Theoretical aspects of the stabilization of atomic hydrogen
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van den Eijnde, J. P. H. W. (1984). Theoretical aspects of the stabilization of atomic hydrogen. Technische
Hogeschool Eindhoven. https://doi.org/10.6100/IR153166
DOI:
10.6100/IR153166
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THEOREnCAL ASPECTS OF THE STABILIZAnON
OF ATOMIC HYDROGEN
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GF2AG VAN DE RECTOR MAGNIFICUS, PROF. DR. S. T. M. ACKERMANS. VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRijDAG 13 APRIL 1984 TE 16.00 UUR
DOOR
JOSEPHUS PAULUS HENDRIK WILHELMUS VAN DEN EI]NDE
0 IT PROEFSCH!UWf IS GOEDGEKEURD OOOR DE PROMOTOREN
Prof.dr. B.J. Verhaar
en
Prof.dr. I.F. Silvera
CONTENTS
CHAPTER O. Summary l
CHAPTER 1. Atomie hydrogen: what, how and why? 3
1.0 Introduetion 3
1.1 Stabilization of atomie hydrogen 4
1.1.1 Expertmental conditions 4
1.1.2 Proton spins: a complication 6
1.1.3 Doubly polarized hydrogen: a new possibility 9
1.2 Exceptional properties of atomie hydrogen 10
CHAPTER 2. Rate constauts derived from collision quantities 15
2.0 Introduetion 15
2.1 Rate equations and constauts 16
2.2 Thermal average with the Boltzmann distribution 18
2.3 Linear-response theory 20
Derivation of an expression for the relaxation rate
2.3.2 Evaluation of the relaxation rate
2.4 The quanturn mechanical Boltzmann equation
CHAPTER 3. Volume processes
3.0 Introduetion
3.1 Principles of scattering theory
20 24 26 31 31 32
3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4 3.4.1 3.4.2 3.4.3 3.4.4 3.5 CHAPTER 4. 4.0 4.1 4.2 4.3 4.3.1 4.3.2 4.3.3 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5
The Hamiltonian for H-H scattering 36
The coupled-channels approach 39
A basis set of eigenfunctions 39
Derivation of the coupled-channels equations 41
Diagonal and coupling terms 43
Selection rules 47
Integration of the CC equatlons 48
Various simpllfications and their validity 52
Introduetion of five assumptions 52
The plane-wave and distorted-wave Born
approximation 53
Evaluation of the T-matrix and the effective
cross section 56
Rigarous check of the five assumptions 59
Results 62
Surface processes 67
Introduetion 67
Problems typical for the surface 68
Magnetic spin-spin interactlans treated in DWBA 71
Study of various surface models 73
Comparison of 2~D model with experiment 73
Comparison of 2~D model with simpler models 75 Results without the high-temperature limit 78
A 3D description of H-H surface collislons 79
General formalism 79
CC scheme with eigenstatea of the surface hamiltoniau
New analytic basis of surface states Results for the 3D model
Interpretation of the results
82 86 89 93
4.4.6 4.5 4.5.1 4.5.2 CHI\PTER 5. 5.0 .5 .1 5.2 CHI\PTER 6. REFERENCES Samenvatting Dankwoord Levensloop
Modifications of the analytic basis set
Other model improvements
The mattress effect and surface dimers Non-adiabatic effects and direct bb recombination
Low-energy scattering
Introduetion
Volume relaxation rate for low temperatures
Two-dimensiona1 effective-range theory
Concluding remarks 98 99 99 102 107 107 108 112 119 121 125 127 128
SU~RY
In 1980 Silvera and Walraven at the University of Amsterdam succeeded for the first time in stahilizing hydragen in lts atomie form. Further important progress has been achieved in several other groups, notably at MIT, Cornell University and at Vancouver. Stabilization is achieved by bringing atomie hydragen in a low-temperature cell with walls covered by superfluid helium and applying a strong magnetic field. At sufficiently low temperatures and high densities it is expected that atomie hydrogen gas will show fascinating properties, in particular the so-called Bose-Einstein condensation and superfluidity.
This thesis describes a theoretica! study of processes leading to recombination of hydragen atoms into molecular form. A relaxation process, due to the transition among the lowest two hyperfine levels of atomie hydrogen, turns out to be of fundamental importance for the recombination rate. Models have been formulated to calculate the relaxation rate by means of quantummechanical scattering theory. ror processes in the bulk of the gas the results of an almost exact coupled-channels calculation have been compared with approximate models. In these roodels first-order approximations are applied, as well as approximations connected with the large distance of closest approach of the colliding hydrogen atoms. The assumptions turned out to be correct to the promille level, except for the so-called high-temperature limit.
Si1nilar approximations have subsequently been applied to processes at the surface. In that case they led to a discrepancy of a factor 50 with experiment. Various model improvements have been studled to explain this discrepancy. In particular, the effects of a three-dimensional description of the atomie motion at the surface have been extensively investigated. Furthermore, the possible existence of surface dimers and the interaction of hydragen atoms with the helium film, have been considered and discussed. The discrepancy has been reduced to one of a factor 35, for which no explanation has as yet been found.
Finally, for low-energy scattering in two diroenslons a new so-called effective-range theory has been formulated. In three diroenslons such a theory is well-known. In two diroenslons complications connected with logarithmic effects up to now preelucled the formulation of a similar theory. Furthermore, the relation has been studled between microscopie two-particle callision quantities and macroscopie
-relaxation constants. Three alternative methods have been considered: a) kinetic gas theory, b) linear response theory, and c) the
CHAPTER 1. Atomie hydrogen: what, how and why?
1.0 Introduetion
Atomie hydragen played a cruelal ro1e in the early formulation of modern physics through the simp1icity of lts atomie states. Quanturn mechanics has since then been applied to many more complicated systerns. A great deal of investigation was devoted to systems exhibiting quanturn phenomena on a macroscopie scale, such as the superconductivity of metals and the superfluidity of 4He. Theoretical
speculations about similar properties of atomie hydragen seemed purely hypothetical, because hydragen does not eKist in its atomie form on earth, it always comes in molecules H
2, or bound to other elements. Once separated, for instanee by an electric discharge, hydragen atoms rapidly recombine again.
In 1959 Hecht (HEC59) mentioned for the first time the exciting possibility of atomie hydragen becoming a superfluid in suitable circumstances, and at the same time indicated a possible way to keep it stab1e against recombination, by polartzing the electron spins in a strong magnette field. In the seventies some more theoretica! papers followed (DUG73, ETT75, STW76), while expertmental efforts to stabilize atomie hydragen remained unsuccessful (WAL78).
