Coherent State Path Integral for the Interaction of Spin ...

Physica Scripta. Vol. 56, 545-553, 1997

Coherent State Path Integral for the Interaction of Spin with External Magnetic Fields T. Boudjedaa,' A. Bounames,' Kh. Nouicer,' L. Chetouani' and T. F. Hammann3 Departement de Physique, Ecole Normale Sufirieure de Jijel, BP 98 Ouled Aissa, 18000 Jijel, Algeria Dbpartement de Physique Theorique, Institut de Physique, Univenite de Constantine, Route Ain El Bey, Constanthe, Algeria Laboratoire de Mathematiques,Physique Mathkmatique et Informatique, Facultb des Sciences et Techniques, Universitb de Haute Alsace, 4 rue des frbres LumBre, F68093 Mulhouse, France Received October 14,1996; accepted in revisedform May 18,1997 PACS Ref: 03.65,03.65Db, 03.65Ca.

(LCT) the problem is reduced to solve the equation of Linear canonical transformation and time transformation are used within motion of a two-level system. The factorization technique is the formalism of path integrals in coherent state representation to solve the used to obtain a formal expression for the transition ampliproblem of an interacting spin with arbitrary time-dependent external mag- tude. Then a general expression for the external magnetic netic fields. The Factorization technique is used to solve the equation of field is derived. In Section 2, the main properties of the motion. Examples of the calculation of the transition probability in terms coherent state representation are briefly exposed. The of elementary functions and special functions for solvable systems are presented. For completeness Berry's phase is also derived under adiabatic formal calculation of the transition amplitude via the (LCT) is performed in Section 3. In Section 4, we present the expansion. explicit calculation of the transition amplitude in the cases of unspecified and specified magnetic fields. Using the adia1. Introduction batic theorem, Berry's phase is also derived. To show the A subject of great importance in physics is the obtainment usefulness of our method, the magnetic fields and the related of analytical solutions to time-dependent systems like the transition probabilities are derived in terms of elementary interaction of a spinning particle with a time-dependent functions in some cases of solvable systems. The method is external magnetic field [l]. For an arbitrary spin, it is also extended to the case of special functions. Concluding known that the main physics is contained in the time evolu- remarks are given in Section 5. tion of the two-level system [2, 31. In general, the two-level system is formulated in terms of two functions of time corre- 2. Coherent state representation sponding to the amplitude or envelope and the detuning fre- In this section we give a brief account of the Glauber coherquency. The two-level system has been useful in modelling ent states (GCS). Let a and at be the bosonic annihilation phenomena in atomic collisions [4], laser spectroscopy [SI, and creation operators verifying the commutation relation magnetic resonance [6] and the dynamics in dissipative [a, ] 'a = 1. (2) environments [7]. The simplest two-level model is the one where the The GCS I Z ) is defined as the eigenstate of the annihilation envelope and the detuning frequency are constant functions. operator a This model has been solved by Rabi [8] in terms of elemen(3) tary functions. The solution with hyperbolic secant envelope a l Q = ZlZ), and a constant detuning frequency was obtained 50 years where Z is a complex number. The above definition is ago by Rosen and Zener [4]. The generalization of this equivalent to generating the state I Z ) from the vacuum IO) work was carried out by many authors [SI. At present, there by exist some new classes of analytic solutions of the two-level system [lo-161. On the other hand various methods have (4) been used to treat the problem, among them the WKB method [l, 171,Magnus approximation [lS], the Riemann- The main properties of these states are Papperitz differential equation [lo] and the Lie algebraic - Non-orthogonality method [13]. In this paper, we shall examine the solution of the problem of a spinning particle in interaction with an arbi(5) trary time-dependent magnetic field governed by the Hamil- Resolution of unity tonian Abstract

H ( t ) = -gJB(t), (1) IZ)(Z, = 1. where g is the gyromagnetic ratio, via the path integral formalism in coherent state representation [191. Defining a The generalization to the multidimensional case is straightnew time and performing a linear canonical transformation forward. Physica Scripta 56

546

T . Boudjedaa et al.

