588

ISSN 0018-151X, High Temperature, 2007, Vol. 45, No. 5, pp. 588–594. © Pleiades Publishing, Ltd., 2007.Original Russian Text © N.M. Kuznetsov, Yu.K. Karasevich, 2007, published in Teplofizika Vysokikh Temperatur, Vol. 45, No. 5, 2007, pp. 654–660.

INTRODUCTION

The method of laser emission analysis gains everwider popularity in the investigations of the atomiccomposition of materials [1–3]. In brief, this methodreduces to the following. As a result of the effect of alaser radiation pulse on a target, an expanding plasmacloud is formed, which has an initial temperature ofseveral tens of thousands degrees. Spectral analysis ofthe radiation of this cloud leads one to inferences aboutthe atomic composition of the target material. Thisanalysis is impossible at the initial instant of timebecause of the presence of a strong background ofbremsstrahlung. Therefore, the spectrum is recordedwith a time delay until the instant when the temperatureof expanding plasma cloud drops to a value below10 000 K. However, the decrease in the cloud tempera-ture occurs rapidly, and the time of exposure proves tobe inadequate for performing qualitative measure-ments. Various techniques are employed for the pur-pose of improving the sensitivity of the method. In par-ticular, a spark electric discharge synchronized with alaser pulse is used [4]. The pulsed action of laser radia-tion on the target causes a number of physical-gasdy-namic effects, in particular, the formation and scatter-ing of gaseous and condensed products of targetdestruction.

Usually, numerous investigations of the effect offocused electromagnetic radiation on the surface of sol-ids have for their object the target proper, its destruc-tion, the transfer of mechanical momentum to the tar-get, nonequilibrium evaporation with the transition ofgas flow through sonic surface [5–10], screening effects[11, 12] etc. (see the list of references in [10]). Plasmaarising in the case of laser stimulation on the surface ofcondensed medium was also considered as the subjectof numerous investigations [6, 7, 10–12]. The theoreti-cal description of the process of formation of such

plasma is a complex problem that has not yet been fullysolved.

The present investigation differs from others by theset of parameters and is devoted to approximate quanti-tative analysis of the processes of formation of theproducts of target destruction, their phase composition,and the size and velocity of scattering of gaseous andcondensed particles. We hope that the thus obtainedresults may be useful for gaining a more completenotion of the operating conditions of laser emissionanalyzer and its capabilities as regards qualitative andquantitative analysis of target compositions beinginvestigated.

ESTIMATIONOF THE INITIAL TEMPERATURE

In performing model calculations of the dynamicsof gas cloud formed as a result of interaction betweeninfrared laser radiation and metal surface, one must firstestimate the initial gas temperature

T

1

. Simple esti-mates demonstrate that, with the laser pulse duration of~10 ns, the optically nontransparent layer of metal,which absorbed the energy that corresponds to a tem-perature of several tens of thousands degrees underconditions of thermodynamic equilibrium of electronand ion subsystems, scatters during a time that is muchshorter than the pulse duration. In the case of sphericalscattering, the layer density decreases, and the layerbecomes transparent to irradiation of the subsequentlayer, and so on. With such dynamics, the temperature

T

1

, which became steady on the process of interactionbetween electron and ion subsystems, is defined by theenergy

E

(

τ

ex

)

absorbed by the layer during the charac-teristic expansion time,

τ

ex

≈

d

/

U

, (1)

The Dynamics of Expansion of Gas and Condensed Particles under Conditions of Local Laser Heating of Metal Surface

N. M. Kuznetsov and Yu. K. Karasevich

Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, 119991, Russia

Received May 31, 2006

Abstract

—A model of formation of the products of interaction between focused electromagnetic radiation andmetal surface is considered. A description of the dynamics of expansion of gas cloud is given; estimates areobtained of the amount, size and velocity of scattering of condensed particles being formed. The obtainedresults enable one to gain a better notion of the possibilities of analysis of emission spectra for qualitative andquantitative investigation of the atomic composition of solid sample under the effect of laser radiation.