However, a new important impulse to the interest in the subject was given by Silvera and Walraven in 1980 (SIL80), who reported for
the first time on the successful stabilization of hydragen in its atomie form under laboratory circumstances. Since then other groups followed (CLI80, YUR83), and atomie hydragen was once again in the center of the attention. Uow the stabilization was achieved will be described briefly insection 1.1. Much effort was also concentrated on the theoretical analysis of the properties of the gas once the so-called Bose-Einstein condensation would have been achieved (Aussois conference, AUS80). The effects to be expected for hydragen were rather unique. We will return to this point insection 1.2.
The relevanee of the stabilization of atomie hydragen now being beyond doubt, theoreticians and experimentalists join forces to eliminate the numerous problems on the way to higher densities necessary to reach the Bose-Einstein-condensation regime. The theoretical side of this venture is what this thesis is about.
First we introduce in chapter 2 the rate equations governlng the development of the hydragen density as a function of time, followed by
-an evaluation of the rate const-ants in terms of microscopical two-particle callision quantities. This is carried out in three different formalisros to resolve the existing disagreement about factors of two in theoretica! expressions. Chapter 2 is partly based on sections III and IV of a paper by Ahn et al (AHN83).
Then in chapter 3 the microscopie cross sections for the so-called relaxatton process are derived for collistons in the bulk of the gas. The approximatlons applied in other theoretica! papers are
scrutinized, and the flnal rate constants are compared with
experiment. A part of this chapter was published as sections II, III and V of a paper by Ahn et al (~HN83). The sameprogram is catried out for surface collislons in chapter 4. A large discrepancy with
expertment is established, and various roodels are examlned, in order to resolve it. This chapter is essentially a compilation of two previously publtshed papers (AHN82, EIJ83).
Ftnally in chapter 5 low-energy scattering theory is applied to atomie hydrogen. In additlon a formalism is proposed for the first time to describe surface collislons in the low-energy limit, including ~eftnitlons for a two-dimenslonal scattering length and effective range. The latter formalism was published in a paper by Verhaar et al (VER84). The thesis is terminared with a set of concluding remarks in chapter 6.
1.1 Stabilization of atomie hydrogen
1.1.1 Experimental conditions
In 1979 Silvera and Walraven (SIL80) succeeded for the first time in stahilizing atomie hydrogen against recombinatlon for periods over 500 seconds at densities of more than 1.8xlQLU m-3 • These samples were called stable on the basis of the criterion that their lifetimes were at least lOb times longer than under usual conditions.
If we let atomie hydrogen, produced by a discharge, enter the stabilization cell, the hydrogen atoms will start to recombine rapidly. Already in 1959 an obvious countermeasure was suggested by Hecht (HEC59), namely the polarization of the electron spins by the application of a strong magnetic field B and a low temperature T. In fact, in molecular hydrogen the electron spins are anti-parallel ( t and
+,
total electron spin quanturn number SsQ). Indeed, the SsO(singlet) potential curve of two hydrogen atoms as a function of the internuclear distance r shows a deep minimum of -4.7478 eV (~55097 K, in temperature units, using the Boltzmann constant kB) at r=0.74116 A (=1.4006 a
0, where a0=0.52917706 A is the Bohr radius), containing a ground state at -4.4776 eV (~51961 K), and numerous vibrational and rotational excited levels (KOL65, SIL80a, HER50). On the other hand, the S=l (triplet) interaction has only a shallow minimum of 6.46 K at r=4.15 A (=7.84 a
0, SIL80a), and contains no bound state at all. See Fig. 1.1 forthese potentials. Therefore, H atoms with electron spins
+
antiparallel to B, denoted by H+, cannot recombine. With values B= 10 T and T=0.2 K an extremely large ratio of the electron Zeeman energy relative to thermal energies 2~eB/kBT~ 13.5/0.2~ 67 can be achieved (~e is the electron spin magnetic moment). This would suggest that recombination is effectively suppressed by an e-6ï~8xlo-30 Boltzmann factor. I
expanded scale
4 > Q)S=1
::.:: w 2 w >, O"l S-Q)1.41
<= 0 Q) ~ I 2 ctS ·~ .p <= Q) -2....,
0S=O
Cl. -20-4}
-4.3 -- -30 -4.75--Fig. 1.1 a) Singlet (S=O) and triplet (S=l) potential curves as functions of the relative distance r between the H-atoms. b) triplet states on enlarged scale at B= 10 T.
There are, however, various mechanisms which may give rise to spin flips in recombination collisions, even at T=O (BER77). In this
respect the first experimental demonstration by Silvera and Walraven (SIL80), that cell walls covered with superfluid helium-4 provide a strongly enhanced lifetime, is to be considered as a major
breakthrough. In the first place this countermeasure (see also STW77) prevents the hydrogen atoms to interact with magnetic impurities on the wall, which may induce spin flips. On the other hand, helium-4 having one of the lowest possible adsorption potentials for hydrogen,
-surface coverage is minimized, thus reducing the number of three-particle collislons necessary for recombination (JON58, WAL82). The apparently successful experimental configuration created in this way, is shown schematically in Fig. 1.2.
\ I
\ cell
1He coating
II
~
H"-' supply
super-conducting
coils
Fig. 1.2 Scheme of experimental configuration for the stabilization of atomie hydrogen.
1.1.2 Proton spins: a complication
Unfortunately, the proton spins introduce a serious complication to the aforementioned stabilization scheme. As the proton Zeeman energy is about nine times smaller than thermal energies, the proton spins are not aligned, but iàstead randomly distributed over up (~) and down (~) states. In a magnette field the triplet state curves of Fig. l.lb actua1ly consist of three closely-lying curves each. In
ltself a randomly oriented proton spin presents no problem, but in the proton spin up atoms H+± the so-called Fermi-contact term in the hyperfine interaction (GRI82, see also section 3.2, Eq. (3.20)) causes a partlal depolarization of the electron spin.
The effective spin hamiltonian for a single hydragen atom in the
+ A 1s ground state in a magnetic field B=Bez reads
1 + +
-4 ao1 .oA + B(v oA p . , z
-v
e o1 ). , z (1. l)+ +
In this expression o
proton A, respectively, while
v
andv
are absolute values of theire P
respective spin magnetic moments. The hyperfine constant a
3 -6
(=8v
0
vevp/3~a0
• 5.8843x10 eV = 68.286 mK) arises from theevaluation of the spatial part of the Fermi-contact term (see (3.20)) with the zero-order ls electronic wave function. The four
eigenfunctions and eigenvalues of hamiltonian Heff turn out to be, in order of increasing energy:
2 -~
E la>= (l+e; ) !H-e:H>,
a lb> lH>, Eb Ie>
(l+e:
2)-~
lt'l+d.t>, ld>!
t.t>' Ed where € = { 2 2 2 ~ B(v e+v
)+ B (v +v) +a /4} p e p \a a {B2( + )2+~2~~
4ve
vp 4 ' a -B(!J-v), 4 e p a 2 2 a2 \ - 4 + {B (V +p ) + 4 } , e P a4
+ B(ve-vp)' a 0.025346 4B(v+v )
e P c"" H-sH K b"" HmK
a"" H-sH B (1.2) (1.3)Fig. 1.3 Rabi diagram (not to scale): four ls hyperfine energy levels of atomie hydragen as functions of the magnetic field B.
