Effect of hyperfine structure on ionization efficiencies in stepwise ionization using broad bandwidth lasers

Sperrrochrmrra Acla. Vol Prmted I” Great Bntam.

46B.

No

II,

pp

1439-1457.

1991 Pcrgamon

Press plc

TOPICS IN LASER SPECTROSCOPY Effect of hyperfine structure on ionization efficiencies in stepwise ionization ming broad bandwidth lasers M. G. PAYNE and S. L. ALLMAN Chemical Physics Section, Oak Ridge National Laboratory,

Oak Ridge, TN 37831-6378 USA

and J. E. PARKS Department

of Physics, University of Tennessee, (Received

14 February

Knoxville, TN 37996 U.S.A.

1991; accepted 13 May 1991)

Abstract-It

has recently been shown that in the stepwise excitation and ionization of an element with broad bandwidth lasers, there are major effects due to the fact that even and odd isotopes have different hyperfine structures. The increase in the number of levels and the decrease in the dipole matrix elements of the individual transitions leads to easily observable differences in ionization probabilities for isotopes with different nuclear spins. Here we describe two methods for eliminating biases due to hyperfine structure on the measurement of isotope ratios using broad bandwidth resonance ionization mass spectroscopy (RIMS).

1. INTR~~XJCH~N IT HAS recently been shown [l-4] that considerable differences occur in the ionization probabilities of the even and odd isotopes of an element when a laser having a bandwidth much larger than the hyperfine splittings is used to carry out resonance ionization spectroscopy (RIS). Related effects were recognized earlier [4] by workers examining the possibility of using stepwise ionization with circularly polarized broad bandwidth light as a source of spin polarized electrons. One set of effects on ionization probabilities have to do with the spoiling of selection rules due to the coupling between electronic and nuclear angular momentum; leading to MJ no longer being a good quantum number. Other effects are related to the increase in the number of levels, the degree of coherence between states induced by the excitation, and the resulting dilution of the oscillator strength of the individual transitions. Under conditions where the discrete-discrete transitions are all saturated, effects related to hyperfine splittings can lead to significant differences in the fraction of the atoms excited. This, in turn, accounts for much of the difference between ionization probabilities for isotopes with and without nuclear spin. In particular, if the product of the hyperfine splitting frequency and the pulse length of the laser is large compared with unity, large differences can occur in the ionization probabilities of different isotopes in the low intensity regions near the edges of the laser beams. This is true even when the ionization probability is close to unity for atoms located in the highest intensity regions of the laser beams. These effects can be particularly large when a poor strategy is used in choosing the excited states to be used in the excitation and ionization. An example of a strategy which leads to large differences between the ionization probabilities of even and odd isotopes, suppose that the symmetries of the states involved in a transition have been chosen so that some of the populated magnetic sublevels would not be excited if MJ were a good quantum number. For instance, suppose that the initial state has .Z = 1 and the selected excited state has .Z = 0. With linearly polarized light, the selection rule on the z component of the total electronic angular momentum quantum number is AM, = 0, so that only one-third (i.e. only the M, = 0 substate) of the population is subject to excitation in the absence of the hyperfine interaction. With a nuclear spin of Z = l/2, the presence of the hyperfine 1439

1440

M. G. PAYNEet

al.

interaction provides a mechanism for breaking the MJ selection rule. In fact, with I = l/2 the ground state can have F = 112 or F = 3/2, while the excited state can only h+ave_F =-l/2. The selection rule is now AMF = 0 for linearly polarized light, where F = I + J. Consequently, 50% of the F = 3/2 population can be excited (i.e. MF = *l/2), while all of the F = l/2 population can be excited. Thus, four of the six magnetic substates can be excited for I = l/2, while only one of three is excited for I = 0. This would lead to a factor of two discrimination against the isotope with I = 0. By choosing an ionization scheme in which J always increases with excitation energy, one avoids really large discrimination effects related to the MJ selection rule. Even when the increasing J rule is followed, the presence of a nuclear spin leads to the dilution of the oscillator strength over a larger number of transitions and to a different fraction of the total atoms being in the excited state when the discrete-discrete transitions are all saturated. Thus, since one is trying to promote a group of atoms in different hyperfine states to a variety of excited hyperfine states which are ionized with different ionization cross sections, differences in ionization probabilities are not surprising. Even with an intelligently chosen ionization scheme, atoms with different nuclear spin will be ionized with different probabilities unless all of these probabilities can be made close to unity. In the present paper we point out two approaches which allow accurate measurements of isotope ratios using Resonance Ionization Mass Spectroscopy with broad bandwidth lasers. In the first method we argue that it is almost always possible to achieve ionization probabilities which are close to unity for both even and odd isotopes in the region of highest power densities. Thus, in the latter region the ionization probabilities for all isotopes are the same. We describe an experimental design in which the introduction of the atoms into the laser beams and the extraction of the ions to the mass spectrometer are such that only ions from the high power density region are transmitted and detected. This first method results in a lower detection sensitivity, but avoids sensitivity to hyperfine structure. In the second method for avoiding the effects of hyperfine structure on ionization probability a single laser is used to carry out two-photon ionization. The single laser is operated at a power density such that when it is tuned to a strong dipole allowed transition, ionization probabilities close to unity are achieved over a power broadened region of several wavenumbers about the resonance. The Rabi frequency of a strong transition is of the order of I&,,] = an&/B, where an is the Bohr radius, and E,, is the laser field amplitude. Thus, a power density of I = 10XW/cm2 gives a power broadened width of = 10 cm-’ and is also sufficient to ionize the excited state with high probability. When the laser is detuned from the resonance by an amount large compared with either the laser bandwidth or the hyperfine splitting, but less than the power broadened width, the ionization probability is close to unity but the two photons are absorbed in a time tabs = l/6,. This time is much less than the precession time for the nuclear spin. Unlike resonance excitation where the product of the hyperfine splitting and the pulse length is large and the amplitudes for the hyperfine levels become completely dephased, the amplitude for the hyperfine levels stay nearly perfectly phased when Iw~~]/]~~]
HFS effectson ionization efficiencies

