EFFECTIVE RANGE THEORY AND MANY-BODY PERTURBATION THEORY APPLIED TO ELECTRON SPIN-POLARIZED ATOMIC HYDROGEN (H+)

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EFFECTIVE RANGE THEORY AND MANY-BODY PERTURBATION THEORY APPLIED TO

ELECTRON SPIN-POLARIZED ATOMIC HYDROGEN (H+)

Y. Uang, W. Stwalley

To cite this version:

Y. Uang, W. Stwalley. EFFECTIVE RANGE THEORY AND MANY-BODY PERTURBATION

THEORY APPLIED TO ELECTRON SPIN-POLARIZED ATOMIC HYDROGEN (H+). Journal

de Physique Colloques, 1980, 41 (C7), pp.C7-33-C7-38. �10.1051/jphyscol:1980706�. �jpa-00220143�

JOURNAL DE PHYSIQUE CoZZoque C 7 , suppZ&rrer,t c.< e 0 7 , Tome 41, j u i z z e t 1980, page C 7 - 3 3

E F F E C T I V E RANGE THEORY AND MANY-BODY P E R T U R B A T I O N THEORY A P P L I E D TO ELECTRON SP IN-POLAR IZED A T O M I C HYDROGEN

(H+)

Y.H. Uang and W.C. Stwalley

University o f Iowa, Iova C i t y , Iowa 52242, USA.

Resum@.- Des valeurs precises pour les longueurs de collision et les port&es effectives sont deter- minees, pour des collisions entre atomes dlhydrog&ne, grace

&

la theorie de la portee effective.+On utilise les calculs precis et ab initio de Kolos et Wolniewicz pour les courbes de potentiel b 3l.u de la molecule H2, ainsi que les coefficients C6, C8 et Clo du developpement a grande distance, de f a ~ o n obtenir une representation precise du potentiel d'interaction

ii

deux corps. La longueur de coll ision pour 1 'onde s ainsi obtenue (A

=

1,33455 a

)

est util isee pour calculer I 'energie

8

tem- perature nulle d'un ensemble d'atomes d'hydrogene po?arises (Ht), en fonction de la densite, grdce

Z

l'utilisation des resultats d'une th6orie de perturbation

ii

n corps valable pour des bosons.

Lorsque l'on neglige des termes petits qui dependent de la forme du potentiel, on obtient l'energie par particule E/N en fonction de la densite, energie qui peut 6tre comparee avec des resultats ana- logues fournis par des calculs de type Monte-Car10 ou variationnel. On donne egalement les longueurs de collision et les portees effectives pour les autres systemes isotopiques de l'hydrogene. Un re- sultat obtenu est que le premier r~iveau lie dans le potentiel b3 c+ (dans le cadre de l1approxima- tion de Born-Oppenheimer) nlexiste que pour un isotope imaginaire ye masse superieure a m=3,258176 g/mole (dans ce cas, la longueur de collision devient infinie). En termes de parametre quantique

rl z

b2/(2v~a2) , 1 'etat lie

ti

deux corps n'existe que si

11 <

0,17094.

La

transition du second-ordre liquide-gaz (correspondant

2,

l'apparition d'un etat lie pour un nombre infini d'atomes) se produit pourlavaleur critique du parametre quantique

17

, determine avec precision

r l c ~ =

0,43891. Dans ces conditions, la longueur de collision s'annulle.

Abstract.- Accurate scattering lengths and effective ranges for hydrogen atomic collisions are determined from effective range theory. The precise ab initio b3ei potential energy curve values of H2 calculated by Kolos and Wolniewicz and C6, C a y Clolong-range expansion coefficients are used in obtaining accurate representation of the two-body interaction potential. The s-wave scattering length obtained for this potential (A