-The approximation in the second step is valid for B>> 0.05 T, the hyperfine field. The five digits shown in the final expression are significant already for B>ST.
The energies of the four hyperfine states are shown in Fig. 1.3, the so-called Rabi diagram, as functions of the magnetic field B. Note that Fig. 1.3 is not drawn to scale, in order to present both the elect.ron Zeeman splitting and the proton hyperfine splitting in one picture in a recognizable way. Note furthermore, that Ea and do not approach asymptotica1ly to Eb and Ed, respectively. The asymptotes of Ea and are not even parallel to the straight-line functions Eb and Ed, respectively, where and Eb diverge asymptotically, and Ec and Ed interseet at B= a(u -IJ )/(4JJ 11 )= 16.683 T. Fig. 1.4 serves as an
e p e p illustration of this behaviour.
0
-10
-20
-30
-40
-50
-60
-70
Fig. 1.4 Energy differences Ea-Eb and magnetic field B.
B (T)
-Ed as functions of the
Obviously, only the a and b levels are occupied at the
temperatures used, because the energy difference between a-b (mainly H+) on the one hand and c-d (mainly Ht) on the other hand corresponds to the original large electron Zeeman energy of about 13 K. The \a> and \b> states are more or less equally occupied in equilibrium, and they have to be used instead of the !+i> and I+~> states in the former incomplete model. One of these states, however, contains an admixture E~1/400 of the wrong electron spin, leading to a singlet state part in two-particle collislons a+a and a+b, and, unfortunately, rapid recombination.
1.1.3 Doubly polarized hydrogen: a new possibility
As was already pointed out by Statt and Berlinsky (STA80), a gas of pure lb> atoms would recombine much more slowly than a gas
containing also la> atoms, as no electron-spin-up component is present before a callision takes place. On the other hand, the la> atoms, being so much more reactive than the lb> atoms, would rapidly recombine, leaving in the end a pure, stable lb> gas. This promising effect once again opens the door to stabilized hydrogen, this time doubly polarized H+~.
tecomb.
filliog
~ooot.
H+l
destr.
01000
2000
3000
t (sec.)Fig. 1.5 Density of hydragen H+ as a function of time. In a schematic way the figure shows the filling of the cell, rapid recombination decay, relative constancy thereafter, and the destruction of the sample by a heat pulse.
The predicted behaviour has actually been observed, by Cline et al (CLI81) and Sprik et al (SPR82), see Fig. l.S. Their measurements on decay curves of atomie hydragen as a function of time, showed a very sharp decrease in the hydragen density as soon as the incoming flux was stopped. After a short period of time (<10 s) this density drop terminated, and the density hardly changed any more, to be interprered as the running out of la> atoms. Cline et al (CLI81) and Sprik et al (SPR82) claim to have reached nuclear polarizations of greater than 99% and 99.8%, respectively, with the mechanism of preferentlal la> recombination. Direct measurements of densities na and nb have been performed by van Yperen et al (YPE83), using electron spin resonance (ESR), whereas Yurke et al (YUR83) acquired some indirect proof for
-the present -theoretica! model, by means of nuclear magnetic resonance (NMR). Both measurements confirm the presence of doubly polarized atomie hydragen H+~.
Thinking that now the aim of a stable atomie hydragen gas has come within reach, again turns out to be a misconception. The doubly polarized H+~ gas appears to be unstable after all, and it is generally believed (STA80, CLI81, SPR82) that relax~tion towards equilibrium by b+a transitions is responsible for this. These transitlans occur in two-particle collislons by means of the weak ~agnetic dipole spin-spin interactions. As a consequence, when almast all la> atoms have disappeared, another decay process can be observed (CLI81), which has a much slower rate (~10-1000 s) than the
reco.-nbination process. In this regime every
I
a> at om formed from a lb> atom by the slow relaxation process, at once recombines into molecular hydrogen.In this way relaxation is the bottleneck for the recombination r11te and is therefore a very important quantity. Experiments have been
-1 carried out to determine the relaxation rate T
1 , both in the volume and at the surface (SPR82, CLI81, YUR83). Also theoretica!
calculations have been performed for the volume case (SIG80, STA80, AHN83), and the surface case (LAG82, RUC82, STA82, AHN82, EIJ83). The determination of these celaxation rates constitutes the main part of this thesis.
1.2 Exceptional properties of atomie hydragen
At this point the question may be raised: why try to stabilize a gas of atomie hydrogen? The answer is, as was already mentioned in the introduction, that atomie hydragen is expected to display a number of most interesting quanturn properties. We shall go into this in somewhat more detail here.
One of the basic postulates of quanturn theory is the fact that systems of identical particles should have a quanturn mechanica! state which is totally symmetrie (for bosons) or antisymmetrie (for
fermions) under all permutations of two particles. It is well-known that the (anti)symmetrization has important repercussions, in particular on macroscopie statistica! properties.
fermions, a proton and an electron. Therefore it acts as a composite boson. Deviations from classical statistics, due to its Bose nature, are expected to show up when the mean distance between atoms r<Àth' where
Àth=(2n~
2/~kBT)\
represents the thermal de Broglie wavelength of the hydrogen atoms, whereasmn
is the atomie mass. For thecorresponding densities
R = ;-3 > À-3 =
(~.k T/2n~
2)
3/
2H th H B (1.4)
for T=0.2 K, the probability for two atoms to occupy the same state is no longer negligible, and the Bose-Einstein condensation is expected to start. As Walraven pointed out (WAL82), the composite nature of the hydrogen atoms, i.e. the rele of the constituent fermions, would only become apparent when r<4
A,
the distance where the difference between the singlet and triplet potentials (see Fig. l.lb), which represents the exchange contribution, becomes significant. This happens at densities ~>1.6xl02~ m-3 , three orders of magnitude higher.In order to obtain a better understanding of macroscopie quanturn systems, it is useful to introduce a generalization of the
thermodynamica! theerem of corresponding states, called the Quanturn Theorem of Corresponding States (QTCS). It was originally introduced by de Boer et al (BOE48), but recently the theory was extended and applied to spin polarized quanturn systems by Nosanow and coworkers (NOS75, NOS80). The QTCS states that for some group of systems the thermodynamica! properties can be derived from the same partition function, if expressed in suitable dimensionless quantities.