PACS numbers: 79.20.Ds. 52.30.-q

DOI:

10.1134/S0018151X07050033

PLASMA INVESTIGATIONS

HIGH TEMPERATURE

Vol. 45

No. 5

2007

THE DYNAMICS OF EXPANSION OF GAS AND CONDENSED PARTICLES 589

where

d

is the layer thickness, and

U

is the sound veloc-ity. By definition, the value of

d

corresponds to the opti-cal thickness of the layer equal to unity. The value of

d

is not included in the end result for

T

1

. It is only impor-tant to have

d

�

h

, (2)

where

h

≈

10

–2

cm is the final depth of funnel (crater).In the process of heating the metal by a laser beam inthe infrared range to temperatures of several tens ofthousands K, the bulk of radiation is absorbed by wayof one-quantum or multiquantum photoionization. Theone-quantum photoionization is energetically possiblefrom excited states, the number of which becomes com-parable with the number of atoms at temperatures of theorder of ten thousand K. The band structure of elec-tronic levels of metal at such temperatures largely losesdeterminacy, and in further estimates we will proceedfrom the individual properties of a separate atom, i.e.,the way it is done in the case of gas. The effective crosssections of photoeffect in gases are

10

–17

–10

–19

cm

2

. Inso doing, the least cross sections from this range (

10

–18

–10

–19

cm

2

) correspond to weakly bound excited states.The mean free path relative to photoionization withsuch effective cross sections and with the density ofexcited state

n

= 6

×

10

22

cm

–3

is

10

–5

–10

–4

cm, whichcorresponds to the absorption coefficient

κ

= 10

5

–10

4

cm

–1

. It is interesting to compare the obtained esti-mate with the mean coefficient of absorption of equilib-rium (blackbody) optical radiation (see Appendix). Theprocesses of multiquantum (multiphoton) ionizationalso occur in the laser radiation field, for which no pre-excitation is required and which lead to an increase inthe total absorption coefficient compared to theobtained estimate. Therefore, inequality (2) is validwith a large margin:

h

/

d

~ 10

3

.

During the time

τ

ex

, the layer absorbs the energy

E

(

τ

ex

) =

β

E

L

τ

ex

/

τ

L

. (3)

Here,

E

L

and

τ

L

denote the laser pulse energy and dura-tion, respectively, and

β

is a coefficient lower thanunity. The difference of

β

from unity is defined by theloss of energy due to reflection of radiation from themetal surface, due to thermal radiation of metal etc. Ifthe energy

Ö

(

τ

ex

)

and the time defined by formula (1)are expressed as functions of temperature and substi-tuted into Eq. (3), we derive the equation defining thetemperature

T

1

,

Sd

ε

(

T

) = (

β

E

L

/

τ

L

)

d

/

U

(

T

), (4)

in which

S

is the area of heated surface, and

ε

(

T

)

is thevolume energy density of the layer for its initial vol-ume. In the ideal plasma approximation, we have

(5)ε T( ) nk32---T 1 αe+( ) αi Ii 1– αi 1– Ei 1–

e+( )i

∑+ ,=

(6)

where

n

is the number of atoms and ions per unit vol-ume;

α

e

=

n

e

/

n

is the relative concentration of free elec-trons;

α

i

=

n

i

/

n

is the relative concentration of ions of

multiplicity

i

;

I

i

– 1

and

denote the ionizationpotential and the energy of electron excitation of ion ofmultiplicity

i

– 1

, expressed in K (for neutral atom

i

– 1 =0); R is the gas constant; γ is the adiabatic exponent forgas; and µ is the atomic weight of metal. Formula (6)corresponds to the “frozen” velocity of sound. Thevalue of γ is calculated accordingly with the fixed com-position of gas and is expressed as

γ = nk[1 + T(1 + αe)]/ε(T). (7)

The concentrations of atoms, ions, and electrons, whichappear in formulas (5)–(7), are defined by the equationsof ionization equilibrium and quasi-neutrality ofplasma. At α3 � 1, these equations in the ideal gasapproximation with the initial number density of heavyparticles of 6 × 1022 cm–3 have the form

(8)

(9)

(10)

(11)

Here, the temperature is in K, and z0, z1, and z2 are elec-tronic statistical sums of atom, singly charged ion, andmultiply charged ion, respectively. The statistical sums

z0 and zi and the energy of electron excitation were calculated in view of the first five electronic levelsby the data from the reference book [13]. The ratio zi/z0at high temperatures (tens of thousands K and higher)depend only slightly on temperature and vary from 0.24to 0.28 with the temperature rising from 40 000 to80 000 K. The electronic statistical sum of doublycharged ion was taken to be equal to the statisticalweight of ground state, z2 = 2.

