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Theoretical study of laser heating and dissociation reactions in solidsusing ultrafast timeresolved xray diffractionH. Ma, S. H. Lin, and P. Rentzepis

Citation: J. Appl. Phys. 72, 2174 (1992); doi: 10.1063/1.351607

View online: http://dx.doi.org/10.1063/1.351607

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Theoretical study of laser heating and dissociation reactions in solids

using ultrafast time-resolved x-ray diffraction

H. Ma and S. H. Lin

Center for Solid State Science, Arizona State University, Tempe, Arizona 85287-l 704

P. Rentzepis

Department of Chemistry, University

of

California, Irvine, California 92717

(Received 6 December 1991; accepted for publication 11 June 1992)

A theory of ultrafast time-resolved x-ray diffraction is presented. We show that for the case in

which vibrational relaxation is fast, the x-ray diffraction intensities in pulse laser heating are

determined by the Debye-Waller factor. We also show how the scattering intensities of the

diffraction spots change with the position of the dissociating atoms in a time-resolved x-ray

diffraction experiment studying dissociation reactions in solids. Our results show that the

scattering intensities change in both the fully dissociated and the partially dissociated cases, and

that time-resolved x-ray diffraction could be a useful tool in probing dissociation reactions in

solids.

I. INTRODUCTION

Recently the application of time-resolved x-ray diffrac-

tion (TRXD) to study rate processes in condensed phases

has begun to attract attention. In previous papers,7-9 we

have applied the TRXD technique to study kinetics of

thermal decompositions of Cd(OH), and CdC03 and

phase transitions in PbO in powdered forms. With recent

improvement of x-ray diffraction instrumentation, we have

demonstrated that reactions in the solid state can be mon-

itored down to time scales of picoseconds, and the weak-

ening of intensities due to laser beam heating has been

observed2- The data obtained in these TRXD experi-

ments are the changes in intensities of the diffraction spots.

In order to study the reactions in the solid state we need to

be able to extract information about the changes of atomic

positions with time from the experimental data. In this

article, we shall present a theory of TRXD and apply it to

pulse-laser-beam heating and dissociation of molecules in

solids. We shall demonstrate how the changes in x-ray

diffraction intensity are related to the changes of position

of the dissociating atoms in a TRXD experiment studying

dissociation reactions in solids. This will facilitate the in-

terpretation of experimental data in the future. We shall

also present a different derivation of the DebyeWaller fac-

tor which is important in probing the local laser-beam

heating. In Sec. II, the general theory of x-ray diffraction

and the new derivation of the Debye-Waller factor will be

presented. In Sec. III, we shall study the case of coherent

dissociation where dissociation occurs in all the observed

unit cells at the same time. In Sec. IV, we shall study the

case of partial dissociation where dissociation occurs only

in a fraction of the unit cells.

II. THEORY

For a system of particles with an internal potential

energy U, the Hamiltonian operator is given in Ref. 10.

i3=ii~+Si,, 9

(1)

where

ti2

G)= c - vT+u+&,

2mj J

(3)

and i;i, denotes the Hamiltonian of the radiation field. Here

pi is the momentum operator of the jth particle, and Aj is

given by

(4)

where ek and Kk are the polarization and propagation vec-

tors of mode

k

with a frequency Ok, and

ak

and

a:

are the

annihilation and creation operators.

The elastic scattering of x rays is a two-photon process

and according to the time-dependent perturbation theory,

the transition rate is given in Ref. 11

(5)

where p(E) is the number of radiation oscillators per unit

volume in which the scattered quantum k can be placed

and is given by

(6)

The initial and final states are given by

I~)\% h,$-l)fioQ nkfiwk),

and

(7)

IF)= Iq,

nk,,hb (nk+ 1)&k)-

(8)

Here kc and k represent the incident and scattered x rays,

and Ei and ef are the energies of the initial and final states.

There are two possible intermediate states,

1M) = 1% nk,,hko, nkfiok>,

(9)

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IW= Ie,, (n%+l)fiOb (nk+l)fiWk).

(10)

Substituting Eqs. (7)-( 10) into Eq. (5) and using

(3)-(4) we obtain

me

c

wfnPmi

m

X

(flp-e^kjm)(mlp*&,,li)

Ei- E,++hQ

(.fjp-~~jm)(ml~-~kli)

+

Ei-E,-&dk

+(fl c

exP[i(K-%)'rj] Ii) ($'&,,) 'p(E),

j

(11)

p is the total dipole-moment operator

p=eCri.

