PHYSICSBUDAPEST INSTITUTE FOR RESEARCH CENTRAL cHungarian ‘Academy of Sciences У iST-ОЯГг

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Р. K Á L M Á N J. S Z I K L A I

У iST-ОЯГг

~ - / / S г'jf '/ 2

KFKI-1979-25

POSSIBLE EFFECT OF FREE ELECTRON AND ION DENSITIES ON THE RESULTS OF

LAMB SHIFT MEASUREMENTS

cHungarian ‘Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

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2017

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KFKI-1979-25

POSSIBLE EFFECT OF FREE ELECTRON AND ION DENSITIES ON THE RESULTS OF LAMB SHIFT MEASUREMENTS

P. Kálmán and J. Sziklai

Central Research Institute for Physics H-1525 Budapest P.O.B. 49. Hungary

Submitted to Phye. Rev. Lett.

HU ISSN 0368 5330 ISBN 963 371 538 5

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ABSTRACT

Corrections to the Lamb shift occurring in plasma surroundings are discussed.

АННОТАЦИЯ

Были определены поправки смещения Лемба в окружении плазмы.

KIVONAT

A Lamb eltolódás plazma környezetben keletkező korrekcióit tár

gyaljuk.

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In the last t e n y e a r s several p u b l i c a t i o n s h a v e

a p p e a r e d d e t a i l i n g L a m b shift m e a s u r e m e n t s i n h y d r o g e n i c s u p 1 2

to Z = 18 ' . The e x p e r i m e n t a l t e c h n i q u e s u t i l i z e d s uggest that the c o r r e c t i o n caused by the m i c r o p h y s i c a l s u r r o u n d i n g s , i.e, by r e a l photons, free e l e c t r o n s and io n s b e i n g p r e s e n t be taken i n t o a c c o u n t . The t e m p e r a t u r e d e p e n d e n t part of the L a m b shift was

3 h

c a l c u l a t e d in the case of b l a c k b o d y r a d i a t i o n ^ ’ . In the a b o v e 1 2

m e n t i o n e d e x p e r i m e n t s ' the p h o t o n s p e c t r u m is not like a b l a c k b o d y one, but is r e m i n i s c e n t r a t h e r of the s p e o t r a r a d i a t e d by p l a s m a s . In v i e w of this, we d i s c u s s c o r r e c t i o n s to the Lamb s h i f t o c c u r r i n g in p l a s m a surroundings, e s p e c i a l l y in l o w d e n s i t y plasmas.

To get the e f f e c t of the pho t o n s w e u s e the n o n r e l a t i - 3

v i s t i c L a m b shift c o r r e c t i o n f o r m u l a of W a l s h o b t a i n e d for the c a s e of b l a c k b o d y r a d i a t i o n

Д Е п Г “ 772 У ^ n n A c ^ [ E - E + h w + E - E - h w ] *

41Г y n n m n m

* N ( w )kdkdi2k ,

w h e r e uj d e n o t e s the f r e q u e n c y a n d к the w a v e n u m b e r of the p h o t o n s , N(io) is the p h o t o n o c c u p a t i o n n u m b e r , e^ is the p o l a r i z a t i o n v e c t o r of the pho t o n s a n d d ß ^ is the u n i t solid a n g l e in the к space. T h e e n e r g i e s of the s t a t i o n a r y states

of the hydrogen-lilce s y s t e m s are d e n o t e d b y E n and the e n ergy of the i n v e s t i g a t e d s t a t e is E , T h e p is the m a t r i x e l e m e n t

m f n m

of the -i h V o p e r a t o r a n d oL is the f i n e s t r u c t u r e constant.

I n s t e a d of the formula of the b l a c k body r a d i a t i o n : N(w) = (eh w / K T _ i ) ”1

we s u b s t i t u t e the o c c u p a t i o n n u m b e r of the p l a s m a r a d i a t i o n N (из) Ír

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- 2 -

into (l), The N (to) is o b t a i n e d f r o m the s p e c t r a l e n e r g y P

d e n s i t i e s of the t r a n s v e r s e a n d l o n g i t u d i n a l w a v e s ( u ^ T a n d u w L ) u s i n g the f o l l o w i n g i d e n t i t y