1441

2. EFFECTSOF HYPERFINESTRUCTURE ON IONIZATIONPROBABILITIES To illustrate the effects of hyperfine structure when an optimum ionization scheme is used, FAIRBANKet al. [l] have considered the resonance ionization of Sn. In particular, he has studied two-photon stepwise ionization by way of the 3Po * 3P, resonance. In this case, the rule of always increasing J with successive excitations has been followed by choosing the ground state to have J = 0 and the excited state has J = 1. The odd isotopes of Sn have I = l/2 and the even isotopes have I = 0. This is the simplest example of hyperfine effects possible. This problem has been discussed theoretically by LAMBROPOULOSand LYRAS [2,3] using a phase diffusion model of the laser. LAMBROPOULOSand LYRAS have applied the model in Refs [2,3] to two different ionization schemes studied by FAIRBANKS[l]. In each case good agreement was obtained with experiment. 2.1. Phase diffusion model of a laser We begin by presenting a short description of the phase diffusion model of a laser field. The statistical properties of a laser are given in terms of the laser field autocorrelation functions. The phase diffusion model is a model of laser bandwidth effects in which the amplitude of the laser field is a smooth function of time, but the phase is a fluctuating quantity. We let the laser field at the position of an atom be of the form E = E,, cos(wt + G’(t)), where E,, is the field amplitude, 6 is the angular frequency at line center, and a(t) is the fluctuating phase. This model arises from the random walk of the phase angle @(t) around the unit circle. In the model d@ldt is discontinuous and successive values are uncorrelated in sign and magnitude. The Langevin equation for the stochastic changes occurring in the phase is

where = (yLn)s(lt-t’l).



The two-time higher order autocorrelation
Et(tf)ecN*(r’)>

function of this field is given by =

f$+(f)e-N2~LI’-“l’2.

The lineshape of the laser follows from the previous equations as the Fourier transform of the field autocorrelation function for N = 1.

wNYLm2 I(O) = (6Y-c3)2 + (yJ2)2

.

Obviously yL is the full width at half maximum in angular frequency The phase diffusion model has one other important property:


iQtr,) &(t,)e@('z)

. , . EO(t2+ l)e-@(f*N-

for this model.

1)Eg(tZN)e’*(fZN)>=

. . . , for t, 2 t2 2 t3 2 . . . 2 t2+, 2 t2N. This multi-time autocorrelation property assumes that the changes in d@/dt are instantaneous and that successive changes are uncorrelated. This is also the property which leads to the Lorenzian lineshape.

1442

M. G. PAYNE et al.

To see how the bandwidth of the lasers enters into the equations of motion consider a two level system. Neglecting spontaneous emission we have the following equations for the amplitudes for being in states 10 > and 11 >.

!&if)

01

dt

A

1,

dA

L = - (6, + d@(t)/dt)A, + i&u;, dt

where 2Q,, is the Rabi frequency, S, is the detuning of the laser from resonance, and a phase change has been made in the amplitude for state II>: A, = u,el@,‘+@@)), where @ is the fluctuating phase of the laser. In the above form LR,,,is a smooth function of time which can be taken as real. The fluctuating phase of the laser has been separated out and absorbed into A,. It is convenient to let W = 2A,a;, = Z2 + iZ3, Z, = IA,/’ - la,,\*.

After the proper averages are taken Z,, Z2 and Z3 will be the elements of the Bloch vector. The connections with the elements of the density matrix are obvious from the definitions. From the equations of motion for the amplitudes we find dW _ = i(6, + dQldt)W + 2iQ,,Z,, dt

d& dt

=

2%,&.

The above equations can be written as W=2i

’ dtlRol(tl)Zl(tl) I -z

exp[i(h(t-4)

+ W

- *(td)l,

When the real part of W is substituted into the second equation z,=-1+

dM(t,ti)Z,(t,),

where,

K&t,) = 4 A solution for Z, can be developed

01(t 2 ) cos (h(t-t2)

+ w

-

Q’(f2)).

by iteration dt3K(t,t2)K(t&

+ ....

At this point the multi-time correlation factorization property of the phase diffusion field can be used on the time ordered products of laser phase factors found in the K(t,,t,) factors to show that the infinite sum can be repeated by the iterative solution of

HFS effects on ionization efficiencies

1443

f



= -1+

I --z

dt, ,

= 4 ’ dtzR,~,(t)n,,,(t2)e-Y~(‘-f2)‘Z cos&(t-t,)).

Using the same features we can show d ___ dt

= 2nc,,
d ~ = i(6, + yJ2) + 2X&,, dt The equations for the diagonal elements of the density matrix are unaffected. However, those for the off diagonal elements are modified by a coherence dephasing term equal to the half width at half maximum of the laser. If these considerations are generalized to a multistate system only equations for off diagonal elements involving states that are coupled by the laser are dephased by the laser bandwidth. The Lorentzian lineshape leads to unphysically large excitation probabilities on the far wing of the line. However, the phase diffusion model is believed to describe resonance excitatation quite accurately. 2.2. Even isotope ionization We follow LAMBROP~ULOS and LYAS [2,3] in using a phase diffusion model for the broad bandwidth laser. For the even isotopes we have two states: 10 >=15p2 3Po> and (1 >= 15~6s 3P, >. If pi, is a density matrix element for the two-state system in the Schrodinger representation, we let a;, = p,,emiwf, i
Above, R,,, = EtJh is the Rabi frequency, where E,, is the laser field amplitude at the time in question and bz is the electronic dipole operator. I, is the ionization rate of the level 1 population, 6, is the detuning of the laser from exact one-photon resonance, and yL is the laser bandwidth. The square of the electronic dipole operator involved in the Rabi frequency is related to the reduced matrix element (lbJ1by 1<1]8,]0>]* = )<1~8,~0>(2 =

