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On the motion of a charged particle in an inhomogeneous magnetic field
Brinkman, H.C. 1959
Physi.ca 25 1016-1020
ON T H E MOTION OF A C H A R G E D P A R T I C L E IN AN I N H O M O G E N E O U S MAGNETIC F I E L D *). H. C. BRINKMAN C e n t r a a l L a b o r a t o r i u m T . N . O . , Delft Summary
The motion of a charged particle in an inhomogeneous magnetic field is treated, employing Hellwig's and Kruskal's expansion of the gyration of the particle in a Fourier series. The relation of various constants of motion, the energy of the particle, Kruskal's action integral and Hellwig's higher order correction of the magnetic moment of the particle, is discussed.
1. Introduction. The motion of a charged, particle in a inhomogeneous magnetic field was t r e a t e d b y A l f v d n 1) for the case of very small deviations from homogeneity. H e l l w i g 2) un d K r u s k a l 3) indicated how the problem can be solved in higher approximation. Generally speaking their t r e a t m e n t s of the problem are equivalent. The motion of the particle is developed in a Fourier series with respect to time. The first term gives the motion of the so-called guiding centre which follows the lines of force in first approximation. The higher terms describe the gyration of the particle around the guiding centre. Hellwig's and Kruskal's t r e a t m e n t s differ in detail. Kruskal's presentation appears to be more systematic. Moreover K r u s k a 1 gives a very general proof for the existance of an adiabatic invariant, valid for a motion which m a y be split up in an oscillating periodic and a slowly v a r y i n g secular part. Kruskal's theorem is analogous to the adiabatic invariance of action integrals 4) introduced b y Ehreniest in q u a n t u m t h e o r y . Hellwig derives an invariant in second approximation, obtained after very involved calculations. I t seemed worthwhile to extend Kruskal's t r e a t m e n t in such a w a y that Hellwig's result fits in his general scheme. The use of Kruskal's invariant and of the energy integral greatly simplifies the calculations.
2. The Hellwig-Kruskal development. The equation of motion for a charged particle in a constant magnetic field reads ~r = Ire(r)]
(1)
Here r is the radius vector of the particle, s is the ratio of its mass and charge (s ---- m/e), while B is the magnetic field strength. *) T h i s w o r k is p a r t of the r e s e a r c h p r o g r a m m e of the " S t i c h t i n g v o o r F u n d a m e n t e e l O n d e r z o e k d e r M a t e r i e " ( F o u n d a t i o n for F u n d a m e n t a l R e s e a r c h on M a t t e r (F.O.M.)). It w a s c a r r i e d o u t a t tile F.O.M. I n s t i t u t e for P l a s m a - p h y s i c s , a t J u t p h a a s .
--
1016-
PARTICLE IN INHOMOGENEOUS MAGNETIC FIELD
10 i7
For a particle in a homogeneous field the solution of (1) reads
r(t) = Co(t) +
S{C1 COS cot
-~ D1 sin cot}
(2)
where co = eB/m is the cyclotron frequency and Co(t) is the radius-vector of the guide centre. For an inhomogeneous field higher order terms should be added to (2) as successive corrections. Avoiding Kruskal's complex vector notation, we write his development
r(t) = ~ = o en{Cn(t) cos no(t)/~ + Dn(t) sin nO(t)/e}
(3)
This development is appropriate if Cn, Dn and 0 are slowly varying functions of t, while sin nO/e and cos nO/e vary rapidly due to the smallness of e. At first sight it seems rather surprising that e can be used as a parameter in the development, as e is not dimensionless. B e r k o w i t z and G a r d n e r S ) have shown, however, that Kruskal's development is an asymptotic expansion of r. For a rapid convergence of the series (3) it is necessary that the magnetic field is approximately constant in a region of the dimension of the cyclotron radius of the particle. For a homogeneous field comparison of (2) and (3) immediately shows 0 = B