[ u dio = ---- л [ fecuN (to) d 3k

J

^(2Г)3

J P ⁽

²

⁾

a n d the d i s p e r s i o n r e l a t i o n s of the t r a n s v e r s e a n d l o n g i t u d i n a l photons. The l o w f r e q u e n c y l i m i t of the plasma r a d i a t i o n g ives the d o m i n a n t e f f e c t and we u s e the s p e c t r a l e n e r g y d e n s i t y f o r m u -

K

lae of this e x t r e m e case . T h e tra n s v e r s e a n d l o n g i t u d i n a l p h o t o n n u m b e r s are

w = --- _K T pT E - E Я

n m

(3) and

N T = pL

K T

(E - E ) X'

n m

2 ^2 fiwp (*)

w h e r e to^ is the plasma f r e q u e n c y , T the e l e c t r o n temperature, К the B o l t z m a n n c o n s t a n t a n d X я ti<V(En “ E m ), Putting (3) and

(*♦) i n t o (l), u s i n g the k,j,dkT = ig<od<o a n d the k ^ d k ^ = — 4ga)d«o

u к 3u 2

i d e n t i t i e s o b t a i n e d from the d i s p e r s i o n r e l a t i o n s 3 (where u я K T /m) a n d the f o r m u l a e

<p „ Ä )‘ d e -=

^ r l o I2 for transverse photons

3 ^nra*

b “ * ь p

*3 IpnnJ Y ° r -*-o n S ^ ^ u d ^ n a ^ P h o t o n s

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we o b t a i n the s h i f t c aused b y the t r a n s v e r s e a n d l o n g i t u d i n a l p h o t o n s present i n the p l a s m a r a d i a t i o n

2 A E t = - 4 * ^ 4 £ |pn j 2 ÍCT f 2

mT 31Г (me f n ' nml

J

¹

-

^{X 2}^{г dX}

^ E m L = “ Or ) Г2 £ fPn mi

о

2 K T

9T / l^nml E - E

(m u) n “

n ш 4

^I(Y)

(

6

)

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where

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- 3 -

i(y )

oo

2 d X

= J

x C i - >x2 )

(

8

)

wit h Y = / ( E - E ). T h e choice o f Y as the l o w e r l i m i t in

p n m

i n t e g r a l (8) e x p r e s s e s the fact tha t all p h o t o n s h a v i n g f r e q u e n c y M ij are a b s o r b e d . T h e integral i n (6) g i v e s zero thus

^ p

Д Е _ = О and the shift is c a used s o l e l y by the l o n g i t u d i n a l m T

photons. In l o w den s i t y p l a s m a s the r e l a t i o n Y « 1 h o l d s , t h e r e f o r e I(y ) can be w r i t t e n a p p r o x i m a t e l y as I = — 2 £nY.

U s i n g the i d e n t i t y P = — im(E - E )x /li, the level s h i f t of the state m is

in (E — E ) , 0

Л Е = £-(i<o T --- n ■- S- T 2 2 In

m 9ir p ^ j-2 '-nm1 E - E

n m

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w h e r e x is the d ipole matrix eleme n t . The c o n c r e t e f o r m s of

— nm

the l e v e l s hifts i n the 2 S and 2 P states a r e ,nlf 2 (n —k

Пá

kn

A E 2S= 3 IR2oI 2 i n ~ ± I + B

й Е гр= (5

+ в -

36r p

RnO 21

2 2 I n2

+ 3 |R 21

n 2 -/*

2 )s!4 t n kn

kn n 2 - I»

R10 21

2 Ц

w h e r e

В = - tn

З Г p n w

,nl

(lO)

(

1 1

)

R n is the p r i n c i p a l q u a n t u m n u m b e r a n d the q u a n t i t i e s ^ 2 0 ’ 1V21 and r" 2 are e x p r e s s e d i n atomic u n i t s ^ . The e n e r g y s h i f t

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)

nO

d i f f e r e n c e

Jl = Ü E 2S- й Е 2Р is J - 22 i3* Й«о Í3 7 Г n 5 (n - 2 )2n~6

1 “ 3 llir P 1 n>2 (n + 2 )2n+ó

4 ♦ 4 i

⁽¹³⁾

and it has no Z d e p e n d e n c e

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- k -

A n o t h e r effect o f the e l e c t r o n d e n s i t y is a s h i e l d i n g of the C o u l o m b p o t e n t i a l of the n u c l e u s . The D e b y e - H ü c k e l p o t e n tial f o r an e l e c t r o n in the s h i e l d e d f i e l d of a n ion of charge Ze is

