I L N U O V O C I M E N T O VOL. X X , N. 3 lo Maggie 1961

Electric and Magnetic Polarizabilities of the Nucleon.

¥ . ~. BARASIIENKOV, ]71. J . K A I S E R a n d A . A . OGREBA

Joint I~stihde ]or 2~uclear Research, Laboratory o] Theoretical Physics - D,ub~ta

( r icevuto il 30 Gennaio 1961)

electric polar izabi l i ty ~ = ( 16 --18). 10 -~3 cm a

In connect ion wi th exper imenta l de te rmina t ions of the of the nucleon the expec ta t ion value of this polar izabi l i ty has been evaluated, theore t i ca l ly ( l ) . In the ci ted paper only one d iagram of t he t y p e A (see Fig. 1) has been ( S t aken in to account , the cont r ibu t ion of the o ther ones has )___ 4 / \ been approx ima ted by the factor 2. Tile magn i tude of . ~ ~ A) the magnet ic polar izabi l i ty ¢/ was not ca lcula ted since esti- mates sh0,,-ed that << L_J

In connect ion wi th recent ly publ ished new exper imentM ~ " ~ da ta i t is in teres t ing to eva lua te ~ and fl more precisely. ( ~ - - - B)

Calculat ions showed tha t

co

4e 2 f ~-4( ~2C04 ~ e2a2 . (1) ~=-v~f-'. d k e x p [ - - k ~ / a 2 ] ~ 2 k 2 - b ] 0 ÷ a4 ] 144~3

0

• dpexp[p wa~] a . ÷ ~_ dk exp[--k2/a '~] • ' o , , 3 o)~(~ + o , , ) "

o o

• 28 k 6 + 22 4- - 3 - - ~ a"-

Here a±(p) are the total cross-sections for t he scat ter-

, 'i,y!, 19)

Fig. 1. - - - the nu- cleon: - - - the g-me- son: ~ the electro-

magnetic field.

ing of ~±-mesons by pro tons : o)p= ~ / ~ ) 2 being the energy of these mesons;

(l) V. S. BAR.k~4H~:NKOV and B. M, ]~AIeB_kSHEV: .Vuel. 1)hys., 9, 426 (1958); O. I, BLOKHINTSEV, V. S. BAR:kSHENKOV a n d B. ~I. B_kRB_~SrI}W: Usp. Fiz . Naulc, 68, 417 (1959).

(*) The electric and magne t i c polarizabili t ics of a nucleon arc defined b y the relat ion H = - - ½ ~E ~ - ½ ~B 2 in the s tat ic l imit , where E and B are cons tan t electric and magne t i c fields respect ively; H is the e~-proportional energT change of the nueieoR (hie to the fields Q'~).

(2) A. M. B2,L])IX: Nucl. Phys., 18. 310 (1960).

594 V. S. B A R A S H E N K ( O , ' , 11. ,1. KAISEI£ a n d A. A. O G R E B A

m=-(%; ]2=0.08 being the (=~P)-coupling cons tan t ; v ( k ) ~ e x p [ - k2/2a ~] being the form factor of the meson field source: a = 5 . 6 . As usual we have set l ~ = c = H = = l .

Eq. (1) contains the cont r ibut ions of the diagrams of the type B, C and D. The ma t r ix e lement corresponding to the shaded regions of these d iagrams is expressed wi th the aid of the dispersion relat ions in terms of ]2 and the cross- sections ~± . (For this it is assumed t h a t the nucleon recoil can be neglected 0)). The e indica ted in (1) contains correct ions of the order Ja and higher, which amount to 5:~ ___ 2.5-10 -~3 cm 3. Diagrams of the t y p e D do not cont r ibu te to a.

