Transient fields acting on heavy ions during slowing-down in magnetized materials

I

3*c

I

Nuclear Physics Not

to be

TRANSIENT DURING

(1971) 413-435;

Al66

@

North-Holland

Publishing

Co., Amsterdam

reproduced by photoprint or microfilm without written permission from the publisher

FIELDS

ACTING

SLOWING-DOWN JENS LINDHARD Institute

ON HEAVY

IN MAGNETIZED

of Physics,

IONS MATERIALS

and AAGE WINTHER University

of Aarhus

and The Niels Bohr Institute,

University

of Copenhagen,

Denmark

Received 2 February 1971 Abstract: This paper treats the theory of transient fields acting on the nucleus of an energetic ion during slowing-down in magnetized materials. The enhancement of density, at the ion nucleus, This implies a strong magnetic of atomic electrons scattered by the ion is x w lo’-103. field from polarized electrons in a ferromagnetic material. Estimates of the enhancement and knowledge of energy loss permit a calculation of the total magnetic precession of a nucleus during slowing-down. Effects of screening are discussed and the sizable relativistic correction is computed. Fluctuation phenomena are also treated, as well as transient electric field gradients due to electron scattering. There is good agreement with the trend of experimental results, and relative values in different ferromagnetics agree within experimental error. But the measurements remain _ 50 % above the formulae. The discrepancy may be due to some of the more complicated corrections, which have not yet been estimated.

1. Introduction

The precession of nuclear spin of ions within a ferromagnetic material was originally studied by immersing radioactive sources in the solid, thereupon measuring the disturbance of angular correlation between successive radiations from decay products ‘). More recently, one has introduced a somewhat different procedure, where excited levels are populated by means of Coulomb excitation, and the recoil motion of the nucleus is utilized to implant it in the ferromagnetic material “). Grodzins and coworkers “) found that the two types of measurement led to different results for the total angle of precession, and suggested that the difference in precession be due to strong magnetic fields acting prior to the settling of the ion in the lattice. They further suggested that these strong magnetic fields were caused by polarized electrons, captured into bound states on the ions during slowing-down. In the present treatise we attempt to show that the transient fields are mainly due to a familiar phenomenon in electron scattering by ions. In fact, when electrons are scattered by an attractive Coulomb field, their density becomes very high at the scattering centre. Polarized electrons from a ferromagnetic must therefore give rise to a strong magnetic field at the nucleus of a moving ion “). This effect is discussed in some detail in the following sections. In sect. 2 we make simple semi-classical estimates of the enhancement of electron density, and of the mag413 May

1971

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J. LINDHARD

AND A. WINTHER

netic field, at the ion nucleus. This is utilized in sect. 3 to evaluate the total magnetic precession of the ion nucleus during slowing-down. The comprehensive formulae are compared with experimental results. In sect. 4 we discuss the effects of statistical fluctuations and of transient electric field gradients. Next, in sect. 5, relativistic corrections to electronic wave functions are calculated in some detail; they are found to be important for heavy ions. Finally, in the appendix, exact solutions of scattering in the screened potential of Hulthen confirm previous results and indicate possibilities of resonances in the enhancement. In the present section we shall, as a preamble, outline a few features of slowingdown of ions and the associated collisions with electrons. On the one hand, knowledge of the stopping of an ion, and of the slowing-down time, are necessary for quantitative evaluation of the effects. On the other hand, an appreciation of the origin of strong transient fields demands fairly accurate ideas as to collisions of electrons with the ion. The description of slowing-down sketched below is treated in more detail in previous publications ‘, “).

Cd in Fe

Fig. 1. The standard nuclear and electronic stopping cross sections 5, s, and in iron as functions of 83. This parameter is proportional to ion velocity, and a* The figure also shows, as a full-drawn curve, the total stopping cross section simple approximation (1.3) to this quantity is indicated by the dashed

s, for cadmium = 2.3 for v = ue. s.+s,, while the curve.