V(r ) (

**1*0**

w h e r e d is the Debye length,

d = (KT/htn e2 )1//2 , (1 5)

n^ a n d T a r e the d e n s i t y and t e m p e r a t u r e of the free e l e c t r o n gas, r e s p e c t i v e l y . The e n e r g y l e v e l s of pure C o u l o m b p o t e n t i a l are s h i f t e d b e c a u s e of t h i s shi e l d i n g . A u s e f u l a p p r o x i m a t e f o r m u l a of t h e s e level s h i f t is o b t a i n e d by S m i t h

7

in t h e form of a p o w e r s e r i e s in a Q/d ( a Q the B°hr r a d i u s )

£1 £1

Д Е п £= П(2~ р - jy(~jp0 2 \ [зп2 - +1)] + h i g h e r order t e r m s ) .(l6)

The l e v e l shift d i f f e r e n c e b e t w e e n the 2S a n d 2P s t a t e s c a u s e d by the s h i e l d i n g is

V - I ( > 2 - < « >

The L a m b shift a n d its c h a n g e by the Stark e f f e c t is d i s c u s s e d by B e t h e and S a l p e t e r f o r the case of w e a k e x t e r n a l

Q

f i e l d s . T h e i r r e s u l t f o r the “ 2i>l/2 s p l i t t i n g is

4E(2S1/S- a P l / 2 ) = I l [i . ( ! -■> <l(na- l ) ( n m ) V e 2a | )l/aj _

w h e r e E is a n e x t e r n a l e l e c t r o s t a t i c field, e is the u n i t c harge, n a n d m are the p r i n c i p a l a n d the m a g n e t i c q u a n t u m

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n u m b e r s r e s p e c t i v e l y , a n d L is the L a m b s p l i t t i n g w i t h o u t e x t e r n a l field. I n our c a s e E is the sum of two terms

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5

E = E , + E .

— — ext — ion

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о

w h e r e Е , is the a p p l i e d e x t e r n a l f i eld a n d E . л s Z er/r*:

— ext — x o n p — p p

is the e l e c t r o s t a t i c f i e l d p r o d u c e d b y a p e r t u r b e r i o n of Z e c harge. The t o tal change o f the L a m b shift c a u s e d b y a l l the ion s pre s e n t c a n be o b t a i n e d by t a k i n g the a v e r a g e of (l8) with the i o n - d i s t r i b u t i o n f u n c tion, n o r m a l i z e d to u nity, g i v i n g the p r o b a b i l i t y of f i n d i n g a p e r t u r b e r i o n at a d i s t a n c e r from

P the h y d r o g e n - l i k e ionо

d p £ ). .-<Vr>

Г г

w h e r e the q u a n t i t y p is d e f i n e d b y

(

2 0

)

kt -3

3 n P= ?

(

2 1

)

a n d n ^ is the d ensity of the p e r t u r b e r ions. T h e c r o s s term f r o m the a v e r a g e v a n i s h e s b e c a u s e of s y m m e t r y c o n s i d e r a t i o n s , and the L a m b s p l i t t i n g takes the fo r m

Г

(°

1

A E ( 2 S 1y2 - 2 P 1^2 )= Jl 1 + (1 + | F + jb I x - 2 /3 e"X dx) (22)

w h e r e x = r/p

b = l6(n2 - 1 ) (nm)2 Z2R2 ( ~ ) * Vl2 , (23)

w h e r e R = me^/2fe2 and

„ 1 / 2 ¹ \ f \2_,2 2 2 / 2 F = R ( n - 1 j(nmj E , e a / L .

— ext o'

( **2*0**

T h u s the c h a n g e of the 2 ^l/2 s p l i t t i n g due to the ion c l o u d is

cf3 = Jb T(|)L (25)

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The total c o r r e c t i o n to the L a m b s hift o c c u r r i n g in p l a s m a s u r r o u n d i n g s is

cf = + </2 + cfj ( 2 6 )

P u t t i n g <o = (^Kn e /m) ' into E q . ( l 3 ) a n d e v a l u a t i n g the

p e

f o r m u l a n u m e r i c a l l y one get s

cf = - 5 . 8 5 8 fn (cm“ ^ ) * lO"”3 MHz. (2 7)

S i m i l a r l y fr o m f o r m u l a (l7) one can o b t a i n

cL= - 1 . 6 6 7 - 1 0 ~ 1 3 n /(ZKT) M Hz , (28)

tS, ©

w h e r e n and K T are e x p r e s s e d in cm -3 a n d eV u nits, r e s p e c -