For the magnet ic polar izabi l i ty one obtains the value

(2)

co

f l = .~. ] k e x p [ - k2/a ~] 2 k ~ - 6 a ~ ] ~ s ÷

0

co co

16 e~-] ~ P k e x p [ - - ( k 2 ÷ P e ) / a 2]k'q) 2 2 -a2 2 - -

0 0

a co

+ 24.-v a p exp [pZ/a'Z] ~- 2 '~- }lk exp [ k2/a2] • • %, j o ) ( o ) + % 0

0 o

• - - - - 1 2 - - ÷ ~ _ - 0 , 2 . 1 0 4 a c r e s . -- 9 a ~ a ~"

In ca lcula t ing fl the d iagrams of the type B, C, D are likewise taken into account . The cont r ibut ion of the diagrams of the type D is compensa ted almost comple te ly by the cont r ibut ion of the corrections in B and C. The to ta l cont r ibut ion of al:l t e rms propor t ional to /t and to higher powers of [ are 0 < Aft ~ 0.1.10 ~aa cm a. (More accurate numerical calculat ions of the.~e corrections are in preparat ion) .

The values ~ and fi are not very dependent on the shape of v(k.) In the papers (2,3) exper imenta l values of ~ are given obta ined from the Compton

effect on the proton. However the coefficient in the formula for the effect ive cross- section which has been named in these papers the electric po la r izab i l i ty of the pro ton is in fact the sum of the two te rms:

Z,,r~> + ~ ,

where M is t i le mass and % / ( r - ) = ( 0 . ~ 0 . 0 4 ) ' 1 0 -la em is t i le mean square radius of the pro ton (*).

The considerat ion of the te rm conta ining <r"-> changes no t i ceab ly the q u a n t i t y ob ta ined from (2,3).

(a) V. I . (~OLI)ANSKY, O. A. KARPUKIIIN. A. V. KUTZENKO a n d \ ' . V. PAVI,OVSKAYA: ~ t t r t t .

l~,ksp. Teor. Fiz . , 38, 1695 (1960) . (*) W e a r c i n d e b t e d to A. M. BALDIN a n d V. A. ]~ETItUNKIN fOF d i s c u s s i n g t h i s p r o b l e m (for

d e t a i l see (%).

(~) H . MIY.~ZAWA: Phys. Per., 104, 17~1 (1956) .

T

> 4

]~LECTI~I(' AND MAGNETIC I ' ( )LAR.IZABIIATIES ~)1" '[ '[ll ~" NUCLEON 595

fl ()btained f rom

l 'ho(olm)duc, t ion of mesons on protons (z)

4 l ' ho toproduc t ion of mesons on protons (v)

12

5.5 :~2,3 (')

7 .5~4 .2

0 < c~ < 200

800 ± 350

0 Pho toproduc t ion of mesons on protons (9)

Compton sca t ter ing by protons (2)

i Considerat ion of the Compton effect toge ther 7 .4--c~ (') wi th the photoptodu(: t ion of n-mesons on the

proton (2)

( 'onsidorat ion of t he ( ' ompton effect toge ther 2 - - 2 ( ' ) wi th the i )holoproduct ion of =-mesons on the

proton (s)

--- Compton sca t ter ing by protons (s)

- Scat ter ing of slew neut rons by heavy nuclei at small angles (~)

I . . . . . . . i . . . . . . . . . . . . .

- Sca t te r ing of slow neutrons by heavy nuclei at small angles (~)

( ' ) In t h i s ease t he u n c e r t a i n t i e s of ~ and ~ dtle to expevim~ 'n ta l e r ro r s i n t he u sed v a l u e s of t he p h o t o p r o d u e t i o n c ross - sec t ions a re no t i n d i c a t e d .

The values of ~ and fi (in units of 10 as cm 3) axe givei~ in the table wi th t h e exper imenta l uncer ta int ies indicated. As is seen the theoret icM values of ~ and fl are close to those of x and fi ob ta ined in (2,3) from the direct a nMysis of the ex- per iment al data.

We are grateful to D. I. BL ~KIIINTSEV for discussing the i)roblem.

(~) I(. 5[. TItALEII: Phys. 12er., 11~, ~27 (1959). (~) V. A. PETRUNKIN: 2~ur,~. Eksp. Teor. Fiz., (in p r in t ) . (7) L. L. FOLDY: (ROt p u b l i s h e d , c i t ed b y (4)). (s) V('. A. ALEK.%tNI)ROV: J~urn. Eksp. Teof. Fi:., 3,3, 294: (1957). (a) L. I . LAPIDU~ a n d CHef" KUAX<;-('H~,O: p r e p r i n t J INl¢ ])-532, p t ' i va t e c o m m u n i c a t i o n .