A TRAJECTORY STUDYOF 32PRECOILIN SODIUMPHOSPHATES
2193
A Trajectory Study of Phosphorus-32 Recoil in Sodium Phosphates by Don L. Bunker and Gregg Van Volkenburgh Department of Chemistry, University oj California, Irvine, California (Received January $8,1070)
We made a classical trajectory study of the recoil of 32P,from (n,y)a2P,in several kinds of sodium phosphate crystal. The chemical effects of this have been extensively studied, but the results are inconclusive, Our calculations strongly suggest that in anhydrous crystals, the primary process is the ejection of either P or P 0 (or P OH, in sodium hydrogen phosphates) from the crystal cage. These fragments traverse only a few neighboring sites before becoming bound. When there is water of hydration, escape of P is severely impeded. It either remains in its original cage or comes to rest in an adjoining one. Preservation of a P-0 bond during the recoil process is extremely unlikely.
+
+
Introduction
with corresponding expressions for y and x , and using
When the nuclear reaction 31P( ~ , T ) is~ carried ~P out by irradiation of crystalline orthophosphates with neutrons, many different phosphorus oxyanions are formed. Analysis of the dissolved crystal after irradiation has shown the presence of orthophosphite, hypophosphate, hypophosphite, isohypophosphate, diphosphite, pyrophosphate, tripolyphosphate, and more complicated polymeric forms. These must arise from the recoil of 32Pon emission of one or more y rays of the several high energies that are possible. Contradictory suggestions have been made about the immediate result of this recoil. It has been proposed that almost none’ and that almost all2of the 32Pescapes its original lattice cage. Intermediate views include half 0, half P ejection3 and that P , 0, and PO2 are the principal ejected fragment^.^ Changes in the results when hydrated crystals are irradiated have been used2 to support the view that recoil out of the cage is important. The present state of understanding of this area of recoil chemistry is not very satisfactory. Our study had two purposes. One was to obtain insight into the nature of the 32Precoil process. The other was to introduce trajectory methods into condensed-phase problems. For the gas phase, trajectory studies have been used as an aid to understanding reaction dynamics for some time. Systems of up to 6 atoms6 have been investigated. A solid-state calculation of radiation damage exists6 in which there were 500 atoms (of Cu) but no molecular interactions, so that simpler computation procedures could be used than are required in a chemical study. We needed a chemically interesting problem of complexity intermediate between these two kinds of calculation. The a2Precoils in sodium phosphate crystals provided an ideal choice.
The Method and the Model We obtained the trajectories of the atoms by solving Hamilton’s equations in Cartesian coordinates dp,/dt
=
-dH/d~,;
dx,/dt = dH/dpzi (1)
p,, = m,(dxi/dt), etc.
(2)
N
T
=
‘/2
c (P,,2 + + Pz,2)/m,; P!A2
a=1
T+ U
=
H
(3)
in which U is the potential energy. The number of atoms N was either 5 or 9. The methods of numerical solution were standard ones.’ The starting conditions were chosen to correspond to a randomly oriented P velocity vector, with a fixed magnitude calculated from a prescribed y-ray energy. All other atoms were initially a t rest. Trajectories were ordinarily terminated if it was determined, from its position and kinetic energy relative to the other atoms, that P had escaped its cage. The complete program for doing all this was written in optimized assembly language for a Digital Equipment Corp. PDP-10. Single precision (27 bit floating fraction, with rounding) was adequate, and several recoils per minute could be simulated. We used 3 general classes of model in order to compare recoil behavior in increasingly complex surroundings. These will be called the free, bound, and hydrated models of the orthophosphate ion. For each of these we also provided the option of replacing one (1) T. R. Sato, P. A. Sellers, and H. H. Strain, J. Inorg. Nucl. Chem., 11, 84 (1969); see also Chem. Ef. Nucl. Transform., Proc.Symp., 1, 503 (1961).
(2) L. Lindner and G. Harbottle, J . Inorg. Nuct. Chem., 15, 386 see also Chem. E,@‘. Nucl. Transform., Proc. Symp., 1, 485
(1960); (1961).