(4)
For an inhomogeneous field the expansion (3) contains an ambiguity. One is free in adding higher order terms to expression (4) for 0, while changing Cn and Dn in (3) in a corresponding way. In fact Kruskal and Hellwig made a different choice for 6. Kruskal maintains (4) in all approximations, while Hellwig adds a first order term, proportional to e. The consequences of Hellwig's choice for C1 and D1 will appear later on. The series (3) is substituted in (1), while the magnetic field is developed in a Taylor series in the neighbourhood of the guiding centre Co 1
B(r) = B(Co) + ~n ~,v. {(r -- C0)" grad}nB(Co)
(5)
In the following, B (Co) will be vcritten as B0. After substitution of (3) and (5) in (1), the factors of cos nO/e and sin nO/e are equated separately for each value of n. This yields a recursion system, the first few equations of which read [CoBol + e{-- Co -- ½0[(C1DI" grad -- D1Cl.grad)Bo]} -= 0(~2)
(6.1)
02C1 + O[D1Bo] + e{-- 2ODl--ODl+[CoCl'grad eo]+[C1eo]}=0(e 2) (6.2)
02D1 -- OEC1Bo] + e(20C1 + OC1 + ECoDl'grad Bo] + [bxBo]} = 0(e 2) (6.3) where 0 (e2) means terms of order higher than e. We shall not need higher order terms or further equations of the recursion system. For our purpose the use of these complicated equations can be avoided by the use of the invariants of (1). Physica 25
1018
H . C . BRINKMAN
In zeroth approximation the solution of (6) is very simple. It reads EdoBo] = CI'Bo = O(e) C12 = D19' + 0(e) O = Bo + 0(e)
D I ' B o ~ O(e) CI" Ol -----0(e)
(7)
In this approximation the particle behaves as if it moves in a homogeneous field. The guiding centre (Co) follows a magnetic line of force, while the particle rotates uniformly around tile guiding centre with the cyclotron frequency co = Bo/e. In first approximation the explicit integration of (6) is already complicated. A few relations m a y be derived, however, comparatively easily. The vector product of (6.1) and Bo gives the well known formula for the drift velocity. Using (7) in the terms of order e and assuming rot Bo = 0, one finds C°± = B~- [no grad Bo] {Cos + -~-02C12} + 0(e 2)
(8)
Here _l_ means the component perpendicular to'the magnetic' field. Co is the velocity parallel to the field and OC1 that perpendicular to the field. From (6.2) and (6.3) three relations will be derived. The first two are found by scalar multiplication of (6.2) by D1 and of (6.3) by C1 and subsequent subtraction of the results. Again using (7) in the manipulation of the terms proportional to e one arrives at d -- (0cl 2) = (9) dt In the next paragraph OC1e will appear to be the first approximation of Kruskal's invariant. Its physical meaning is the magnetic flux through the cyclotron orbit of the particle. The same procedure leads to a further relation by introducing the scalar product of (6.1) and C0 into (6.2).D1 -- (6.3).C1. One finds d {Co2 + 0 C12} = (10) dt In the next paragraph this will appear to be the first approximation to the energy integral. The third relation is found by considering (6.2). Cl + (6.3). D1. The result takes an elegant form by an appropriate choice ot 0. As stated before, the higher order terms in 0 may be chosen freely. We adopt Hellwig's choice 0 -- Bo + ew* + 0(e 2) (11) where co* =
1
Bo(C12 -4- DI 2)
Co'{[Cl(Cl'grad)eo] + [Dl(Dl.grad)eo]}
(12)
P A R T I C L E IN I N H O M O G E N E O U S M A G N E T I C F I E L D
1019
In this w a y our result becomes CI"D1 -- D I ' C 1 = 0(e)
(13)
or using (7) D I ' C 1 = 0(e)
(11"D1 = 0(s)
(14)