We substitute Eqs. (5) and (6) into (4) to derive theequation

(12)

which, along with Eqs. (8)–(11), forms the set of equa-tions for the calculation of temperature T1 and compo-sition of gas. In the problem under consideration, S ~10–4 cm2, n = 6 × 1022 cm–3 (aluminum), I0 = 6.94 ×104 K, I1 = 2.18 × 105 ä, and EL/τL ≈ 1014 erg/s. Thenumerical solution of the set of equations (8)–(12) withthese parameters in the β = 1 approximation gives

(13)

U γRT 1 αe+( )/µ,=

Ei 1–e

αeα1/α0 8.1 10 8–× T3/2 z1/z0( ) I0/T–( ),exp=

αeα2/α1 8.1 10 8– T3/2 z2/z1( ) I1/T–( ),exp×=

αe α1 2α2+ ,=

α0 α1 α2+ + 1.=

Ei 1–e

Sε T( ) γRT 1 αe+( )/µ EL/τL,=

T1 6.6 104 ä, α0× 0.699, α1 0.285,= = =

α2 0.016, αe 0.317.= =

590

HIGH TEMPERATURE Vol. 45 No. 5 2007

KUZNETSOV, KARASEVICH

The possible error of temperature T1 is associatedwith a number of unaccounted-for factors of complexoptical and gasdynamic processes. The chief draw-backs of model calculation include the use of idealplasma approximation and disregard of the dynamics ofexpansion of absorbing layer (in deriving Eq. (4)) andof energy loss (β = 1 approximation). The effect ofthese errors is partly compensated by one another; nev-ertheless, the temperature T1 may be overestimated by15–20%.

EXPANSION OF GAS

If the products are not subjected to external effect(for example, of electric discharge), they are scatteredalmost adiabatically (within the loss due to radiation,which is low compared to total energy of expandingmatter). The decrease in temperature with decreasingdensity of scattering (expanding) products under localthermodynamic equilibrium is defined by isentrope T =T(ρ)s. In so doing, the entropy S is defined by the initialtemperature T1 and initial density ρ1 (we will expressthe density as the number of atoms and ions per cubiccentimeter). In making quantitative estimates, we willassume that the target is made of aluminum. In thiscase, ρ1 = 2.7 g/cm3 and the total number of atoms andions n1 = 6 × 1022cm–3. According to Eq. (13), we haveT1 ≈ 6.6 × 104 K with a possible error of ~15–20%. Inaccordance with this value of T1, in estimating theparameters of isentrope of expanding gas, we will pro-ceed from the fact that this isentrope on the n, T (den-sity, temperature)-plane passes through the initialpoint: T1 ≈ 6.6 × 104 K, n1 = 6 × 1022 cm–3.

The segment of isentrope from the initial point toapproximately n = 8 × 1020 cm–3 corresponds to denseplasma and may be calculated only approximately. The

rest of the isentrope up to any degree of expansion maybe calculated with a high degree of accuracy by solvingthe set of equations of statistical physics for ideal ion-ized gas with the calculation of electron-ion composi-tion, energy, and pressure at every point of {n, T}-plane. Leaving aside the question of exact calculationof isentrope, we will estimate the dependence of tem-perature on density of expanding gas proceeding fromthe approximate analogy of thermodynamic propertiesof ionized vapors of aluminum and air. The results ofcalculation of isentrope T(δ) passing through point{n1, T1} of air plasma, obtained using thermodynamictables [14], are given in the figure (δ is the ratio of den-sity of expanding gas to density of gas under normalconditions). For densities in the range n1/75 ≤ n < n1

beyond the limits of maximal density nmax ≈ n1/75 in thetables [14], the isentropic dependence of temperatureon density was calculated approximately by the Pois-son formula for ideal gas,

T = constinγ – 1

with the adiabatic exponent γ = 1.4 and γ = 1.3 at

n1/10 ≤ n < n1 (14)

and

n1/75 ≤ n < n1/10. (15)

respectively. The subscript i, which takes the values of1 and 2, indicates that the values of const in the densityranges (14) and (15) are different (defined by the initialcondition {n1, T1} and conditions of joining of temper-ature on the boundary between ranges (14) and (15)).