1.l. SLOWING-DOWN

An ion moving through matter is subject to two types of collisions. In the so-called nuclear collisions the Coulomb force between ion and atom serves to transfer momentum and energy to translatory motion of the struck atom. In electronic collisions,

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electrons are excited or ejected from atoms along the particle path. The nuclear collisions are relatively rare, but violent, and at higher ion velocities they determine merely the multiple scattering, i.e. the deflections, whereas electronic collisions in this case completely dominate the slowing-down. At velocities below - v0 * Zf’, electronic stopping is approximately proportional to velocity. Here, and in the following, the suffix 1 refers to the ion and the suffix 2 to an atom of the substance, so that e.g. Z, and Z, are the atomic numbers of ion and atom, respectively. When the ion velocity v decreases below the above-mentioned velocity u,Zf, the nuclear stopping keeps increasing with decreasing velocity so that, at a velocity of order of a0 = e2/h, the nuclear stopping curve crosses the electronic stopping curve. Below this point, the nuclear stopping dominates, but goes through a maximum and, when the velocity is small compared to uO, finally decreases with decreasing ion velocity, approximately as ~8. These features are illustrated by the example shown in fig. 1. The above general picture becomes somewhat distorted in the extreme case of Z,/Zz < 1, where electronic stopping becomes large in a relative sense. For the present purposes, however, the case of Z,/Z, < 1 does hardly occur, since we are usually concerned with ions of atomic number above 20. In order to describe stopping at low velocities, it is convenient to use reduced variables, E and p, for the energy E and range R of the incoming ion. The reduced variables are derived from Thomas-Fermi concepts, and are defined by a

E4

E-

A,+A2

Z,Z2e2’

R4na2N

(A,

AI P

The mass numbers

=

A2 +A2j2

are A, and AZ, the Thomas-Fermi a = a,0.8853Z-*,

(1.1) .

radius is

z = (zz+zp,

(1.2)

where a, = h2/me2, and N is the density of atoms. Fig. 1 indicates the behaviour of electronic, nuclear and total stopping in s-p variables for Cd ions in Fe, at moderate and low velocities. As to the general behaviour of stopping and scattering cross sections, appropriate formulae and curves are presented elsewhere ‘, “); they may be used for numerical evaluation of the formulae to be introduced below. By and large, agreement of the curves with experimental results is within - 20 %. The detailed behaviour and individual accuracy of the slowing-down curves is, however, not decisive for most of our purposes, since the dominating uncertainties in the formulae in the following presumably stem from phenomena other than slowing-down. In order to obtain approximate comprehensive formulae, we can therefore use the following simple expression for slowing-down, chosen with a view to the special case of estimates of precession

416

J. LINDHARD

AND

A. WINTHER

in ferromagnetics (& .X 26, or 2, = 64), for ions of not quite low atomic number, (1.3) where so is a constant, s,, NNO-4-0.5, and k is the reduced constant in velocity-proportional electronic stopping 5), usually of order of 0.15. For Cd ions in Fe, the formula (1.3) is compared with the more accurate curves in fig. 1. The major defect of (1#3) appears at quite 1,owvelocities, which region will turn out not to be critical for the present purposes. f.2. ELECTRONIC

