004 Nuclear Physk 19. C.M.G. Lattes, GP.S. Occhialini and C.F. Powell, Nature, 116, 453 (1947). 20. P. Freier, E.J. Lofgren, E.P. Ney, F. Oppenheimer, H.L. Peters, Phys. Rev. 74, 213 (1948). Problems 1. The radius of the solar system is 1.2 x 1013 m and the within it is 10T*T. What is the maximum energy of the be confined within the solar system? 2 Calculate the kinetic energy of the muon emitted in the decay rest assuming m« = 139.58 MeV and «p = 105.66 MeV. 3. Calculate the maximum kinetic energy of the electrons in aP' 4. A positron of kinetic energy 4 moc collides with an photons are emitted at equal angles (0) w.r.t. to the direction positron. Find 0. 5 Prove that in Anderson’s experiment on the discovery of panicle emerging from the lead plate is a proton (of ! then its kinetic energy should be 0.3 MeV. What will be its if it is a positron ? 6 Calculate the radius of the earth in Stoermer unit for protons energies 1,10,59.6 and 100 GeV. The magnetic dipole moment is A# = 8.1 x 1022 J/T. Radius of the earth = 6378.16 km. 7 Show that the radiation unit of length in the electronic theory has the values 0.52 cm and 330 m respectively m air 8. What is the percentage change in the mean life of muons 500 MeV ? APPENDIX A-I ELECTRIC QUADRUPOLE MOMENT itial at a point P (r) due to an arbitrary charge distribution ') is given by <*(r)= » rAlr lr-r'l AntJ r, •(AI‘1 r 1 is the distance of the field point P from the element of the source region (see Fig. A-l). We can expand the where we have used the relation ^ o I. 2 and 3 each corresponding to Here i,j can have the values x, x2=y, x3 = z. We then have
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(1947). 20. P. Freier, E.J. Lofgren, E.P. Ney, F. Oppenheimer, H.L.
Peters, Phys. Rev. 74, 213 (1948).
Problems
1. The radius of the solar system is 1.2 x 1013 m and the
within it is 10T*T. What is the maximum energy of the be confined within the solar system?
2 Calculate the kinetic energy of the muon emitted in the decay rest assuming m« = 139.58 MeV and «p = 105.66 MeV.
3. Calculate the maximum kinetic energy of the electrons in
aP'
4. A positron of kinetic energy 4 moc collides with an photons are emitted at equal angles (0) w.r.t. to the direction
positron. Find 0. 5 Prove that in Anderson’s experiment on the discovery of
panicle emerging from the lead plate is a proton (of ! then its kinetic energy should be 0.3 MeV. What will be its
if it is a positron ? 6 Calculate the radius of the earth in Stoermer unit for protons
energies 1,10,59.6 and 100 GeV. The magnetic dipole moment
is A# = 8.1 x 1022 J/T. Radius of the earth = 6378.16 km. 7 Show that the radiation unit of length in the electronic
theory has the values 0.52 cm and 330 m respectively m air
8. What is the percentage change in the mean life of muons
500 MeV ?
APPENDIX A-I
ELECTRIC QUADRUPOLE MOMENT
itial at a point P (r) due to an arbitrary charge distribution ') is given by
<*(r)= » rAlr lr-r'l AntJ r, •(AI‘1
r 1 is the distance of the field point P from the element of
the source region (see Fig. A-l). We can expand the
where we have used the relation ^ o
I. 2 and 3 each corresponding to
Here i,j can have the values
x, x2=y, x3 = z. We then have
1006 Nuclear Physic
The first and second terms in the above expansion give the potential* due to the electrip monopole (total charge) and the electric dipole IocuKhI
at the origin of the coordinate system which are not of interest for us if) the casp Of the nuclear charge distribution. The third term gives the potential due to the electric quadrupole located at the origin and is given
by
Since the integration is over the source coordinates xwhile th#
derivatives of 1/r are with respect to the coordinates of the field point (*,),
these can be taken out of the integral so that we can write
d>2 = -r- \ z s 3-4- Nip ('') *1 x'j ...(Al-5) 2 47teo 2 i j dxjdxj l r JJ 1
...(Al-3)
Now since 1/r is a solution of Laplace’s equation for all points except r = 0, we can write
HU - = ZX^ V [rj-rd^rjij^dx^r
We multiply the above expression by
1 -8,1 = 0 ..(Al-6>
and subtract it from Eq. (AI-5) which does not change the expression fo* 02. So we get
IX Z -4- (-|/p 00 Ox',x) - /2 8,y) dx' ...(Al-7) 2 4 hEq 6 i j axt axj [r )J 1
If we write
We get
Qij = J P (r') (3 x\ x/j - /2 8tj) dx'
._1_1 _ _ n ^ i = 4 JtEo 6 f j ^ Bxj d Xj
...(AI-8)
...(AI-9)
The nine quantities Qy constitute a tensor of rank 2 known as the
quadrupole moment tensor which can be written as
011 012 013
{Q}=' 02i Q22 Qn' — (AI-10)
Qi\. 032 033.
It is a symmetric tensor so that Qy = Qjt which reduces the number
of components to six. It can be shown that the sum of the diagonal elements (trace) is zero:
XQ„. = xjp(r') (3.x'2- r'2) dx'
leptric Quadrupole Moment 1CK)7
= Jp(r') (3X^-3j/W
\ w = J p (r') (3r'2 - 3r'2) dx’ = 0
\y<j|iave <2,, + 222 + G33 = 0 ...(AMI)
This means that only two of the diagonal components are dependent. This reduces the number of independent components from
ix to'five. v
Notv if wfc refer to the quadrupole moment tensor to the principal es, all the off-diagonal components vanish; Qtj = 0 for i *■ j. So we are
left with dnly fwo independent components of the quadrupole tensor. Finally if the charge distribution is axially symmetric as in the case
of an ellipsoid of revolution (spheroid), then Qu = Q22. This gives
033 =“ (0i 1 + 022) = - 2 Qu = - 2 e22 ...(AI-12)
Thus we are left with only one component of the quadrupole moment tensor Q = Q33 which is called the quadrupole moment of the charge
istribution. It is given by
0 - 033 = j P (O 0 Z.'2 - r'2) dx’ ...(AI-13)
For a system of point charges qa located at the points r 'a we have
Qij = ZqaOx'aix'aj-/2aSy) , r ...(AI-14)
For an axial distribution of charges we have
Qv = ZqaOz'2a-r'2a) ...(Al-15)
The above discussion is purely classical. Eq. (AI-14)dr (AI-15) gives
he intrinsic quadrupole moment Q0 of an axially symmetric charge
istribution. The measured value of the .quadrupole moment Q is usually
ifferent from Q0 because of the rules of space quantization in quantum
techanics.