The usual sealing parameters for length and energy, the so-called cellision diameter a and the well-depth E of the two-particle
interaction potential, respectively, suffice in the classical case to
* * * *
define the dimensionless Helmholtz free energy F =F (T ,V ) as a
*
*
function of the redu'Ced temperature T and volume V • In quanturn theory, however, the "quantum of action" ~ represents an extra sealing parameter, and requires the introduetion of an additional
dimensionless quantity, the quanturn parameter n, which is defined as
n=~2/~Eo2=(A/2n)2, A being the de Boer parameter. Essentially only in
this way the hamiltonian of the system H=T+U (T and U are the kinetic and potential energies, respectively) can be written in dimensionless ferm
-*
H TlT
*
+ U ,*
*
*
*
where H :H/~, U =U/~ and T =T~a2/A2,
(1.5)
As macroscopie quanturn effects are expected to show up more clearly as the kinetlc energy in the lowest-energy macrqscopic states lncreases relative to the interaction energy, equation (1.5) suggests that the parameter Tl would indeed be a good measure for the
"quantumness" of the system. In Table 1.1 the values of Tl are shown for some substances to which the QTCS is applied, together with the parameters ~· a and ~ (NOS80, STW76). Note that there is of course a dtfference in behavtour between bosonic and fermtonic substances.
substance type ~ (amu) ~ (K) (J ( ~) Tl
f - - - -
---H<- boson 1.008 6.46 3.69 0.547 D<- fermion 2.014 6.46 3.69 0.274n
boson 3.016 6.46 3.69 0.183 3He fermion 3.016 10.85 2.643 0.212 4He boson 4.003 10.85 2.643 0.160 boson 2.016 37.o
2.92 0.0763 D2 boson 4.028 37.0 2.92 0.0382 zoNe boson 19.99 42.0 2.764 0.00750 40Ar boson_______
39.95 142.1 3.351 0.000761...__
____
Tab1e 1.1 Various corresponding substances, with the type of statistles they fol1ow, their mass
mn·
we11 depth ~ and colliston diameter a of their pair potentia1, and quanturn parametern.
The table suggests that atomie hydrogen possesses by far the most pronounced quanturn character, while spin-po1arized deuterium still has more quantumness than its well-known fermion successar 3ne.
The aforementioned extension of de Boer's QTCS by Nosanow and coworkers consisted in consirlering n as an additional thermodynamic
variable which can vary continuously. In their formalism they define a generalized characteristic thermadynamie function, the free energy
* * * *
F =F (T ,V .~). Then it is possible to derive generalized phase diagrams, phase transitions and critica! points (NOS75, MIL75, MIL77). The zero-temperature critica! point for bosons turns out to lie at
~=0.46, so H~ appears to be the only material to remain gaseous down to absolute zero. In fact, H~ is so light and weakly-interacting, that it is probably an almest ideal Bose-gas, thus offering an opportunity to cbserve Bose-Einstein condensation almest in its pure form. Also the possibility to reach an arbitrary low temperature with H~ as a cooling refrigerant has been mentioned (NOS80).
Apart from an expected peak in the specific heat and possible superfluidity, the paramagnetic character of atomie hydrogen enables one to study spatial condensation, when a streng magnetic field gradient is applied (WAL82), and possibly the phenomenon of spin waves. As for practical applications of H~ its unequalled specific potential energy suggests its use for energy storage and fuel. The extremely large densities, however, being necessary for this purpose (~loze m-j, JON58) will not be feasible in the near future.
-CHAPTER 2. Rate constauts derived from collision quantities
2.0 Introduetion
In this chapter we will introduce the rate equations governing the development of the densities na and ~ of a and b atoms as functions of time, in section 2.1. Both volume and surface processes are included. Then the relaxation rate constants GV and G
5 for volume and surface collisions, determining the decay rate of doubly polarized hydrogen Hfl, are expressed in microscopie scattering cross sections and cross lengths, respectively.
In section 2.2 this is carried out in a straightforward way using kinetic gas theory. There has been considerable controversy between several groups about factors of two in theoretica! expressions, probably due to uncareful treatments of identical-particle aspects. The final expression for the relaxation rate constant GV of Statt and Berlinsky (STA80) agreed in the first instanee with ours (AHN83), but was a factor of 2 lower than that of Siggia and Ruckenstein (SIG80). In a private communication to us, however, the former authors "corrected" their results in agreement with the latter authors. A simtlar discrepancy arose for the surface relaxation rate constant G
8, as three papers (LAG82, RUC82, STA82) agreed on a theoretica!
expression, which was a factor of two larger than ours (AHN82). This prompted us to carry out two alternative derivations of the rate constauts GV and G
5, which are presented here in sections 2.3 and 2.4. In both derivations identical-particle aspects are treated using
the more fundamental second-quantization formalism, which facilitates considerably the handling of these aspects. Section 2.3 is essentially based on linear-response theory according to Kubo and Tomita (KUB54), also used by Ruckenstein and Siggia (RUC82); the related formalismof correlation functions (ABR61) is used by Lagendijk and van Yperen (LAG82, YPE84). The latter, however, does not incorporate the treatment of identical particles. Moreover, it is not clear how it could be included, without ending up with a formalism of the type of the linear response theory or the quanturn mechanica! Boltzmann equation, to be considered below. Without such a more fundamental formalism one is forced to resolve the "factors of two" problem by normalizing the theoretica! relaxation formulae against classical expressions.
In section 2.4 the quanturn mechanical Boltzmann equation for
-identical bosons is discussed and applied to the relaxation problem, offering at the same time a very general and rigorous formalism for deducing rate constants from microscopie quantities in similar problems.
Note that sections 2.2 and 2.3 are essentially based on sections III and IV of a paper by Ahn et al (AHN83).
2.1 Rate equations and constauts
In a situation, where volume V and temperature T are considered to be kept constant, the time evolution of the macroscopie system of atomie hydrogen may be descrlbed by the densities na(t) and nb(t) of
la> and lb> atoms, respectively. These densities are determined by the rate equations
dn 2
+ nanb) - 2 2K 2
~ G.j.(nb Gt(na + nanb) - Kabnanb - aa a n (2 .l) and
~
2nanb) + 2 + nanb)
-G .j.(nb + Gt(na - Kabna0b'
dt (2 .2)
the total hydrogen density being equal to the sum
left out terms descrihing the filling flux, thermal leakage, and impurity effects (SPR82). In the equations the recombination rate constants Kab and Kaa appear, from a+b and a+a collisions,
respectively; a Kbb term is missing for reasoos mentioned in section 1.1. The K terms refer to recombinatton of surface-adsorbed H-atoms, converted to effective volume terms. We come back to this point shortly. Three-particle recombination is neglected (WAL82, JON58; see also chapter 6). The G terms represent relaxation processes,
incorporating a b+a transition (G.j., exothermal) or an a+b transition (Gt, endothermal). Transitions b+b-a+a, requiring a double spin flip, are negligible (see section 3.4.3). Note that, for instance, the reactlons b+b+a+b and a+b+a+a share the same rate constant G.j., since 1) thermal collislons outnumber relaxation collislons by far, so that
la> and lb> atoms have the same energy distribution, and 2) the spin interactions can be treated to first order (see section 3.4).