~2m2

3J(2J+ 1)(23+2)

ll~l12L0,m, 7AJ=O,J#O,

(2)

52 - m*

3J(2J- 1)(2J+ 1) ll~]]*~m,,.m,~IA.4 = 1, where m = m,, is the magnetic quantum number of IO> and J is the larger of the total angular momentum quantum numbers for the two states. The reduced matrix element is related to the absorption oscillator strength of the transition by (3)

M. G. PAYNEet al.

1444

Equation (3) can be used to replace @j[* in Eqn (2) in terms of F,,,. In Eqn (3) m, is the electron mass, w is the resonance frequency for the transition, e is the charge of the electron, and g,, is the degeneracy of state IO>. The effect of hyperfine structure on the photoionization cross sections is far more complicated. LYRAS and LAMBROPOULOS [2,3] have discussed the effects of nuclear spin on photoionization cross sections for different hyperfine levels. 2.3. Odd isotope ionization Since this paper is intended for experimentalists who are involved in the use of stepwise ionization techniques in the measurement of isotope ratios we will briefly review some well known material related to effect of hyperfine structure on the Rabi frequencies involved in excitation with a broad bandwidth laser. In the case of the odd isotopes the pertinent quantum numbers which replace J and MJ are the total angular momentum F and its z component quantum number mF. Thus, the ground state has F = l/2, while the excited state is split into two states with F = l/2 and F = 312. The state function in the presence of the nuclear spin is related to that in the absence of nuclear spin by b,F,m+

= c C(J,I,F;m,m,,m,)Ja,J,m m.ml

> II,ml >,

where I is the nuclear spin angular momentum, (Yis the for the state in question, and C is the Clebsch-Gordon the Rabi frequencies without nuclear spin can be related state and the two hyperfine levels. For instance, for Sn < 1/2,1/2~&~1/2, l/2 >

= = = =

(4)

rest of the quantum numbers coefficient [5]. Using Eqn (4) to those between the ground we get

~0, Ojlj,[l, 0X(1,

l/2,3/2; 0, l/2, l/2), 0 > C(1, l/2,3/2; 0, -l/2, -l/2), co, 0\&11,0 > C(1, l/2, l/2; 0, l/2, l/2), co, OlB,Jl, 0 > C(1, l/2, l/2; 0, -l/2, -l/2),

~0, OJ&Il,

(5)

as the only non-zero matrix elements. For brevity, we have suppressed the (Y in presenting the results. The first two matrix elements in Eqn (5) are equal, as are the last two. We see from Eqn (5) that the matrix elements which enter into the Rabi frequency differ from those for the even isotopes by exactly a Clebsch-Gordon coefficient. The oscillator strength splits between the two hyperfine transitions as the ratio of the squares of the corresponding Clebsch-Gordon coefficients. There are two special values [5] of Clebsch-Gordon coefficients which are of interest here: C(J,, l/2, Ji -t l/2; ml, l/2, m) =

J,+m+1/2 J

C(J,, l/2, J, - l/2; ml, l/2, m) = -

2J,+1

(6)

’

J,-m+1/2 J

2J,+1

.

From Eqn (6) for J, = 1 and ml = 0 we find that the oscillator strength splits so that two-thirds of the total is associated with the l/2+3/2 transition and one-third is associated with the l/2+1/2 transition. The sum of the squares of the Rabi frequencies for the two hyperfine transitions is equal to the square of the Rabi frequency in the absence of nuclear spin. In discussing the ionization of the odd isotopes, we note that with linearly polarized laser light the two magnetic substates in the ground state only couple to excited states with the same rnP Further, the Rabi frequency coupling one of these magnetic substates to hyperfine states with corresponding mF is the same for mF = l/2 and for mF = -l/2. This means that the two populations behave identically and we need only deal with a three state system. Considering that there are two lower states and six excited

HFS effects on ionization efficiencies

states, this is a considerable have found[2]

simplification.

1445

For this system LAMBROPOULOS and LYRAS

(7) d2

= [ia, - (l/2) (r, + y&J,0 - f

du,z __ = _[iw 12+ dt

(l/2) (r, +

i~,,,(U,j-UIH,)-i:~,'l,U,,,j,j'

r2)l U12

=

1,2yo'fj')7

+ i~n,,,U,,2-i:~,,2U,,,.

We have followed LAMBROPOULOS and LYRAS[~]in neglecting the terms in the equation doi2 for r arising from interferences due to the two pathways to ionization by way of the states ]I > or (2 > [6-81. These terms are very small in the special case of this sequence of transitions in Sn. However, in general such terms are necessary, particularly of the wings of the line where they are responsible for the interference terms arising in a time dependent perturbation theory treatment at large IS,]. The nature of there terms will be displayed explicitly in Section 3.2. There are six of these equations, with three of the six being differential equations for complex quantities. Thus, Eqns (7) are equivalent to nine simultaneous real differential equations. Above, 10 > is the groundstate, 11 > is the excited F = l/2 level, and 12 > is the excited F = 312 level. 6, and a2 are detunings of the line center of the laser from resonance with the 10 > -11 > and the 10 > t, 12 > transitions respectively, 0 12 is the hyperfine splitting, I, and I2 are the ionization rates for states 11 > and (2 >, and yL is the laser bandwidth. The quantities R, are the Rabi frequencies between 10 > and b >, j = 1,2. The conditions for Eqns (7) to reduce to rate equations are well known. However, since the effects of hyperfine structure have only been worked out for a few special cases we will repeat the considerations which lead to rate equations as an approximation to the more fundamental equations for the elements of the density matrix. There is nothing compelling about the use of rate equations, but in a situation where N levels are involved there would be P simultaneous real differential equations to be solved for the elements of the density matrix, and only N to be solved for the diagonal elements remaining in the rate equation approximation. Nz can be quite large in cases where two resonance steps are used in the ionization method and the nuclear spin is much larger than l/2. Further, when each laser is tuned between the centroids of the hyperfine levels for each discrete-discrete step (i.e. the case where the broad bandwidth lasers pump all hyperfine transitions resonantly) the rate equations can be reduced to having constant coefficients if all lasers have the same pulse shapes and pulse lengths. The resulting linear first-order differential equations can then be solved by standard eigenvalue and eigenvector methods. Correspondingly, the reduction in effort can be substantial for the user who is not set up to do numerics on large systems of stiff differential equations with variable coefficients. In the case of Sn the hyperfine splitting is o12 = 0.25 cm-‘. Correspondingly, with laser pulses having pulse length r = 10 ns the relative phase of the probability amplitudes for states ]I> and 12~ oscillate many times during a laser pulse. When yL B ]fl,,j] we have a situation where at least two of the off-diagonal elements of the density matrix adiabatically follow the diagonal elements, and Eqns (7) or Eqns (1) reduce to a smaller number of simpler equations. Usually, IQ,,] Z+I,. We also assume yL B I,. Thus,