To illustrate this we consider a very simple system consisting of two
qual charges q placed alo^g the 2-axis at the distance / from the origin
_n either side of the latter, z* is the body fixed axis of the charge system.
[Since r' = l for both the charges, the intrinsic quadrupole moment calculated with the help of Eq. (AI-15) comes to he
0q = QO l2 -l2) + q (3 l2 - /2) = 4 ql1
We now calculate the quadrupole moment Q referred to the space-fixed axis z.
Let / be the spin so that the magnitude of the angular momentum,
ording to quantum mechanics, is V7(/+ l) which iS>:along z'. Its
projection along some preferred direction z in space (wfiicft may be the
direction o' an applied electric field) is MU, where the (21+ 1) possible
yalues of M are /, 7 - 1, / - 2.— /. If (3 is the angle between z' and z, then (here are (21+ 1) possible values of B given by
Nuclear Physics
cos ft = |-^= ...(AI-16) W+ 1)
Since the maximum value of M-l, the minimum angle ftm between
and z is given by _
o / a/ /
Pm V/(/+1) V/+1 ...(AI-17)
Eq. (AI-15) can be used to find the quadrupole moment Q referred
to the space-fixed axis z for a given orientation ft of z' w.r.t. z :
Q = q (312 cos2 p - P) + q(3 ? cos2 0 - Z2)
= 2qf 3 M2
./(/+1) . ...(AI-18)
j£_I
i(i+1) 3 M2 — / (f + 1)
/(/+1)
For M = I, we get
It can be shown that Eq. (AI-19) can be applied to any chargr
distribution with an axial symmetry.
So we get, using Eq. (AI-13), •
...(AI-20)
Eq. (AI-20) is semi-classical since it does not take into account the
nuclear wave function. If we consider a nucleus in the shape of an
ellipsoid of revolution rotating in space, the angular momentum / has the
projection K upon the symmetry axis of the nucleus (see Fig. 2.9 tit
Ch. II). It is then possible to connect the electric quadrupole moment Q
referred to a space-fixed axis along which the nuclear spin / has the
component M - ! with the intrinsic quadrupole moment Qq referred to the
axis of symmetry from a knowledge of the wave functions of the rotating
ellipsoid. The result that one gets is*
...(AI-20)
3Ez-/(/+l) e = Qa ...(AI-21) u Wo(/+l)(2/+3) ■
This is the result quoted in Ch. II (see Eq. 2.12-12).
Energy of a charge distribution in an external electric field
Let $ be the potential from which the field E is derived f
E = - V <)>. If p denotes the charge density, then the energy of the charge
distribution in the field is
See Physics of the Nucleus by M.A. Preston, Ch IV See also Theoretical Nutt*#* Physics by Blatt and Weisskopff.
Electric Quadrupole Moment
^ U=jp(r)ty (r) dx ...(AI-22)
If we expand * (r) about the origin of the coordinate system, we can
'' 1 (d£.l X =<HO)-£x,(Ei)0--i:ix^k^ ••••
' i L t j \axi Jo
where v E,=- fr/dxj \ < * We then have .
’ • r T 1 I dEj l tU = \p(r) <KO)-i:x,(E,)0--IIx1.x.^
Writing
...(AI-23)
(7 = X Url we have
l/0 = Jp(r)<|>(0)<#t' = <fr (°> J P (r) dx’ = q if (0) ...(AI-24)
where q = \p(f)dxJ is the total charge of the distribution assumed
concentrated at the origin.
U, = - J p (r) X x, (Ei)0 <fx = -Z (£,)„ J p (r)xtdx
= -IPi(£<)o-~P ' ..(AI-25)
where p, = j p(r) x, dx is the i th component of the electric dipole moment
of the charge distribution, assumed concentrated at the origin.
Thus U0 and 17, are the energies of the point charge q and the dipole
of moment p located at the origin respectively.
The third term in the expansion gives
U2=-±lp(r)21x,Xj dx z I j V1 Jo
2 ' 1 I4**- n
...(AI-26)
Since the sources producing the field E are far removed from the
given charge distribution, we have
3E, 3Ei „ V-E-Zy1* E£^5jy=0
i oXj i j oXj
If we multiply this by p (r) ? 5,/6 and subtract from the integrand
in Eq. (AI-26), the value of U2 remains unchanged. Thus we get
Nuclear Physics
- _ uU
1 3£, f J p(r) (3 X'Xj-SS^dx ...(AI-27)
v 1 Jo In view of Eq. (AI-8), this can be written as
1 (be'
t'2 = ‘6ffe«('^J0 •<AI-28'
where QtJ represents the quadrupole moment tensor of rank 2. U2 gives the
quadruple contribution to the energy of the charge distribution, the quadrupole being assumed to be located at the origin.
p h "> *
APPENDIX A ll ?
THEORY OF SECTOR FOCUSED CYCLOTRONS
The principle of azimuthally varying field (AVF) or isochronous cyclotrons which finally led to the construction of sector focused cyclotrons discussed in § 12.15 was first proposed by L.D. Thomas in ___ ..... « l .• n.. *!__tkaon; r»f frtniQina
by the azimuthally varying magnetic field.* Referring to Fig. 12.17 in § 12.15 we note that if there are N hills
and N valleys and the magnetic field variation is sinusoidal, we can
express the field B (0) it an azimuth 0 by the following equation :
B (0) = < B > (1 +/cos NQ) (AW-1)
where < B > is the average guiding field and / is the peak flutt^r amplitude. B (0) = Bz is along z in the medium plane. This fluttering of the
field gives rise to a force qv < B >/cos NQ on the orbiting particle which distorts the particle orbit. Instead of a closed circular orbit of radius 7? in a field which is the same at all azimuths, the orbit is now a curved
polygon as shown in Fig. 12.17c. Denoting by x the radial displacement from the constant radius orbit
due to the fluttering field, we can write the equation of motion as
mx = -qv<B >/cos NQ tAu-Z)
It has a solution of the form
x = (fR/N2) cos N 0
Thus x has the same periodicity a» the magnetic field variation. The orbit extends beyond the ipean circle in the hill regions and lies within the
circle in the valley as can be seen in Fig. 12.17c.
The radial velocity vr of the particle is
vr = (0 • dx/dQ = - (vsfR/N) • sin IV 0 (AH-3)
where 0) is the angular velocity.
Since VxB = 0, we get
(V xB)r = (1/r) ■ dB /39 - dB^/dz = 0 (AII-4)
Hence the azimuthal component of the magnetic field is given by (writing
R for r)
* See The Principle of AVF Cyclotrons by C. Ambasankaran, S. Chatterjee and A.S.
Divatia in Physics News, Vol. 8, No. 3 (1977).