If we would omit recombination for a moment then the relaxation processes a-b would drive the densities na and nb exponentially to the
equilibrium state na= ~G+/(G++Gt), nb= ~Gt/(Gt+G+)' like
exp{-~(G++Gt)t}, from any initial state, the total density ~ being a constant. The characteristic rate for this exponentlal process is the relaxation rate
On the other hand, when the recombination terms are included a rapid depletion of the fa> state takes place. Soon the density na is very small compared to nb, and knowing that G+~t and Kab~Kaa' but Kab>>G+ (CLI81), the rate equations reduce to
and
dn dta
dnb dt
The relaxation-bottlenecked-recombination regime starts when
)/(~)~
n /n >>1, dt b a so and~
dt (2 .3) (2.4) (2.5)The salution to this equation, nH(t)= n
0/(2n0G+t+l) has a typical rate
T~
1=
2n0G+, and it is essentially this rate that can be measured in actual experiments. The calculation of the important quantity G+ will be the main topic of the remaining chapters. The subscript "+" will be dropped in the following.
Although Eqs. (2.1) and (2.2) refer in the first instanee to volume densities, particles bound to the surface with an energy E
8 and surface density n
8 play an important role. The gas and surface phases, each with their own set of rate equations, are continuously exchanging atoms. Since it is believed that the exchange rate is much faster than relaxation and recombination rates (SPR82), the surface coverage can be thought to be in constant thermal equilibrium with the volume density, according to n
8
/~= Àthexp(Es/kBT), where Àth is the thermal de Broglie wavelength. This implies that the surface decay can be effectively included in the volume rate equations as (dn8/dt)(A/V), where A/V is the surface to volume ratio of the cell. Relaxation rate constauts are then built up out of a volume and an effective surface constant, according to
-G (2.6)
On the other hand, the recombination constant has only an effective surface contribution, reading
K (2.7)
In the remaining part of this chapter we shall go into the derivation of the relaxation rate constants, dealing with GV and G
5 sirnultaneously.
2.2 Thermal average with the Boltzmann distelbution
In this section we will introduce a relatively simple way to e~press both volume and surface relaxation rate constauts in
microscopie callision quantities. To this end we use a simple Maxwell-Boltzmann distribution to average over all veloeities of individual atoms before a possible relaxation collision, thereby assuming that the densities are still far from the Bose-Einstein regime.
Foc individual particles we have the following velocity distribut ion:
(2.8)
where d is the dirneusion of the system (surface: d=2, volume: d~3). Thus the velocity distribution on the surface is simply assumed to equal the part of the volume distelbution parallel to the plane. In scattering theory only relative veloeities are relevant, and the two-particle distribution functlon for these is found to be
2
t
~ )~d -\~v +2nk T exp{~)dv,
B B
in which ~=\~ is the reduced mass.
Now let us evaluate the rate of change for the nb density caused by b+b+a+b transitions. Consider a single b atom. Then the
differentlal number d 2N of b atoms with relative velocity in an
+ +
interval dv around v that cross a given area (ar line segment for d=2)
d-1 +
~ perpendicular to v, in a time interval dt, equals 2 d + d-l d + +
d Nb(v) = vdt ~ ~ P (v)dv. (2.10)
Hence the number of pairs splitting up again as an a+b pair with a
1\ 1\
relative velocity direction in an interval dv' around v' turns out to be
3 d 1\ + d + + /\1 d + +
d Nab,bb(v',v) = nbvdt oab,bb(v',v)dv P (v)dv. (2 .11) d + +
In this equation oab,bb(v',v) represents the so-called differentlal colliston cross section ( .. cross length .. for d=2, for which we also introduced the symbol À, see AHN82) for the bb+ab transltion. It is essentially defined as an area (line plece for d=2) perpendicular to
+
the incoming partiele flux with velocity v which scatters into the +
direction of v' with the internal state transition indicated. For a more explicit definition see section 3.1. The d=3 cross sections will be calculated in chapter .3, the d=2 cross lengths in chapter 4.
The destred rate of change follows sirnply by integrating over all initial and final veloeities and summing over all b particles:
(2 .12)
where the quantity Gab,bb(T), defined as
Gab,bb(T) +
1\ d + d + +
:= \Jdv fdv' vP (v) oab,bb(v',v), Rd ~,d
(2 .13a)
turns out to match the quantity Gf from Eq. (2.2), if one identifles (dnb/dt)ab,bb as the first term on the right hand side of (2.2). Note that the factor \ appears in (2.13a) to avoid double counting in the summation over the b atoms befare col1islon.
The other terms in (2.1) and (2.2) can be delt with analogously, and we find for a general relaxation rate constant due to a transition
-s+s' between two-particle spin states s and s'
J
+ A d + d + +\; dv Jdv' vP (v) os's(v',v). Rd ~,d
(2.13b)
Since bb-aa transitlans are negligible, the remairring relevant transitlans are bb-ab and aa-ab. In case the final state contains particles in identical spin states, the factor \; avoids double-counting due to the integration over all final directions.
Finally, defining an effective cross section as (d=2 or 3 only):
1 J A J / I d + +
d- 1 - dk' dk os's(k' ,k), 2 11
(2 .14)
where we have identified ~ (and ~·) with the direction of the wave
+ + +
vector k=~v/~ (and k' accordingly), we arrive at the eKpression
(2.15)
We made use of a transition to the asymptotic relative kinetic energy
E=\;~v
2Note that
o~ff,s's(E)
vanishes for E<Es'-Es, if s+s' is an endothermal reaction. The equations (2.14) and (2.15) provide the necessary conneetion between the microscopie quantitieso~ff
and the macroscopically measurable constauts G.2.3 Linear-response theory
2.3.1 Derivation of an expression for the relaxation rate
As a further support of the foregoing result, we use a starting point based on Kubo and Tomita's approach (KUB54). It was also briefly mentioned by Ruckenstein and Siggia (RUC82). Our aim is to corroborate the expression (2.15) for the relaxation constant, in view of the disagreement about factors of two between several groups (AHN82, AHN83, LAG82, SIG80, RUC82, STA80). The present derivatiön, which makes use of an accupation number representation, allows a more
straightforward way of handling the identical-particle aspects. On the other hand, this derivation is lengthy and complicated. Therefore we introduce a number of approximations, which may only be valid in some
idealizing limit, but which have no bearing on our discussten about factors of two. The approximations are:
(1) The spin energies involved are neglected compared to thermal energies; this assumption, called the high-temperature limit (HTL), will be discussed in chapters 3 and 4.