If we assume approximately that the data given inthe figure are true of aluminum as well, the conditionsfor the condensation of vapors during the expansion ofaluminum vapors into vacuum from the initial point ofinterval {n1, T1} are attained at δ ≈ 10–6, i.e., in the caseof expansion by a factor of approximately 109 com-pared to the initial density of solid. This degree ofexpansion on the average corresponds to the increase inthe sphere diameter d (which defines the bulk ofexpanding gas) to 1260‡, where a is the initial lineardimension of “target” heated to temperature ~T1; wewill further assume that a = 0.01 cm.

A qualitatively different pattern corresponds to theexpansion of “explosion” products if this expansionoccurs in an air medium rather than in vacuum. In thecase of expansion into air of normal density, its massforced out in part by the flow of metal (aluminum)vapors and partly mixed with these vapors becomesapproximately equal to the mass of evaporated matter atd ≈ 10a ≈ 0.1 cm. Corresponding to this is the expansionof vapors by a factor of approximately 103 and thedecrease in temperature to ~9000 K. Further expansionoccurs under conditions of intensive mixing of alumi-num vapors with air. The mixing is promoted by the ini-tial nonuniformity of distribution of temperature and

60

50

40

30

20

10

0–6 –4 –2 0 2 4

T/1000, K

logδ

Isentrope for air passing through point δ = 2240, T = 60000 K.

HIGH TEMPERATURE Vol. 45 No. 5 2007

THE DYNAMICS OF EXPANSION OF GAS AND CONDENSED PARTICLES 591

density of evaporating matter and instability of contactdiscontinuity at the vapor-air interface, as well as by thecombustion of vapors with the formation of aluminumoxides. By the instant of time when the cloud of alumi-num-air mixture reaches a size of (25–30)a, the mass ofair in the mixture is 10 to 30 times that of metal vapors,the mixture temperature drops below 1000 K, and thecondensation of oxides with the formation of fine parti-cles becomes possible. The transient pattern of the pro-cess of expansion of cloud, defined by the smallness ofits initial size (a = 0.01 cm), prevents the formation oflarge particles.

In so doing, the average density of mixture is closeto the initial density of air, and the rate of increase in thesize of gas mixture cloud U ≡ dr/dt is ~0.1 of the aver-age rate of expansion into vacuum, i.e., of the order of1 km/s. With another increase of diameter d by a factorof ten, namely, to ~300a, the rate U decreases to~102 cm/s. After a further increase in the diameter by afactor of two or three, the cloud in fact ceases to grow.Corresponding to this size of cloud is approximatelythe distance from the heated target, at which the diag-nostics of gas are performed in the analyzer. The spec-trum of cooled gas containing molecular componentsand particles of condensed phase is very complex,“dense” and, accordingly, unsuitable for extraction ofthe desired component (useful signal) from this spec-trum. Measurements at an earlier stage of expansion areperformed at much lower values of d, where there isstill neither condensation nor combustion and wherethe spectrum is more convenient for analysis, and entailother difficulties (small field for analysis, high densitygradients, high rates of variation of all parameters, andbremsstrahlung background).

These difficulties of analysis are largely eliminatedowing to the fact that the expanding gas cloud on itsway to the region of diagnostics is subjected to theeffect of electric discharge which increases its energysuch that the gas mixture in the region of diagnosticsturns out to be heated to 10 000 K. In so doing, the pos-sibility of condensation in the region of observation isruled out completely. However, this in no way impliesthe absence of condensed particles. The thing is that theeffect of laser radiation on the target does not involveevaporation alone. The peripheral layers of heated tar-get volume do not evaporate; however, their tempera-ture may be sufficient for melting. During melting, thetarget material loses its strength and is discharged to theambient space under the effect of expansion wave trav-eling from the central part of evaporated matter deepinto and to the sides of the resultant cavity. One mustbear in mind that this does not require complete melt-ing. It is sufficient that a two-phase solid-liquid mixturewith a relatively low concentration of liquid wouldform. Accordingly, it is to be expected that the scatter-ing condensed particles consist of two phases.