=

1/33455 ao) is used to calculate the zero temperature energy as a function of density of electron spin-polarized ("spin-aligned") atomic hydrogen (Ht) using previous many-body perturbation theory results for bosons. Neglecting small potential shape dependent terms, one obtains E/N (energy per particle) as a function of density, which may be compared with similar results obtained from Monte-Carlo or variational calculations. The scatte- ring lengths and effective ranges for the other hydrogen isotopic systems are also presented. It is shown that the first bound state for the b3c: potential (within the Born-Oppenheimer approxi- mation) only exists for a hypothetical isotopic species with mass larger than a value m

=

3.258176 g/mole (where the scattering length becomes infinite). In terms of the quantum parameter, (?pea2), two-body binding occurs in the b3c+ potential only for

II <

0.17094. The second order

n

- ^b2/

l~qu~d-to-gas transition (corresponding to ! h e onset of infinite-body binding) occurs at the accu- rately determined critical value of the quantum parameter ncB

=

0.43891. This corresponds to zero scattering length.

1. INTRODUCTION.- During the last few years, at- ing H4 have been investigated by several groups /4- tempts to achieve a stable bulk electron-spin-po- lo/. Recent results /lo/ (see also other papers in larized ("spin-a1 igned") atomic hydrogen (symbo- this colloquium) have provided definitive evidence lized Ht) has been a subject of considerable in- for bulk H4, albeit at densities well below those terest, both theoretically and experimentally. Be- needed for Bose-Einstein condensation. A recent cause of the high energy storage possible and the detailed close-coupled calculation on low energy predicted strong quantum mechanical behavior, e.g. Df - H+ scattering /11/ has shown the importance of Bose-Einstein condensation /1-3/, Ht seems to be an a Df impurity destruction mechanism in Ht that was important fundamental system of great promise for previously postulated

141.

Another crucial issue, improving our basic understanding of quantum pheno- the explosion limit (density) of Hf, has also been mena. The stability conditions needed for prepar- discussed /12/ based on the viewpoint of microsco-

pic kinetics of a recombination event.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980706

J O U R N A L DE P H Y S I Q U E

I n t h e above mentioned low energy s c a t t e r i n g process, i t was shown t h a t t h e e l a s t i c cross sec- t i o n , i n t h e b3zu p o t e n t i a l , converges t o a f i n i t e t value a t t h e zero k i n e t i c energy l i m i t /11/. Ac- c o r d i n g t o e f f e c t i v e range t h e o r y /13/, t h i s a1 lows us t o determine t h e s c a t t e r i n g l e n g t h o f t h e c o r r e s - ponding c o l l i s i o n process. The p r e s e n t work i s

t h e r e f o r e devoted t o a c a r e f u l d e t e r m i n a t i o n o f c e r - t a i n e f f e c t i v e r a n g e t h e o r y parameters: t h e s c a t t e r - i n g length, A, and t h e e f f e c t i v e range, ro ( f o r a l l i n t e r a c t i o n s between hydrogen and i t s i s o t o p e s ) .

One o f t h e i n t e r e s t i n g a p p l i c a t i o n s r e l a t e d t o our c a l c u l a t i o n o f A and ro i s u s i n g these r e s u l t s t o c a l c u l a t e t h e energy per p a r t i c l e E/N o f b u l k H+

through many-body p e r t u r b a t i o n t h e o r y (e. g. Refer- ence 14) on i n t e r a c t i n g bosons. These c a l c u l a t i o n s o f E/N a r e s e r i e s expansions i n terms o f t h e p a r t i - c l e d e n s i t y and t h e s-wave s c a t t e r i n g l e n g t h o f t h e two-body i n t e r a c t i o n . The t h e o r e t i c a l i n t e r e s t i n e v a l u a t i n g E/N f o r H+ and o t h e r " s p i n - a l i g n e d " sys- tems i s evidenced by previous work u s i n g e l a b o r a t e Monte-Carlo /15-17/ o r v a r i a t i o n a l methods /18-19/.