(2) The central interactions are omitted. The resulting plane-wave approximation (see sectien 3.4.2) gives rise to a diverging scattering cross sectien in two dimensions. However, it is assurned that in the resulting expresslon for G more exact approximations for the cross section may be subst.ituted.
(3) Transition probabilities are calculated using first order time-dependent perturbation theory.
(4) 8nly small deviations from the equilibrium situation na=nb are considered (linear response).
(5) The gas is considered to be dilute. (6) Only single spin flips are included.
(7) For the surface case the gas is treated purely two-dimensionally.
Agaln the surface relaxation (d=2) is discussed along with the volume case (d=3). Consider a system of NH atorns in a "volume" Ld
~.Yith
corresponding periodic boundary conditions for the free (assurnption-'!id
+ +(2)) single-partiele wave functions I~>= ~ L exp(ik .r). The state
z;a a
~r is a one-atom internal spin state (z;a=a or b). For d=2 a wave "a.
function for the motion perpendicular to the surface is omitted, according to assumption (7). The Hamiltonian of the system consists in
the first place of an unperturbed part H
0, which is the sum of a (free) translational part H~r with single-partiele eigenvalues
2 2 int
ea=~ ka/2~, and an internalpart H
0 with single partiele
eigenvalues E =E or Eb. In second-quantized form we have (see ROB73, a a
GRE82)
H
=
Htr+
Hint0 0 0 (2 .16)
K
where a and at are destructien and creation operators of the
single-K K
partiele mode K, respectively. The perturbing spin-spin part is correspondingly
-V (2.17)
V standing for a matrix element of the interaction among a pair
KÀ,IJ\1
of atoms. Formally, a system with thls hamiltonian is known to be equivalent to a fictitious spin-\ gas (obeying Bose statistics) in an ex:ternal homogeneaus magnetic field. In the following we shall use terms like spin temperature, etc., on the basis of this analogy.
The eigenstates of the unperturbed hamiltonian are I~>= +
i"
1,"2,N3,, •. '>, Na being the number of atoms in state a (=ka,ça)· We now describe the translatlonal subsystem by a canonical ensemble with a certaln temperature (S=l/k
8T), and the spin subsystem by one with a spin temperature ( =1/k T ). The latter temperature is in one-to-one
B s
correspondence with the a and b occupations. tn general non-equilibrium states S
8tS the time-dependent density operator is given
by
p( t) (2.18)
tn thls ex:pression, Z is the partition functlon. In the spirit of the high-temperature limit (assumptlon (l)) we neglect the heat capacity of the spin system relative to that of the translational system, and consider. the relaxati.on of Bs to B, taldng the latter to be constant.
According to flrst-order time-dependent perturbation theory (assumption (3)), the transition probabillty from a state IN> to IN'> in a time interval T is 2
I
<N tI
V I N>I
f ( T' I.I)N' N ) ' (2.19) with f(T,w) (2.20) and ( 2. 21)provided the interval T is longer than a typtcal callision time, but smaller than the time between collisions. Note that we introduced the short-hand notation EN=ÏN e and EN=LN E • The change in the average
total spin energy in the time interval T equals
With assumption (4) this change ~<e> is approximated to first order in ~B= Bs-B' giving
~<e>
T (2. 23)
In this equation the quasi-5-function
·a·
has been introduced in anticipation of the replacement+ 2'1fÖ(w), (2.24)
valid if we take T much larger than a typical callision time. We used the narrowness of the "ö"-function to get rid of the zero-order term in (2.23) in particular, through the (anti)symmetry in N' and N.
In turn we can express ~B in the distance of the average total spin energy <e> from lts equilibrium value <e>00, once again using a linearization procedure, as well as the fact that the individual spin systems are not correlated to each other (low density, assumption (5)) and to the translational system. These assumptions lead to the
relation
<e> - <e> 00 2
exp(6Eab) HTL
-NHM Eab jl+exp( BEab)
f2
( 2.25)where E
8b= Eb-Ea is the energy difference between the la> and lb>
states. Combining (2.25) and (2.23) we have (T<<T 1!) in which d<e> ~~ 4'1f '\ \'
NTZ
L L H (a6)(yö) N ~<e> T 1 T (<e>- <e>".), 1 (2.26) 23-(2.27)
This is the destred relaxation rate, because <e:> depends linearly on the difference nb-n , i.e. the longitudinal magnetization of the
a 2
fictitious system of spin-\ bosons. The sum LN' I<N' IV IN> I was
evaluated with the use of (2.17), noting that at most four N' numbers, labeled by a, 8, y, and o, can differ from the corresponding N values fora non-vanishing contribution. For fixed {N/ a sequence {N'} is defined by gi1ring the two unordered pairs (afl) and ( yo); equalities among these labels lead to negligible contributtons for low denstties. Next, the KÀlJ\1 summati.on was found to reduce to four terms. Finally,
the neglect of b+b .. a+a transitlans in assumption (6) allowed us to use the equality E:N'-e;N=Eab for all transiti.ons N->N', in Eq. (2.27). The expressten for the relaxation rate will be worked out in the next subsection.
2.3.2 Evaluation of the relaxation rate
In order to evaluate the relaxation rate, we first convert the pair summatton (afl)(yo) over one-particle states into four independent summations, at the same time inserting a factor 1/4 for compensation. Next the four summati.ons over the wave veetors are transformed into lntegrals, taking the limit L->=. Then the integrations over wave
+ + + +
veetors ky and k
0 (before collision), as wellas ka and k8 (after collisi.on) are replaced by integrations over relative wave veetors
+ + + .• ·• + + + +
k=\(ky-k
0), k'=\(ka-kB)' and center-of-mass wave vector.s K~k/k
0
,+, + + f
K =ka+k
6• There ore we have
Writing out the two-particle matrix elements with the same transformation of wave vectors, we get
(2.29)
where the so-called identical-particle T-matrix (see also section 3.1) in the plane-wave Born approximation (cf. sectien 3.4.2) is defined as
id + +
T~ ~ ~ ~ (k',k)
ex
B'
y óThe labels l and 2 stand for a pair of 11 atoms, and P
12 is a permutation operator. Substituting (2.28), (2.29) and (2.30) into (2.27) we find
(2.31)
where we carried out the substitution lzwN'N= E'-E. The summatien over N is now included in the ensemble average < ••• >. This average is calculated with the use of the grand-canonical ensemble, so that the accupation numbers are uncorrelated, and we arrtve at
(2.32)
For the low densities considered (assumption (5)) <N +1> and <N +1>
a B
have been replaced by 1, while for the evaluat.lon of <Ny> and <N 6> an
ordinary Boltzmann distribution is sufflcient. Evaluatlng some integrals we finally arrive at
25
d
The cross section (or cross length) a is related to the T-matrix in (2.31) by the definitions that will be used in sections 3.1 and 4.2:
(2.34)
d The effective cross section creff is deflned ln terms of cr
accordlng to (2.14). ln the plaue-wave Born approximation plus HTL all cross sections for the eight contributing processes (assumption (6)) aa-ab, aa-ba, bb-ab, and bb-ba are equal. The summation over ~a' ~S' r,y, and ç
0 is therefore dropped and replaced by a factor of eight in equation (2.33).