Now we turn to the bosonization of the spin operators following Schwinger's recipe. The operators .Ti are given in term of a bosonic 2D-oscillator (7)

where U(t)is assumed to verify Ut(t)U(t) = 1, det U(t) = 1,

(16)

and q(t) is a new complex variable describing the dynamics of the spin. Making the substitution of (15) in the transition amplitude (13) gives

+

Due to the constraint 5 ' = (N/2)[(N/2) 11, where N is the total occupation number of the 2D-oscillator, the vector state I $) is projected in Hilbert's sub-space where the total number N is fixed to the value 2j

x (1

By using this model of spin, the Hamiltonian (1) is easily written as

H

= -o(t)(ab

- btb) - u(t)a% - u*(t)b'a,

(17)

Knowing that in the path integral formalism the infinitesimal action is expanded only to first order in time, we write

(9)

where the time-dependent functions o(t)and u(t) are given in terms of the magnetic field components

]

+ isQ(I))U(l- 1) ei'lql-l .

dU

dt (I),

(18)

+ ieQ(I))U(l- 1)= 1 + iEQ1(I),

(19)

U(1 - 1) = U(I)- E and

9 o(t)= B3(t) = S 3 ( t ) , 2

(10)

9 2

u(t) = - (Bl(t) - iBz(t))= X l ( t )- iXz(t).

Ut(l)(l where

(11)

In the next section we are interested in the calculation of the transition amplitude of the system governed by the Hamiltonian (9) using linear canonical transformation (LCT).

dU

QiN = ut(I)Q(I)U(I)+ iut(I)dt (0-

(20)

So the transition amplitude conserves the same form

3. Formalism and method The transition amplitude between the state 12,) at time ti = 0 and the state 12,) at time t - T is given as a matrix element of the time dependent evolution operator

1

+ q/(l + ieQl(I))eiA1ql- .

f:

Kj(zf

zi

3

T ) = (2, I T exp (-i

b(t)

dt)pj

(21)

I zi>, (12) Further simplifications can be brought by introducing a

new time s defined by where Z is a pair of complex variables describing the spin dt evolution and T is Dyson's time ordering symbol. - = f (4, Discretizing the time E = T/(N l), using the Trotter ds formula and inserting N-times the resolution of unity (6) wheref(s) is a function assumed to be monotonic. At the between each pair of the evolution operator at time E, we discretized level, we have obtain the discretized form of the transition amplitude

+

E

KXZ,, Zi; T ) = lim

1=1

= zf(s,),

S

z =-

N+1'

Then, the problem is reduced to the evaluation of the transition amplitude (13) with a time-dependent interaction term given by

2

Qz(O

=f(sJQi(I), (24) where Qz(s(t)) is assumed to have a simplified form. The equation (20) is now transformed to Q(t) =

(

u*(t)

Qz(0 =f(si)Ut(I)Q(I)U(I)+ iU'(2)

-w(t) '

Physica Scripta 56

(0-

(25)

Equation (25) plays the role of an auxiliary equation where U(t)andf(s) may be chosen in such a way that the problem under investigation reduces to a simple or a known one. We point out that a similar technique has been used in phasespace path integrals using generalized canonical transform(15) ations (GCT) to solve time-dependent systems [20].

and the formula ei'(ata+btb) 12) = I e"Z) has been used. Now, we go on with the reduction of the problem to a simple one by introducing linear canonical transformation (LCT) defined by Z(t)= ~ ( t ) r l ( t ) ,

dU

Coherent State Path Integral for the Interaction of Spin with External Magnetic Fields

4. Explicit calculation of the transition amplitude

In order to illustrate the usefulness of the formalism devel-

547

tude in the coherent state representation, K , ( Z f , Zi;T ) , is given by

oped in Section 3, let us consider the calculation of the transition amplitude relative to the case of an arbitrary K,(mf, mi; T ) = time-dependent magnetic field B(t). Other main applications will consist in calculating Berry's phase and finding classes of magnetic fields where the transition amplitude can be Using the expansion of the coherent states 12) = 1 a, /?> on evaluated explicitly in terms of elementary functions and the basis of the spin states I j m, j - m ) special functions.

jd: dF +

4.1. Unspecified magnetic field 4.1.1. Formal calculation of the transition amplitude. Our aim is to reduce the problem with an unspecified timedependent magnetic field B(t) with Hamiltonian (9) to the free one, H = 0. Let us consider the following LCT, Z(t) = U(t)tl(t),

(26)

where

I a, B>

= exp

(- -- -)

12 B12

m=+j

x jc

c

m=-j

J

+

(j m ) ! ( j - m)!

the matrix element ( m 12)is easily derived

(- q) +

aj+m

< m l z > = exp

J(j

Bj - m

m ) ! ( j- m)!