(3) W. F. Libby, J . Amer. Chem. SOC.,62, 1930 (1940). (4) V. C. Anselmo, Thesis, University of Kansas, 1961 (University
Microfilms, Ann Arbor, Mich.). (5) D. L. Bunker and M. D. Pattengill, Chem. Phgs. Lett., 4, 315 (1969). (6) J. B. Gibson, A . N. Goland, Phys. Rev., 120, 1229 (1960).
IM.Milgram, and G. H. Vineyard,
(7) The Runge-Kutta-Gill method was used to start the integrations: 8 . Gill, Proc. Cambridge Phil. SOC., 47, 96 (1951). The faster but not self-starting Adams-Moulton integration was then applied: e.g., J. B. Scarborough, “Numerical Mathematical Analysis,” Johns Hopkins Press, Baltimore, Md., 1962, pp 318-325. Volume 74, Number 10 May 1.6, 1070
2194
DONL. BUNKER AND GREGG VAN VOLKENBURGH
or two of the 0 by OH, so that the series Na3P04, Na2HP04,NaH2P04could be studied. Free phosphate was, in effect, a gaseous ion. It had 4 0 arrayed in a regular tetrahedron about P, each one bound to it by a Morse potential. The 0 repelled one another by virtue of parabolically rising repulsion functions which began at prescribed values of the 0-0 distances. The possibility of O2 formation was omitted, and it was later verified that objectionably small 0-0 distances did not occur in the trajectories. The potential energy for this model was 4
+ + + + + +
U=CD[l-
ro))I2
i=l L’lZ
u 1 3
u 2 3
u 1 4
u 2 4
11’34
(4)
where ut, =
k(st,
- sop
(54
if si! < so or
ut, = 0
(5b)
in all other cases. The P-0 distances are Ti, the 0-0 distances are st,. These are easily expressed in cartesian coordinates for the purposes of eq 3. The estimateds value of D was 100 kcal. Spectroscopy9 gives 1.54 d for ro and 2.13 A-l for p. The repulsion parameters, whose values are not very critical, were arbitrarily k = 332 kca1/8 and so = 1.2 A. With this k there is 118 kcal of repulsion at sil = 0.6 8. Since we are uncertain of how the charges in the fragmented ion are distributed, we have adopted these simplified interactions. ‘The results do not seem unduly sensitive to this. Bound phosphate represents the anhydrous sodium phosphate lattice, with the unit cell simplified to a cube. I n this model alternating corners of the cube were used as fixed anchor points, to which the 0 were bound. This involved adding to the U of eq 4 the additional Morse terms 4
D‘[1 - exp(-p’(r,’ i- 1
-T~’))]~
(6)
in which rt’ are 0-anchor distances. From the known density of the Na3P04crystal, ro’ = 2.57 d. To make our simplified lattice electrically neutral, we would need to associate 6 elementary positive charges with each anchor point and 3/4 electron charge with each 0. The electrostatic interaction energy for each 0-anchor bond, calculated on this basis, is 580 kcal. This is not an unreasonable value for such a quantity, and was used for D’. The unknown p’ was arbitrarily set equal to p. For this model we tested the sensitivity of the results to the assumed values of the parameters. The most critical assumptions are probably thoPe about D and D’. We tried a variant model (D = 125, D‘ = T h e Journal of Physical Chemistry
435). As will be shown, no qualitative differences arose when we did this. Hydrated phosphate was limited by computer economics to the fictitious form Na3P04.4H20. The 4 H 2 0 were each single particles of mass 18 amu, symmetrically introduced above the tetrahedral faces of the PO4ion at a distance of 2.83 from P. The additional repulsions, to be added to eq 4 and 6, are
These are of the same form as in eq 5, including the cutoff feature. The P-H20 distances are p i ; p o = 2.5 A2; k, is such that there is 8 X lo3kcal of repulsion energy at pi = 1.25 A. The sum over i , j gives the O-HzO repulsions. There are 12 of these. Each H 2 0 interacts with the 30 at the corners of the nearby tetrahedral face. The fourth 0 interaction for each H 2 0 and the inter-H20 repulsions may be omitted. We let k and so have the same values as in eq 5. None of these parameter values is very critical, since in any event we will surely obtain a valid lower limit to the complexity of NaaPOl.12H20. For the hydrogen phosphates, there are the following changes, arising from different charge distributions and unit cell volumes. In Na2HP04, D’ = 296, YO‘ = 2.99. In NaH2P04,D’ = 126, rot = 2.64. In these cases we take OH as a single particle of mass 17 amu, and we delete the potential term between it and the corresponding anchor point. The unit cell remains