The p r e s e n t use o f many-body t h e o r y provides a sim- p l e b u t accurate a l t e r n a t e method. F r i e n d and E t - t e r s /20/, i n t h e i r most r e c e n t work, have a l r e a d y o u t l i n e d a s i m i l a r c a l c u l a t i o n . They, however, o n l y consider t h e expansion s e r i e s up t o t h e second term and as discussed below use a l e s s accurate r e - p r e s e n t a t i o n o f t h e two-body p o t e n t i a l .

I n t h e f o l l o w i n g , we w i l l f i r s t b r i e f l y de- s c r i b e our r e p r e s e n t a t i o n o f t h e two body poten- t i a l , our use o f e f f e c t i v e range t h e o r y t o c a l c u - l a t e A and ro, and our many-body boson E/N r e s u l t s . We then discuss these r e s u l t s and compare them w i t h those p r e v i o u s l y reported.

2. THE b3z: POTENTIAL ENERGY CURVE.- The ab i n i t i o c a l c u l a t i o n s by Kolos and Wolniewicz /21/ o f t h e

p o t e n t i a l energy curves o f

H,

a r e w i d e l y regarded t o be t h e most accurate a v a i l a b l e f o r any n e u t r a l molecule. O f primary concern here i s t h e b3x; po- t e n t i a l curve a r i s i n g from two ground s t a t e H atoms, p a r t i c u l a r l y i n t h e r e g i o n o f t h e shallow van d e r Waals w e l l . The a b s o l u t e accuracy o f t h e p o i n t s c a l c u l a t e d i n t h i s w e l l should be

-lo-'

a.u.

(-3 x K), as evidenced by t h e agreement be- tween t h e l a r g e s t d i s t a n c e p o i n t c a l c u l a t e d by Ko- 10s and Wolniewicz and t h e l o n g range p o t e n t i a l en- e r g y (-C6/r6

- c 8 / r 8 -

~ ~ ~ / r l O ) c a l c u l a t e d a t t h e same d i s t a n c e /22/. One can f i t t h e e n t i r e s e t o f p o i n t s c a l c u l a t e d by Kolos-Wolniewicz t o a s i n g l e multiparameter f u n c t i o n , e.g. t h e s o - c a l l e d S i l v e r a f i t given by F r i e n d and E t t e r s /20/. T h a t f i t , h o w - ever, decreases t h e accuracy o f t h e f i t consider- a b l y as shown i n Table I. We have used a piecewise

Table I. Comparison o f H-H b31: p o t e n t i a l parame- t e r s from d i f f e r e n t f i t s t o t h e ab i n i t i o values o f Kolos and Wolniewicz.

- -

t h i s work r e f e r e n c e 20 piecewise c u b i c g l o b a l form s p l i n e p l u s l o n g range a n a l y t i c a l form

E (a.u.) 2046.96 x 2006.55 x

lo-'

cubic s p l i n e f i t instead. T h i s f i t reproduces

ex-

a c t l y every Kolos-Wolniewicz p o i n t and i t and i t s f i r s t d e r i v a t i v e a r e continuous. A t l a r g e d i s - tance, t h e f i t merges smoothly t o t h e l o n g range p o t e n t i a l energy given above. I n t h i s way, we f e e l we have maintained most of t h e accuracy i n h e r e n t i n

t h e Kolos-Wolniewicz c a l c u l a t i o n and have represen- A-G80-259-2

I ^{I I} I

t e d the two-body p o t e n t i a l much more a c c u r a t e l y than have F r i e n d and E t t e r s . Note i n p a r t i c u l a r t h a t o u r more accurate p o t e n t i a l r e p r e s e n t a t i o n i n comparison t o t h e i r s has a w e l l depth (E) 40.4 x

lo-'

a.u. g r e a t e r (-0.13 K), a hard sphere r a d i u s ( a ) 33 x a. smaller, and a w e l l minimunl p o s i - t i o n (, ,,r

) 47 x a, smaller. Even t h e values o f

n

= f i 2 / ( 2 u ~ a 2 ) d i f f e r by over 1% (p i s t h e reduced mass).