COLLISIONS

We may next discuss a few aspects of electronic collisions, because they are of basic importance to the appearance of transient fields. Consider an ion moving through a degenerate gas of electrons with constant density in space. The rate of change af velocity of the ion is quite small and there will be a stationary flow of electrons about the ion. The field surrounding the ion is a Coulomb field shielded in a self-consistent manner both by electrons following the ion in bound states, and by electrons being scattered by the ion. The ion carries no net charge, being completely screened by electrons at large distances. Since scattering and spatial distribution of the various electrons is determined by the total field, and the field is determined by the charge distribution, one may derive the field in a self-consistent manner, much as a static Hartree field. The field is expected to change smoothly and slowly if the velocity u of the ion is changed. When v is low, the field is similar to a static Thomas-Fermi Geld for a neutral atom, with screening distance ct given by (1.2), where 2 x! Z1 if the Fermi velocity uF of the gas is low. If the Fermi velocity is high, the effect of the electron gas density can be included in the screening length a in (1.2) by writing Z” z Zf + u&,, the effective atomic number of a Fermi gas being w (v,/u~)~. In the stationary flow there is a steady current of electrons towards the ion, and a steady current of scattered electrons away from it. If one includes higher order effects, electrons can by mutual collisions jump between various states of excitation, The total balance in stationary flow replaces the often used concept of competing capture and loss processes. This picture of interaction with electrons, belonging to a Fermi gas of constant density, is, in the main, applicable also when an ion moves through an actual solid, where the density of electrons varies considerably in space. A major change from the free Fermi gas occurs because the strong Coulomb fields around atomic nuclei alfow an increase in the rate of processes of electrons jumping into and away from bound states on the ion. The above description of balance should still apply. Other effects due to atoms, such as those of outer atomic shells, or band structure in solids, may often be disregarded, particularly if 2, is not small, or if the ion velocity u is large. In spite of being qualitative, the picture can be used in estimating many aspects of scattering phenomena and is, as we shall see, particularly useful for the present pur-

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poses. Suppose now that an ion of medium or high atomic number moves through a solid with energy in excess of, say, 50 keV. The closest distance of approach to atoms is then less than a,. Apparently, the ion is able to move rather freely in the lattice and the more loosely bound atomic electrons can readily be scattered in the ion field. One may be interested in estimating various scattering quantities as functions of velocity 21 and of the atomic numbers 2, and 2,. As an example, one might consider the rate of momentum loss to electrons, i.e. the electronic stopping cross section S,. This quantity requires care in calculation, because it depends somewhat on resonances in phase shifts in scattering of electrons by the ion field. Still, the stopping cross section remains less sensitive to screening than, for instance, the total cross section for electron scattering. In the following, however, it turns out that we are interested merely in the density of scattered electrons at the position of the ion nucleus. The density is determined mainly by the well-defined unscreened Coulomb potential close to the nucleus, and less by the screening of the ion potential. Accordingly, resonance effects are only of secondary importance in the above picture of scattering by a self-consistent screened ion potential, cf. appendix. Let us return to a detail concerning the free Fermi gas of electrons which, with average velocity I), is streaming past a screened stationary ion. If u is small compared to t+, low energy scattering states will be filled, as will also the bound states on the ion. But if D is large compared to +, the low-energy scattering states are empty, and the upper bound states on the ion are not filled. One would expect, corresponding to the arguments of N. Bohr *), that the bound states with binding velocities less than N (l-2)” are only partially occupied by etectrons. Thus, for fast ions in solids the bound electrons on the ion have a greater freedom than for slow ions.

2. Magnetic field at ion nucleus In this section we attempt to make straightforward estimates of the magnetic interaction, based on semi-classical concepts. We discuss the phenomena in the rest system of the ion, and suppose that the scattered polarized electrons give a stationary magnetization density proportional to the probability density, and can then compute the resulting magnetic field at the origin. Since the electronic motion is disturbed only little by the magnetic interaction, and since very many electrons contribute, the semiclassical stationary calculation should not be far in error. This is verified in the more detailed estimates in sect. 5. We consider a ferromagnetic solid, with atomic number Z, and N atoms per cm3. The number of polarized electrons per atom is denoted as [, so that for Fe we have !: = 2.2 in the case of saturation magnetization, If an ion moves through the solid with velocity v, having random probability distribution with respect to the lattice, it meets an average density &V of polarized electrons. As we shall see, the average density of polarized electrons is enhanced to a maximum density, ClvQ, at the ion nucleus,

41s

J. LINDHARD

AND

A. WINTHER

where Q is the enhancement factor, The electron spin magnetic moment is ft = e~l2?~~, and let p have the direction of the polarization. We can now easily estimate the total magnetic field B at the ion nucleus due to all polarized electrons. In fact, if a magnetization density is spherically symmetric around the origin, where it has a maximum, and if the density vanishes at infinity, the magnetic field at the origin is 843 times the density of magnetic moment at this point. In the present case, this situation is actually obeyed if the electrons move isotropically with respect to the nucleus. Allowing for some anisotropy because of the motion of the ion nucleus, we therefore find B =

$cQ[Np -(1-i-i).