1012 Nuclear Physics
R „_i<^ <B>Nfz . 6 dz'Z~rdd'Z~-R~Sm NQ CAD-S)
Due to the action of Be on the radial component vr of the velocity,
the particle experiences a force Fz in the vertical direction (see Fig. 12. l<vr in § 12.15). This is given by
Fz = -<!VrB0--q<B>uif2zsm2Ne Since B = mto/q, we then get
Fz = ~ nuo2f2z sin2 N 0 ...(AII-6)
^vanishes at the centres of the valleys (0 = Jt, 3jt,5ji etc.) and the hills
ft"0;-,2”,’4" ^dc.). ft is maximum at the hill-valley boundaries where , n/I’3K/2etc- This force ts always focusing. Since the oscillation frequency of the particles is small compared to the frequency of variation
of the field, we can take the mean value of sin 2 AO equal to - in the
expression for Fz so that we can write
< Fz>~ 2 m<£f2z ...(AII-7)
Apart from the action of Be on v, the radial component Br of the magnetic
field acting on the velocity component ve = r to also exerts a force F on
the particle in the vertical direction given by
F\ = gvBBr = qnoBr * ...(AH-8)
. dfl, dB where Br = —Z = — Z .(AH-9)
in view of the vanishing of the curl of B. If n denotes the average field index we can write
< Bz > - < B0 > ^ J ...(An-10)
where n > 0 since the field increases outwards (see § 12.8). n is given by
d<Bz>
We then get
<BZ>
dBz rz - qrta z = n Bzooqz = nmus2z
where we have put S.a) = mu>2/q
The equation of motion along z is then given by
2 + j^--«]a>2z = 0
...(AII-11)
...(All-12)
...(All-13)
We saw in § 12.7 that a magnetic field decreasing with increasing radius produces a net focusing of the beam in the z-direction. Hence the magnetic field Bz increasing outwards has a defocusing action on the
theory of Sector Focused Cyclotrons 1013
beam However, the combined actions of the azimuthal component Be and
ihe rad^l component Br of the field will produce net focusing due to
vertical oscillations if in Eq. All-13.
' /2/2 > n ...(All-14)
This means that for focusing, flutter amplitude/of the magnetic field
.ast be large enough to fulfil the condition (All-14). Since the angular [velocity (0 of the particle must remain constant for resonance condition to [hold the average magnetic field < Bz> at all azimuths must increase
according to the following relation
<iz> = <B0>Wl-P2 =<B0>Y ...(All-15)
_- where T=l/-Vl -p*.
Further to = v/r= Bc/r = constant = Mq (say)- ■
|So we have r= « “b I'Ve then get
d<Bz> d<B,>dB <«o>P
dr " ap dr (1-P2)3/2
Hso using Eqs. (All-15) and (AII-16) we have
<BZ> <Bo1<B0>
We then get from Eqs. (AII-17) and (AH-18)
r d<Bz> pc oibp<g0>ir?
<BZ> <%7 <Bo> = P2r2
So the focusing condition (AH-14) reduces to
f /L> .
.(AII-16)
...(AII-17)
...(All-18)
...(AH-19)
...(An-20)
The frequencies of the vertical and horizontal betatron oscillations depend not only on n but also on the azimuthal configuration of the magnetic field. The deperfdence of the frequency ratios V, - U)/co and
v =a>/0) on the structure of the magnetic field is very complex and
rigorous calculations of the orbits in the AVF machines can only be done
numerically with the help of computer. Writing the magnetic field at an arbitrary point in the median plane as
we can obtain the
0,0) = B0 j d>(r, (
he function <1> (r, 0) from the condition
2n Jo d> (r, 0) dQ = 1
...(AII-21)
...(AII-22)
Tlris leads to Eq (All-19) for the variation of the average field
< B, > vith r as given earlier.
Nuclear Physic#
We now introduce the/luster function F(r) defined as ' ^
,.(An-2.<
Further 'f ^ (r) denotes the angle made by the tangent to the spiral with the radial direption then it can be shown that to a first order of approximation*
V'2=1+" ...(All-24)
vl = -n + F (R)(l + l/2tan2Q ...(AII-25)
Thus for the radial oscillations the dependence is similar to that for a conventional cyclotron (see Eq. 12.7-10). The vertical focusing increases with F and with - ..'.s _ *
The function «!> (r, 6) for sinusoidal contours in purely radial sec tori' can be written as (see Eq. All-1)
<J>(r)=l+/cosWe ...(AII-26)
In this case, the flutter function F=f2/2 is a constant where /is the flutter amplitude. J
For spiral sectors and sinusoidal field variation d>(r, 0) can be written as
<I>(r,e)=l+/c0s[tf{e-g(r)}] . ...(AII-27) where g(r) depends on the shape of the spiral. In this case also it is found F=f /2. ' ^ . , Tw°kmds of sPiral are generally used, the logarithmic spiral and the
Archimedian spiral, for which g{r) has the following forms :
For logarithmic spiral : g(r) = tan £ In (pr) ...(AII-28)
For Archimedian spiral : g(r) = p r = tan £ . .(An-29)
where p. is a constant.
The condition for isochronism of the particle motion and the accelerating voltage requires that the average field index n be a function ot the radius r. This variation can be expressed in terms of the energy E for any radius r as follows.** 62
- ~(An-3<»
a . .For better vertical focusing it is desirable to increase C and use the Archimedian spiral for which £ increases linearly with r (see above). For radial oscillations vr * y which shows that \r changes appreciably with
energy and leads to instability of the orbit for the integral resonance Vf z. this limits the energy of these cyclotrons to about y=2 or the kinetic energy T=E0 which is 940 MeV for protons.
A.V.F. cyclotrons have been built to produce intense meson beams Cmeson factories).
* ?ee PnnciPks of Particle Accelerators by Perisco, Ferrari and Seere ** See ibid.
1 i i ! lilillll i in i
m APPENDIX A-III
vTHEORY OF ALTERNATING GRADIENT (AO) FOCUSING USING QUADRUPOLE
' LENS SYSTEMS
(We shall consider the motion of a charged particle through an _zctrostatic guadrupole lens which consists of four electrodes with sections in the’shape of rectangular hyperbolas arranged symmetrically as shuvvn in Fig. 12.20a. The alternate electrodes are at (he potentials + V
End - V- The beam passes along the axis of the electrode system (z-axis). ■Tie transverse field Ez has the form Ex = -Gex in this region, G being I constant. Since the field has no Z-component, the condition V • E-0 Eves Ev = + Gey. This makes it possible to focus the beam strongly in the
Ez plane. At\he same time there will be strong defocusing in the Ez plane. If the beam now passes through a second e.s. quadruple lens Eith reversed gradients in which Ex = + Gex and Ey = -Ge y, there will be
Eefocusing in the x-z plane and focusing in the y-z plane. However there Is net focusing in both the planes for the beam passing through the two
nses successively (see Fig. 12.20b). The above field pattern is derivable from a potential of the form
V=(Gt/2)(x1-y2) ...(AIII-1)
he equipotentials are rectangular hyperbolas.