These relations mean that the vectors C1 and D1 do not rotate around Bo, only their length varies. This result is obtained b y giving the cyclotron frequency the small correction co*. So, the difference between Kruskal's or Hellwig's choice of 0 amounts to the choice between admitting a rotation of the vectors Ct and D1 around the lines of force or incorporating this rotation in the cyclotron frequency. 3. Invariants. From the equation of motion one immediately sees t h a t i 2 = V z is a constant of motion. Expressing ~2 b y means of (3) in terms proportional to sin n0/e and cos n0/e with n = O, 1, 2, ..., one concludes that the terms with n = 0 should be equal to V 2, while the fluctuating terms, obtained for different values of n, should give zero. In this w a y one finds V 2 = ~'0 2 "-{- ½02(C12 + Di e) %- eO(CI"D1 -- e l ' i ) 1 ) o = -
ODl.do +
e{d0.di +
+ 0(e 2)
(15.1)
O z ( c ~ . c 2 + D~.D2)} %- 0(e 2)
(15.2)
0 = OCl. Co %- e{Co'D1 -Jr- 02(D2"C1 -- C2"D1)} %- 0(e 2)
(15.3)
etc. Substitution of (13) in (l 5.1) leads to V z = 0o 2 %- -~-0z(C1z %- Dt z) %- 0(e z)
(16)
The simplicity of (16) is only apparent, of course, as the terms of order s are hidden in 0 (cf. (11)). The zeroth order terms of (15.2), etc., yield results, already known from (7). Kruskal has derived an adiabatic invariant K
K =
f ]=~ Or(t, O) p(t, O) oO dO
(17)
which is a constant of motion if r a n d p m a y be written as periodic functions of 0. As 0 is a function of time, equation (15) is purely formal. Its usefulness depends on the question if an appropriate choice of 0 can be made. This will be the case if the motion of the particle consists of a rapidly fluctuating periodic part and a slowly varying aperiodic one. Expression (17) is an average over a fluctuation, while the slowly varying part of r is kept constant during the averaging process, Such an average has a physical meaning only if the motion can be split in the way, discussed above.
1020
PARTICLE IN INHOMOGENEOUS MAGNETIC FIELD
By substitution of r from (3) in (17), one easily derives, using p = mr -5 eA, where rot A = B, that apart from a constant factor K = OV2/OO -5 (1/:~e~) fo~" AOr/O0 dO.
(18)
Therefore, using (15) K = (0 - -
½B0)(C12 - 5 D12) - 5 e ( C l ' D 1
-- CI"DI)
-5 0 ( e 2)
(19)
Again using (13), one finds K = (0 -- ½B0)(C12 -5 D12) -5 0(e 2)
(20)
Substituting 0 from (11) one obtains
K = ½Bo(CI 2 + DI 2) -5 eo~*(C12
-5
DI 2) -5 0(e~)
(21)
After very laborious calculations, not reproduced in his paper, Helhvig found an invariant, analogous to (21). Our derivation of (21) was made relatively simple by the use of Kruskal's invariant (17). Kruskal does not express his invariant in the'slowly varying quantities Cn, Dn and 0. He introduces the local value of the field B(r) and the velocity of the particle. As Kruskal does not report his method of deriving his expression, a comparion is difficult. Finally we shall compare (21) to the magnetic m o m e n t # = mv L2/2B
(22)
which is introduced as a constant in the approximate theory. In higher order v J contains rapidly fluctuating terms. Omitting these terms and calling the slowly carying part v12, one easily derives v± 2 = ½02(CI 2 -5
DI 2) -5 0(e~)
(23)
The invariant (21) becomes K = v±2
B0
+ 0(~2).
(24)
We shall not give a discussion of the usefulness of these results in problems of high temperature plasma's in magnetic fields. In a subsequent paper the theory will be compared to results of numerical calculations performed for the evaluation of an experiment. Reeeived 29-5-59 REFERENCES 1) 2} 3) 4) 5)
Alv6n, H., Cosmical Electrodynamies, Oxford 1950. H e l h v i g , G., Z, Naturforschung 10a (1955) 508. K r u s k a l , M., Proe. of the 3rd Conf. on ionised gases, Venice 1957. B u r g e r s , J. M., Ann. Physik 5:1 (1917) 195. B e r k o w i t z , J. and G a r d n e r , C., report NYO-7975, Inst. of math. Sciences, New York Univer sity 1957.