(2.1)

The quantity .E_ we call the asymmetry correction. It vanishes if the density distribution in space is spherically symmetric, or if we can take the average over the direction of the velocity U.The asymmetry correction is a tensor of type of E_= ~a(3uu--v2)~u2, where the number a may be a function of velocity. We shall usually disregard the asymmetry correction in the following, but for completeness we estimate its magnitude at the end of this section. 2. I. ENHANCEMENT

FACTOR

In order to estimate the basic part of the enhancement factor Q contained in (2.1), let us consider a free electron being scattered by a screened Coulomb potential, the electron having relative velocity D, at large distances. We consider at first an unscreened Coulomb potential with charge Z,e. Let the density of the incoming wave be unity. The density at the scattering centre is then, according to the non-relativistic Schrodinger equation [cf. e.g. ref. “)I, x=

27-v ----w-------, 1 _eezxq

2nZ, ve a,

(2.2)

where the approximation to the right demands that Q 5 Z1 uo/u, 2 4. The enhancement 2 increases strongIy with decreasing u,. If ur = v. we find x % 6Z,, and x can be of order of 102-103. According to (2.1) we may therefore in Fe be concerned with fields B in the megagauss region. In the above estimate of scattering of free electrons we disregarded the screening of the ionic field. The effect of screening is, however, expected to be comparatively small, because the increase of density occurs at distances N a, Z; 1 from the nucleus. The screening is dominating at much larger distances, corresponding to r > a x % a,Z-*. Since the density of, say, a WKB wave function is inversely proportional to the square root of the local kinetic energy, we expect that the behaviour of the potential in the outer regions of the ion is of minor importance for the density calculation. This result for the density may be contrasted with phase shifts, which are sensitive to such potential changes. Although the qualitative arguments put forward

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here are quite appropriate, we have in the appendix tried to make more accurate estimates for screened Coulomb fields, by exact quantum mechanical solution of a special case. The density enhancement x in (2.2) is expected to be the major part of the total enhancement Q in (2.1). Since there are various correction terms to be looked into, we introduce Q in such a way that the corrections are explicit, i.e.

Q =xRCP.

(2.3)

In this formula, 2 is simply the average of (2.2) over the distribution of the relative velocities of polarized atomic electrons with respect to the ion, to be discussed presently. The correction factor R is due to relativistic effects in the electronic motion close to the ion nucleus, as well as to possible effects of finite nuclear size. The factor R is expected to increase from 1 for the lightest ions to somewhat larger values for heavy ions. It is discussed in sect. 5. The factor C = C,,,,Cr, in (2.3) contains Cat_ i.e. effects of the atomic binding of the polarized electrons, which was disregarded in (2.2), where the electrons were considered as forming a free Fermi gas. Next, C comprises Ci,,, i.e. partly the abovementioned resonance effects in the ion potential, cf. appendix, and partly the contribution from partially filled bound states on the ion. Finally, the factor P in (2.3) is the relative probability of meeting polarized electrons, i.e. at high energies P = 1, since the ion moves randomly everywhere within the lattice. At low energies, the ion fails to meet some of the polarized electrons because it cannot come sufficiently close to atoms, so that P is considerably less than 1. We may therefore assume simply that P is unity above a certain cut-off energy E,, and that P is zero at low energies. We shall find that the results are not sensitive to Ea. Note that in practice the initial ion energy is in the MeV region, i.e. large compared to Eo. From (2.2) we can estimate 2 as a function of ion velocity. If the ion collides with free electrons, with velocities ui in the laboratory system, their relative velocity in (2.2) is U, = In-oil. For a given value of the electron velocity ui we find, averaging over direction of Vi, that the average of l/v, is l/v, = llu, 1l/Vi

9

for for

V >

Vi)

V <

Vi.