We now analyze the motion of the particle in such a field. The
quation of motion in the x-z plane in the first lens can be written as
mix-- qGex, x + p2x + 0, |52 = qG/m ...(AIII-2)
i he solution of this equation is of the form
x = Xq cos Pf + (xo/P) sin Pt
= x0 cos kz + (xo/to) sin kz ...(AIII-3)
x0 and x0 are the initial position and transverse velocity respectively. Here
we have used the relation z = vt, v being the velocity of the ion beam along
k = (qGe/ mv2)1/2
We then get for the exit point z = a from the field
Also *=-‘bc«*a + (V*v)sinfez -.(AHI-S) so x/v = - kxQ sin ka + (Xq/v) cos ka (Amftl
I he two equations (AIII-5) anH rAm , u- l. *’ transverse velocity at the exi* i . w^,c^ S,ye the position aritl yield the * —*■ fon,»
I* U Xl/^slntoit^ f l*/lT M«n*B cosfai| ji^v --(AIU-7)
In the second field region since the tion g-~ ■ solution of Rn /aih - ’ nCC th sgn of G' ,s ^versed, the
sinusoidal functions so that"we fU"Ct'0nS in Place of ***
_ cosh ka (l/k) sinh jta! 1 Jd j I | k sinh Jta cosh Jta j 1 Vv| ...(AIII-8)
regions successively become 4 t •
x \ -\ cosh ka (i/k) sinh fail I
— J l-*s= M,^q |£| *b (cosh ta cos ka — sinh Jta sin /ta) +
, = . . . , (V*v)(cosh fci sin fci + sinh Jta cos fai) * (sinh ka cos ka- cosh Jta sin Jta) +
(Vv Kcosh ka cos ka + sinh far sin ka) j
Similarly we g« in the „ plane, * fo||0„ing . ~ 'AnM>
yo (cosh ka cos ka + sinh ka sin ka) +•
L/J= . a. . . , 6-(/*vXcosh *a sin + Sinh ka cos ka) \yv I *yo(smhfaj cos ka- cosh ka sin ka) +
* ^ Wcosh ka cos ka- sinh ka sin ka)
qu.d™p'oleate„VL1^“°S ^ T™* »*.» t^e'tie discussed above. In this case the quadrupole Ienses product of the particle velStv ® grad,ent is by the AG accelerators, the particles na« ^eld 8rad*ent In actual elements. The matrix thr°Ugh many ^ands of AG yield any meaningful result^ h beCOmes to° complicated to
S °f is
z:*: %*£
coska (I /A) sin ka x0 * sin ka cos ka Xq/v
i
FTheory of Alternating Gradient 1017
transformation matrices for a magnetic quadrupole lens as given below. If
we write'’" .
k = (qGJmv)w2
cos ka sin (ka)/k
v ~ -k sin ka cos ka
...(AIII-11)
...(AIII-12)
cosh ka
...(Ain-14)
...(Am-15)
..(AIH-16)
| ' , . . sinhta v I i and, i = C°Shka k ■ * ...(AIII-13)
^ y v k sinh ka cosh Jta
r for motions in the focusing and defocusing planes respectively. For the basic triplet of lenses as stated above, the transformation
matrix for finding the final position and transverse velocity at the exit end in the case when the first element is focusing is given by
cos(Jta/2) [sin (Jta/2)]/Jt coshta [sinh (ka))/k x — Jfc sin (Jta/2) cos (Jta/2) k sinh ka coshta
cos (ka/2) [sin (ka/2)]/k /A ITT 141 -k sin (Jta/2) cos (ka/2) '' ;
After multiplication of the three matrices, the resultant matrix becomes
cosh Jta cos ka l/k (sinh ka + cosh ka sin ka) (Affl-15) k (sinh ka — cosh Jta sin ka) cosh ka cos ka
This matrix can be written as
cos [i (1/if) sinp ..(AIH46) - K sin p cos [t
provided the product cosh ka cos Jta lies between - 1 and + 1. i.e.,
— 1 < cosh Jta cos Jta < +1 ...(AIII-17)
In Eq. (Am-16) we have written cos p = cos Jta cosh Jta „.(AIII-18)
and K2 = Jfc2(l - cosh*Jta cos2 Jta)/(sinhJta + cosh ka sin ka)2 ...(AIII-19)
As long as the condition (AIII-17) given above is fulfilled, both p and K are real and the matrix (AIII-16) is the proper transformation matrix for obtaining the exit position and transverse velocity. Comparing with the focusing matrix (Am-12), we conclude that the matrix (AIII-16) is also a focusing matrix so that there is net focusing of the beam of particles in going through the basic triplet of lenses considered here.
If, then, a large number of the sequences of the basic triplets as above are added one after another, we shall get net focusing provided the condition (AIII-17) is fulfilled. This is known as the criterion of stability.
The above analysis applies to both linear and circular systems. In the circular system, like the AG synchrotron, the betatron oscillations are along the radial direction (x) about the closed orbit of radius R and in the
APPENDIX A-IV 1018 Nuclear Phytlyt
vertical direction (y) about the median plane. For a magnetic field variation with a field index
\n\ = ^ B dr
the equations of motion in the radial direction for two consecutive sector* are*
d2x/ds2 + (n- l)/n k2 x = 0 a , , ...(AIII-20) and d x/ds2-{n+\)/n k2x = 0 ...(A1II-2I) where k has been defined above, j is the distance measured along the
In the AG machines n is large (n ~ 100 or more). So the above equations reduces tf>
cfx/ds2 + k2x = 0 ...(AIII-22) since n and the field gradient have opposite signs in the two cases Thl equatmn of motion in the vertical direction has the exact form given by tq. (AIII-22). Thus the oscillation wavelengths are equal in the tw# cases : k.. — k
Now the quantity p given in Eq. (AIII-18) is the phase shift in th# betatron oscillation per pair of quadrupoles. If there are N pairs
SmnSh 6 u°F 2N AG-quadruP°les, then the total phase shill I ound the orbit will be Mp. If this is an integral multiple of 2n, there will
be an integral number of betatron oscillations per revolution which leads to orbit instability. Such instabilities are avoided by calculating theoretically the locations of these resonances and then taking necessary steps to operate in regions between the resonance lines.