Physi.ca 25 1016-1020
ON T H E MOTION OF A C H A R G E D P A R T I C L E IN AN I N H O M O G E N E O U S MAGNETIC F I E L D *). H. C. BRINKMAN C e n t r a a l L a b o r a t o r i u m T . N . O . , Delft Summary
The motion of a charged particle in an inhomogeneous magnetic field is treated, employing Hellwig's and Kruskal's expansion of the gyration of the particle in a Fourier series. The relation of various constants of motion, the energy of the particle, Kruskal's action integral and Hellwig's higher order correction of the magnetic moment of the particle, is discussed.
1. Introduction. The motion of a charged, particle in a inhomogeneous magnetic field was t r e a t e d b y A l f v d n 1) for the case of very small deviations from homogeneity. H e l l w i g 2) un d K r u s k a l 3) indicated how the problem can be solved in higher approximation. Generally speaking their t r e a t m e n t s of the problem are equivalent. The motion of the particle is developed in a Fourier series with respect to time. The first term gives the motion of the so-called guiding centre which follows the lines of force in first approximation. The higher terms describe the gyration of the particle around the guiding centre. Hellwig's and Kruskal's t r e a t m e n t s differ in detail. Kruskal's presentation appears to be more systematic. Moreover K r u s k a 1 gives a very general proof for the existance of an adiabatic invariant, valid for a motion which m a y be split up in an oscillating periodic and a slowly v a r y i n g secular part. Kruskal's theorem is analogous to the adiabatic invariance of action integrals 4) introduced b y Ehreniest in q u a n t u m t h e o r y . Hellwig derives an invariant in second approximation, obtained after very involved calculations. I t seemed worthwhile to extend Kruskal's t r e a t m e n t in such a w a y that Hellwig's result fits in his general scheme. The use of Kruskal's invariant and of the energy integral greatly simplifies the calculations.
2. The Hellwig-Kruskal development. The equation of motion for a charged particle in a constant magnetic field reads ~r = Ire(r)]
(1)
Here r is the radius vector of the particle, s is the ratio of its mass and charge (s ---- m/e), while B is the magnetic field strength. *) T h i s w o r k is p a r t of the r e s e a r c h p r o g r a m m e of the " S t i c h t i n g v o o r F u n d a m e n t e e l O n d e r z o e k d e r M a t e r i e " ( F o u n d a t i o n for F u n d a m e n t a l R e s e a r c h on M a t t e r (F.O.M.)). It w a s c a r r i e d o u t a t tile F.O.M. I n s t i t u t e for P l a s m a - p h y s i c s , a t J u t p h a a s .
--
1016-
PARTICLE IN INHOMOGENEOUS MAGNETIC FIELD
10 i7
For a particle in a homogeneous field the solution of (1) reads
r(t) = Co(t) +
S{C1 COS cot
-~ D1 sin cot}
(2)
where co = eB/m is the cyclotron frequency and Co(t) is the radius-vector of the guide centre. For an inhomogeneous field higher order terms should be added to (2) as successive corrections. Avoiding Kruskal's complex vector notation, we write his development
r(t) = ~ = o en{Cn(t) cos no(t)/~ + Dn(t) sin nO(t)/e}
(3)
This development is appropriate if Cn, Dn and 0 are slowly varying functions of t, while sin nO/e and cos nO/e vary rapidly due to the smallness of e. At first sight it seems rather surprising that e can be used as a parameter in the development, as e is not dimensionless. B e r k o w i t z and G a r d n e r S ) have shown, however, that Kruskal's development is an asymptotic expansion of r. For a rapid convergence of the series (3) it is necessary that the magnetic field is approximately constant in a region of the dimension of the cyclotron radius of the particle. For a homogeneous field comparison of (2) and (3) immediately shows 0 = B