This implies that x is proportional to v-r at high velocities v, whereas 2 tends towards a constant, determined by (vi’>, at low velocities u. We therefore define a velocity factor f(o) by f=:

Y(u),

(2.4)

where up1 = (vTl> is an effective reciprocal velocity of the polarized electrons.

420

J. LINDHARD

AND

A. WINTHER

The above results also suggest the following approximation

forf(v),

3

f(u) =

3 = Ep u i 1,

(I &

’

for

v>uP

or

s>sp,

for

v
or

s<.sp,

(2.5)

where E is the energy measured in the dimensionless unit (l.l), and ap the value corresponding to velocity v,,. The formula (2.5) is expected not to be much in error, because simple estimates show that the transition between high and low velocity regions is rather sharp. In (2.4) and (2.5) we have expressed 2 in terms of one parameter, up, characteristic of the polarized electrons. In the case of Fe one would expect that up is somewhat less than uO. 2.2. ASYMMETRY

CORRECTION

The magnitude of the asymmetry correction < in (2.1) is easily estimated in the basic case corresponding to (2.2). In fact, when a free electron with relative velocity u is scattered by an unscreened Coulomb potential, the probability density p is constant on any paraboloid with the nucleus as focal point and axis parallel to “) u, i.e. p = p(r - r * u/u). The electron is assumed everywhere to have polarization in the fixed direction p, and if one then integrates over one parabolic shell, the corresponding contribution to the field at the focal point averages to zero. The field, therefore, is determined only by the boundary conditions. At infinity appears the usual contribution from macroscopic magnets; this term is very small in a relative sense, and we can disregard it. There remains the contribution at r = 0. It is proportional to the density at Y = 0, and contains an asymmetry of purely geometric origin. In fact, if u is perpendicular to p, the field is $ of that belonging to spherical symmetry. This means that in the general formula

(2.6) the non-relativistic scattering of electrons originally at rest in the laboratory system leads to c1 = 1. Consider next the case of an ion, with fixed velocity vector u, colliding with an electron having fixed velocity ui, but random direction. We therefore replace u in (2.6) by u - Vi, multiply by 1u - vi 1-I, and average over direction of Vi.This leads to a total asymmetry correction of type of (2.6), but with

(2.7) Averaging finally over the magnitude of vi, we expect that in first approximation coefficient a in (2.6) becomes fx = f&I~,)~ where v, is defined in (2.4).

the (2.8)

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421

The result (2.8) shows that the asymmetry correction is usually quite small. In the low velocity region, which gives a major contribution to precession, the asymmetry is absent. In the high velocity region, e.g. a2 > 2v$ the asymmetry is present, but here the precession is quite small anyway. 3. Precession of spin of ion nucleus We are now able to give simple estimates of the total magnetic precession of the spin of an ionic nucleus during slowing-down. We disregard fluctuations, which turns out to be a good approximation. Effects of fluctuations and of electric field gradients are discussed in sect. 4. 3.1. THEORETICAL

ESTIMATES

Consider therefore a definite magnetic field at the ion nucleus, with magnitude depending on the ion velocity, B = B(v), and with a fixed direction in space. During a time dt, the differential angle of precession of the nucleus is dq = w(E)dt,

(3.1)

where run = eh/2Mc is the nuclear magneton, M being the nuclear mass unit, while g is the gyromagnetic ratio of the nucleus, and w = o(E) is the average rate of precession. In order to integrate (3. l), we must transform the time variable to a velocity variable. As indicated above, we assume that slowing-down is a continuous process, i.e. that dE

dt = d_R= V

VNS(E)’