See Particle Accelators by M.S. Livingston and I P Rl™,»
D VATION OF RARITA - SCHWINGER
1 EQUATION
i For a mixture of S and D states, the wave function is
:•/ ’ t j- <p = q>5 + <j>D
a where the S-state wave function is
<t>s = v(r)
'101
and the D- state wave function is
<j> =^My> D r 121
The Hamiltonian operator for the two body systeip can be written as [ (for M] <= N2 = M)
Hsd-^--fV2 + V M
: n H i 3 a r2 3 r dr ?
where S12 is the tensor operator. )
? 3 V 1 a = ——r sin 0 +
■te¥cW+ Vf(r)Sl
.‘f j t
30 sin 2 0 3 ® sin0 90
It can be easily seen that 1 3 2 3 v(r) v" (r) o -V ' -v
r2 dr dr r (AIV-1)
, 1 3 7 3 (0 (r) _ o)"(r) r1 dr dr r r
(AIV-2)
The operator a2 operating on the spin angular part of the wave
function gives
a 2 yljLs = I‘(L+ 1) //ls (AIV-3)
where L can have the two values L = 0 (S—state) and L — 2 (D— state). Substituting in the wave equation, we get
1019
1020 Nuclear Physics
,®L_V i ) . ^2 to(r) 2(2+1).., A# r )I<JI ^ 121 +T7—' ,+ M r
Because of the orthogonality of the spherical harmonics, we then gel
~ M di* + Vc W “ Ev = ~ Vr(r) ® ...(AIV-5)
T^fd2 (a 6ti)'l A/ j_2 j +^c(r)^-2Vr(r)(o-Fxo = —'I^VT(r)v
... (AIV-6) where )ve have made use of the results (see. Eqs. 17.12-16 and 17-12-17)
^12 ^ioi - ^ Yl: ... (AIV -7)
...(AIV - H) ^12 ^121 - ^ioi - 2y,2, ...(AIV - K
Proof of the relations (17.12-16) and (17.12-17\
Let Sn y/o, = a + b Y'2\
It may be noted that does not contain 0 or O. When averagctl
over the angles < $i2 > d>=0 and the average on the l.h.s. <Sf}
K!oi > = r.h.s. since Y^ does not contain 0 and 4>, the average
of the first term is not zero and hence we should put b = 0. The above relationship is valid for all 0 and <b. In particular, if we choose 0 = 0, we
get y22(0 = O) = y2<e=°) = 0, while y°(e = 0) = V^ 2 4n
On the other hand
V0 = 0) = 3 (of x rQ ) (^ • r0 ) - of-6^
^t2 ^101 - (30lz ~ ^
= ^-(3 «. «2-a, “2) for =0
Againy,1L (0 = 0)= Y v~'Y r° + Y i 121 ’ M0 »*1 10 21Z' V10 20?
...(AIV-4
Derivation of Rarita - Schwinger Equation IUZ1
l)ct,«2
V 2tyx2
V321T _a/3jt M VT 2a,«2
1 1 20t2«2
5,2y,ot=>/8 y.'j, for 0 = 0
.-. b = >IS and
5.,y.!».=V8 y,'21 for all 0 J12 101
...(AIV-10)
'To find Sy, y|21 we choose 0 = n/2 and <t> = 0 (x - direction).
cos 0 = 0, sin 0=1, cos 0= 1, sin <t> = 0
S12 = 3 (o| - r0) (a2 r0-) — o|, o2
= 3<Jlx Ofr -
A = «n2 e exP (») =
= sin 0 cos 0 exp (i d>) = 0 oTC
. (AIV - 11)
(AIV - 12)
... (AIV-13)
...(AIV-14)
,.JJT 2~ V8*
r/xi'-VI’
=^/m»^*“^10'^16n Wiaj
yl r 12*
(AIV-15)
5|2 y,21 " (3®U ®2* ®l ’ **2)
» - _
-«|«2 3 P2
l32^ + l^TJ v > 3P,P2 «1*2 9a1<X2 3 P1P2
l^+l327 + l327~l32^
10 a,otj 6fi,P2
=^32r_^r
8 0,04
V32 n + 2
$12 ^121 =V8 y' -2y;21
j" 0,02 3 P, p2"
(V32 n V 32 it (AIV-16)
...(AIV-17)
APPENDIX A-V ■ '!* i V •
VECTOR ADDITION OF ANGULAR MOMENTA: CLEBSCH GORDAN
COEFFICIENTS
angular S£i“m v««o" “ ?<»"P°'"’d W“ of the system Thus thp nrhif,1 eJhe resultant total angular momentum
electron combine to producethl ?otal?u“ 2*^”^ *”L“°”“ gives rise to the fine structure of the affeiiic snectral lines i„rh+ S Wh'uh
55a3F»^»^=SS Consider two angular momenta /, and J2 which separately satisfy the
commutation r„,es [7,./,) =0 etc.. and which mutually commit,eu!
m' 6 the slmultaneous eigenfunction of the operators j] and J,
m “fvV0 thC reSpCCtive e*£envalues 7, (/, + I) and «, where and J .™8f be ,ntegers or half-°dd integers such that -j,<m <+j Similarly Y^ ,S the simultaneous eigenfunction of ;22 and Ju belonging
to the respective eigenvalues j2(/2 + 1) and m2 where -;2 <*,.< + /,
The product of the wave functions Yy. (ll| , y is an eigenfunction of
ZJS3ZLZZtotal angular m“m * with the
* ~ «-«■ «-
two angular momenta J, and J2 „ is, how.™ VosllbleTform ££ combinations of the products y y * • , proaucts Yjy Wj which are simultaneous
ergcnfunctions of / and 7, wMf'dla eigenvalues 7(7+1, and «
respectively. Denoting these eigenfunctions by ip"^ we then have
j 1 72
<AV‘I)
1022
109^ Vector Addition of Angular Momenta
’*vhere C; j (J, M ; m,, m2) are called the Clebsch-Gordan coefficients also
denoted by (/, A tn2l JM). The C-G coefficients are real numbers and
yabish unless M = m, + m2. So the double sum in Eq. (AV-1) is essentially
a single sum over m, with m2 determined by m2 = M-m,. A it may be noted that the bilinear combinations T „ can be
ex firessed*in terms of the eigenfunctions ^ . :
J kJ-7
...(AV-2)
where the values of J extend from \j\ —fal to (/i +J2)- The C-G coefficients satisfy the symmetry relations given below
Cj j2(J, M ; m, mf) = (- 1 )J,+Jl~7[c/2j1 (J>M ; m2mi)l
THEORY OF GEOMAGNETIC EFFECTS OF V THE COSMIC RAYS
Consider the motion of a charged cosmic ray particle of relativistic mass m carrying a charge q coming from infinity towards the earth.