(4)
For an inhomogeneous field the expansion (3) contains an ambiguity. One is free in adding higher order terms to expression (4) for 0, while changing Cn and Dn in (3) in a corresponding way. In fact Kruskal and Hellwig made a different choice for 6. Kruskal maintains (4) in all approximations, while Hellwig adds a first order term, proportional to e. The consequences of Hellwig's choice for C1 and D1 will appear later on. The series (3) is substituted in (1), while the magnetic field is developed in a Taylor series in the neighbourhood of the guiding centre Co 1
B(r) = B(Co) + ~n ~,v. {(r -- C0)" grad}nB(Co)
(5)
In the following, B (Co) will be vcritten as B0. After substitution of (3) and (5) in (1), the factors of cos nO/e and sin nO/e are equated separately for each value of n. This yields a recursion system, the first few equations of which read [CoBol + e{-- Co -- ½0[(C1DI" grad -- D1Cl.grad)Bo]} -= 0(~2)
(6.1)
02C1 + O[D1Bo] + e{-- 2ODl--ODl+[CoCl'grad eo]+[C1eo]}=0(e 2) (6.2)
02D1 -- OEC1Bo] + e(20C1 + OC1 + ECoDl'grad Bo] + [bxBo]} = 0(e 2) (6.3) where 0 (e2) means terms of order higher than e. We shall not need higher order terms or further equations of the recursion system. For our purpose the use of these complicated equations can be avoided by the use of the invariants of (1). Physica 25
1018
H . C . BRINKMAN
In zeroth approximation the solution of (6) is very simple. It reads EdoBo] = CI'Bo = O(e) C12 = D19' + 0(e) O = Bo + 0(e)
D I ' B o ~ O(e) CI" Ol -----0(e)
(7)
In this approximation the particle behaves as if it moves in a homogeneous field. The guiding centre (Co) follows a magnetic line of force, while the particle rotates uniformly around tile guiding centre with the cyclotron frequency co = Bo/e. In first approximation the explicit integration of (6) is already complicated. A few relations m a y be derived, however, comparatively easily. The vector product of (6.1) and Bo gives the well known formula for the drift velocity. Using (7) in the terms of order e and assuming rot Bo = 0, one finds C°± = B~- [no grad Bo] {Cos + -~-02C12} + 0(e 2)
(8)
Here _l_ means the component perpendicular to'the magnetic' field. Co is the velocity parallel to the field and OC1 that perpendicular to the field. From (6.2) and (6.3) three relations will be derived. The first two are found by scalar multiplication of (6.2) by D1 and of (6.3) by C1 and subsequent subtraction of the results. Again using (7) in the manipulation of the terms proportional to e one arrives at d -- (0cl 2) = (9) dt In the next paragraph OC1e will appear to be the first approximation of Kruskal's invariant. Its physical meaning is the magnetic flux through the cyclotron orbit of the particle. The same procedure leads to a further relation by introducing the scalar product of (6.1) and C0 into (6.2).D1 -- (6.3).C1. One finds d {Co2 + 0 C12} = (10) dt In the next paragraph this will appear to be the first approximation to the energy integral. The third relation is found by considering (6.2). Cl + (6.3). D1. The result takes an elegant form by an appropriate choice ot 0. As stated before, the higher order terms in 0 may be chosen freely. We adopt Hellwig's choice 0 -- Bo + ew* + 0(e 2) (11) where co* =
1
Bo(C12 -4- DI 2)
Co'{[Cl(Cl'grad)eo] + [Dl(Dl.grad)eo]}
(12)
P A R T I C L E IN I N H O M O G E N E O U S M A G N E T I C F I E L D
1019
In this w a y our result becomes CI"D1 -- D I ' C 1 = 0(e)
(13)
or using (7) D I ' C 1 = 0(e)
(11"D1 = 0(s)
(14)