ESTIMATION OF THE AMOUNT AND AVERAGE SIZE OF SCATTERING CONDENSED PARTICLES

We will estimate the mass of particles, which (pre-sumably) scatters in the form of condensed particles.We assume that the initial thickness ∆ of scatteringlayer of condensed particles is small compared todimension a (the validity of this assumption will beconfirmed by further calculations). This enables us totreat the problem on the heating of the layer as a two-dimensional problem of heat conduction correspondingto approximation ∆/a = ∞.

The upper temperature limit í* of the layer underconsideration, below which the specific thermal energyis insufficient for complete evaporation, is defined bythe equation

(16)

where T0 ~ 300 K is the initial temperature, and CV and∆H(T0) denote the isochoric heat capacity and the heatof evaporation of metal, respectively. For aluminum,∆H(T0) = 320 kJ/mol [15]. According to the Dulong andPetit rule, the heat capacity of crystal lattice of metal athigh temperatures is about 3R = 24.9 J/(mol K). Thisrule is adequately valid in the case of aluminum. Dataon isobaric heat capacity CP, which is close to heatcapacity CV in solids, are available from referencebooks. At temperatures of 400, 600, and 1000 K, theheat capacity CP is 25.7, 28, and 32 J/(mol K), respec-tively [16]. The latter value relates to liquid aluminum.

The heat capacity difference may be estimated usingthe well-known thermodynamic relation

(17)

and the data on the Grüneisen Γ and on the velocities oflongitudinal and transverse sound waves. The deriva-

tive is related to heat capacity CV as

(18)

The isothermal derivative appearing in Eq. (17)

is expressed in terms of isentropic derivative and

further in terms of volume velocity of sound U,

(19)

where ρ = µ/V is the density. According to the theory ofelasticity, the volume velocity of sound is expressed in

CV Td

T0

T*

∫ ∆H T0( ),=

CP CV– T∂P∂T------⎝ ⎠

⎛ ⎞V

2 ∂V∂P-------⎝ ⎠

⎛ ⎞T

,–=

∂P∂T------⎝ ⎠

⎛ ⎞V

V∂P∂T------⎝ ⎠

⎛ ⎞V

ΓCV

µ----------.=

∂V∂P-------⎝ ⎠

⎛ ⎞T

∂V∂P-------⎝ ⎠

⎛ ⎞S

∂V∂P-------⎝ ⎠

⎛ ⎞T

∂V∂P-------⎝ ⎠

⎛ ⎞S

CP

CV

------ µρ2U2------------

CP

CV

------, U2 ∂P∂ρ------⎝ ⎠

⎛ ⎞S

,≡–= =

592

HIGH TEMPERATURE Vol. 45 No. 5 2007

KUZNETSOV, KARASEVICH

terms of velocity UL of longitudinal sound waves andUS of transverse sound waves [17],

(20)

Equations (17)–(20) yield

(21)

Finally, we substitute into the right-hand part of Eq. (21)the data for aluminum Γ = 2.09 [18], µ = 27 g/mol,UL = 6.26 km/s, US = 3.08 km/s [16], and the heatcapacity in the Dulong and Petit approximation CV =24.9 J/(mol K) to derive

1 – CV/CP ≈ 0.15(T/1000). (22)

It follows from Eq. (22) and from the foregoing dataon CP at temperatures of 400, 600, and 1000 K that CV ≈24, 25, and 27 J/(mol K), respectively. The electroncomponent of heat capacity increases with temperature;however, at temperatures of at least up to 104 K itremains relatively small. It is to be expected that theheat capacity CV of heavy particles after the melting ofmetal would decrease with increasing temperature andapproach the heat capacity of ideal gas; however, up toa temperature of ~104 K this decrease is low. Based onthis reasoning, the temperature T* in Eq. (16) may beestimated within ~10–20% assuming the total heatcapacity to be CV = const ~ 25–30 J/(mol K). In sodoing, Eq. (16) yields T* = (11–13) × 103 K.