1446

M. G. PAYNE et al.

If we retain 6r and s2 in Eqns (8) a Lorentzian lineshape is predicted for the low intensity excitation. This is an artifact arising partly from the crude phase diffusion model of the laser used in deriving Eqns 7 and partly from the use of the rate approximation on the wing of the line. In general it is safest to use the rate approximation only when the laser is tuned to resonance. We are interested mostly in the case where the laser full width at half maximum (FWHM), yL, is very large compared with CQ. In this limiting case, and with the laser tuned midway between the two hyperfine lines, we obtain

Above, r = (%21@M As long as yL + I~oll, yL 9 IsrI, and yL B 10~~1Eqns (9) can be used even if two different lasers are used for the excitation and ionization steps. An appealing ionization scheme would be obtained if the second laser were much more powerful than the laser used to pump the discrete-discrete transitions. This would avoid extreme power broadening of the discrete-discrete transition when the intensity of the second laser is increased to the point where the ionization probability is close to unity. In the case of Sn, we also have o127 % 1, so that cr12can also be replaced in terms of diagonal elements. The physical reason for this is that the relative phase of the probability amplitudes for the two hyperfine levels oscillates many times during the pulse so that on the average there is no persistent phase relation between the two amplitudes. This means that the off diagonal element of the density matrix cr12is small. Mathematically, it also adiabatically follows the diagonal elements so that it can be eliminated from the equations of motion. In this limit, a set of rate equations results for the diagonal elements of the density matrix. However, in a case where the hyperfine splitting is very small, the full set of four equations must be retained. In the case of small hyperfine splitting (i.e. o127 4 1) we can show from Eqns (9) that the populations of (l> and 12> evolve in time so that they are always in the ratio does not require saturation of the transitions. That is, these u22/u11 = r2. This property populations are always in the ratio of the oscillator strengths for the different transitions. Saturation occurs with half of the population left in the ground state. The other half of the population is distributed so that u1 1 = (l/2)/( 1 + r’) and u22 = (?/2)/(1 + 9). With the two level system involving Sn, we have r = d2, so that ull = 116 and U 22 = l/3. That is, when there is no precession of the nuclear spin during the laser pulse, the number of excited atoms and ions produced is identical to the case where the hyperfine structure does not exist. This bit of insight was pointed out clearly in reference 4 and is mentioned in Refs [2, 31. Consider the case of Sn where ml27 s 1. On eliminating u12 the following set of rate equations results from Eqns (9).

(10)

HFS effects on ionization efficiendes

1447

where

(11)

Equations (10) can also be derived as a function of I$, so that a lineshape is predicted. The lineshape becomes inaccurate once IQil becomes larger than yL (due to power broadening), while the 6r = 0 result continues to be a reasonably accurate approximation, We note that the dependence on the power density of the R,, and of the I, are identical. This suggests letting

g(t’)dt’,

h=

where g(t) is the pulse shape function which is normalized so that at t = 0 it has the value 1. On changing the independent variable to h, we obtain a set of three linear homogeneous equations with constant coefficients which can easily be solved analytically when the ionization rates are small compared with the rates R,,. The results are ’ s(t)

=

@-l(O) + rd”),(t) 3

M?(t)

7

(13)

= -h(t)R,,(O)k+,

Such solutions can also be carried out for the resonance case for more complicated cases. However, the eigenvalues and the eigenvectors for the matrix of constant coefficients and its adjoint may need to be determined numerically for cases where the number of levels is considerably larger. Such solutions are still useful since an analytical function of t can be written for each element of the density matrix. We see from Eqns (13) that when the discrete-discrete transition is strongly saturated the three levels have equal populations, Thus, two thirds of the population is excited, while for an even isotope only one half of the population is excited. It is not surprising that the odd isotopes are ionized more efficiently at power densities which have not yet saturated the ionization step. In more general cases each hyperfine state has equal

1448

M. G. PAYNEet al.

I (10’ W/cm’) Fig. 1. Ionization probability versus laser power density for the ionization of tin atoms when a laser with bandwidth 0.4 cm-’ is tuned to the 5~’ ‘P,, o 5p6sJP, resonance. The smooth curves are based on rate equations, while the discrete points are calculated from the full density matrix equations. At the very highest power densities, the rate equations deviate from the solution to the density matrix equations by about 1% for odd mass isotopes. In the case of even isotopes the rate equations agree with the density matrix equations to 0.1% for all power densities shown.