3. CALCULATION OF SCATTERING LENGTH AND EFFECTIVE

RANGE.-

E f f e c t i v e range t h e o r y (ERT)

,

o r i g i n a l l y used i n a n a l y z i n g low energy n u c l e a r two-body s c a t - t e r i n g data /23/, has been w i d e l y adopted a l s o f o r low energy atomic c o l l i s i o n s . Spruch and co-wor- k e r ~ 1241 have shown i t s v a l i d i t y when t h e poten- t i a l f a l l s o f f more r a p i d l y than r-5 as r + m/25/.

Since t h e H-H i n t e r a c t i o n p o t e n t i a l energy curve f a l l s o f f as r-6 a t l o n g range, ERT i s c e r t a i n l y expected t o be a p p l i c a b l e t o t h e case o f H+

- H+

c o l l i s i o n s . The b a s i c expression i n ERT i s t h e r e - l a t i o n /26/ between t h e s-wave phase s h i f t Eo and t h e wave number k:

k c o t 60 =

-*

1

⁺

^{$,k2 +} 0 ( k 4 ) . o r a l t e r n a t i v e l y

t a n 60 = -Ak

- T

1

~

+

A

0 ( k 4 ) ,

~ ~ ~

where A and ro are c a l l e d t h e s c a t t e r i n g l e n g t h and t h e e f f e c t i v e range r e s p e c t i v e l y . Both parameters A and ro have u n i t s o f l e n g t h . They a r e a l s o inde- pendent o f energy b u t do depend on t h e shape o f t h e

- O X ) - EFFECTIVE RANGE THEORY H?- H?

kcor8,= - + + + r O k 2

-

^{- 0 71 -}

-

10 0

-

^{- 0 7 2 -}

0 - 0 7 3 - 0 1

-0.74 -

LEAST SOUARE FIT

-0.75 - --

A

= - 0 74931436 f 8.2 X 1 ~ ~ 0 ; :

- 0 76 - ro = 3 2 3 2563 f 0 0788 a,

- 0 7 7 - A = I 3 3 4 5 5 0 ~

F i g u r e 1. V a l i d a t i o n o f t h e e f f e c t i v e range expan- s i o n of t h e s-wave phase s h i f t 60 f o r low energy

H+

-

H+ c o l l i s i o n s .

range 0.001 a;' 5 k 5 0.0i5

A-' .

S j m i l a r r e s u l t s

0

a r e a l s o obtained f o r t h e o t h e r hydrogen isotopes.

L e a s t - s q u a r e - f i t r e s u l t s f o r A and ro f o r t h e v a r i - ous i s o t o p i c combinations a r e l i s t e d i n Table 11.

Table 11. The least-square f i t r e s u l t f o r s c a t t e r - i n g l e n g t h A and e f f e c t i v e range ro f o r t h e parame- t e r r, . h 2 / 2 U ~ 0 2 corresponding t o v a r i o u s i s o t o p i c species.

species 11 A (a,) ro (ao)

H, 0.55262 1.33455 323.26

HD 0.41457 -0.44755 5090.87

HT 0.36861 -1.59671 533.30

D2 0.27644 -6.90697 75.39

DT 0.23052 -15.52503 39.84

T2 0.18460 -81.54518 23.22

p o t e n t i a l energy curve. Accurate numerical calcu-

l a t i o n o f t h e phase s h i f t 6, i s done u s i n g t h e Nu- We have a l s o p l o t t e d A and ro a g a i n s t t h e v a r i a b l e merov method o f s o l v i n g t h e r a d i a l SchrSdinger n-l i n Figures 2 and 3. Note t h a t ro + as A + 0,

equation 1271. although A 2 r remains we1 1 behaved. The quantum

0

As expected, t h e l i n e a r i t y between k c o t 60 parameter

n

(2 T i 2 / 2 u ~ a 2 ) i s used f o r convenience, and k 2 a t low energy i s q u i t e s t r i k i n g , as shown i n s i n c e i t should be t h e c r i t i c a l q u a n t i t y i n F i g u r e 1 f o r t h e case o f H+

-

H+ s c a t t e r i n g i n t h e

JOURNAL DE PHYSIQUE

7-'

Figure 2. Scattering length A ( i n ao) f o r low en- ergy atomic c o l l i s i o n s i n the b3zU potential a s a

+

function of the reciprocal of the quantum parameter rl ( 5 n 2 / 2 ~ & 0 2

1.

describing t h e present quantum systems /2,28/. For the present calculation we have used the E and o values f o r our f i t t o t h e b3zu potential curves gi-

+

ven i n Table I .