(3.2)

where dR is the differential path length, and NS(E) = dEjdR is the specific energy loss, S being the stopping cross section per atom. From (3.1) and (3.2) we can directly learn about the qualitative behaviour of the precession. At high velocities, B is proportional to l/v according to (2.5), and S is proportional to v, so that J drp cc J dv ve2. This means that the total angle of precession, q, has an upper limit, and that it saturates at moderate velocities. At low velocities, B is constant, so that the precession J dqpis proportional to slowing-down time, which again, cf. (3.2), varies as s dv/S(E), wh ere S is approximately constant, except at extremely low velocities, cf. fig. 1. We find from (3.1) and (3.2) the total average precession, q(u), suffered by an ion with initial velocity v q(v) = ‘2

A, M

s

” B(v)dv

oNS(E)’

(3.3)

422

It

Y. LXNDHARD

is here convenient to introduce

AND A. WJNTHER

the ~mensionle~

range”ene~~

wiabIes, p and q

cf. (f,ff, We obtain from (X.1), (2.3), (2.4) and (X3),

where

(3.4)

Wherr the approximatjon (2.5) is used, we havef = I for E < spt andf = (a&)* for E > cp, where ap is the value of E for v = I+ The formula (3.4) has several simple properties. The precession q(u) is for instance independent of N, the density of atoms. It contains a factor y depending exclusively on the atomic numbers and the mass numbers of ion and atom. This factor is approximately ~r~~ortionaI to the atomic nnmber ZI af the ion, and has values in the region q ru f@-iU3. A further increase with Z1 is contained in the relativistic COCT~O~, R = R(Z1), which depends only on Z,, behaving approximately as R FZ 1 i- (X,/84)‘, cf. sect. 5. A possible oscillatory resonance effect depending on Z1 is discussed in the appendix, and is contained in Ci, in (3.4) The properties ofthe ion nucleus appear in (3.4) through the ~omagn~~i~ mtio, g. The properties of the substance enter mainly through 5, the number ofpolari%d electrons per atom, and less through vP and C,,,,. The dependence on up ’ is partly compensated by an opposite variation of I with lip.The parameter v, should not he essentially different for different ferromagnetics, but its magnitude is diicult to assess accurately. As to the integral J(i>) in (?.Sjl. its sa~ra~o~ value I(W) changes very little With Z,. If zip = oO,and in Fe, one gets 1(co) G I I 1whereas in Gd the value is lower, r(w)> z 7. Values of I(U) may be obtamed by numerical integration. Still, it is usually preferable to use a simple approximation to ds/dp, allowing analytical estimates of 1. A formula of this kinct is given by ( 1.3), cf. also fig_ 1. In that connection we need at present only note that the factor P in (3.5) introduces a cut-of? at very low values of E=In the same region, actual values of dsjdp in (3.5) begin to decrease as rv c+. As a total result, it will be a tir approximation to disregard P in (3.5) and to assume that (1.3) is valid at all velocities. This leads to the formula, valid for u > z$,

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SLOWING-DOWN

x = ke*/s, and x, = k&;/s,, E,,being the reduced energy belonging to velocity up. As indicated in connection with (1.3) we may puts,, M 0.40 in Fe, and s0 = 0.45 in Gd. It may be added that, for x < x,, eq. (3.6) is replaced by I = k-l log (x+(1+x’)*). The quantity Z(u)/Z( cc ) as illustrated in fig. 2 as a function of v/v, for different values of xP, while the asymptotic value Z(cc) can be obtained from fig. 3. It is noted that for values of x 2 3, Z(v) is approximately given by

where

I(w) NNI@)3.2. COMPARISON

WITH

;

for

x>>l.