(12)

if G~>E,,, - ei, which is true for x-ray scattering,

t two terms in Bq. ( 11) can be neglected. So Eq.

wFI=

x (~k*$$*/&%

(13)

=K-K~ If the incident wave is unpolarized, we

to average the polarization term and obtain

($k-Q2=f( 1 +cos2 6),

(14)

is the scattering angle. Substituting Bqs. (6) and

into Bq. ( 13) and rememberi ng that the incident flux

we obtain the differential scattering cross sectionI

da

~=($)2i(l+C0S2B)l (fl C exp(rXr,)lf) f2.

i

Using the adiabatic approximation

$f= 3i= *Bav 9

(16)

2

i( 1+cos2 6) c exp(iK*Rj)

I

I

XI ~/Iexp(,X.r~)(e,,Imp(iK.u,)Ie.,))1,

(17)

is the origin of the jth unit cell, rY is the position

yt and uY is the displacement of atom y due to

Equation (17) represents the cross sec-

the vibronic state (av). The atomic scattering factor

is given by

drp,(r) exp(ik-r),

(18)

where p,(r) is the charge density for atom y. In order to

obtain the experimentally TRXD cross section, it is neces-

sary to take into accoun t the distribution of the system

among the vibronic (av) states, i.e.,

-(I+cos*~) 1 exp(iK-Rj) CPav(t)

I

X/ ~f~enp(~~.r,)(B.,lexp(jK.4)/8,)/2, (19)

Y

where the summation over u is over all the vibrational

levels. P,,(t) denotes the distribution function and its time-

dependent behavior de pends on the dynamical processes

under consideration. In the case of laser-beam heating,

Bloembergen et al. 13-15

pointed out that the absorbed laser

energy is essentially thermalized after a time of the order of

a picosecond. Absorption of laser pulses comparable to or

longer than a picosecond can therefore be regarded as a

rapid heating process. A number of basic experiments on

picosecond laser heating of semiconductors done by

Bloembergens groupG8 provided evidence in support of

the thermal description. In this case, we can assume that

local vibrational equilibrium is established and the idea of

local temperature T(t) can be used. The temperature 7(t)

may be considered to decay exponentially,g

T(t) - Tq= CT,- T,,)exp( -f/7,,),

(20)

where

T,

is the maximum temperature due to the laser-

beam heating and rv denotes the vibrational-relaxation

time. In this case, the Boltzmann distribution can be used

for P,,(t),

(21)

p,,=e-[U/+(l/2)]fwJkT

I

C e-[u+(l/2)IfioJk.

(22)

u

Now we are ready to evaluate the term in Eq. ( 19)

involving the thermal average of the atomic displacement

factor

(23)

Previous evaluations2021of this term involved moving the

summation into the square and thus reduced the problem

to the evaluation of the following term

(24)

which can be easily evaluated. However we think this is not

rigorous and we present a new derivation in the following

which does not use this assumption. Notice that

K-u,= c KxQ= c Kx; +&k= ; by&k t

(25)

x x

where &s are normal coordinates and that

J. Appl. Phys., Vol. 72, No. 6, 15 September 1992

Ma, Lin, and Rentzepis

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%= n xavk(Qd,

k

(26)

where for harmonic oscillators,

xav,=Nu,Ho,( &Qk)exP( --t&Q ;).

Using Eqs. (25)-(27) yields

(e,, I exp(i K-u,) I W

(27)

= v exp( -g) (l+uk@-$k) +vk~~~l) (hz2,,)2

+

uk(uk--1) (ok-2)

(31)2

(Gyk,>3

+-+$$ Gyk)uk

)

(28)

Using the contour integral relation

1 1

s

dz

-=-

n 27ri

e flexp(d,

Eq. (28) becomes

(%lexp(z~yJ 1%)

= j exP(?)(&) , g$exp(r)

(29)

(30)

x [z+ (Gyk) +*

(31)

Substituting Kqs. (21)-(22) and (31) into Eq. (23) yields

exp[z~-(ry-ry)] bPkW,

k

x (zl+h22,k)(z2+hZ2y,k)]-1,

where

(33)

tk= fiOk/kT.

