New classical inversion formulas for centrosymmetric electric and magnetic fields; focusing potentials

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New classical inversion formulas for centrosymmetric electric and magnetic fields; focusing potentials

I . V. Bogdanov and Yu. N. Demkov

Leningrad State University (Submitted 30 December 1981)

Zh. Eksp. Teor. Fiz. 82, 1798-1806 (June 1982)

New inversion formulas are obtained for the classical scattering of a charged particle by a spherical or axisymmetric electric or magnetic field at a fixed impact parameter or angular momentum. For different cases, focusing fields are obtained similar to those previously considered for scattering by an electric field at a given energy, viz., of the backscattering (cat's eye), Maxwell fish eye, or Luneberg lens type. A magnetwlectric analogy is formulated, namely the existence of equivalent axisymmetric electric and magnetic fields that scatter charged particles in identical fashion.

PACS numbers: 41.80.

-

^y

1. INTRODUCTION F o r scattering of a particle by a magnetic field, the analog of a spherically s y m m e t r i c a l potential is a field The trajectory of a particle s c a t t e r e d by a spherically

H directed along the z axis and s y m m e t r i c a l with symmetric potential f l y ) is determined in the c l a s s i c a l

r e s p e c t to rotations about this a x i s , while the trajec- approximation by two p a r a m e t e r s , e.g., the particle

t o r i e s of the incident charged p a r t i c l e s a r e a s s u m e d t o velocity v, a t infinity and the impact p a r a m e t e r b, o r

lie in the ( x , y ) plane.

e l s e the total energy E and the angular momentum I.

The deflection angle

x

^isa function of these two param- An inversion formula f o r this problem could be ob- e t e r s , and when solving the inverse problem we m u s t tained only f o r the c a s e of a fixed generalized momen- reconstruct V(Y), given

x

and c e r t a i n values of these tum Po

=

I.

parameters.

Inversion formulas can be used to obtain fields having T h e r e a r e a t present two quadrature solutions of the various focusing properties.= F o r the Hoyt c a s e , in inversion problem: the F i r s o v algorithm,' in which the e l e c t r i c and magnetic fields, lenses w e r e obtained of angle

x

i s specified a s a function of the impact param- the Luneberg type (which focuses a parallel particle e t e r b while the energy E i s fixed, and the Hoyt algo- beam onto the edge of the lens), of the Maxwell "fish- rithm; in which 1 i s fixed and the deflection angle is eye" type (in which the s o u r c e and the focus a r e a t specified a s a function of energy. diametrically opposite points), and of the "cat's-eye"

type (in which the particles a r e s c a t t e r e d strictly back- We shall show h e r e t h e r e e x i s t s a l s o a thirdalgorithm

ward). The possession of a p a r t i c u l a r focusing proper- with an explicit solution, with the impact parameter

ty by a potential is undoubtedly connected with the inter- fixed and the deflection angle a l s o specified a s a func-

nal s y m m e t r y in the corresponding c l a s s i c a l o r quan- tion of energy.

tum-mechanical problem.¶

The Firsov algorithm is most useful for scattering by

In Sec. 3 we obtain the focusing potentials f o r this a microtarget, for in this c a s e the only quantity that case. In Sec. 4 we d e r i v e an inversion formula for a we can fix i s the energy. Next, knowing the intensity of _fixedimpact Section deals with the in- scattering Of a wide beam with the target) Of version problem for an axisymmetric magnetic field at incident particles through various angles we can r e - a fixed angular momentum, and the magneto-electric construct the function x(b) (if certain uniqueness condi- analogy. The focusing magnetic fields for this case tions a r e satisfied).

a r e obtained in Sec. 6.

When a macroscopic spherically o r axially s y m m e t r i c force field i s probed, the value of ~ ( b ) can be obtained by scanning the field with a particle beam that i s narrow compared with the field dimensions. This measure- ment calls f o r the s o u r c e to be moved. Another pos- sibility is to fix the s o u r c e position (and accordingly fix the impact p a r a m e t e r ) and v a r y the particle energy, with x(E) measured a t constant b. This method may prove to be m o r e convenient in s o m e cases.

It appears that t h e r e exist a l s o inverse-problem formulations other than the three mentioned above.

The deflection angle can in the general c a s e b e specified on s o m e line located in the plane of the p a r a m e t e r s E and 1 o r v, and b. The question i s : what conditions must this line satisfy to make a solution of the inverse problem possible?