The magnetic field of the earth is equivalent to that due to a
magnetic dipole of moment M = 8.1 x 1022 J/T located near the centre of the earth, the dipole axis pointing south. Using spherical polar coordinates r, X and <|> shown in Fig. 19.4, we can write the velocity of the particle in terms of the components vr = r, = r X and = r cos X ■ d> and the unit
vectors er ex and along r ,X and <b respectively. We get
v='rer + rXex + r cosXOe^ ..(AVI-1)
The magnetic vector potential due to the dipole of the earth is
4 Jt _
Mo M cos X a ...(AVI-2)
The magnetic induction is Eg UnM
B=VxA=--j (- 2 sin X er + cos A. • e...(AVI-3) 4 it r
The horizontal component of the earth’s magnetic induction at the equator is Bt=0.31 x 10“4 T. If p denotes the relativistic momentum and jp the
radius of curvature of a particle of unit electronic charge e in this field we can write *
Be v = m v2/p
B p = e e
pc = Bpec ’ ...(AVI-4)
The momentum of the particle for which p = re (the earth’s radius) is
then given by
pc = Bre ec
- 0.31 x 10" 4 x 6.378 x 106 x 1.6 x 10"19 x 3 x 108
= 59.6 GeV
1025
Nuclear I’hwo *
Thus pc is large compared to the rest energy of the particle, be it,#
proton (rest energy 0.938 GeV) or an electron (0.51 x 1(T3 GeV). Hcne# pc can be taken to be equal to the total energy (or kinetic energy) of ih« particle. More accurate estimates give the kinetic energies of the electron and the proton to be 59.6*GeV and 58.5 GeV respectively for the abov# value of pc.
In a static magnetic field there is no change in the velocity v of th« particle or in its energy E and momentum p. The Lagrangian of such a particle is*
L = -m0c2 Vl -p2 +e(A ■ v) ...(AVI-S)
Using Eqs. AVI-1 and AVI-2 we get
L = - m0cz P2 +'C p0 Me cos2 A.
...(AVI-ft)
Since L is independent of <)) we get from the Lagrangian equation of motion
= = 0 ...(AVI-7)
so that
dt di> d <j> dL
P* = 3^ ~ constant
...(AVI-7)
...(AVI-8)
dL 2 d ,.r. FF 3v *VoMe 7 ^ = -m0e^(V1^P)^ + —cos X ...(AVI-9)
We get finally
a<T ^ 19(p
v2 = r2 + r2 X2 + r2 cos2 A, <j>2
9v r2 cos2 A, d>
d<|> v
...(AVI-10)
...(AVI-11)
V, ...... r2 cos2 A. .i Po Me cos2 A. . P* = P -+ - ...(AVI-12)
* L v 4nrp J
Since p^ and p are both constant, we get
P» r2 cos2 A. ; Po Me cos2 . —* =-<b +---= constant ...(AV-13) p V 4 71 r p
Let 0 be the angle made by the trajectory of the particle in space with the meridian plane. 9 will be taken positive if the orbit crosses the meridian plane in the direction of increasing <|> so that sin 0 is equal to the <j)-component of a unit vector in the direction of the velocity. Hence
„ v* r cos A. • (|) sin0 = -I =-31 ...(AVI-14)
...(AVI-12)
...(AVI-14)
Then from Eq. (AVI-1 we get
P* - . „ Po Me cos2 A. - - r cos A sin 0 +---; p 4 n rp
...(AVI-15)
Theory of Geomagnetic Effects of the Cosmic Rays
[phVe b is a constant which in the equatorial plane is f ■qua) to the impact parameter / ks can be seen from Fig. A-2. ■r< -^q. (AVI-15) is known ' jfe Stbermeir’s theorem. P yte now introduce a new knit oV length C called f* iStoermer’ defined as ** ■follows Let \ Fig. A-2 Motion in the equatorial plane (X= 0).
bef *in0
..0 o viiiSN!. at 'II i.i(AVM6) • j 4 7t mv
t has the physical significance of being the radius of a circular orbit of Particle of rigidity mv/c in the equatorial plane of a dipole of strength Wt\ Since the field due to this dipole in the equatorial plane (A. = 0) is
= p0 Af/4 n r we get
mv2 „ PoMev
9Me .'i In Thte emhktant Y is proportional t6 the Component of the ntthfar tnomentum of the particle parallel to the magnetic moment of the earth at
infinite distance. We define the meridian plane as the plane through the position
Vector of the particle and the geomagnetic polar axis (z-axis). Obviously this plane has different orientations at different points. The motion of a particle in the field of a magnetic dipole can be separated into the motion in this meridian plane and the motion of the meridian plane about the dipole axis. Eq. (AVI-18) gives one of them viz., the ^-component of the
Theory of Geomagnetic Effects of the Cosmic Rays
Where b is a constant which In the equatorial plane is t fcqual to the impact parameter /
■s can be seen from Fig. A-2.
I; \Eq. (AVI-15) is known
I^Stoemter’s theorem. *5
p JliWe now introduce a new •* •
nnit ‘of 'length C called ^ ' - i t?
pStoermer’ defined as '$
follows. LetV Rg- A-2 Motion in the e I s • ' ;
V c2 » — r.-«, 6 vs;Ai!- ttt n ;..(AVM6) • ^ 4it mv
fc has the physical significance of being the radius of a circular orbit of «IS particle of rigidity mv/c in the equatorial plane of a dipole of strength ■if. Since the field due to this dipole in the equatorial plane (X = 0) is
?B = |i0 M/4 n r3 we get
mv2 „ \ioMev ‘
Orbit b=f sin6
ivhich gives r =-- ...(AVl-i i) f 4 7t mv
Substituting C in place of r gives the above interpretation for C.
If the lengths are expressed in Stoermer units Eq. (AVI-15) becomes
... i • n . cos2X._ - r*VT_ 1
,Z 1 ” Ho Ate ’
p i The constant y is proportional to the Component of the angular
momentum of the particle parallel to the magnetic moment of the earth at
infinite distance. We define the meridian plane as the plane through the position
vector of the particle and the geomagnetic polar axis (z-axis). Obviously
this plane has differenborientations at different points. The motion of a
particle in the field of a magnetic dipole can be separated into the motion
in this meridian plane and the motion of the meridian plane about the
dipole axis. Eq. (AVI-18) gives one of them viz., the ^-component of the
motion. This equation can be solved to give sinG : ' »i3 -
*».~2xizssg . • ...(avi-20) r, cos A, C
; 1 J •; •• 'i „ ■' .' Eqs. (AVI-18) and (AVI-20) are valid for positive particles. For
negative particles, the second term on the l.h.s. in Eq. (AVI-18) is
negative which makes the second term on the r.h.s. in Eq. (AVI-20)
' Nuclear Phyxitl
positive. For a given y the regions of the meridian plane for which
I sin 8 I > 1 are forbidden for the particles. Only the regions for which
I sin 8 I < 1 are allowed. Even in these allowed regions the particles cannot
be everywhere. The allowed regions which extend to infinity are available Mi
the cosmic ray particles. However, those allowed regions isolated from
infinity by forbidden regions are not available to the cosmic ray particle*
Only high energy particles shot from the earth can circulate in these region*
For r, = 1 the radius of the particle trajectory p - the radius of the
earth. This happens for the particle momentum p = 59.6 GeV/c (sec Bq,
AVI-4). A close look at Eq. (AVI-18) or (AVI-20) shows that for
*> 1 the particle can come to the earth from any direction.