(3.7)

EXPERIMENTS

In fig. 4 the expression (3.4) with Z(v) given by (3.6) has been compared with available experimental data. The full-drawn and dashed curves correspond to up = v. and VP= +vg, respectively, and we suppose that the number of polarized electrons per atom is c = 2.2 in Fe and 5 = 7.0 in Gd. Although the theoretical curves follow the general trend of the experimental points rather well, there remains an average discrepancy by a factor of less than 2. In the discussion of the origin of this discrepancy it should be noted that the ratio between the experimental precession in Fe and Gd agrees with the theoretical ratio within 10 ‘/& Also the measurements which have been made on the velocity dependence of the precession angle ‘) show the same constant deviation from the theory

0.6

0

0

0.5

1.0

1.5

2.0

2.5

Fig. 2. The ratio I(o)/I(oo), according to eq. (3.6), as a function of u/u, for various values For values of o/o, larger than those given on the figure, one may use eq. (3.7). The quantity can be determined from fig. 3.

of xp. I(w)

424

J. LINDHARD

AND A. WINTHER

XP

I I -0.5 1 1,s Fig. 3. The quantity kI(co), according to eq. (3.6), as a function of x, = ke,+/so.

as do the points in fig. 4. These comparisons between different experiments for a given ion are independent of the nuclear g-factor, which in most experiments has been assumed to have the value 0.3. The deviation between calculated curves and experiment may correspond to C - 1.5, which is not an excessively large value. A calculation of C should for instance include the fact that the polarized electrons are bound in the Fe or Gd atoms and have to be released prior to the scattering process. We have not made any quantitative estimate of this effect. In the comparison between experiments for different recoil ions there may be an indication of some oscillations of the experimental points about a mean curve as a function of 2,. As is discussed in the appendix, it is conceivable that the transient magnetic field shows some shell effects, giving rise to oscillations in the precession. A number of other effects which we have looked into seem in the present experiments to be of minor importance: Firstly, the asymmetry correction (2.7) implies a

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HEAVY XONS SHOWING-DOWN

precession around the vector 3( u - p)u/v’ -p, which precession vanishes for u < 0,. With the usual experimental geometry, where u is perpendicular to I(, this correction leads to a small decrease in the precession angle by an amount of less than IO %. It should be noted, however, that in a different geometry one might also observe a total mr

cp(v)

100

9

in

Fe

from

for

recoils

33 MeV

016

50

I

t

1

I

I

,

I

I

I

10

20

30

LO

50

60

70

80

90

mr F

in Gd from

for

recoils

45MeV

0’”

50 t

I

I

10

I

I

0

I

I

I

I

I

20

30

LO

50

60

70

60

90

Fig. 4. The precession angle q(o) divided by the nuclear g-factor, for implantation in iron and gadolinium. The curves are calculated from eq. (3.6), assuming Y,,= u0 (full-drawn curve) and up = 4u0 (dashed curve). The experimental points ‘f correspond to recoils from (a): 33 MeV I60 ions in Fe, and (b): 4.5 MeV I60 ions in Gd.

precession about the velocity u, of magnitude almost half the total precession (3.4). Secondly, one should take into account that the slowing-down of low-energy ions is not a continuous process. As we shall see in sect. 4, such effects, although in principle large, are not important under present experimental conditions.

J. LINDHARD

426

AND

A. WIN-l-HER

Thirdly, we also consider in sect. 4 the effects of quadrupole interactions between a possible nuclear quadrupole moment and transient electric field gradients during slowing-down. As will be shown, these field gradients are quite large and could give rise to a significant precession of the nuclear spin about the velocity D. Since, however, in present experiments the nuclear polarization is perpendicular to o, the precession cannot be detected as a rotation, but only as an attenuation of the angular distribution of the y-quanta, which effect is usually of minor importance. 4. Fluctuations in precession, transient electric field gradients In the magnetic statistical tions like effects.

previous discussion we have derived average precession due to transient fields. These are two kinds of deviations from the simple picture: firstly, the fluctuations in magnetic precession, and, secondly, non-magnetic interactransient electric fields, which can give rise to fluctuations as well as to other