In estimating the thickness ∆ of a layer whose topand bottom surfaces are heated to temperature T = T*and to melting temperature T = Tm, respectively, we willassume that the heating of the layer occurs largely afterthe outer hotter layers evaporate during a time approx-imately equal to half width of laser pulse τL. We willfurther assume that, during a time ~ ~τL/4 after that, thetemperature of the surface under consideration irradi-ated by the “tail” of laser pulse remains constant. In thisapproximation, the value of ∆ may be estimated usingthe known solution (given below) of the problem on theheating of semi-infinite volume at a preassigned tem-perature on its plane boundary.

The distribution of temperature as a function ofcoordinate x and time t in a semi-infinite volume of ini-tially cold matter (T = 0) defined by a plane surfacex = 0, on which a constant (high) temperature t is pre-assigned beginning from some instant of time T1 = 0, isdefined by the formula (see for example, [17])

T = T1[1 – erf(Z)], (23)

where Z ≡ x/[2 ], χ is the thermal diffusivity, and

erf(Z) = dξ is the error integral. The

sought layer thickness ∆ = x is defined by Eq. (23) if we

U2 UL2 4/3( )US

2.–=

1CP

CV

------–Γ2TCV

µ UL2 4

3---US

2–⎝ ⎠⎛ ⎞

--------------------------------.=

χt2

π------- ξ2–( )exp

0

Z∫

substitute the melting temperature of aluminum Tm =833 K and time t = τL/4. We represent by Zm the root ofEq. (23) at t = τL/4 and T = Tm and derive

(24)

The numerical solution of Eq. (23), which is tran-scendental relative to Z at T = Tm and T* = 1.3 × 104 K,gives Zm = 1.31. The coefficient χ may be calculatedusing the reference data on the thermal conductivitycoefficient κ, specific heat capacity CV/µ, and density ρof matter by the formula

χ = κµ/(ρCV). (25)

The substitution into Eq. (25) of the values for alumi-num [16] κ = 2.30 W/(m K) (this value of κ correspondsto í = 600 K and to approximately the average value ofκ in the range from room temperature to the meltingtemperature), ρ = 2.7 g/cm3, CV ≈ 25 J/(mol K), and µ =27 g/mol gives χ ≈ 23/25 cm2/s. With this value of χ, aswell as with Zm = 1.3 and τL = 10 ns, in accordance withEq. (24) we obtain ∆ ≈ 1.3 × 10–4 cm. In the case of ahemispherical funnel of diameter ‡, the volume of mat-ter escaping from the target in the form of condensedparticles is πa2∆/2. The volume of hemisphere of evap-orated matter is πa3/12. The relative fraction of con-densed particles is expressed in the form (1 + a/6∆)–1. Ata = 10–2 cm and ∆ = 1.3 × 10–4, this fraction is about 0.07.

In addition to estimation of the fraction of con-densed particles, it is of further interest to determinetheir sizes and total number. The solution of this prob-lem involves calculations of dynamic force field arisingin the target under the effect of laser pulse and goes farbeyond the limits of this approximate analysis. As afirst approximation, we can assume that the lineardimensions of condensed particles are approximatelythe same in all three dimensions and take the particlesto be spherical. In this case, the layer thickness ∆defined by formula (24) is the upper limit of the particlediameter. If we assume that the real dimensions of par-ticles are close to this limit, their total number isN = π/2(a/∆)2 ≈ 104.

ESTIMATION OF THE VELOCITYOF SCATTERING PARTICLES

The time of heating of scattering condensed layer(condensed particles) and the scattering time are com-parable in duration, because the thickness of this layeris determined from the condition of equality of thesetimes. Under these conditions, the heating of the layeroccurs simultaneously with its thermal expansion.Then, in making an approximate estimate of the veloc-ity of scattering of the layer, we simplify the problemby separating in time the processes of heating and scat-tering. We assume the heating to be instantaneous andwill consider it for the constant density of layer (isoch-oric process).

∆ Zm χτL.=

HIGH TEMPERATURE Vol. 45 No. 5 2007

THE DYNAMICS OF EXPANSION OF GAS AND CONDENSED PARTICLES 593

The energy

∆E = CV(Tm – T1) (26)

is spent in isochoric heating of one mole of metal frominitial temperature T1 to melting temperature Tm. WithT1 = 293 K, Tm = 833 K, and CV = 25 J/(mol K), Eq. (26)yields ∆E ≈ 1.35 × 104 J/mol.