populations under conditions of saturation. Thus, the relative excited state and ground state populations depend on the number of excited hyperfine levels relative to the total number of levels. 2.4. isotope effects in the ionization of Sn LAMBROPOULOS and LYRAS [2, 31 have studied the full set of Eqns (3) and Eqns (7) numerically. We will demonstrate that these results can be mostly understood from Eqns (10) and the rate form of Eqns (3). This shows that the presence of a large hyperfine splitting and a large laser bandwidth destroys any observable coherence between the two excited states, and brings about the validity of a rate analysis. We first show both the ionization predicted from the rate equations and from the full density matrix. We use the parameters calculated by LAMBROPOULOS and LYRAS [2, 31. For the odd isotopes, the two Rabi frequencies are written as Cl<,,= c,di, where the power density is in W/cm’ and R,, is in radians/s. Here, c, = -3.57 x 10’ and c2 = -5.05 X 10’. The single Rabi frequency for the even isotopes is Ro, = 6.18 X lO’d/1. The ionization rates for the excited states of the odd isotopes are written as I, = k,Z. The values are: k, = 2.74 and k2 = 3.08 in units of cmY(W-s). The ionization rate for the even isotope excited state is I1 = 2.961, with I, in units of s-’ and I in W/cm’. The hyperfine splitting for the odd Sn isotopes is wol = 0.25 cm-‘. We have used yL = 0.4 cm-‘, and we assume the laser is tuned midway between the two resonances. The pulse shape function was taken to be g(t) = exp(-2(t/r)*), with T = 4.0 x lo-’ s. Figure 1 compares the calculated ionization based on rate equations to that determined from Eqns (7). Note that the agreement is good over several decades of power density for both the even and odd isotopes. It is only at power densities I = lo8 W/cm* that a difference of the order of 1% appears. This may be due to the Rabi rates becoming larger than the laser bandwidth, so that the laser can produce coherent changes in the density matrix over a time period much less than the coherence time of the laser light. Some evidence is provided for this view by noting that when Eqns (9) are solved, and used to calculate the ionization the agreement with the fully density matrix is within about two parts per thousand. Note that in the case of the even isotopes the rate analysis stays accurate even at the highest power densities. It is important to point out that the agreement between the rate analysis and the full density matrix treatment is

HFS effects on ionization efficiencies

1449

t/r Fig. 2. Ionization probability versus time for odd mass isotopes of Sn subjected to several different power densities of the ionizing laser. A first laser is tuned to the 5pZ 2P,, 9 $6.~ ‘P, resonance and its power density is held fixed at 2 x 105 W/cm*. The first laser has a bandwidth of 0.4 cm-‘. The second laser is assumed to dominate the ionization rate, but to have no effect on pumping the discrete-discrete transition. The full lines are the diagonal element of the density matrix calculated in the rate approximation, while the discrete points are based on the full set of equations for the density matrix. The predictions of the rate equations are within 0.1% of the results obtained by solving the equations of motion for the density matrix.

reasonably close throughout the entire pulse, not just when the discrete-discrete transitions are saturated. A better ionization scheme is achieved by using different lasers for the discrete-discrete transition and for the ionization step. In that way, the power density used to resonantly pump the discrete-discrete transition can be kept several orders of magnitude lower than the I = 5 x lOa W/cm* required to saturate the ionization thoroughly. At a power density where the ionization is just beginning to be saturated, the line is power broadened to width of = 6 cm-’ FWHM in cases where a single color of laser light is used. To show how well the rate analysis can work for the improved ionization scheme, we imagine that the first lasers power density is kept at I, = 2.0 x lo5 W/cm2, while the intensity of the second laser is varied. We take yL = 1.0 cm-’ and o12 = 0.1 cm-‘. Figure 2 shows the two populations as a function of time. Even though we have used the same Rabi frequencies as for Sn, the new o i2 means the results no longer describe the ionization of this element. In this model the difference between the density matrix solution and rate equations never gets larger than 0.1%. Clearly, when 0~~7 + 1, and Ihjl 4 yL this phenomena can be described accurately by rate equations. The value of g(t) was taken to be the same as defined in the previous paragraph. With nearly all ionization schemes the rate at which the discrete-discrete transition is pumped is far larger than the rate of ionization of the excited states. Thus, in any case, where appreciable ionization is occurring, the discrete-discrete transitions are strongly saturated. In the case of Sn, this means that each of the three states involved for the odd isotopes has a population equal to one-third of the fraction of the atoms that have not been ionized at most times during the laser pulse. In the case of the even isotopes each of the two states involved has a population of one-half the fraction of the atoms that have not been ionized earlier in the pulse. This means that the probability of ionization for each isotope is given for all except very low intensities (where PO and P, are small anyway) by

PO = 1 - exp -~(11(0)+12(O))T/3 i

1

,

1450

M. G. PAYNEet ai.

&=l-exp(-&,(O)T/2). where P, is the probability of ionization of the odd isotopes and P, is the probability of ionization of the even isotopes. The peak ionization rates I’,(O) and l?,(O) for the two excited states of the odd isotopes and I’,(O) is the peak ionization rate of the excited state of the even isotopes. We have assumed above that the time dependence of the laser intensity is L(t) = I(O)exp(-2(t/r)*). The approximate ionization probabilities shown in Eqn (14) are also appropriate for discussing the effect of the radial intensity profile of the lasers on the total yield of ions. Assume that the radial intensity profile of the laser is

where Z(O,O) is the peak power density at beam center. The yield of ions is given by

N,(y) = 2NnLo

pdp [ 1 -exp( -ye-fplR)z)],

(16)

where for the odd isotopes y = y0 = G(l?,(O,O) + r2(0,0))7/3, and for the even isotope y=y, = ~r~(O,O)~/2. Ab ove, r,(O,O), T,(O,O), and r,(O,O) are ionization rates at t = 0 and at p = 0, N is the concentration of the Sn isotope in question (assumed to be constant over the beams), and Lo is the length of the region over which ions are to be extracted from the unfocused beams. With either choice for y, we can evaluate the integral to find for either even or odd isotopes (according to the choice of y) N,(y) = nR2Nt#%y)

+ Y +

ld.Y)k

(17)

where,

is the exponential integral, and y = 0.577 . . . is Euler’s constant. We follow Ref. [3J in defining a parameter p by

Now, N,(y) = 0.9y for y I 1. Thus, at the lower power densities y 9 1 El is exponentially small, and 2 ln(1.31) ’ E 2rn(10-xf)’

p ^- 0.25. When

(20)

= ~_ OS4 1.15 f 2 ln(lO-V(O,O))’ Even when y = 10 we still have p = 0.1. This logarithmic decrease in p indicates that it is not practical to get rid of the isotope effects by simply making the ionization probability at beam center extremely close to unity. This is an indication that a lower intensity region always exists where the ionization step is not saturated, and a significant fraction of the ions come from this region.