These relations mean that the vectors C1 and D1 do not rotate around Bo, only their length varies. This result is obtained b y giving the cyclotron frequency the small correction co*. So, the difference between Kruskal's or Hellwig's choice of 0 amounts to the choice between admitting a rotation of the vectors Ct and D1 around the lines of force or incorporating this rotation in the cyclotron frequency. 3. Invariants. From the equation of motion one immediately sees t h a t i 2 = V z is a constant of motion. Expressing ~2 b y means of (3) in terms proportional to sin n0/e and cos n0/e with n = O, 1, 2, ..., one concludes that the terms with n = 0 should be equal to V 2, while the fluctuating terms, obtained for different values of n, should give zero. In this w a y one finds V 2 = ~'0 2 "-{- ½02(C12 + Di e) %- eO(CI"D1 -- e l ' i ) 1 ) o = -
ODl.do +
e{d0.di +
+ 0(e 2)
(15.1)
O z ( c ~ . c 2 + D~.D2)} %- 0(e 2)
(15.2)
0 = OCl. Co %- e{Co'D1 -Jr- 02(D2"C1 -- C2"D1)} %- 0(e 2)
(15.3)
etc. Substitution of (13) in (l 5.1) leads to V z = 0o 2 %- -~-0z(C1z %- Dt z) %- 0(e z)
(16)
The simplicity of (16) is only apparent, of course, as the terms of order s are hidden in 0 (cf. (11)). The zeroth order terms of (15.2), etc., yield results, already known from (7). Kruskal has derived an adiabatic invariant K
K =
f ]=~ Or(t, O) p(t, O) oO dO
(17)
which is a constant of motion if r a n d p m a y be written as periodic functions of 0. As 0 is a function of time, equation (15) is purely formal. Its usefulness depends on the question if an appropriate choice of 0 can be made. This will be the case if the motion of the particle consists of a rapidly fluctuating periodic part and a slowly varying aperiodic one. Expression (17) is an average over a fluctuation, while the slowly varying part of r is kept constant during the averaging process, Such an average has a physical meaning only if the motion can be split in the way, discussed above.
1020
PARTICLE IN INHOMOGENEOUS MAGNETIC FIELD
By substitution of r from (3) in (17), one easily derives, using p = mr -5 eA, where rot A = B, that apart from a constant factor K = OV2/OO -5 (1/:~e~) fo~" AOr/O0 dO.
(18)
Therefore, using (15) K = (0 - -
½B0)(C12 - 5 D12) - 5 e ( C l ' D 1
-- CI"DI)
-5 0 ( e 2)
(19)
Again using (13), one finds K = (0 -- ½B0)(C12 -5 D12) -5 0(e 2)
(20)
Substituting 0 from (11) one obtains
K = ½Bo(CI 2 + DI 2) -5 eo~*(C12
-5
DI 2) -5 0(e~)
(21)
After very laborious calculations, not reproduced in his paper, Helhvig found an invariant, analogous to (21). Our derivation of (21) was made relatively simple by the use of Kruskal's invariant (17). Kruskal does not express his invariant in the'slowly varying quantities Cn, Dn and 0. He introduces the local value of the field B(r) and the velocity of the particle. As Kruskal does not report his method of deriving his expression, a comparion is difficult. Finally we shall compare (21) to the magnetic m o m e n t # = mv L2/2B
(22)
which is introduced as a constant in the approximate theory. In higher order v J contains rapidly fluctuating terms. Omitting these terms and calling the slowly carying part v12, one easily derives v± 2 = ½02(CI 2 -5
DI 2) -5 0(e~)
(23)
The invariant (21) becomes K = v±2
B0
+ 0(~2).
(24)
We shall not give a discussion of the usefulness of these results in problems of high temperature plasma's in magnetic fields. In a subsequent paper the theory will be compared to results of numerical calculations performed for the evaluation of an experiment. Reeeived 29-5-59 REFERENCES 1) 2} 3) 4) 5)
Alv6n, H., Cosmical Electrodynamies, Oxford 1950. H e l h v i g , G., Z, Naturforschung 10a (1955) 508. K r u s k a l , M., Proe. of the 3rd Conf. on ionised gases, Venice 1957. B u r g e r s , J. M., Ann. Physik 5:1 (1917) 195. B e r k o w i t z , J. and G a r d n e r , C., report NYO-7975, Inst. of math. Sciences, New York Univer sity 1957.