h e o r y [in Russian], Gostekhizdat, Moscow (1955).

10. 11. 12.

EFFECT

OF

ELECTRICAL B. V.

LATTICE

OXYGEN

PROPERTIES

M. M o g i l e v s k i i , F. Tumpurova,

OF

ON T H E R M A L

AND

FLUORITE

V . N. R o m a n o v , and A. F. Chudnovskii

UDC 536.21 +537.311 +537.22

The e l e c t r i c a l conductivity, t h e r m a l conductivity, and t h e r m o - e m f of CaF 2 c r y s t a l s with an oxygen i m p u r i t y a r e m e a s u r e d . Values of the defect t r a n s f e r p a r a m e t e r a r e obtained. The existence of t h e r m a l diffusion of o x y g e n - v a c a n c y c o m p l e x e s in fluorite is p r o p o s e d . One of the g r e a t e s t difficulties in synthesizing fluorite c r y s t a l s is elimination of oxygen, which is always 0 r e s e n t in the original i n g r e d i e n t s . This p r o b l e m a r i s e s b e c a u s e of the d i r e c t connection between optical p r o p e r t i e s of CaF 2 and 02- concentration. In p a r t i c u l a r , the p r e s e n c e of oxygen produces u n s a t i s f a c t o r y r a d i a t i o n stability in the c r y s t a l s , low t r a n s p a r e n c y in the ultraviolet, existence of l i g h t - s c a t t e r i n g phases, etc. At the s a m e t i m e , in growth of CaF2 c r y s t a l s activated by r a r e e a r t h metals for use as volume r e g i s t r a t i o n d e v i c e s oxygen plays an i m p o r t a n t r o l e , since it e n s u r e s stability of the nonequilibrium t r a n s i t i o n of the r a r e e a r t h ion f r o m the t r i v a l e n t to the bivalent state [1]. Thus, it is of p r a c t i c a l i m p o r t a n c e to study means for d e t e r m i n a t i o n and control of oxygen content in fluorite s p e c i m e n s . In c o n t r a s t to univalent cation i m p u r i t i e s which a r e able to produce both v a c a n c i e s and i n t e r s t i t i a l ions (depending on the mode of solution in the fluorite lattice), the p r e s e n c e of 02- leads only to v a c a n c i e s [1]. T h u s , the coincidence of kinetic defect p a r a m e t e r s in oxygen-containing c r y s t a l s and s p e c i m e n s of CaF 2 + MeF is additional and independent c o n f i r m a t i o n of the vacancy nature of the defects produced by a univalent m e t a l i m p u r i t y (as in the case of B a F 2 [2]). I n t e r e s t in the anion i m p u r i t y in a n t i - F r a n k e l s y s t e m s is caused by the fact that in this c a s e conditions exist for t h e r m a l diffusion of the i m p u r i t y ion or its c o m p l e x e s . The p r e s e n t study c o n s i d e r s the e l e c t r i c a l conductivity, t h e r m a l conductivity, and t h e r m o - e m f of a n u m b e r of oxygen-containing CaF2 s p e c i m e n s . The m e a s u r e m e n t method is d e s c r i b e d in [4, 5]. The s p e c i m e n s w e r e produced by the S t o k e b a r g e r method in a graphite c r u c i b l e . Doping with oxygen was effeeted by a d m i s sion of a weak a i r flow through the v a c u u m c r y s t a l l i z a t i o n c h a m b e r over the c o u r s e of the entire c r y s t a l growth p r o c e s s . As a r e s u l t , the t r a n s m i s s i o n s p e c t r a of the c r y s t a l s (Fig. 1) show intense a b s o r p t i o n bands at ~155 nm and ~205 nm, connected with the p r e s e n c e of 02 [3]. F o r c o m p a r i s o n , Fig. 1 a l s o shows the t r a n s m i s s i o n s p e c t r u m of pure CaF 2. In both s p e c i m e n s (Nos. 1 and 6, Fig. 2a) the oxygen content was d e t e r m i n e d to an a c c u r a c y of 20% by the v a c u u m fusion method (No. 1, 0.22 mole % 02-; No. 6, 0.64 mole % O2-). Results of conductivity m e a s u r e m e n t s on s e v e r a l CaF 2 + O2 s p e c i m e n s (Nos. 1-6, Fig. 2a) r e v e a l that the t e m p e r a t u r e dependences ~ = f(1/T) a r e s i m i l a r to each other and to analogous functions for c r y s t a l s of CaF 2 + NaF (No. 7, Fig. 2a). On the c u r v e s a = f(1/T) one c a n distinguish s e g m e n t s of natural conductivity ( h i g h - t e m p e r a t u r e region), d i s s o c i a t l o n (middle region), and a s s o c i a t i o n ( l o w - t e m p e r a t u r e region). The slopes of t h e s e s e g m e n t s c h a r a c t e r i z e the activation e n e r g i e s , which, within the limits of e x p e r i m e n t a l e r r o r , coincide with the following values for CaF2 + NaF [4]: in the d i s s o c i a t i o n r e g i o n h_ = (0.5 ~ 0.03) eV, in the a s s o ciation r e g i o n (h_ + (l/2)ha-) = (0.7 ~ 0.05) eV. Hence, the bonding energy of the O ~ - - - v a c a n c y h a - c o m p l e x i s A g r o p h y s i c a l S c i e n t i f i c - R e s e a r c h institute, Leningrad. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i c h e s k i i Zhurnal, Vol. 30, No. 3, pp. 483-487, March, 1976. Original a r t i c l e submitted January 20, 1975.

[

I

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, N e w York, N.Y. 10011. N o part | o f this publication m a r be'reproduced, stored in a retrieval system, or transmitted, in an), form or by any means electronic, mechanical, photocopying, microfilming, recording or otherwise, ~vithout written permission o f the publisher. A copy o f this article is available yrom the publisher f o r $7..~0.

318

l