HFS effects on ionization efficiencies

1451

3. AVOIDINGISOTOPEBIASES IN RESONANCEIONIZATION MASS SPECTROSCOPY 3.1. Method I If separate lasers are used for the discrete-discrete

and ionizing steps it is relatively easy to achieve a situation where the ionization of both even and odd isotopes is very close to unity in the region near the center of the laser beams. One does not have to know accurate oscillator strengths since, for all except the weakest transitions, onephoton discrete-discrete transitions saturate with a power density in the range 103-lo6 W/cm*. This assumes a pulse length of lo-% and a bandwidth of 0.5 cm-‘. Absorption oscillator strengths were allowed to range from lop3 to 1 in this estimate. These power densities require (for the highest power density limit) 1 mJ laser output with a beam diameter of 3 mm. For the ionization step it should usually be possible to choose an upper state with a photoionization cross section larger than 10-i’ cm*. This frequently requires avoiding s states as the upper most excited state because of their tendency to exhibit Cooper minima in the region near ionization threshold. This reduction in the photoionization rate often keeps s states with moderate principle quantum numbers from having a photoionization cross section larger than 10-l* cm2 for any photon energy. If careful choices are made in choosing the resonance states, a fixed frequency source in the infrared (i.r.) portion of the spectrum can usually be used for the ionization step. With such a source, the laser output can easily be 100 mJ without causing much decrease in the output of the dye lasers pumped with harmonics of the same source. The 100 mJ output permits power densities up to 10s W/cm* with a 3 mm beam. This output gives strong saturation even with a lo-i8 cm* photoionization cross section. However, there is always a lower intensity region of the laser beams where the discrete-discrete transition is saturated, but the ionization step is not. It is from this large volume region of lower intensities that odd isotopes will usually be ionized more efficiently than even isotopes, leading to very appreciable biases in the ionization probabilities. In order to achieve unbiased measurements of all isotopes, we design the atomizer for the mass spectrometer (see Fig. 3) so that the neutral atoms are introduced into the laser beam as a narrow collimated beam moving nearly orthogonally to the direction of the laser beams, and to the direction the ions will be extracted. The laser beams are chosen to have diameters which are about three times larger than either the diameter of the neutral atom beam, or the diameter of the aperture through which the ions are extracted into the mass spectrometer. In the design shown in Fig. 3, ions can only be extracted into the mass spectrometer if they are created near the center of the laser beams where the ionization probability is close to unity for all isotopes. The advantage of this method is that the power densities of all short wavelength lasers can be kept relatively low, while a powerful infrared laser can be used for very efficient ionization. If needed, more than one discrete-discrete transition can be used so that high elemental selectivity can be retained. In this method, the analysis of the ionization probabilities can be carried out by a rate analysis. Thus, the analysis of the ionization probability is somewhat simplified. The success of the method can be investigated by studying the magnitude of the signals as a function of laser power densities. To be consistent with ionization probabilities being very close to 1 near beam center, the observed ion signal should be independent of laser intensities, even when the intensity of any of the lasers is reduced a factor of five. The disadvantage is that by introducing the atoms as a collimated beam moving orthogonal to the direction of ion extraction a great deal of sensitivity is lost. Fortunately, when accurate isotope ratios are being measured sensitivity is usually not an issue. If lasers with good beam quality are used, it should be possible to eliminate isotopic biases by making use of this ionizer design. 3.2. Method II The use of lasers in this method is somewhat similar to the ionization scheme studied by LYRAS and LAMBROPOULOS [2]. That is, one tunes a single laser very near a strong

M. G.

1452

rumple

PAYNE et al.

coUinu&.edbeam ..

:* . . . . .. ‘. 1. . * .,:

of neue

-.

*.:

fromhere collimfdr II

’

normaltopage

Fig. 3. Proposed modification of a RIMS mass spectrometer to remove isotopes biases due to hyperfine structure from the observed isotope ratios. The key effect of this arrangement is that ions only enter the mass spectrometer if they are produced in the high intensity region of the laser beams where the Ionization probability is very close to 1 for all isotopes. If all ions were transmitted, the biases due to hyperfine are significant even when ionization probabilities are very close to unity at beam center.

one-photon resonance with the power density such that the ionization probability is close to unity at the center of the laser beam. With power densities of I = lo8 W/cm2 the Rabi frequency for a strong transition is CIi2 = lO’*/s. Thus, the power broadened width of the ionization signal is of the order of 10 cm-‘. Instead of tuning the laser directly to resonance, we detune by an amount which is very large compared with either the hyperfine splittings or the laser bandwidth. Ionization at a large detuning (i.e. 161 + YL + 1~121) re q uires that the second photon be absorbed in a time l/i&, where 6, is the detuning of the single laser from the centroid of the hyperfine resonances. Since the two photons must be absorbed in a time which is short compared with the precession time of the nuclear spin, we expect that the presence of hyperfine structure will not affect the ionization probability. However, at a power density of lo8 W/cm2, the power broadening is so large that the ionization probability should not be small even at a detuning of 5 cm- I. At such large detunings the atomic response is adiabatic [9]. One expects to be able to use adiabatic (or multiple time scale) perturbation theory in order to derive the atomic response. To illustrate this effect, we imagine a pulsed laser with a Gaussian line shape and bandwidth 0.4 cm-’ tuned near the one-photon resonance between the 5p2 3Po and the 5~6s 3P1 levels of Sn. The parameters relating the power density to the Rabi frequency and the ionization rates are the same here as in the earlier discussion of Sn. The pulse length parameter is chosen to be T = 8 X lop9 s for the single dye laser used in this study. We consider ionization probability versus detuning for atoms at the center of the beam, where the power density is I = 1.0 x lo8 W/cm2. The laser model described earlier cannot be used unless the phase diffusion model is modified to suppress the overlap of the Lorentzian wings of the model laser with the resonance when line center for the laser is widely detuned from resonance. Without modification