cubical. Results and Discussion The most immediately interesting data are the distributions of fragment types that escape the recoil site. These are presented in Tables I and I1 for all the cases that were studied. L4ctual numbers of computed events are shown; the uncertainty will be roughly the square root of each figure. The four y-ray energies’o that were used sample the 32Precoil spectrum in a representative way. This spectrum has four intense transitions, of which we used 6.79 MeV (the highest of the four), 3.90, and 2.10 (the lowest). These occur with relative probability 10: 14: 13 and together make up 44% of all transitions. The very weak 7.85MeV emission is 0.09 MeV below the maximum observed; the only transition lower than 2.10 MeV is a weak one at 1.60. A rough estimate of the probable combined results for the full range of transition ener-
*
(8) By Pauling’s rule (electronegativity formula), 85 kcal; from Pauling’s bond length-bond order and Johnston’s bond energybond order (“BEBO”) formulas, 99 kcal, using the known bond length; by comparison with known P-0 bond strengths, perhaps 120 kcal. (9) D. W.J. Cruickshank and E. A. Robinson, Spectrochim, Acta, 22, 557 (1966). (10) The corresponding P velocities are 7.95, 6.87, 3.95, and 2.12 X 106 cm sec-’. The P kinetic energies are 2.44 X lo4, 1.82 X lo‘, 6.00 X 103, and 1.74 X 103 kcal.
2195
A TRAJECTORY STUDYOF 32PRECOILIN SODIUMPHOSPHATES
Table I : Fragment Distribution in **PRecoils, Anhydrous Cases -No. Y
Model
Free Po43Bound
Po43-
Variant of bound Pod3Bound HP042Bound H2P04-
a Also 3(P OH).
only
7.85 3.90 6.79 3.90 2.10 6.79 3.90 2.10 6.79 3.90 2.10 6.79 3.90 2.10
62 16 70 62 81 68 63 60 61 54 48 46 41 33
3.90 Mev
Bound PO":
6.79 Mev
P
of events with ejection ofP +
P
energy, MeV
Free Poi"
P
+0
P + OH
o+
38 84 30 38 19 32 37 40 30 34 35 25 33 27
... . .. ... ... .,. ... ...
...
..,
OH .,.
.*.
...
P
...
... ...
P+ 0
-.
9 12 15 29 26 28
*L
0 0
P i 0
2 0 0 4a
2.10 Mev
+ 20H), 2(P + 20), 1 each OH,20H, and (0+
gies can be obtained by adding the numbers in the tables. A striking result is that P-0 bonds are almost never carried away from the recoil site." The sole mild exception to this rule is the messy situation that developed a t the lowest energy in NaH2P04.4H20. I n free phosphate, if P recoils in a direction where there is no 0, it escapes; if there is an 0, it too is carried away. The chemical bonds might as well not be present. When P is moving fastest, it is most likely to escape alone, as would be expected. Adding the anchor points, in the bound model, makes 0 more difficult to dislodge but does not change the character of the recoil process. The hydrogen phosphates produce (non-P) fragments in amounts intermediate between what was observed for free and bound phosphate, which is reasonable. The ratios between (P 0) and (P OH) ejection are in proportion to the numbers of 0 and OH, which again indicates that the initial recoil direction largely determines the outcome. A very feeble exception to this, for anhydrous crystals, occurs at the lowest energy for H2P04-, where there are a few complex trajectories leading in four cases to retention of P. This effect is magnified, however, when the 4Hz0 are present. The increasing clutter of the unit cell leads to more complicated trajectories, even a t high energies. (Note the substantial number of cases in which more than two fragments were ejected.) At low energies the retention of P has become very appreciable, and might even be the dominant feature if we were able to put in all 12 waters of hydration. These observations suggest that most of the penetrating power of the P is exhausted in the process of its escape from its original cage. To discuss this more fully, we need the energy spectra of the ejected frag-
+
n 2 P
k P+O *
P+O
b
I
of IO0 initial energy o/o
Figure 1. Smoothed energy spectra, on a scale of 0 to 100% of original recoil energy, of ejected fragments in anhydrous crystals. The energies are for P atoms, unless marked (*) for 0 atoms. The type of ejection process is listed under each spectrum. Vertical units are arbitrary. Hydrated PO;*-
6.79 MeV
A P
H2O
A P t 0 H20
+
2.10 -M re
p*yo
G P*O* H& Hydrated H2P0; 2.10 -V M e
0%
o f 0 initial energy
P * H20
b
initial Y@ ofenergy lob
Figure 2. Same m in Figure 1, but for hydrated crystals. The spectrum marked ( * * ) is for HzO.