The pressure increment ∆P in the case of isochoricheating under consideration is related to ∆E as

∆P = Γρ1∆E/µ, (27)

where Γ is the Grüneisen coefficient equal to 2.09 foraluminum [19], ρ1 = 2.7 g/cm3 is the density of alumi-num on the isochore under consideration (i.e., undernormal conditions), and µ = 27 g/mol is the molecularweight of aluminum. The substitution of these valuesinto Eq. (27) gives ∆P = 2.8 × 104 atm.

In the process of adiabatic relief of pressure ∆P to1 atm (almost to zero) with the density varying from ρ1to some value ρ2 which will be found below, the workR is performed which causes an increase in the kineticenergy of particles,

(28)

Here, ∆PS is the isentropic dependence of pressureon molar volume V.

Work (28) is a part of energy (26), with the restremaining in the form of thermal energy of particles. Inthe acoustic approximation, we have

∆PS = U2(ρ – ρ2), ∆P = U2(ρ1 – ρ2), (29)

R = ∆P(V2 – V1)/2, (30)

where U is the volume velocity of sound expressed interms of velocities of longitudinal (UL) and transverse(US) waves,

For aluminum, UL = 6.26 km/s, US = 3.08 km/s [16]and, accordingly, U = 5.15 km/s.

We substitute ∆P = 2.8 × 104 atm, ρ1 = 2.7 g/cm3,and U = 5.15 km/s into the second one of formulas (29)to derive ρ2 ≈ 2.59 g/cm3 and, consequently, V2 ≈10.4 cm3/mol. Equation (30) gives R ≈ 580 J/mol.Finally, we equate the work R to the kinetic energy ofone mole (1/2)µU2 and derive the sought velocity ofscattering condensed particles U ≈ 0.21 km/s. Thisvelocity is an order of magnitude lower than the rate ofgas expansion. The actual value of the velocity of scat-tering of particles is somewhat lower than the foregoingestimate obtained in the approximation of instanta-neous heating of the layer.

R ∆P( )S V , Vi µ/ρi≡ , id

V1

V2

∫ 1 2.,= =

U2 UL2 4

3---US

2.–=

CONCLUSIONS

As a result of approximate analysis, a number ofquantitative estimates have been obtained of the phasecomposition and dynamics of formation of a cloudcaused by the effect of laser radiation on the surface oftarget. Estimates have been made of the initial temper-ature of gas and of the dynamics of variation of temper-ature and density of expanding cloud, a model of for-mation of condensed products has been suggested, andthe parameters of condensed particles have been esti-mated. The results of analysis enable one to gain a bet-ter idea of the composition and energy characteristicsof the products of interaction between laser radiationand the surface of solid sample. On the one hand, theestimates made by us may be useful in developingmethods of elemental analysis using a laser emissionanalyzer; on the other hand, these estimates may serveas a first approximation for theoretical description ofcomplex optical and gasdynamic processes which havean effect on the cloud structure. A rigorous solution ofthese problems calls for inclusion of a number of fac-tors, which were not considered in our model study, andis associated with the need for abandoning the idealplasma approximation and performing complex calcu-lations of dynamic force field arising in the target dur-ing interaction with laser radiation.

APPENDIX

The average coefficient κ of absorption of opticalradiation, which characterizes the integral emissivity ofthe heated region of metal, is estimated by the order ofmagnitude in the Kramers–Unsold approximation [14,19, 20],

(A1)

where α = = 0.96 × 10–7 cm2 K2; c and h

denote the velocity of light in vacuum and the Planckconstant, respectively; ni is the number of ions ofdegree (multiplicity) of ionization i; and Ii is the ioniza-tion potential of ion of multiplicity i. The substitutionof concentrations αi from (13), n = 6 × 1022, and T = 6 ×104 K into Eq. (A1) gives the value of κ = 6.1 × 105 cm–1

which is 6 to 60 times that of the absorption coefficientfor infrared laser radiation.

ACKNOWLEDGMENTS

This study was supported by the International Sci-entific-and-Technical Center (project no. 1292).

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κ 45a

π4T2-----------n 2.5 i+( )2αi Ii/kT–( ),exp

i

∑=

8πe6

3 3chk2----------------------

594

HIGH TEMPERATURE Vol. 45 No. 5 2007

KUZNETSOV, KARASEVICH

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