For rs < 1, pc < 59.6 GeV, the particles cannot reach the cailli
from all directions. Detailed calculation shows that if y > - 1 for positive
particles, the particles are still observable. On the other hand for
y<+ 1, the particles cannot reach the earth from any detection. The
observable values of sinG are determined by the condition (for y>- |)
■ a cos A 6 7^1 7~ ...(avi-21)
€ ' e
The limiting value of the angle is then given by (for positive particleii)
• n 2 cos A , sme‘=^X—pr ...(avi-22)
f 'e f
The complement of the critical angle 0C is equal Jo the semivertical angle
of a cone known as the Stoermer cone with axis perpendicular to the meridian plane such that no particle is able to arrive from infinity at the observation point along directions within this cone. Figs. 19.5a and b show the Stoermer cones for positive and negative particles respectively The planes are the horizontal planes at the point of observation P. For positive particles all directions east of the cone such as PA in (a) are forbidden. Similarly for the negative particles all directions west of the cone in (b) such as PA are forbidden. Thus the directions such as PB in both the figures are allowed. The limiting values of the momenta for given 0 and A are determined as follows.*
Since the radius of the earth is 6.378 x 10^ m its value in Stoermci unit can be written as (using Eq. AVI-16)
6.378 xlO6 6.378 xlO6^ .
e >7C V (p 0 Me/A re) °'
where pc is in GeV. This gives pc = 59.61*.
The inequality (AVI-21) gives
^ 1 — Vl - cos a cos3A f > . cos a cos A ...(AVI-23)
See Nuclear Physics by Enrico Fermi.
Hiiiiiiii.iiminiunriiHiiMii
Theory of Geomagnetic Effects of the Cosmic Rays
where a = Jt/2-0. Hence using Eq. (AVI-23) we get _. _ -.9
„ , 1 - V1 - cos a cos? A :> 59.6 -r-
cos a cos A ...(AVI-24)
For particles in the meridian plane (sin 0 = 0), this reduces to
■t{ ' pc > 14.9 cos4 A GeV ,..(AVI-24a) <t ■%
Table AVI-1 gives the minimum momenta in GeV/c for positive
particles which can reach the earth from different directions at different
latitudes. * » ’ Table AVI-1
Direction j of arrival
c pm in in GeV at X cos a
3° 30° 45° 60° 90°
West -i 10.2 6.4 3.1 0.9 0
Zenith 0 14.9 8.4 3.7 0.93 0
East +1 59.6 13.2 4.6 1.0 0
0 2 4 t 8 10 12 It It II 20
BRxNf’tesla-m J Fig. A-3. Variation of the half angle a of the Stoermer cone with the magnetic
rigidity Bp,
Fig. A-3 shows the variation of the half angle a of the Stoermer cone
with the magnetic rigidity Bp which is measure of the particle energy. For
example at A = 20°, a = 180° for Bp = 28T-m. So all directions are
forbidden for particles of the above magnetic rigidity or less. On the other
hand for Bp = 78T-m, a = 0°, so that all directions of approach are
possible for particles of this magnetic rigidity or greater.
The general conclusion that one can reach is that the positively charged
primary cosmic rays should arrive at a place in smaller numbers from the east
than from the west, the Stoermer cones pointing east for them. For negatively
charged primary cosmic rays just the opposite is true. They should arrive in
greater number from the east than from the west. ,
APPENDIX A-VII
SOME FUNDAMENTAL CONSTANTS
Velocity light in vacuum (c)
Permeability of free space (po)
Permittivity of free space (e-o)
Electronic charge (e)
Avogadro number (No)
Specific charge of electron (e/m,)
Planck's constant (A)
Ti =(h/2n)
Boltzmann constant (k)
Compton wavelength for electron (A,c)
Electron volt (eV)
Atomic mass unit (u)
Electron rest mass (m,)
" .? " :i
Proton rest mass (Mp)
Neutron rest mass (M„)
Proton to electron mass ratio ,
Bohr magneton (p«) * •* ’ ,.
Nuclear magneton (py)
Classical electron radius (r,)
Bohr radius (do)
Rydberg constant for infinite mass (/?_)
Fine structure constant (a) g-factor for proton (gp,)
g-factor for neutron (g„)
Gravitational constant (G)
1030
2.997925 x 108 m/s
4itx 10'7 H/m
' 8 85419 x 10 '12 F/m
1.60219X 10'I9C
6.02205 x 1023 molecules per mole
1.758805x 10" C/kg
6.62618 x 10”34 Js
1.05459 x 10'34 Js
1.3807 x 10“23 J/K
2.42631 x 10'12 m
1 eV = 1.60219x 10'I9J
lu= 1.6606x10'27 kg = 931.50 lyieV
9.10953 x 10'3lkg
5.48580 x 10'4u
5.11003 xift^eV
1.67265 x 10'27 kg 1.0072765 u
. i. 938.280 MeV
' 1.67495x10' 27 kg 1.008665 u 939.573 MeV
1836.15
9.2741 x 10' 24 J/T
5.05082 x 1(T 27 J/r
2.81794 x 10'15 m
5.29177 x 10'" m
1.0973732 x 107m~‘
1/137.0360 5.586
- 3.8262
6.6720 x 10'11 Nm2/kg2
The modified long form periodic table.