Making use of the binomial formula and the identity Following step by step the method of Section 3, we get the following expression for the transition amplitude in the new basis

(37) it is easy to get the result

KAmf mi; T ) 9

x E 1 =e 1 x p

[-

lr111'

],

+ In-11' + q/ eiAfql2

(28)

+ +

1 (j mi)!(j - m f ) ! (mi - m f ) !J(j m f ) !(j - mi)

with the auxiliary equation (20)

x [A(T)]""J[A*(T)]'-"'[ - B*(T)]"'-"f

dU Q(t)U(t) i -= 0

x 'F1( -j

+

dt

'

+ m i , -j - m f , mi - mf + 1; -

which is equivalent to the system

dA _ - i(oA dt

(38)

(30) This expression is exactly the one obtained by the authors in previous papers [21] after the identification of the timedependent coefficients A(T) and B*(T),which can be derived dB* -- i( - oB* - u*A), B*(O)= 0. (31) perturbatively from the system (30) and (31), with the matrix dt elements Ri,XT). The same equations have been obtained by Kochetov [22] 4.1.2. Berry's phase. Another method to solve the auxvia SU(2) coherent state path integral and fractional canon- iliary system (30) and (31) is to use adiabatic expansion. In ical transformations. A similar set of equations can be this order we are going to extract Berry's phase from the obtained for a general SU(1, 1) Hamiltonian using fractional asymptotic solution of the auxiliary system. We adopt the canonical transformation [23] or by direct path integration Euclidian technique to path integral formalism [25] in at the discretized level [24]. which the real term of the exponent appearing in the propaA direct integration of (28) yields gator constitutes the usual dynamical part while the imaginary term is identified as the geometrical part. The Euclidian transition amplitude is obtained by making the substitution t -+ -it. After this substitution the auxiliary Inverting the LCT (26), we obtain the transition amplitude system becomes in the old basis dV = QV, (39) 1 (2: U(T)Zi)'j exp - IzfIZ+ I Z ~ I ' ] dt K,(Z f , Zi ; T ) = 7 2 (23)' (33) where Q is given by (14) and V = From this result we see that the main dynamics of a multiFor convenience we rewrite the magnetic field in polar level system is essentially contained in the time evolution of coordinates. Then, a two-level system. The relation between the transition amplitude in the spin state representation, K,(m,, mi; T), and the transition ampli- uB*),

A(0) = 1,

[

(-;*)

Physica Scripta 56


548

where X ( t ) is the magnitude of the field. To find the asymptotic solution of eqs (30) and (31) we introduce a scaled time t s=T'

and

Q$!g(s) = Uf(s)Q")(s)U~(s)*

(54)

(41) Knowing that all terms of order O(l/T1),12 2, are ignored in the adiabatic approximation, it is easy to show that U,@, 1 2 3, tend to unity and

OGsGl.

Now eq. (39) reads as

Qi:~~(s) = (sys)

-d"(s) - TQ(s)V(s),

+ 2Ti

(55)

ds Therefore, the auxiliary equation is reduced to

Let us adopt the following ansatz

r

1 B(z)a3 dz C

rs

1 Jo

V(s)= U(s) exp T

1

(43)

where where

U&) are matrices, j I ( z ) scalar functions and C a constant vector. First we choose U,@) such that it makes Q(s) diagonal, namely

(57)

By identification of terms of order 0(1)and O(T), respec(45) tively, we get i p0(s) = x(s),pl(s) = (d, COS e $hi)). (58)

+

Then, the adiabatic solution of the auxiliary equation is given by

and

Qdiag(s)= s ( s ) 0 3

-

(47)

Using the ansatz (43) and multiplying on the left the auxiliary equation by Uf(s)-weget and hence, the linear canonical transformation (26) is easily derived. Substituting this latter result in (33), we obtain an expression for the transition amplitude in the adiabatic approximation,

KXZ,

9

zi; T )

= Y $ ( Z f ,H(T))Y?(Z,,H(0))

