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consider the e\ufb00ect of inhomogeneity of magnetic \ufb01eld on gyrating charged particle. The x component of Lorentz force F L = qv ×B perpendicular to the magnetic \ufb01eld (z direction) and the magnitude B of the magnetic \ufb01eld near the guiding center are FLx = qvyB = \u2212|q|v\u22a5 cos \u3b8B B = B0 + \u2202B \u2202x \u3c1\u3a9 cos \u3b8 + \u2202B \u2202y \u3c1\u3a9 sin \u3b8. The time average of x component of Lorentz force is given by \u3008FLx\u3009 = 12(\u2202B/\u2202x)(\u2212|q|)v\u22a5\u3c1\u3a9 and the y component is also given by the same way, and we \ufb01nd (see \ufb01g.2.5) \u3008F L\u3009\u22a5 = \u2212 mv2\u22a5/2 B \u2207\u22a5B. Next it is necessary to estimate the time average of z component of Lorentz force. The equation \u2207 ·B = 0 near the guiding center in \ufb01g.2.5 becomes Br/r + \u2202Br/\u2202r + \u2202Bz/\u2202z = 0 and we \ufb01nd \u3008FLz\u3009 = \u2212\u3008qv\u3b8Br\u3009 = |q|v\u22a5\u3c1\u3a9\u2202Br \u2202r = \u2212mv 2 \u22a5/2 B \u2202B \u2202z , 11 12 2 Plasma Characteristics Fig.2.5 Larmor motion in inhomogeneous magnetic \ufb01eld. since r is very small. Thus the necessary expression of g\u2207B is derived. 2.5 Magnetic Moment, Mirror Con\ufb01nement, Longitudinal Adiabatic Constant A current loop with the current I encircling the area S has the magnetic moment of \u3bcm = IS. Since the current and encircling area of gyrating Larmor motion are I = q\u3a9/2\u3c0, S = \u3c0\u3c12\u3a9 respectively, it has the magnetic moment of \u3bcm = q\u3a9 2\u3c0 \u3c0\u3c12\u3a9 = mv2\u22a5 2B . (2.10) This physical quantity is adiabatically invariant as is shown later in this section. When the magnetic \ufb01eld changes slowly, the magnetic moment is conserved. Therefore if B is increased, mv2\u22a5 = \u3bcmB is also increased and the particles are heated. This kind of heating is called adiabatic heating. Let us consider a mirror \ufb01eld as is shown in \ufb01g.2.6, in which magnetic \ufb01eld is weak at the center and is strong at both ends of mirror \ufb01eld. For simplicity the electric \ufb01eld is assumed to be zero. Since Lorentz force is perpendicular to the velocity, the magnetic \ufb01eld does not contribute the change of kinetic energy and mv2\u2016 2 + mv2\u22a5 2 = mv2 2 = E = const. (2.11) Since the magnetic moment is conserved, we \ufb01nd v\u2016 = ± ( 2 m E \u2212 v2\u22a5 )1/2 = ± ( v2 \u2212 2 m \u3bcmB )1/2 . When the particle moves toward the open ends, the magnetic \ufb01eld becomes large and v\u2016 becomes small or even zero. Since the force along the parallel direction to the magnetic \ufb01eld is \u2212\u3bcm\u2207\u2016B, the both ends of the mirror \ufb01eld repulse charged particles as a mirror re\ufb02ects light. The ratio of magnitude of magnetic \ufb01eld at open end to the central value is called mirror ratio: RM = BM B0 . Let us denote the parallel and perpendicular components of the velocity at the mirror center by v\u20160 and v\u22a50 respectively. The value v2\u22a5 at the position of maximum magnetic \ufb01eld BM is given by v2\u22a5M = BM B0 v2\u22a50. If this value is larger than v2 = v20 , this particle can not pass through the open end, so that the particle satisfying the following condition is re\ufb02ected and is trapped in the mirror \ufb01eld:( v\u22a50 v0 )2 > B0 BM = 1 RM . (2.12) 12 2.5 Magnetic Moment, Mirror Con\ufb01nement, Longitudinal · · · 13 Fig.2.6 Mirror \ufb01eld and loss cone in v\u2016 - v\u22a5 space. Particles in the region where sin \u3b8 \u2261 v\u22a50/v0 satis\ufb01es sin2 \u3b8 \u2264 1 RM are not trapped and the region is called loss cone in v\u2016 - v\u22a5 space (see Fig.2.6). Let us check the invariance of \u3bcm in the presence of a slowly changing magnetic \ufb01eld (|\u2202B/\u2202t| \ufffd |\u3a9B|). Scalar product of v\u22a5 and the equation of motion is mv\u22a5 · dv\u22a5dt = d dt ( mv2\u22a5 2 ) = q(v\u22a5 ·E\u22a5). During one period 2\u3c0/|\u3a9| of Larmor motion, the change \u394W\u22a5 of the kinetic energy W\u22a5 = mv2\u22a5/2 is \u394W\u22a5 = q \u222b (v\u22a5 ·E\u22a5)dt = q \u222e E\u22a5 · ds = q \u222b (\u2207×E · n)dS where \u222e ds is the closed line integral along Larmor orbit and \u222b dS is surface integral over the encircled area of Larmor orbit. Since \u2207×E = \u2212\u2202B/\u2202t, \u394W\u22a5 is \u394W\u22a5 = \u2212q \u222b \u2202B \u2202t · ndS = |q|\u3c0\u3c12\u3a9 \u2202B \u2202t . The change of magnetic \ufb01eld \u394B during one period of Larmor motion is \u394B = (\u2202B/\u2202t)(2\u3c0/|\u3a9|), we \ufb01nd \u394W\u22a5 = mv2\u22a5 2 \u394B B = W\u22a5 \u394B B and \u3bcm = W\u22a5 B = const. When a system is periodic in time, the action integral \u222e pdq, in terms of the canonical variables p, q, is an adiabatic invariant in general. The action integral of Larmor motion is J\u22a5 = (\u2212m\u3c1\u3a9\u3a9)2\u3c0\u3c1\u3a9 = \u2212(4\u3c0m/q)\u3bcm. J\u22a5 is called transversal adiabatic invariant. A particle trapped in a mirror \ufb01eld moves back and forth along the \ufb01eld line between both ends. The second action integral of this periodic motion J\u2016 = m \u222e v\u2016dl (2.13) is also another adiabatic invariant. J\u2016 is called longitudinal adiabatic invariant. As one makes the mirror length l shorter, \u3008v\u2016\u3009 increases (for J\u2016 = 2m\u3008v\u2016\u3009l is conserved), and the particles are accelerated. This phenomena is called Fermi acceleration. 13 14 2 Plasma Characteristics Fig.2.7 Probability of collision of a sphere a with spheres b. The line of magnetic force of mirror is convex toward outside. The particles trapped by the mirror are subjected to curvature drift and gradient B drift, so that the trapped particles move back and forth, while drifting in \u3b8 direction. The orbit (r, \u3b8) of the crossing point at z = 0 plane of back and forth movement is given by J\u2016(r, \u3b8, \u3bcm, E) = const. 2.6 Coulomb Collision Time, Fast Neutral Beam Injection The motions of charged particles were analyzed in the previous section without considering the e\ufb00ects of collisions between particles. In this section, phenomena associated with Coulomb collisions will be discussed. Let us start from a simple model. Assume that a sphere with the radius a moves with the velocity v in the region where spheres with the radius b are \ufb01lled with the number density n (see \ufb01g.2.7). When the distance between the two particles becomes less than a + b, collision takes place. The cross section \u3c3 of this collision is \u3c3 = \u3c0(a+ b)2. Since the sphere a moves by the distance l = v\u3b4t during \u3b4t, the probability of collision with the sphere b is nl\u3c3 = n\u3c3v\u3b4t since nl is the possible number of the sphere b, with which the sphere a within a unit area of incidence may collides, and nl\u3c3 is the total cross section per unit area of incidence during the period of \u3b4t. Therefore the inverse of collision time tcoll is (tcoll)\u22121 = n\u3c3v. In this simple case the cross section \u3c3 of the collision is independent of the velocity of the incident sphere a. However the cross section is dependent on the incident velocity in general. Let us consider strong Coulomb collision of an incident electron with ions (see \ufb01g.2.8) in which the electron is de\ufb02ected strongly after the collision. Such a collision can take place when the magnitude of electrostatic potential of the electron at the closest distance b is the order of the kinetic energy of incident electron, that is, Ze2 4\u3c0\ufffd0b = mev 2 e 2 . The cross section of the strong Coulomb collision is \u3c3 = \u3c0b2. The inverse of the collision time of Fig.2.8 Coulomb collision of electron with ion. 14 2.6 Coulomb Collision Time, Fast Neutral Beam Injection 15 the strong Coulomb collision is 1 tcoll = ni\u3c3ve = nive\u3c0b2 = ni\u3c0(Ze2)2ve (4\u3c0\ufffd0mev2e/2)2 = Z2e4ni 4\u3c0\ufffd20m2ev3e . Since Coulomb force is long range interaction, a test particle is de\ufb02ected by small angle even by a distant \ufb01eld particle, which the test particle does not become very close to. As is described in sec.1.2, the Coulomb \ufb01eld of a \ufb01eld particle is not shielded inside the Debye sphere with the radius of Debye length \u3bbD and there are many \ufb01eld particles inside the Debye sphere in the usual laboratory plasmas (weakly coupled plasmas). Accumulation of many collisions with small angle de\ufb02ection results in large e\ufb00ect. When the e\ufb00ect of the small angle de\ufb02ection is taken into account, the total Coulomb cross section increases by the factor of Coulomb logarithm ln\u39b \ufffd ln ( 2\u3bbD b ) \ufffd \u222b \u3bbD b/2 1 r dr \ufffd 15 \u223c 20. The time derivative of the momentum p\u2016 parallel to the incident direction of the