2. HOYT'S FORMULA

We consider in the c l a s s i c a l approximation three types of inversion formulas that reconstruct a centro- s y m m e t r i c field V(Y) f r o m the deflection angle x of the particle: a t a given particle energy E (Ref. I), a t a given angular momentum I (Ref. 2) and a t a given impact parameter b. We a s s u m e h e r e a f t e r V(r) is a r e - pulsive field, i.e., V> 0, and that

v(-)

= 0 , although many r e s u l t s hold a l s o in a m o r e g e n e r a l case.

We consider f i r s t the second inversion-formula type, i.e., a t a given angular momentum I. F o r the polar angle q, in the field V ( Y ) we have (see, e.g., Ref. 5) the e x p r e s s ion

1037 Sov Phys. JETP 55(6), June 1982 0038-5646/82/06 1037-05$04.00 O 1982 American Institute of Physics 1037

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s p h e r e of radius R. This focusing condition corresponds to the so-called Maxwell's fish eye (see Ref. 3). H e r e

2 arc sin ( b / R ) , b t R

r ~ ) - ( O, b > ~ ^a

where h 3 I/- is fixed.

Introducing now a new variable U a

v

+ A'/? and chang-

ing from r to U, we obtain Using the tabulated formulas7

z sin z& n I-cosa

1

i o s z z (cosa z-cos2 =-- 2 cos2 a where r(U) is the function inverse t o U(r).

Formula (1) should be regarded a s an integral equation f o r the function l/r(U) when the left-hand s i d e i s given. This is an Abel equation and can be solved analy tically6:

and putting 5 = b / ~ 3 c o s x and a m A/RlJ'I2 a cos a , we obtain f r o m the inversion formula (3)

F r o m this we get the potential An inversion formula a t fixed A in the form (2) was

f i r s t obtained by Hoyt.'

Thus, the Coulomb potential (5), cut off a t r = R , Using now the formulas I =mv,b and E = m&/2, we

focuses particles leaving a n a r b i t r a r y point on a s p h e r e introduce the impact parameter b and note that under

of radius R unto the diametrically opposite point of the conditions of our problem, when A i s fixed and E i s

this splere, if the particle orbital is varied, the impact distance is a variable quantity and z = x ~ . Just a s in the c a s e of the "fish-eye" poten-

='/' O r The Hoyt (2) can then be tial, we can continue this potential analytically f a r t h e r rewritten by introducing it in the impact p a r a m e t e r b

explicity. Changing in (2) f r o m E t o b and introducing than the r a d i u s R; the focusing property will be p r e - served a l s o f o r trajectories that e m e r g e outside the the particle deflection angle

x

=2q,

-

^n,we g e t ulti-

mately s p h e r e r =R. This focusing property of a Columb field

reflects the known fact5 that a bundle of Coulomb t r a - jectories passing through two points diametrically opposite relative t o the f o r c e center corresponds t o a constant angular momentum.

U ( r ) =V ( r ) +A2/?

2) L e t R , = = and R, = R , i.e., a plane-parallel beam i s focused onto the s u r f a c e of a s p h e r e of radius R . Such a focusing system i s usually called a Luneburg lens.= In this c a s e

Equations (3) constitute in fact the solution of the clas- s ical inverse scattering problem if 1 =A$&- is fixed and t h e deflection angle is specified a s a function of the impact parameter.

arc sin ( b / R ) , b<R

x O ) =

I

^0, ^b>R

3. FOCUSING POTENTIALS

Using the cited tabulated values of the integrals, we obtain f r o m (3)

We construct now with the aid of the inversion formulas (3) spherically s y m m e t r i c s y s t e m s that have a radius R and focus particles with a specified angular momentum, in the s a m e manner a s F i r s o v ' s formula

1 was used in Ref. 3 to construct s y s t e m s having spherical s y m m e t r y and focusing particles with specified energy.

f r o m which we determine the potential. If a centrifugal potential A'/? is added to the potential V and is analytically continued to the singularity r =2R, the potential becomes s y m m e t r i c a l about the point r =R. This yields, e.g., an additional s y m m e t r y property for bound s t a t e s of the corresponding quantum problem. We have The focusing condition (see Ref. 3) c a l l s h e r e f o r a l l

the particles with specified A, emerging f r o m a certain point on the s y m m e t r y axis a t a distance R , f r o m the f o r c e c e n t e r , to land a t another point of the s a m e axis and located a distance R, f r o m the center.