(34)

Using the relation

z1z2--e

-k(zl+~ykk) tz2+b;,k)

= (l--e-k)(zl+;lyk)(Z2+jly~k) -C,

where

(35)

+k)

h= - (ek- 1)

(36)

2176

J. Appl. Phvs.. Vol. 72. No. 6. 15 September 1992

(32)

and

(37)

we obtain

co

pkl,~=exp(-~,k-~,,k+~~~k+fh,k) c

1

n=O m

x [nk(nk+1)(Liz2yk)(hzzy,k)l,

(38)

where

iik= I/(ek- 1).

Equation (38) can also be written as

(39)

x [ (+$+2ik(&+l) ]n.

(40)

The high-order terms in Kq. (40) are very small compared

to the first term, which is equal to one, so we can approx-

imate the summation as one. Finally, we obtain for the

TRXD differential cross section

2$(l+cos2B)~ ~Jexp(iK*Rj)~2,

i

(41)

where

Fj=

xf,exp( --M,)exp(i K-r,)

Y

(42)

is the unit-cell scattering factor. Here the Debye-Waller

factor M,, is given by

(43)

In a single-atom crystal the expansion of the thermal

displacements of any y th atom with respect to the dimen-

sionless normal coordinates Qk,, characterized by wave

vectors k and branch number n (n = 1,2,3), may be written

in the form:2223

l/2

eLnXdRr)Qkn 3

(44)

where

XdRy) =

cos(k*RJ, k,>O,

Mk-R,),

k, < 0,

(45)

and m is the mass of an atom. Substituting Eq. (44) into

Kq. (43) and remembering that

x:V$J +,&(R,) = 1,

(46)

and for cubic crystals

c (K*ek,)2=K2,

n

(47)

Ma, Lin, and Rentzepis

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4-

3-

F

ir

2-

1-

v,--- ,

100 200

3;10

4lio

600

(8)

T W)

60

QC

100

200

300

400

500

(bt

T (to

FIG. 1. (A) The function F( 7) =0.25+ (T/B) @(O/T) is plotted as a

function of temperature for different va lues of Debye temperature 0. (B)

The calculated diffraction intensity for the (220) spot of Si crystal is

plotted as a function of temperature for different values of Debye tem-

perature 0.

yields

(48)

By means of the Debye model, Eq. (48) can be trans-

formed into the following integral

(49)

where 8 is the so-called Debye temperature and

(50)

Now we define the sum of terms in the brackets of Eq. (49)

+Cfxsin[2?r(hxx+kyx+lz,)l+Y2(hkZ)}2.

as (56)

(51)

F(T) is plotted as a function of temperature T for different

values of 8 in Fig. 1 A). It can be seen that the Debye-

Waller factor is approximately a linear function of temper-

ature and temperature has less effect on larger values of 8.

It has been shown that t he scattering intensity, I, has

appreciable values only at the reciprocal lattice points of

the crystal. So the scattering factor of the unit cell can be

written as

I;itkl= ~fyexp[2~i(hx,+ky,+Zz~) 1,

Y

(52)

where hkl are the index of the reciprocal lattice points, xyz

are the fractional coordinates of the atoms, and the factor,

exp( -My), is absorbed into

f

.,, for convenienc e. In Fig.

1 B), we plot I for the (220) diffraction spot of the Si

crystal as a function of temperature, T, which is a function

of time in laser-beam heating. This behav ior has been ob-

served experimentally.4 Figure 1 B) also shows that tem-

perature has less effect on materials with higher Debye

temperatures.

Ill. COHERENT DISSOCIATION

In this sect ion we study the case where dissociation

occurs simultaneously in all the unit cells under observa-

tion. According to this assumption the position of the dis-

sociating atom, X, in each unit cell will be the same at any

moment and the unit-cell scattering factor, Fhkb is given by

Fj,kl=fX

exp[2riTTi(hXX+kyx+lzx)

n-1

+ c fy expCkW,+b,+&)

I.

Y

The sec ond term of Es. (53) can be expressed as

n-1

F fyexp[2~i(hxy+kyy+Izy)l=Y*+iY2,

where Y, and Y2 are given by

n-1

Y,(hkO= 2 fy

cost2~(hXy+kyy+ky)l,

Y

and

n-1

Y2(hkZ)= 2

fysin[2~(hxy+kyy+lz,)l.