HFS effects on ionization efficiencies

1453

the ionization probability calculated with this model with 6r = 8 cm-’ is nearly equal to that with S1 = 0. This is a result of the fact that a substantial intensity of laser light still overlaps the resonance, and even with more than a factor of lo3 less intensity, the transition is still saturated. Instead of modifying the phase diffusion model we use a Monte Carlo simulation model which includes multimode effects (and thereby both amplitude and phase fluctuations). We describe the excitation and ionization of both isotopes by the time dependent Schrodinger equation. However, for each laser pulse the amplitude and phase of the laser is a very complicated function of time, which changes randomly from laser pulse to laser pulse. We now derive the equations of motion. We present the time dependent wave function as follows

+

J

2

dwcu,(wc,i,t)e-iwc’(i,w~>,

where, states IO>, II>, and 12~ are as defined earlier for the ionization scheme in Sn. The i in the continuum states represents angular momentum quantum numbers for the ion and spin and orbital quantum numbers for the photoelectron. The continuously varying photoelectron energy in the continuum is tiw,. We use = S,,,



= 0,

>i,w$,wc>

=

SijS(O,-O:.),

%T&> = Rw,li>,

and the time dependent Schrodinger absence of the laser field)

-

= iRoleiShz,

f$

= iflloe-‘~l~aao+ (i/2b) c

dt

+ iR,,,e%

& = ilR e-iS 2 ra,, 20 dt da, $w_i,t)

equation to show (%& is the Hamiltonian

+

in the

23

1 do,

ac(w,,i,t)e’(w-wc+wl)‘,

(i/28) 2 dw, <2(Eofi,(i,w,> I i

uc(wc,i,f)ei(w-“ctw2)‘,

= (i/2R)

i

e-‘(“-“c+“l%,

+ (i/25) ec-r(w--wc+“&72.

The Rabi frequencies appearing above contain both the amplitude and the fluctuating phase of the laser field. Thus, they are complex quantities with both the relative size of the real and imaginary parts and the absolute value fluctuating many times during a pulse. However, through our averaging over pulses we will obtain an average response throughout the pulse, as well as information about pulse to pulse fluctuations. ac(wc,i,t) can be expressed in terms of a time integral over a, and u2 by using the fourth of the above equations. Substituting this expression for the continuum amplitude in the other equations and assuming the discrete-continuum wavefunctions are very slowly varying functions of the energy of the photoelectron

dao= ifl dt

dA 1

dt

=

01

A

1

+ iQ

02

i& + iI’J2)A,

A

23

+ iQloao - yA2,

@lb)

1454

M. G.

PAYNE et al.

where A, = u,elV,

A2 = a,e*s2’ _ .

Above, the integral over o, has been replaced by the sum of a delta function in time and a principal value integral. The delta function in time gives rise to the usual ionization rates and the real part of P,~. The real part of k involves cross terms between CO, = 2w + wo, @),E,,Il> and <21~,&(w, = 2w + w,,, i>. We will presently see that it leads to interference terms between the two paths to ionization. Some of the principal value integrals lead to the imaginary part of ki2 and others lead to a.c. Stark shifts which have been neglected (since at the power densities to be used here they are always much smaller than IQ,/ and the laser bandwidth). This procedure for removing the continuum amplitudes from the problem is an old and well established procedure [lo]. We will now demonstrate the role of p.i2. The ionization probability, P,, satisfies s

= IJlA,12 + I’2]A2]2+ 2%[p12]%[A2A;].

In cases where the laser is tuned to resonance and the bandwidth is large (so that rate equations apply) the cross term in A,AF is small. Also, A, and A2 are coupled more strongly by way of their mutual interaction with a0 than by 8(p), so that in the region near one-photon resonance the terms in k are seldom of importance with broad bandwidth lasers. The expression for dP,ldt was arrived at by considering

dP, _ d _dt = %(l - lao(t - IAdOl’- kb(Ol”), and using Eqns (21) on the right hand side. If we assume we are on the far wing of the line where A, = &/61 and A2 = C12,J8i, we get

where g(f!.cc < w, = 20 + w~JLS,E+-. This is exactly the form arrived at from second order time dependent perturbation theory under the assumption that two resonances dominate the sum over intermediate states. The advantage of this method is that nothing needs to be modified about the mass spectrometer. The ionization from regions of lower power density obeys the adiabatic approximation just as well as the signal from the high power density regions. Also, only one laser is required for the measurements. The disadvantage is that there may be background ionization from elements that are much more abundant than the element selected by the laser ionization scheme. If multiphoton ionization background signals come from isotopes of another element having the same mass as an isotope of the chosen element, biases will result. The same type of disadvantages occur if one attempts to avoid these effects by using a picosecond laser (as suggested by LAMBROPOULOS). In fact, in order to achieve ionization probabilities near unity at beam center the power densities with a picosecond laser would have to be much larger due to the photons being delivered in a time interval nearly 1000 times smaller, causing even more danger of MPI for other elements. To show some degree of model independence of the adiabatic behavior on the far wing of the line we have carried out a computer simulation of the ionization probability

HFS effects on ionization efficiencies

1455

based on another model for the laser. In this model the positive frequency part of the laser pulse is taken as