ments. Our data on this are too extensive for full presentation, so we have displayed selected examples in (11) The parameters of the variant model of anhydrous NaaPOa were chosen to provide the maximum opportunity for preservation of the P-0 bond.
Volume 74, Number 10 Mag 14, 1970
2196
DONL. BUNKERAND GREGGVANVOLKENBURCH
Table I1 : Fragment Distribution in S2PRecoils, Hydrated Cases
Model
pOra-(HzO )4 H~POI-(H~O)~
,--------
P only
HzO only
6.79 3.90 2.10 6.79 3.90 2.10
3 1 0
0 0 22 0 2 16
+
1
0 0
P+O+ P + O
P+HzO
HzO
3
67 60 58 53 38 17
27 38 11 15 31
1 1
4 0 2
+
10
Also PO 2 (non-P) fragments, 3; POH 1or 2 (non-P) fragments, 11; P(0H)z ments, 2; assorted (non-P) fragments not listed above, 17. 5
Figures 1 and 2. (We have omitted intermediate energies, cases in which OH behaves like 0, and minor ejection modes.) In the hydrated crystals, Figure 2, the degradation of the initial P energy is very apparent; the P will probably be unable to escape the next cage it enters. Even in the anhydrous cases the energy loss is not negligible. Most P leave their original sites, but very many must become entangled with the 0 in the neighboring cells and stopped. We have, then, the following description of the P recoil process. At first P behaves as if it were a free particle, without chemical bonds. It moves until it hits something, and in this first encounter the chemical nature of the struck object is relatively unimportant. The chance of P escaping its original lattice cell depends mostly on the number of things there are for it to hit. I n an anhydrous crystal it is likely to get away, in our fictitious tetrahydrate escape is considerably harder, and in real dodecahydrated Na3P04 it must be very difficult indeed. A large fraction of the P energy is lost at every encounter, especially a t low recoil energies. The final bonding of P to the crystal occurs within a very few cell lengths (in hydrated crystals, one at most) of its original site. These chemical bonds are very seldom the same ones the P had before its recoil. Xevertheless, the individual ions recovered from the final solution must be largely made of atoms that mere near neighbors in the crystal before it was irradiated. We might therefore speculate, on the basis of our results for anhydrous crystals, that oxygen-poor mono-
The Journal of Physical Chemistry
-___-
No. of events with ejeotion of-
Y
energy, MeV
0
+ Ha0 0 0 8 0 0 3
P
+ OH .*.
----.
+ OH + HzO
P
...
... ...
...
4
23 28 125
1
1
.*.
+ 1 or 2 (non-P) fragments, 8; P + 3 frag-
meric ions (such as orthophosphite and hypophosphite) arise from P 0 ejection followed by reflection of P back to its original site by the surrounding lattice. We might also suppose that dimeric species represent migration of P to the next lattice cage and that polymeric species such as tripolyphosphate require traversal of more than one neighboring P site. Hydration of the crystal would then tend to suppress polymeric forms and encourage monomeric ones, as the experimental results apparently show.2
+
Conclusions Our findings most nearly support the original suggestion of Libbya that P and 0 are the most important fragments ejected from the lattice unit where the recoil occurred. In hydrated crystals at least one H20 is also displaced. Retention of P is only important at low energies in hydrated crystals, and P-0 bonds rarely survive the P recoil. Nevertheless, the region of disruption of the crystal is relatively small; only a few unit cells are involved. We also conclude that trajectory studies of condensed-phase chemical dynamics are both feasible and useful.
Acknowledgments. We are highly indebted to F. S. Rowland for arousing our interest in this problem. The project was supported by the National Science Foundation, whom we thank. We acknowledge the valuable assistance of Barbara Jacobson in the data processing phase of the work.