Equation (6) specifies in f a c t the potential that r e a l i z e s a Luneburg lens f o r particles with a given angular momentum.

In this c a s e , as shown in Ref. 3 , arc sin ( b / R i ) +arc sin (b/R.) , b<R

x ( b ) -

{

_0, _b>R.

The inversion problem has a l s o a n exact solution in the c a s e of scattering a t a constant angle, i-e.,

x

=cn, where c is a n a r b i t r a r y r e a l positive quantity.

We consider two interesting particular c a s e s of (4), in which a l l the calculations can b e c a r r i e d through to conclusion.

The inversion algorithm (3) yields f o r this c a s e 1) L e t R , =R, = R , i.e., the "source" and "sink" of the

particles a r e a t diametrically opposite points on a

1038 Sov. Phys. JETP 55(6), June 1982 I. V. Bogdanov and Yu. N. Dernkov 1038

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If the field V(Y) is specified in a l l of s p a c e , R m u s t be extended to infinity. The potential is then

At c = 1 we obtain the so-called c a t ' s eye (backscattering). The corresponding potential is

F o r a field cut off a t ^Y= R i t i s e a s y to obtain f r o m (7) f o r V a quadratic equation that becomes linear a t c =1 and yields f o r the potential the expression

The cut-off field (10) i s thus capable of reflecting particles incident on i t with an angular momentum 1 = ~fi.

4. INVERSION FORMULA FOR A GIVEN IMPACT PAR AM ETER

We obtain now a s i m p l e inversion formula a t a fixed particle impact parameter b. We a s s u m e that the deflection angle

x

is given a s a function of the energy E.

From Ref. 5 we have bdr

"(')=I rZ[l-Jr'/e-b2/?l".

r m

-

_bdr

V,/E"~=

J

r ( ~ - @ ) ' ~ [ ~ - f v / ( ? - b 2 ) ^{] j h} ^'

.I

We introduce now I J ( ~ ) ~ ? ~ V ( Y ) / ( T ~ - b 2 ) and change f r o m

Y to U in the integral. Then

I,,,

- bdr dU d

q,lE'"= =J---- arc sin

-

b

r ( ? - b Z ) " [ E - U ( r ) ] ' " ( E - U ) ' " d U r ( U ) ' (11) This is an Abel integral equation f o r the function a r c s i n (b/r(U)) a t a given angle p0(E). Introducing the particle deflection angle x ( E ) and using Eq. (6), we obtain

b 1

"

x - x ( E ) arc sin

-

⁼^-

r ( u ) 2 n ! k m d E . F r o m this we get ultimately

The inversion formulas (12) solve in f a c t the foregoing inverse problem.

I t is thus known, f o r example (Ref. 5, p. 72) that f o r a field V = a /?2 with CY > 0 the deflection angle is

We apply t o this c a s e the inversion algorithm (12). Sub- stituting (13) in (12) and carrying out elementary inte- gration, we obtain

U ( r ) = a / ( ? - b z ) , V = a l f , a s it should be.

5. INVERSION FORMULA FOR A MAGNETIC FIELD Consider the scattering of a classical nonrelativistic particle of m a s s m and charge e in an axially symmetric

field H directed along the z axis. The vector potential is

A= ( 0 , A , ( p ) , 0)

-

^{( 0 ,}^A^{( P )}^,^0).

The component of the vector H a r e H,=O, H,=O, Hz--H=rot, A-A ( p ) /p+aJ ( p ) .

The Coulomb gauge condition div A = 0 is found to be automatically satisfied. We consider only planar (at p, = m i = 0) motion of the particle, and replace p with Y.

F r o m the Lagrangians of such a particle in a magnetic field of the indicated configuration (c =1)

L = - ( rn

2 ^i8+P@')+eripA ( r )

we obtain two integrals of motion: the generalized angular momentum

l=p,=mr"@+erA ( r ) and the energy

Now, using these two conservation laws, we easy construct a q u a d r a t u r e expression f o r the particle t r a - jectory and the deflection angle. We have = ( I - m ~ ) / m P and from the energy integral we get

Eliminating dt with the aid of the angular-momentum conservation law and integrating, we obtain ultimately

In analogy with the c a s e of a central electric field, such a t r a j e c t o r y i s s y m m e t r i c about a straight line drawn f r o m the symmetry center to the closest trajectory point. The polar angle is

Using (15), we construct now an inversion algorithm that allows us to reconstruct the potential A(?-) given the function x(E), where

x

^{= 2 p 0}

-

^n,a t a fixed particle angular momentum 1. We a s s u m e hereafter that

rA ( r ) +O, r - t m .