(49)

x exp [jT l d z { X ( r ) - Ti ( 1 - cos @))

and

H")=

(

- - d,

;T

i i sin 8, - 8, X(s) - d, cos 8 2T 2T

+

where Y$(Z,H(t))is the ground state wave function

We note that the first and second components of H")are of order O(l / T ) . Proceeding in the same manner, we define U&) as a matrix which diagonalizes Q(')(s), and the imaginary part of the exponent is Berry's phase [26] where $(')(s) and P ( s ) are given by tan tan

8

= --

d, sin e'

sin2 e + e2 2 T P ( s ) + i$ cos 8

e(ys) = -i J

Physica Scripta 56

4.1.3. Solution of the equation of motion. As we have shown in the previous section, the reduction of the problem leads to an auxiliary equation which in the case of two-level systems is equivalent to a second order linear differential (53) equation. Indeed, decoupling the system of eqs (30) and (31)


4.2. SpeciJiedmagneticfield 4.2.1. Constant envelope and variable frequency. We now

gives d2A dA 9(~) ds2

+

549

ds + Q2(s)A= 0,

use a particular time transformation followed by an LCT to reduce the problem to that of a two-level system where the with the initial conditions A(si) and (dA/ds)Isi = i(o(si)A(si) envelope connecting the levels is constant and the detuning - u(si)B*(si)).The functions 9 ( s ) and Q2(s) are given, respec- frequency is time-dependent. The formal continuous exprestively, by sion for the transition amplitude (13) is found by taking the limitN+co

w Q2(s) = i - o(s) + 02(s) + I u(s) 1' 4s)

K,(Zf, Zi; T )=

- ih(s).

(65)

The resolution of this differential equation is carried out using the factorization technique. Indeed eq. (63) is written as a sequence of differential equations of lower order

+ 25(s),

[ U:

dlDZtDZ exp i

+ ZtQ(t)e'")

- (ZtzA- ZZX)

dt],

(73)

where Q(t) is given by (14),

and

where the functions &) and q(s) are solutions to 9 ( s ) = ?(S)

s

(67)

(74)

9 + -1 9 2 = Q2 + -1 $(s) + -1 -. dtl(s) -I dLet us first introduce the following time transformation 2ds 4 4 2 ds (75) Equation (68) is a Riccati equation whose solution is not s( t) = B( t) - 2a(t). generally known. Then from (66) the solution of eq. (63) is We assume that the t+s(t) transform exists and is invertgiven by

A(sJ

ible. The functions B(t) and a(t) are defined by

+ i[(o(si) - iS(si))A(sJ- u(si)B*(si)lh(s)]

c7f3

COS

b(t) = -,

c7f

sin B(t) = >,

S T

c7fT

c7fT

=/,

(76)

sd

a(t) = - c7f2(z) dz, a(0) = 0. From eq. (31) we derive a similar expression for B*(s)

1

(77)

Within this transformation, the transition amplitude becomes

s [ 61'(i

si)= DlDZtDZ K,(Zf, s f ; Zi, x exp i

( z t ~-, zi1)

+ Zt dB

Q(s(t)) eiAZ)ds]. da -(s) - 2 - (s) dt dt

where h(s) is given by

(78)

Following the method of Section 4.1.1 we obtain the following equations of motion To end this section, we evaluate the transition amplitude from the spin state mi to the spin state mf and the transition probability from - j to m - j . For the transition amplitude the expression remains the same as eq. (38) except that A(s) and B*(s) are now given by (69) and (70). Using the property 2Fl(a,0, y, x) = 1, the transition probability from - j to m - j is derived from eq. (38)

In the following we shall make use of eqs (67) and (68) to derive explicit transition probabilities in some cases of solvable systems.

dAds

i db

dt (4 - 2 dB* dS

-

( o A - UB*), A($ = 1,

da

(79)

(3)

i (oB*+ u*A), B*ls,) = 0. -dB.. ,do:.. - (S) - z - (S) dt dt \-,I

180) ,--,

Now we introduce the LCT defined by

+ By(s) exp

[

-i

1%

dr] sin 5, B

(8 1) Physica Scripta 56

550


[I: 1

-B*(s) = A,($ exp i

derived

D(r) dz sin -

_ -da dt '

where the function D(s) is given by

D @ ) ( g - 2 $)cos

8)#

(89)

We note that, in our classification, D(s) must be a real function. The case of q(s) = i m l . After a long but straightforward calculation we obtain

For this choice the expressions for D(s) and ((s) are given by 1

D(s) = 2

with the initial conditions A,(si) = cos (&/2) and BT(si)= sin (BJ2). This is the desired system of equations describing a twolevel quantum system with a constant envelope and a variable detuning frequency. The transition amplitude is then given by eq. (32). Equations (84)and (85) may be combined to yield the following second-order linear differential equation d2A, ds2

dA, ds

+

1 4

- 2iD(s) -+ - A,

i 2

[

-2i

I. 1 D(t) dz

-

Jr+l).