Y

(53)

(54)

(55)

From Eq. (41) we see that the scattering intensity for a

perfect crystal is proportional to the square of the unit-cell

scattering

factor,

1FhklI 2, which is

given

by

IFhkrlz=CfXcOS[2rr(hXX+ky~+Izx)] + Y,UnW2

So the intensity of each diffraction spot will change with

the position of atom X according to Eq. (56).

As a demonstrati on we consider a simple cubic crystal.

For this purpose, we choose Y,(hkl)=Y,(hkl)=l.O, fx

=0.2, and the atomic position of X changes from (0.5,0,0)

to (0.9,0,0) during dissociation. The intensities of diffrac-

tion spots ( l,O,O), (2,0,0), and (3,0,0) as a function

atomic position of atom X are then calculated and plotted

in Fig. 2. So the time de pendence of the dissociating atom

can be obtained through the change of intensities of dif-

fraction spots.

2177

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0.5 0.6

0.7 0.8

position of atom X

0.9

FIG. 2. Square of the unit-cell scattering factor as a function of the

atomic positions of the dissociating atoms. The solid, dotted, and dashed

lines are for the diffraction spots, 100, 200, and 300, respectively.

IV. PARTIAL DISSOCIATION

In this section we consider the case of partial dissoci-

ation, i.e., dissociation occurs only in a fraction of unit cells

and the dissociating atoms departs coherently. In this case,

different unit cells may have different scattering factors and

the scattering intensity is now proportional to the follow-

ing term:

N

2 FTFj, exp[@(Rj-Rj,)],

jJ=l

(57)

where Fj is given by the following equation for the case of

partial dissociation:

Fj=F+c~fx[exp(i~r~)

-exp(iqQ)],

(58)

where F is the unit-cell scattering factor of the crystal, cj

has values of 1 and 0 depending on whether the atom X in

that unit cell is dissociating or not. r, is the initial position

of atom X and ri is the atomic position of atom X at any

instant during dissociation.

Neglecting small fluctuations, we may replace Eq.

(58) by its mean value calculated for a random distribu-

tion of dissociating atoms. This averaging proces& re-

duces to an averaging over the quantity cj which takes the

value 1 with a probability c equal to the fraction of disso-

ciating atoms, and value 0 with a probability l-c. After the

averaging Eq. (57) reduces to

2

exp ( i K.Rj) ,

(59) L. A. Lompre, J. M. Liu, H. Kurz, and N. Bloembergen, Appl. Phys.

Lat. 44, 3 (1984).

where the unit-cell scattering factor for the dissociating

5. M. Liu, R. Yen, H. Kurz, and N. Bloembergen, Appl. Phys. Lett. 39,

system FLkl is given by

755 (1981).

F&[=

~fyexp[2~i(hx,+ky,+lz,)l

Y

19R. J. Sension, S. T. Repinec, and R. M. Hochst rasser, J. Chem. Phys.

93, 9185 (1990).

+cfx[exp(iK*rjy)-exp(iK*rx)].

(60)

So in the case of partial dissociation, the scattering inten-

sity is also proportional to the square of the unit-cell scat-

tering factor, which is given by Eq. (60), and is a function

20M. A. Krivoglaz, Theory of X-ray and Thermal-neutron Scattering by

Real Crystals (Plenum, New York, 1969), Chap. VI.

G. L. Squires, Introduction to the Theory of Themal Neutron Scattering

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23R. Peierls, Quantum Theory of So/ids (Oxford University Press, Oxford

1955).

of both the atomic positions of the dissociating atoms and

the fraction of unit cells in which dissociation takes place.

V. CONCLUSIONS

In this article, we have presented a microscopic theory

of TRXD. To show the application of the theory, laser-

beam heating and dissociation of molecules in solids have

been treated. In dissociation reactions in the solid moni-

tored by the TRXD, the scattering intensities of diffraction

spots change in both the fully and the partially dissociated

cases. Our results show that TRXD could be a useful tool

in probing dissociation reactions in solids.

ACKNOWLEDGMENTS

This research was supported in part by grant No. DE-

FGO2-87ER45305, US Department of Energy, Basic En-

ergy Sciences, Division of Materials Sciences. One of us

(Rentzepis) was supported in the past by W. M. Keck

Foundation and AFOSR grant No. F49620-89-0104.

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2178 J. Appl. Phys., Vol. 72, No. 6, 15 September 1992

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