We introduce in (15) new variables

With allowance f o r (16), we have

Expression (17) i s a nonlinear integral equation with a boundary condition for the function g(s) a t a specified po(x) [or X(x)]. I t can be solved by various means. One is to transform (17) into

where f (s) = 2 / g 2 ( s ) , and changing f r o m the variable s t o f . The resultant Abel-type equations in t e r m s of the antiderivative of the function l / g ( s ) is inverted, the

1039 Sov. Phys. JETP 55(6), June 1982 I. V. Bogdanov and Yu. N. Demkov 1039

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r e s u l t of the inversion is differentiated with r e s p e c t t o s, and a f i r s t - o r d e r differential equation in g ( s ) is obtained and c a n b e d i r e c t l y integrated.

Another possible solution p r o c e d u r e is c o n s i d e r e d i n Ref. 5. T h e final solution of the p r o b l e m (17) i s

1

g ( r ) =exp

-

^{( E )}arch ( l / r g 1 2 m E ) d ~ ,

0

g ( r ) = l / ( l - e r A ( r ) ) =bu/rvv, w h e r e v is t h e p a r t i c l e velocity.

We note now that under o u r conditions, when the angular momentum of t h e p a r t i c l e is s p e c i f i e d , the i m p a c t d i s t a n c e b is the v a r i e d quantity and E =h2/b2, w h e r e X~ 3 l 2 /2 m.

We s h a l l need below a n inversion f o r m u l a that con- tains the i m p a c t p a r a m e t e r explicitly. Changing in (18) f r o m E t o b and integrating by p a r t s , we obtain t h e v a r i a n t of i n t e r e s t to u s

T h i s equation is mathematically equivalent t o the known F i r s o v formula,' s i n c e it coincides with the l a t t e r when g ( r ) is r e p l a c e d by n ( r ) =[I

-

e @ ( r ) / ~ ] " ~ , w h e r e 9 ( r ) i s the c e n t r o s y m m e t r i c s c a t t e r i n g field.

F r o m the physical point of view the equivalence of t h e s e two f o r m u l a s l i e s in the f a c t that if charged p a r - t i c l e s with specified energy E move in a n e l e c t r i c field + ( r ) along c e r t a i n t r a j e c t o r i e s , then the s a m e p a r t i c l e s with specified angular momentum I will move in the magnetic field

along the s a m e t r a j e c t o r i e s . This s t a t e m e n t c a n be called the magneto-electric analogy.

The weak-field approximation c a n be considered in (19) when e r ~ / l < < 1 and x<< 1 (in analogy with the F i r - sov problem). In f a c t , expanding the exponential in a s e r i e s , we have

w h e r e g is replaced by unity in the integrand and a t the l i m i t s . Next, i n t h e left-hand s i d e g = (1

-

e r d / l ) - '

= 1 + e r A / l , and we ultimately obtain a s i m p l e explicit f o r m u l a i n t h e weak-field approximation:

6. FOCUSING MAGNETIC FIELDS

We s h a l l use the inversion algorithm (19) f o r the magnetic field t o c o n s t r u c t planar focusing s y s t e m s of r a d i u s R , of the type a l r e a d y considered in Sec. 3 of this paper and i n Ref. 3 f o r t h e c a s e of a s c a l a r field

V(7 ).

Corresponding to one s u c h exactly solvable s y s t e m s i s the focusing condition (4a), w h e r e the "source" and

"sink" of P a r ~ l c l P s of c h a r g e e l i e on a c i r c l e of r a d i u s R. T h i s is a s y s t e m of the Maxwell "fish-eye" type.

Substituting (4a) in (19) we obtain ( s e e Ref. 3)

s o that t h e magnetic potential is

I t is e a s y t o verify that this v e c t o r potential c o r r e - s p o n d s t o a homogeneous magnetic f i e l d H =

-

l/eR2, and a l l the t r a j e c t o r i e s a r e thus c i r c l e s in this c a s e .