(22) where for each laser pulse the frequencies wi are generated anew in such a way that they are randomly distributed in the region ij - 5I 5 6 + 5T. A set of phases @, are also generated for each laser pulse. Our model is one in which the modes do not repeat from shot to shot and the phase of different modes are randomly distributed. The constant K is chosen so that cell is the mean power density of the pulse at the position of the atom and at time t. This model is a chaotic field model with a Gaussian lineshape. The lineshape for a laser obeying this model is I(w) = q9-2(w-WW, providing (w-01 < 5I’. The field autocorrelation

(22)

function is

G(“)(t) = <(E+(T+t))“(E-(T))“>/()”,

(23)

= n!exp(-n(I?)2/8). The <> indicates an average over a very large number of pulses having different sets of W, and @i* In the simulation the time dependent Schrddinger equation was applied to the three states (or two states for even isotopes) system appropriate for Sn, with the laser field for a particular pulse being given by Eqn (22). E,(t) was chosen to agree with the pulse shape used in the phase diffusion model calculations described above. The equations for the probability amplitudes of the states at large times were then solved for a large number of pulses and a large number of detunings for both even and odd isotopes. The ionization probability was then calculated from

P = 1 - < 2 ]c(@)]2>,

(24)

where the ai are the probability amplitudes for being in states Ii> at time t. The elements of the density matrix can be calculated from aij(t) = ,

where the <> represents an average over a large number of laser pulses. Unlike the phase diffusion model, this laser model exhibits amplitude fluctuations which enhance the ionization probability of the far wing of the line by a factor of 2!. The results of a simulation with a mean peak power density of 10H W/cm2, a pulse length parameter 7 = 8.0 x lo-” s, a number of modes N = 30, and a bandwidth 0.4 cm-’ is shown in Fig. 4. The results for the even isotopes were averaged over 50 pulses, while those for the odd isotopes were averaged over 15 pulses. The Rabi frequencies and ionization cross sections used were those of LAMBROPOULOS and LYRAS [3]. As in the latter studies, p12 has been neglected since its real part is small. We have demonstrated this effect for Sn, but with many elements it will work even better. This is because many elements have smaller hyperfine splittings. Also, many transitions are stronger than the transitions used in Sn so that the detuning can be made even larger without large losses of signal. The root mean square deviation of the ionization probability near line center was small, but became as large as 30% on the far wing of the line. With the small number of cases averaged there are still some fluctuations in the points generated for the odd isotopes. However, it is clear that the difference in the two results becomes relatively small on the far wing of the line.

M. G.

1456

PAYNEet

6,

al.

(cm-‘)

Fig. 4. Ionization probability versus detuning from the position of the line in the absence of hyperfine structure for the near resonant two-photon ionization of Sn. In this case a single laser generates a beam with the characteristics of a chaotic field and a Gaussian lineshape. The laser has a power density of I = 1.0 x lox W/cm’, a bandwidth of 0.4 cm-‘, and a pulse length of r = 8 x lo-%. The laser is tuned near the 5p*‘P,, o 5~6s “Pi resonance. The smooth line is for even isotopes, while the X’S are a few points for the odd isotopes. Note that the power broadening of the transition is so large that the ionization signal is substantial at a detuning of 8 cm-‘.

4. CONCLUSION

We have shown that in cases where the discrete-discrete transitions are all pumped with power densities such that Rabi frequencies are much less than either the hyperfine splitting or the laser bandwidth, rate equations provide a very accurate description the effects of hyperfine structure on stepwise ionization probabilities with broad bandwidth lasers. Thus, considerable effort can be saved by using a simplified treatment. General rules were suggested to avoid effects that are due entirely to the modification of angular momentum selection rules due to the hyperfine interaction. We have suggested two means of avoiding isotopic biases when stepise ionization with broad bandwidth lasers are in resonance ionization mass spectroscopy (RIMS) to measure isotope ratios. The effects of nuclear spin cause some subtle problems, but these effects can be dealt with in carefully designed experiments. When the proper precautions are taken RIMS remains a powerful tool for both the measurement of isotope ratios and for sensitive elemental analysis. Finally, we believe there is a need for further calculations of the type carried out by LAMBROPOULOS and LYRAS[2, 31. With more experience it may become possible to classify situations where isotopic bias effects are of little importance. This, however, requires more than two examples for the simplest possible case of nuclear spin l/2 and initial J = 0. Acknowledgements-Research sponsored by the Office of Health and Environmental Research, U.S. Department of Energy under Contract No. DE-AC05-840R214OO with Martin Marietta Energy Systems, Inc.

REFERENCES [l] W. M. Fairbank, Jr., M. T. Sparr, J. E. Parks and J. M. R. Hutchinson, Phys. Rev. A40, Phys. Rev. A40, 2199 (1990).

[2] P. Lambropoulos and A. Lyras,

2195(1989).

HFS effects on ionization efficiencies

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(31 A. Lyras, B. Zorman and P. Lambropoulos, Phys. Rev. A42, 543 (1990). [4] E. H. A. Granneman, M. Klewer and M. J. Van der Wiel, J. Phys. B9, 2819 (1976); E. H. A. Granneman, M. Klewer, K. Nygaard and M. J. Van der Wiel, J. Whys. B9, L87 (1976); M. J. Van der Wiel and E. H. A. Granneman, “Polarization Effects in Two-Photon Ionization” In Mulriphoron Processes, (Eds J. H. Eberly and P. Lambropoulos), pp. 199-213. John Wiley, New York (1978). [5] A. R. Edmonds, Angular Momentum in Quantum Mechanics. Princeton University Press, Princeton, NJ (1969). [6] P. Knight, Opt. Commum. 31, 148 (1979). [7] P. Zoller and P. Lambropoulos, .I. Phys. B12, L547 (1979). [8] S. N. Dixit, P. Lambropoulos and P. Zoller, Phys. Rev. AU, 318 (1981). [9] G. S. Hurst and M. G. Payne, Principles and Applications of Resonance Ionization Spectroscopy, pp 91-108. Adam Hilger, Bristol and Philadelphia (1988).