-1 Comp!'lring (2.33) with (2.15), and using T
1 =2nHG, we have thus confirmed the derfvation of section 2.2 wlth respect to factors of two.
2.4 The quanturn mechanica! Boltzmann equation
We wi11 uow brlefly go lnto the derivatlon and app1ication of the quantnm mechanic'll Boltzmann equation, as a third way of expressing GV and G
5 in callision quantities. Once again the second-quantization formallsm is used to enable an elegant and straightforward way to treat ldentlcal-partlcle aspects. The present approach is very generally applicable, and easily allows for generalizations, such as the inclusion of more internat energy levels, or the use of the Bose-Einstein thermal distribution. For an extensive derivation of various quantummechanical Boltzmann-type equations we refer to a paper by Hess (HES67). In the following we will glve only the outlines of this derivation. In addition, we will point out an error in HES67, and modify the T-matrix definition, in order to match the usual convention. See also TR083 for these corrections.
The system of atomie hydrogen under consideration is agaln described by the hamiltonian, introduced in Eqs. (2.16) and (2.17). For a description of the time evolution of the system the Heisenberg pleture is introduced, so that the operator fields become time dependent. Matrix elements of all one-particle observables can be expressed in terros of the one-particle distributton matrix Fkk'~
Tr(pa~
1
~),
the diagonal elements k=k1 of which equal the expectationvalues of the accupation numbers Nk. The time dependenee of Fkk1 is governed by the equation
(V F(Z) -F(Z) V ) k~ mn mn k 1
" k ~ mn mn k 1 R. '
' ' ;c,. ' '
(2.35)
in which Ek is the total energy of a partiele in mode k. In this equation the so-called two-particle distributton matrices F(k~,mn 2)
=
Tr(pa:a:akat) come into play, in which the matrix elements of two-parttcle observables can he expressed. The time-evolution equation for(2) . (3)
F in turn contains matreces F , and so on. This endless, so-called BBGKY, hierarchy is not solvable in practice, unless some additional physical assumption is put in.
At this point it is conventent to introduce the correlation operators CkR.:=~at. In the present case the aforementioned physical assumption consistsof the neglect of ataC terms relative to C terms in the time evolution equation for Ck~· This is justified by the fact that in the matrix elements each additional
a:a~
couple represents a factor I(Nk+l)/Nt' leading to small contributtons for lo~ densities. The approximation turns out to be equivalent to the neglect ofthree-(3) (2)
partiele contributtons F in the equatton for F . With the definition
we find the equation for the Ckt to be
. ._de èt'(it 1\ (E + V)C, (2. 36) (2.37)
in matrix notation. This equation, resembling the ordinary Schrödinger equation, may be understood to describe the colltsion of two particles k and ~. taken from the gas by the destruction operators ak and a~. The idea is then to relate the solution after colliston C(t) and thus the distributton matrix
F~~~n(t)=
Tr(pc;nckt)' to their values long before colliston (long compared to typical callision times). These values being unknown, a second supposition, the so-called molecular chaos approximation, enters the derivation: it is assumed that two-particles are uncorrelated before collision, due to frequent scattering processes by other particles.
(2 0) .
The F ' matrix for uncorrelated particles can easily be worked out, using a grand-canonical partition function. The summation over all states IN> can be factorized into summations over single-state accupation numbers
~a·
The matrix element<Nia~,ai,~a~IN'>
is found to be NkN~(ök'kö~'t+ök'~ö~,k), so the Nk and N~ summations reduce to <Nk>=Fkk and <N~>=F U' respectively. All other sums in the numerator and the denominator cancel. In this way the F(2) matrix reduces to(2.38)
for some time t
0 befare the collision. In the derivation of Hess (HES67) there appears a factor ~ on the right hand slde, which might seem natural, consiclering the right-hand side of Eq. (2.38) as a symmetrized expression, but which is nevertheless incorrect.
The salution C(t) to Eq. (2.37) may be expressed in the salution C(O)(t) of the same equation without interaction V, by means oE the operator (l-2:iT(-)), i f t-t0 i,; large compared to the colltsion t.ime. The operato~ T(-) in turn is connected to the transition matrix T for distlnguishable particles (see sectlon 3.1) according to
0
Jdr exp(+rJT/h)exp(tf?r/h) T exp(-ii?r/h). (2.39) ±<»
The factor exp(+nr/h) has been inserted to suppress contributions from other collislons for large lrl values, while nis chosen toyieldan exponentlal value of about l at the time of the colliston under considerat ion.
(2)
The abovementioned expresslons enable us to express F (t) after the colliston in F(2 ,O) (t
0) befare the collis ion, and the latter quantity in turn in one-particle distributton matrices with (2.38).
(2) .
Substituting this F (t) into equation (2.35) g~ves the general quanturn mechanica! Boltzmann equation (in matrix notation):
. dF
l
j
{
-id id t id id t~hdt = E,F + Tr2 rriT FFT + (T FF-FFT ) ), (2.40)
assumed F to be diagonal in the energy. The identical-particle id
matrices (T corresponds to the quantity indicated by the same notatien in section 2.3) are defined by
and
(2.41)
rid =r +r =rid +rid +rid +rid C2 42)
kt,mn kt,mn kt,nm (-)kt,mn (+)kt,mn (-)kt,nm (+)kt,nm' • id
The deftnltion for T , applied i.mplicitly in HES67, apparently contains an additional factor ~. which is rather unusual. Together v.~ith the error in Hess' equation corresponding to our (2.38), this leads to an incorrect additional factor of two under the trace in Eq. (2 .40).
The Boltzmann equation (2.40) can be brought into a somewhat more familiar form, the so-called Wang Chang-Uhlenbeck-de Boer equation (WCUB; see WANSl), with the use of the assumption that F is also diagonal in its momenturn indices, whlch can be proved to be valid for homogeneaus systems. Summations are converted to lntegrals as L"""'·
...