We now extend the homogeneous magnetic field H = H z o v e r a l l of space. T h e c h a r g e d p a r t i c l e s move i n s u c h a field in c i r c l e s a t constant velocities proportional t o the r a d i i of t h e s e c i r c l e . On the b a s i s of the magneto- e l e c t r i c analogy obtained in t h e preceding s e c t i o n we conclude that this c a s e c o r r e s p o n d s t o the Maxwell

"fish eye" problem c o n s i d e r e d in Ref. 3. C l a s s i c a l p a r - t i c l e s of e n e r g y , in a "fish-eye" field

specified in a l l of s p a c e , have c i r c u l a r t r a j e ~ t o r i e s . ~ If t h e s e c i r c l e s p a s s through a point r , they p a s s a l s o through the point -R2r/r' obtained f r o m the initial point a s a r e s u l t of inversion in a s p h e r e having a r a d i u s R and a c e n t e r a t z e r o , as well a s reflection a t the o r i g i n , with r a n a r b i t r a r y point i n s p a c e . C h a r g e d p a r t i c l e s with specified generalized angular momentum I a l s o m o v e s along the s a m e t r a j e c t o r i e s in a homogeneous magnetic field H = - l / e ~ ~ . T h e c i r c l e s passing through d i a m e t r i c a l l y opposite points ( r e l a t i v e to a n a r b i t r a r y chosen f o r c e c e n t e r ) c o r r e s p o n d t o one and the s a m e angular momentum. T h e particle-focusing p r o p e r t y in s u c h a magnetic field c a n b e e a s i l y proved by a purely g e o m e t r i c method.

Another s y s t e m f o r which a l l the calculations c a n be c a r r i e d through t o conclusion i s defined by condition (5a). In the s c a l a r c a s e i t c o r r e s p o n d e d to the so-called Luneburg lens. A plane-parallel b e a m of charged p a r - t i c l e s is focused h e r e into a c i r c l e of r a d i u s R .

The inversion p r o c e d u r e (19) yields f o r this c a s e

and the potential is

A ( r ) = - 1 ( 1 - 1 / [ 2 - r Z / R Z ] ' " ) , r<R.

er (22

T h e inversion p r o b l e m is solved just a s a c c u r a t e l y in the c a s e of backward reflection of c h a r g e d p a r t i c l e s with a given o r b i t a l momentum I. H e r e

and we obtain f r o m (19)

Hence the magnetic c a t ' s - e y e l e n s

1040 Sov. Phys. JETP 55(6), June 1982 I. V. Bogdanov and Yu. N. Demkov 1040

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The magnetic potentials (21)-(23) can be obtained by the indicated magnetoelectric analogy from the corresponding scalar potentials V(7) obtained in Ref. 3.

7. CONCLUSION

The new inversion formulas and new focusing fields obtained here extend greatly the earlier results in this field. They can be of interest for research in electron optics, as well ^astor the heretofore little-investigated connection between the focusing properties of electric o r magnetic fields and their internal symmetries, the existence of additional integral of motion etc. in the problem of particle motion in these field (this question has been elucidated fully enough only for the Maxwell

"f ish-eye" problem, Ref. 4).

As already noted, it is of interest to ascertain how many different explicit inversion formulas exist. I t should be noted, finally, that when using a fixed angular momentum we come closest to the quantum formulation of the inverse problem of reconstructuring the potential from the scattering phase shift. It appears that i t ^is

precisely here that it is easiest to track the transition from the classical to the semiclassical and further to the quantum inverse problem.

b. B. FLreov, Zh. Eksp. Teor. Flc. U, 279 0953).

z ~ . Hoyt, Phys. Rev. 66,664 @939).

=YU. N. Demkov, V. N. ~ s t r o v s l d , and N. B. Bere*, Zh.

Ekep. Teor. Fiz. 60, 1604 (1971) [Sov. Pbys. JETP S8.867 0971)l.

~ Y U . N. ~ e m k w aad V. N. oetrovskir, zh. ~krrp. Teor.

ma.

a,

ail g s 7 i ) I&V. ~av. JETP

n,

i m 3

os7tn.

6 ~ . D, Landau and E. M. L W t r , lrdekhadka ~ c s ) , Nauka, 1973 [Pegamca, 19761.

%. I. Smirnov, Kurs v y ~ s h d matematlld (Course of Hlgimr Mathematics), Vol. 2, Nauka, 1874, p. 250 [Pergamoaj.

'I. S. ~radehtek and I. M. Ryzhlk, Tables of Integrals, Sums, Series, and Products, Academlrr, 1965.

OL. D. Landau and E. M. Litehite, Claesical Theory of FieIda, Pergamcm, 1975.

Translated by J. G. Adashko

1041 Sw. Phys. JETP W6). June 1982 I. V. Bogdanov a d Yu. N. Demkw 1041