APPENDIX A-IX
PROPERTIES OF STABLE ISOTOPES*
The atomic masses are given in 12 C scaie. The star (*) marked isotopes arc radioactive
Element Symbol z A Relative Abundance (%)
Atomic Mass i
Neutron n 0 1 1.008 665 Hydrogen H 1 I 99.99 1.007 825
2 0.01 2.014 102 Helium He 2 3 1.3 x 10_< 3.016 030
4 100 4.002 603 Lithium Li 3 6 7.4 6.015 123
7 92.6 7.016005 Beryllium Be 4 9 too 9.012 183 Boron B 5 10 19.6 10.012939
11 80.4 11.009 305 Carbon C 6 12 98.9 12.000000
13 1.1 13.003 355 Nitrogen N 7 14 99.6 14.003074
15 0.4 15.000 109 Oxygen O 8 16 99.76 15.994915
17 0.04 16.999133 • '• ' 18 0.20 17.999 160
Fluorine F 9 19 100 18 998 405 Neon Ne 10 20 90.9 19.992 440
N (1688) 1/2 (5/2* ) 1670 -1690 120 -145 Na 60 Naa 40
N (1700) 1/2 (l/2‘ ) 1660 -1700 m Na 50 AK - 10 Naa 30
N (1700) 1/2 (3/2' ) 1660-1710 80 -120 Na - 10 Naa 90 AK - 1
N (1780) 1/2 (1/2* ) 1650 -1750 100 -180 Na 20 IK 10 Nt) 2-20 Naa >50
N (1810) 1/? (3/2* ) 1650 -1750 100 -300 Na 20 Naa 70 AK 1-4 IK 2
* /-isotopic spin, G-C-parity, /-spin, /’-space parity, C-charge parity. Question mark in brackets denotes quantum numbers that have not been established. Intervals of mass, width and relative decay probabilities are shown for short-lived (resonance) particles.
Collective model of nuclei 412 coupling of particle and collective
motions 422 deformed nuclei 416
energy distribution 971 experimental methods 967 from extra galatic region 999 interactions 975 latitude effect 964 longitude effect 1025 nature of primaries 971 Rossi transition curve, 979 Variation belts 999-1000 shower 895 special techniques 886-887 Stormer’s theory 936
Coulomb corrections 148 Coulomb excitation 217 Cross section of scattering
differential 6, 190 total 6, 7
Curie 72 Curium (Z = 96) 773 Cyclotron 554
focusing of beam 558 ion sources 546 limitations 559 sector-focused 582 superconducting 585 synchro-cyclotron 560
rotational spectra 420 vibrational spectra 4l4
Collisions elastics 440 high energy 878 non-elastic 445
Compound nucleus hypothesis 487-490 Angular distribution 513-514 Breit-Winger formula 490 charged particle resonances 499 disintegration 507 entropy of residual nucleus 511 evaporation model 509 Ghoshal experiment 504 independence hypothesis 490 level density 511 resonances 495
Deuteron excited suite 793 ground state 785 magnetic moment 828 photo disintegration 830 quadrupole moment 827 radius 793-795 . wave equation 788 wave functions 792
Direct reactions 514 pick up 516 stripping 516
Direction focusing by electric field 318 by magnetic field 315 Double focusing principle 319
Heavy ion reactions 529 back bending 535 features 530 nuclear molecules 532 rotational spectra 534 super-heavy elements 533 techniques 529 transfer processes 534
Hyperons 894
discovery 894 properties 899, 901 A - hyperons 899 I - hyperons 899 S - hyperons 900 Q - hyperons 901 strangeneus 894
Inelastic scattering of electrons 218 Interactions
electromagnetic 901 electro-weak 903, 948 graviton 902 hyper-nuclei 903 strong nuclear 903 types 904 W and Z bosons 903, 949 weak nuclear 903-949
Linear accelerator 601 accelerating system 609 focusing of beam 604 for heavy ions 608 phase focusing 605 power requirement 607
Mass defect 20 Mass dispersion 317 Mass measurement 314
double method 333 peak matching technique 338
Mass spectroscope Aston type 323 Bainbridge type 328 Bainbridge and Jordan type 331 Dempster type 322 double focusing type 319, 330 helical path type 335 mass synchrometer 336
Mattauch-Herzog type 331 omegatron 336 quadrupole type 339 resolving power 341 using cyclotron principle 334
synthesis of 782 Mossbauer effect (recoil - less transi¬
tion) 222-223 applications 227 chemical shift 228 gravitational red shift 229 hyperfine splitting 227
Muons decay 891 discoveiy 897 interactions 892 mass 901 atom 892 muonium 893 production 890 spin and magnetic moment 992
Neptunium series 68 Neutrino 139
antineutrino 180 detection 181 double beta decay 179 helicity 176-179 in muon decay 891 mass 151 properties 141, 892, 910
1052
Neutron age determination 667 age equation 665 charge 617 cross-section 639 decay 617 detectors 639 diffusion 653, 656 discovery 613 energy classification 620 log decrement 650 magnetic moment 618 mass 615 moderating ratio 651 monochromators 641 slowing down 647, 651 sources 621 spin 617
Neutron-proton capture 833 Neutron scattering 34
by neutrons 840
by ortho arid parahydrogen 815 elastic 474 effects of chemical finding 813 liquid mirror experiment 820
n-p scattering 813 cross-section 797 effective range theory 809-812 high energy 852 limits of energy 797 low energy 795 low energy parameters 821 partial wave method 796 phase shifts 798 reduced mass 813 r-
islands of isomerism 213, 404 shape isomerism 693 Nuclear magnetic moment 38,
400 atomic beam method 351 Bloch’s method 356 determination 351 electron paleo magnetic resonance
357
hyperfine splitting 341 microwave method 359
{ndex
nuclear induction method '358 Rabi’s magnetic resonance
method 352 \ resonance magnetic absorption *355
. Zeeman effect method 345 NufMhr mass 15
'determination 338 unit 18
Nuclear matrix element 160 Nuclear model ^
alpha decay 377 applications 377 Bdthe-W^eizsacker formula 372 configuration mixing 407 collective model 414 evaporation model 569 Fermi gas model 383 individual particle model 410 isomerism 404 liquid drop model 371 mass parabola 376 mirror nuclei 32. 381 optical model 520 quadrapole moment 406 Schmidt lines 403 shell model 407 single particle shell model 407 spin-orbit interaction 393 super fluid model 427 unified (Nilsson’s) model 424
Shell structure of nuclei 386 evidence 388 islands of isomerism 404 magnetic moment 394 nuclear spin 388 quadrupole moments 406 single particle states 389 spin-orbit interaction 393
Solid state track detectors 285 Spark chamber 279 Statistics of counting 304 Storage rings 597 Straggling of range 122 Strange particles 896 Strangeness conservation 918 Super-fluid model 427 Superheavy elements 781 Symmetry classification 933 SUz symmetry 933
Technetium (Z = 43) 783 Thorium series 68 Transutanic elements 769
discovery 769 electronic configuration 780
Unified model 424 uranium-radium series 67
Voltage discrimination 292
Weak interaction 948
Zero-Zero transition 216
v--"' t W S.CHAND
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The logo shows waves/bars that are converging towards a dot. The dot signifies "goar and the waves/bars denote “focus'' that is very much needed in preparation of Competitive Exams. The logo conveys the message that the competition and reference books by S. Chand will help the student to converge their attention towards their ambition.
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A di vision of S, Chand & Company I ' !An ISO 9001 : 2000 Company)