We note here a l s o t h a t our program does repro- duce the previous r e s u l t /20/ f o r the H+

-

H4 s c a t - t e r i n g length i f the representation of the b3z: po- t e n t i a l reported i n reference 20 i s used; we calcu- l a t e A = 1.3701 a. = 0.7250

8,

versus the reference 20 value of 0.72

i.

Figure 3. Effective range ro ( i n ao) f o r low en- ergy atomic c o l l i s i o n s in the b3yu potential as a t

function of the reciprocal of the quantum parameter rl (- n2/2u€02

1.

GASES.- The quantum-mechanical many-body treatment of the energy of an interacting Bose-Einstein gas has been derived by many authors /14,29-32/. The expression f o r energy per p a r t i c l e , E/N, i s gener- a l l y in terms of p a r t i c l e density p , the two-body s c a t t e r i n g length, A , and, in higher order terms, other potential dependent parameters. They may be reduced t o the following equation:

where the higher order terms with potential shape dependent c o e f f i c i e n t s (e.g. a r e a1 1 neglec- ted. B i s a defined s c a l a r f a c t o r of in the

l o g a r i t h m i c term. I n our view, t h e s c a l a r f a c t o r

B

i s somewhat a r b i t r a r y . Since I n (bpA3) = I n 6

+

I n pA3, t h e I n

B

term can be grouped i n t h e pure pA3 t e r m i n t h e s e r i e s expansion which has been o m i t t e d i n t h e present c a l c u l a t i o n . Using A = 1.33455 a.

(0.70621

W ) ,

t h e E/N values o f s p i n - a l i g n e d hydro- gen are g i v e n i n F i g u r e 4 for B = 8n, from Sawada

F i g u r e 4. E/N, energy p e r p a r t i c l e ( i n K), as a f u n c t i o n o f d e n s i t y p . The present c a l c u l a t i o n i s shown by t h e s o l i d l i n e . Those r e p o r t e d p r e v i o u s l y a r e :

---

by F r i e n d and E t t e r s ( r e f e r e n c e 20); A by Mi 1 le r and Nosanow (reference 18) ;

p

by L a n t t o and Nieminen ( r e f e r e n c e 19) ; + by E t t e r s , Danilowicz and Palmer ( r e f e r e n c e 17).

and Brueckner /31/. A1 t e r n a t e values o f 6 (e.g. 64 n3 i n r e f e r e n c e 14) do n o t s i g n i f i c a n t l y change E/N

i n t h e d e n s i t y regime shown i n F i g u r e 4.

5. DISCUSSION.- The r e s u l t s o f t h e present c a l c u - l a t i o n a r e a l s o shown i n F i g u r e 4 as compared w i t h those p r e v i o u s l y reported. Since o u r accurate H+

-

H f s c a t t e r i n g l e n g t h i s s m a l l e r than t h a t o f

r e f e r e n c e 20 and s i n c e t h e c o n t r i b u t i o n o f t h e l o - g a r i t h m i c term i s n e g a t i v e f o r t h e d e n s i t y range shown, a s m a l l e r o v e r a l l r e s u l t i s expected as com- pared t o t h e values o f F r i e n d and E t t e r s 1201. I t should be noted t h a t s i g n i f i c a n t d i f f e r e n c e s a r i s - i n g from t h e h i g h e r o r d e r terms occur o n l y f o r p >

lOZ0/cm3. O f course, a t h i g h e r d e n s i t i e s (say p = 0.1 Am3 = 3.5 x 1022/cm3) t h e expansion ceases t o be a p p r o p r i a t e .