Wf, i = [Re (A(t))12+ [Re (-B*(t))12,

(90)

(91)

where A(t) and B*(t) are given, respectively, by eqs (81) and (82). A straightforward calculation leads to the well known Rabi formula

4

s.

Now, fixing the parameter y and the time s(t) we obtain the transition probabilities related to some known magnetic fields [l5].

)

.

;(Ji

If, for convenience, we specify the initial and final states as the eigenstates of c 2 , the transition probability from the initial state I i) to the final state I f) is given by

I

stitutions in eqs (69) and (70): u(s) = - exp

5(s) =

wf, . = ("yy l) sin2

= 0.

To solve this equation we have to make the following subw(s) = 0,

&,

a. y = (( 0 4 0'wo)2

and s(t) = (w - oo)t.

Hence we have the following expressions for A,(s) and BT(s), Then D(s) = q / ( w - coo) and the transition probability is respectively given by

(87) If we put P(t) = wt we arrive at the Rabi magnetic field

B:(s)

[il 1

= 2 exp 2i

D(z) dz

B = ( B , sin ut, -Bo, B , cos ut),

(94)

where Bo = wo/g and B, = 2w,/g.

The function D(s) remains the same and the transition probability is where &s) and q(s) are solutions of eqs (67) and (68) with the substitutions 9 ( s ) = 2iD(s) and sZ2 = $. Examples of the detuning frequency D(s) can be derived from eqs (67) and (68). In the following we consider the case of spinj = 3. 4.2.2. Examples of transition probabilities. We present here the derivation of explicit expressions for the transition probability in terms of elementary functions. First we choose an expression for the function q(s). Then from (68) we derive the detuning frequency D(s) and from eq. (67) we finally get the function ((s). The transition probability is derived after returning to the coherent state IZ).From eqs (76), (77) and (83), a general expression for the magnetic field is easily Physica Scripta 56

(95) Putting, again, p(t) = wt we get the damped Rabi magnetic field

(96) where Bo = oo/g and B , = 2w,/g.


551

Assuming again si = pi = n, we obtain the following transition probability

1 i sin (s/2) The case of q(s) = 2 sin (s/2) i cos (s/2)

- +

1 cos2 (s/2fi) = 4cos2 (n/2$)

For this case the solution of eqs (67) and (68) is

wf9

(97) Using the eigenstates of o,, the transition probability is given by Wf,i = I B*(t) 1' where B*(t) is given by eq. (82). where h(s) is given by Assuming in the following that si = pi = n, we obtain

W,, i =

1 + cos2 (s/2) 4 dh(s)

1 sin (s/2)

h(s)) cos

f/',

As a particular case we consider and

where h(s) is now given by exp [ -i lntan (z/4)]

h(s) =

The constant a is such that cot (a) = n/$. field corresponding to this case is given by (99) a2 e - 2 At 1 ot sin 1 + a2 e-2At 1 02t2 '

dz.

( ++ )

Choosing s(t) = 2 arcsin

1

and jl(t) =

J1.;i";"

ot

1

+ W2t2

+

n,

we obtain the following magnetic field:

1

2B

+ A2t2

(

a2 e - 2 L t

B

_-_

The magnetic

1

+ a'

))

cos e-2Ar 1+ 0 2 t 2 +

with B = A/g and Bo = o / g . In the limit o = 0 the transition probability reduces to

1 - 02t2 Bo (1 + o2t2)2'