©

tively. If К Т = О . О25 eV, c o r r e s p o n d i n g to r o o m temperature, then Zcf_ b e c o m e s equal to cf, at n = 10 cm

-3

e l e c t r o n

2 1 e

density. It f o l l o w s f r o m (27) a n d (28) that |cfJ » |zcf,,|

■JO e Q

if n < 10 c m ” J ( s u p posing K T = 0 . 025 eV). E v e n low e l e c t r o n Ö

d e n s i t i e s produce cf^ c o r r e c t i o n s for the L a m b shift w h i c h are in 5 — 3 the o r der of the e x p e r i m e n t a l errors, e.g. n = 1 . 6 6 * 1 0 cm

©

g i v e s ct^= - 0.02 MHz, w h i c h is equal to the e x p e r i m e n t a l error of the Lamb shift m e a s u r e m e n t of A n d r e w s a n d Newton^-0, B e c a u s e of the s t r o n g Z d e p e n d e n c e of L the r e l a t i v e S t a r k e f f e c t

c hange cf^/L, d e c r e a s e s for h y d r o g e n i c s of i n c r e a s i n g Z, as can be seen fr o m E q s , ( 2 3 ) a n d (2 5), thus we o n l y d e a l w i t h the Z = 1 case. U s i n g the e x p e r i m e n t a l v a l u e L = 1 0 5 7 . 8 6 2 M H z ^ a n d

E q s . (2 1), (2 3) a n d (2 5) w i t h Z^= 1 one get s

cf / L = 9,877* l O ” 18 n 4/ 3 . (29)

P

a b o v e m e n t i o n e d experiment'*'^, and s u b s t i t u t i n g it into (29) we

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o b t a i n the n u m e r i c a l v a l u e s of cT^/L at d i f f e r e n t b e a m current d e n s i t i e s i, e x p r e s s e d in p A /mm u n i t s (see Table l).

The a b o v e c a l c u l a t i o n s s h o w that the most i m p o r t a n t c o r r e c t i o n to the Lamb shift is l i k e l y to a r i s e from the e l e c t r o n densi t y , t h e r efore we sug g e s t that it be t a k e n into a c c o u n t a n d that it be i n c l u d e d in the s y s tematic c o r r e c t i o n s w h e n e v a l u a t i n g the L a m b shift m e a s u r e m e n t s .

The a u t h o r s a r e very g r a t e f u l to B . K a r d o n for hie v a l u a b l e r e m a r k s on this subject.

T a ble 1.

T h e d e p endence of the r e l a t i v e S t a r k e f f e c t change 5 ^ / L, on the p r o t o n b e a m cu r r e n t d e n s i t y i.

i ( p A / m m ^ )

V L

1 **.5 2.10“7

2 l . l ^ ' l O " 6

5 3 , 8 6 « 1 0 ~ 6

1 0 9 . 7 3 ' 1 0 “ 6

2 0 2 . 4 5 ' 1 0 “ 5

50 8.32*1 0 " 5

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R E F E R E N C E S

^H„ W, Kugel and D. E. Murnick, Rep. Prog. Phys. 4 0 f297 (l977 ).

2 H. G o u l d a n d R.Marrus, P h y s . Rev. L e t t . *11,1^57 (1978 ).

3J.E. Walsh, Phys. Rev. Lett. 27.208 (1971).

^G. Barton, Phys.Rev. (1972).

3T . B i r m i n g h a m , J, D a w s o n a n d C. O b e r m a n , Phys, F luids 13,297 (1965 ),

^ H . A . B e t h e a n d E , E . S a l p e t e r , Q u a n t u m M e c h a n i c s of O n e — and T w o - E l e o t r o n A t o m s (Springer, B e r l i n 1 9 5 7 )»P P . 252- 2 6 0 .

7 C.R. Smith, Phys.Rev. 1 3 ^ A . 12 35 (l964 ).

^See (55.8) in ref.6.

^ J . C ooper, Rep. Prog. Phys. 2_2, 35 ( 1 9 6 6 ).

10D.A.Andrews and G.Newton, Phys.Rev.Lett.37.125^(1976).

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Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Szegő Károly

Szakmai lektor: Kardon Béla Nyelvi lektor: Harvey Shenker

Példányszám: 120 Törzsszám: 79-429 Készült a KFKI sokszorosító üzemében Budapest, 1979. május hó