We a l s o n o t e t h a t here we have concentrated o u r a t t e n t i o n on t h e case o f i n t e r a c t i n g bosons.

However, s p i n - a l i g n e d deuterium, D+, a p r e d i c t e d Fermi gas /28/, i s a l s o a v e r y i n t e r e s t i n g system.

The formula f o r many-body i n t e r a c t i n g fermions has a l s o been developed by Baker /33/. H i s theory, however, r e q u i r e s a d d i t i o n a l e f f e c t i v e range para- meters, i n c l u d i n g t h e p-wave s c a t t e r i n g l e n g t h ; we hope t o c a r r y o u t such c a l c u l a t i o n s i n t h e f u t u r e . F i n a l l y , we conclude by n o t i n g some i n t e r e s t - i n g r e s u l t s shown i n Figures 2 and 3. As

n-'

ap- proaches 5.85008, t h e r e i s a jump o f IT f o r phase s h i f t 60 a t k = 0. The s-wave s c a t t e r i n g l e n g t h then changes from

--

t o rn. T h i s i s t h e evidence f o r having a bound s t a t e according t o Levinson's theorem /34/. T h i s i n d i c a t e s t h a t t h e f i r s t bound s t a t e f o r H f

-

^{H f} b 3 z u p o t e n t i a l curve ( w i t h i n t h e

+

Born-Oppenheimer approximation) w i l l o n l y e x i s t f o r a h y p o t h e t i c a l i s o t o p i c species w i t h mass l a r g e r than m = 3.258176 g/mole (c, 5 0.17094).

Note a l s o t h a t t h e low d e n s i t y l i m i t f o r E/N

i f and o n l y i f A = 0. Since i t has p r e v i o u s l y been shown /35/ t h a t t h e c r i t i c a l

n,

f o r b i n d i n g o f t h e i n f i n i t e - b o d y system i s determined by t h i s same de- r i v a t i v e r e l a t i o n s h i p , our d e t e r m i n a t i o n o f n(A= 9)

= 0.43891 i s e q u i v a l e n t t o t h e d e t e r m i n a t i o n o f

C7-38 JOURNAL DE P H Y S I Q U E

t h e

nm

z

nCB

discussed by o t h e r s /2,28/ p r e v i o u s l y ; i . e . f o r systems governed by t h e two-body Kolos- Wolniewicz p o t e n t i a l , t h e sysi.?m w i l l be a gas a t a b s o l u t e zero i f q 2 qCB = 0.43891 and w i l l be a l i q u i d o r s o l i d otherwise. We f e e l t h a t t h i s i s t h e most accurate a v a i l a b l e d e t e r m i n a t i o n o f qCB, which may be compared w i t h t h e value 0.45576 ob- t a i n e d by M i l l e r e t a l . /28/ (using t h e Lennard- Jones approximation t o t h e Kolos-Wolniewicz poten- t i a l ).

6. ACKNOWLEDGMENT.- Acknowledgment i s made t o t h e Donors o f t h e Petroleum Research Fund, administered by t h e American Chemical Society, f o r support o f t h i s work. We a l s o wish t o thank Professors Richard E t t e r s and Manuel de Llano f o r i n t e r e s t i n g discus- sions.

References

/1/ Stwalley, W. C . and Nosanow, L. H., Phys.

Rev. L e t t .

36

(1976) 910; and references t h e r e i n .

/2/ Nosanow, L. H., i n "Quantum F l u i d s and So- l i d s " ( e d i t e d by S. B. Trickey, E. D. Adams and J . W. D u f t y ) Plenum, New York (1977), p.

279.

/3/ B e r l i n s k y , A. J., Phys. Rev. L e t t .

39

(1977) 359.

/4/ Stwalley, W. C., Phys. Rev. L e t t . 37 (1976) 1628; a l s o Stwalley, W. C., J. de Physique

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