Then F is replaced by a functlon f(p) that is normalized as a partiele
...
density. A final approximation applied is the diagonality of f(p) in the internal coordinates as well. According toMoraal (MOR75) this holds as long as the difference ~E between internal energies is not too small, i.e. ~E.T>>h, where T is the time between collisions. This can be made plausible by noting that h/~E is the time it takes an atom to "forget" it has been in a superposition of internal states due to the previous collision. The WCUB equation reads
...
(Jfr;(p)=
1at
q2 q•2 xo ( - - -
+ 2u 211 (2.43)In this equation the r; indices refer to internal (spin) coordinates. The E term represents the difference in spin energies before and after
+ + + + + +
the collision. The veetors q~~(p-p
1
) and q'=\(p'-pi) are relative momenta. Note that for one internal state the WCUB equatlon takes the forrn of the classica! Boltzmann equation (BOL72), except for a factor-~ appearing in the former. This factor sterns from the fact that the identical-particle quanturn mechanica! cross section can be written as the sum of two equal "classica!" cross sections for distinguishable particles, besides two interference terms.
Substituting a Maxwell-Boltzmann momenturn distribution times the ç-state density nç for the f functions on the right-hand side, and carrying out the integrations, we obtain the rate equations. We found that our results in section 2.2 were confirmed once more.
GRAPTER 3. Volume processes
3.0 Introduetion
In the last chapter we have seen how the macroscopically measurable quantities, such as volume and surface relaxation rate constants, GV and G
5 respectively, can be expressed in terms of a microscopie effective cross sectien oeff or cross length Àeff'
respectively. In turn, the latter quantities can be derived from a theory of two-particle collisions. From this point on we have to separately treat the processes taking place in the volume and these where the atoms involved are bound to the surface. In this chapter some models are introduced to describe collislons in the volume.
In sectien 3.1 the basic ingredients of scattering theory are reiTiewed briefly, and it is pointed out how the solutions 'I' of the time-independent Schrödinger equation are connected with the scattering amplitude f, the T- and S-matrices, and finally the differentlal cross sectien o, both for distinguishable and identical particles. In sectien 3.2 we introduce the Hamilton operator and focus on the terms which are relevant to our problem. Then we show in sectien 3.3 how the Schrödinger equation can be solveil in a rigoreus way, using the so-called coupled-channels or close-coupling (CC) method. The wave function 'I' is expanded in a complete set of suitable basis functions containing all partiele coordinates, except for the relative distance r between the two colliding hydrogen atoms. The result is a set of second-order coupled differentlal equations. The diagonal and coupling matrices are simplified using symmetry
arguments, and some difficultles arising in the practical integration of the CC equations are delt with. In sectien 3.4 various reasonable-looking approximations will be discussed, which significantly decrease computation time. Next, these approximations are checked using the more exact CC calculation. This thorough check was our main motivation
for starting the CC attack of the problem in the first place. Finally, in sectien 3.5, we give the numerical results for oeff and GV' and compare with other theoretica! results and with experiment.
Note that the present chapter is partly based on sections II, III, and V of a paper by Ahn et al (AHN83).
-3.1 Principles of scattering theory
In quanturn mechanics a pure state of a physical system can be described by a wave function W, depending on all coordinates of the system. The time evolution of the system is governed by the so-called time-dependent Schrödinger equation, reading
a'!'
i~ =H'I'(t). (3.1)
ln this equatlon H is a hermitean operator corresponding to the total energy of the system.
Scattering theory is concerned with the descrlptlon of a small number of particles, interacting with an external field and/or with each other only for a flnite pertod of time in a llmlted region of space. lt aims at establishing the relation between the state of the particles long befare and long after their mutual interactlon. In this thesis we restriet ourselves to two-particle collislons in an external field. The proper way to treat these collislons is in the
time-dependent formalism, using wave packets for the particles with an in-and out-asymptote, connected to each other by the so-called S-operator (TAY72). We shall adopt here, however, the more commonly used and simplee t !me-independent formalism (i·1ES6 t), where the incoming and outgoing particles correspond to a plane wave and a spherically scattered wave, respectlvely. It is always possible to construct wave packets Erom a superpaaition of stationary waves.
-iEt/fi
Substituting an e time dependenee in (3.1) we obtain the time-independent Schrödinger equation:
+ +
HW (~,5_}
=
E'l' (~,5_)· (3.2)ln this equation E is the total energy of the system. The salution ~ depends on all spatlal coordinates ~ and spin coordinates
5.·
The symbol ~ stands for a vector in the 3N-3 dimensional configuration space of relative coordinates, N being the total number of"elementary" particles involved in the collision. The center-of-mass motion is left out of consideration here.
The following considerations are restricted to a situation in which a} the total system formed from the colliding systems can only separate lnto two clusters of partlcles, b) these clusters are
electrically neutral, and c} identical-particle aspects are not taken into account. The third restrietion is dropped at the end of this
section. The following boundary condition at infinity is imposed on '1':
(3.3)
where the sign ~ refers to asymptotic behaviour. ln this way qr+ is the salution of (3.2) containing an incoming plane wave with given wave
+
vector ka and initia! two-particle internal state Wa· The second term in (3.3) represents the spherical scattered waves for all possible final states
w
8, called channels, and corresponding wave numbers kb, each present with an amplitude fB - the so-called scattering
a A
amplitude - depending on the scattering direction rB. l t is pointed out here that only veetors in three-dimensional space are denoted with an arrow on top, contrary to general vectors, which are underlined. Furthermore, the channel indices a and
B
are composed of the subindices (A,a) and (B,b), respectively, a capital letter labeling the composition of both colliding fragments out of elementary particles, and a lower case letter indicating the internal state of+ +
both fragments. Finally, rB (rA) is the relative vector between the centers-of-mass of the scattered fragments, and ~B (~A) stands for the internal coordinates of the fragments, the remainder of ~ in channel
+ +
B
(a) after remaval of the coordinates of rB (rA).A
At large values of 1~1 and fora given direction x=~/ I~
I
only terms corresponding to at most a single composition B contribute to the right-hand side of (3.3). Projecting the sum of such terms on the internal statew
8 defines the Expanding it in
• A
spher1cal harmonies Y~'m'(rB)
I;B(tB, ka)
2
Y~'m' (~B)Y~mC~a)
(41Ti~j~(kbrB) öaBö~, ~öm'm+
rB+"' ~· ~
m'm
i kb rE
+fB~'m',a~mCka) ~"7B)·
(3 .4) Definitions of Y~m and the spherical Bessel function j~(x) can be found in Morse and Feshbach (MOR53). Taking the inner product of (3.4)*
A Awith Y~'m'(rB)Y~m(ka) and substituting the asymptotic form