4.2.3. Variable envelope and constant Pequency. We come now to consider the case where the transition probability with B = A/g, Bo = o / g . can be derived in terms of special functions. To this aim we The choice take profit from the recursion relations between special functions like the hypergeometric functions, Bessel functions s(t) = 2 arcsin cosh (oo t/2) and the parabolic cylinder functions . . . etc. In our case we just consider the recursion relation for the hypergeometric with the same p(t) leads to the following magnetic field function. ot B 1 0 2 t 2 Let us consider a two-level system where the envelope B = -sin (1 + 0 2 t 2 ) ' - cosh (aot/2) Bo (1 + 0 2 t 2 ) 2 ' connecting the levels is time-dependent and the detuning frequency is constant. Here we are going to use the same method as before with some changes. gcos( 2 1 + W2t2 ' From eqs (30) and (31) we consider the case w(t) = 0 and u(t) = D(t) e-iKtwhere IC is a constant with B = wo/g,Bo = o / g . In the limit o = 0 the transition probability reduces to the simple formula

(:

))

and we recover the magnetic fields given in [16]. S

The case of q(s) = The solutions of eqs (67) and (68) are 1 cot D(s) = - 2$

(5) ,

i2Jz + l tan ((s) = -

(102) dB* -= -iD(t) eiKrA. dt To simplify the problem we introduce the time scale t/To and the time transformation t --+ z = z(t).Then we get = -if(z)

dz

(G).

dB* = (103) dz

2

e-irTOt(z)B*(z),

-q(z)5 eiKTOf(s)A(Z), Z

(109) (110) Physica Scripta 56

T. Boudjedaa et al.

552

where we have used the following dimensionless parameters z. =d- z

dt ’

DWO f(4= -, Bo =;

S

and S =

BO

(118)

Here Bo plays the role of a coupling constant. S is the area of the envelope. Decoupling the system (109) and (1 10) leads to d’A dz2

-+

Writingf(z) and i in terms of z it is easy to see that A(z) and B*(z) can be written in terms of the hypergeometric function

(dldt) In i -

(flf) + ilcTo dA Bif’ +-A dz

i

i2

= 0.

B*(z) =

’‘BO ixT0

Z(iKTo+ 1)/2F

+

+

iKTO 1

(111)

Applying the results of Section 4.1.3 with the identifications o ( z )= 0 and u(z) = Bo[f(z)/i] and imposing the initial conditions A(z,) = 1, B*(zi)= 0, we obtain the solutions

+1

iuTO 1 +Bo, 2

2

(1 19) The steady-state transition probability of the atom to the excited state is given by Wf, = 1 B*(z = 1) .1’ Using the property that [27]

(112)

F(c - a, c - b, 1

- c + U + b) + c ; 1) = r ( l +r ( c)T(l l + a)r(l + b)

9

(121)

and the relations between the Gamma functions with complex arguments, we get finally the celebrated result of Rosen and Zener [4] 1

(113) where ((z)and q(z)are solutions of eqs (67) and (68) with the following identifications

+

(d/ds)In i - (flf) iKTO = %4, i

f ’(z) - gp(z). pi 7

Wf,= sin2 (S) sech2

(“2”) . 7c

We point out that the method used here, subjected to the existence of solutions to the auxiliary equations (67) and (68), constitute a powerful tool to find new envelope functions which lead to exact solutions of time-dependent twolevel systems.

5. Conclusion

In the following we are concerned by the Rosen-Zener case In this paper we have investigated the problem of a spinning particle interacting with a time-dependent external magnetic [4] in which 1 dr, f ( z ) = cosh t ‘ From the first equation it is easy to get an expression for

t(4, 1 2

t(z) = - In -

(l:z)-

The derivation of the function t ( z ) proceeds as follows. Considering the second factor of eq. (66) and knowing that the solution of the Rosen-Zener problem is epxressed in terms of the hypergeometric function and with the help of the property d

- F(a, b, C; dz

Z)

ab

= - F(a C

+ 1, b + 1, c + 1; z),

we suggest the following expression for ( ( z )

and from (67)

field via coherent state path integrals. The problem has been reduced to a two-level system by using linear canonical transformation and time transformation. The factorization technique has been used to derive a formal solution for the transition amplitudes. As an illustration of our formalism, magnetic fields and related transition probabilities have been calculated in terms of elementary functions and special functions for solvable systems. The advantage of our method is the fact that it depends only on two arbitrary time-dependent functions, t(t)and q(t). The problem with time-dependent envelope and detuning frequency is under investigation.

References

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I. 8. Physica Scripta 56

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