4.1. FLUCTUATIONS

IN MAGNETIC

PRECESSION

We shall try to single out some of the dominating causes of fluctuations and their approximate contribution. It should be emphasized, however, that present experimental methods do not allow direct measurements of these fluctuations in precession. In fact, one is concerned with quite small average angular shifts - e.g. by an angle of one degree - of angular correlation curves, obtained as averages over very many individual events. Even if the fluctuation in precession for the individual ion were larger than the average precession, Acp > cp, it could hardly be seen in such measurements. We shall find that actual fluctuations are negligible, being considerably less than (p. In one respect, however, there is an indirect effect of fluctuations, in that they may change estimates of the average precession. Suppose first that ions are slowed down in a continuous way, so that the energy is a unique function of the distance travelled by an ion, E = E(R). In this case fluctuations in cp would arise only because the number of polarized electrons encountered by the ion can fluctuate. If the collisions with electrons are independent, with average number V, we iind that the relative square fluctuation (zlq): in cp, due to polarized electrons, is (dq)$cp2 = (~v)~/v~ = l/y. As an example, consider slowing-down from initial velocity v = up, where aP x v,,, and assume that in this region the stopping cross section is constant, ds/dp = s0 z 0.5. An electron may be supposed to give rise to precession if its angular momentum with respect to the ion is less than h. The number of electrons encountered by the ion is then easily seen to be

For Cd ions in Fe this number will be ij - 600, and (zlq),/Cp - 4 %. Even though this is a crude estimate, we need not improve upon it, since we shall find that other fluetuation contributions dominate, so that (dq); may be disregarded.

We may next, in more detail, discuss the contribution from fluctuations in energy loss during slowing-down. Let us, for simplicity, consider the example of an ion which, with initial energy E, is slowed down entirely within the ferromagnetic solid. Suppose that the previous fluctuation, Aq,, may be neglected, so that at a given velocity v there is tl steady precession, dq/dt = u(E), as given by eq. (3.1). Let the probability for a total precession between cp and ~0tdp, during the total slowing-down process, be Pkcp,E)dfo. Then, jOmp(~,E)d@ = 1, and q-(E) = fF@‘(p, E)drp. The differential cross section for collision with atoms is do, and the total energy loss in the collision is T (including inelastic and elastic energy transfers). We can assume that collisions are random, and find then the following integral equation for P(q, E), of familiar type, (4.2) Eq. (4.2) is compIetely analogous to e.g. eq. (3.1) in ref. 5)9 where the derivation is given in detail. The reader is also referred to ref. ‘) pp. 18-21, for a more complete discussion of integral equations for fluctuations. From (4.2) one obtains the integral equation determining the average precession @(E)>

If small energy transfers are preponderant, the bracketed term is w -T@?(E), so that in first approximation ~‘(~)~~~(~) = o(E), where S = f doT is the stopping cross section. This is usually a satisfactory approximation and it leads directly to the result (3.3). Suppose therefore that $5(E) has been derived from (4.3). We can then introduce the mean square fluctuation in precession, Aq’(E) = @(E)-(p’(E). From (4.2) we obtain an equation for A#,

where the right-hand side, according to (4.3), is a known source term, Consider now a simple case where (4.3) and (4.4) can be solved directly. Suppose that the initial ion velocity is moderate, so that w(E) is constant, cf, sect. 3. Suppose further that da = KdT/(ET3)*, 0 *r: T < E, corresponding to nuclear stopping being constant, electronic stopping being negligible, and to A, 3 AZ. Solving (4.3) and (4.4), we find, firstly, that the relative square fluctuation in cpis

AP2 --.-= rc-3 -2 cp

!z 0.14.

(45)

This fluctuations though considerably in excess of the previous one resulting from (4.1), is moderate in the sense that it can not be seen in the usual precession measurements. Note here that (4.5) overestimates the fluctuation in question.

428

J. LINDHARD

AND

A. WINTHER

Secondly, we can in the present example find a precise solution of@(E) in (4.3), and compare it with (3.3). It turns out that the solution @p(E) in (4.3) is (n-2)-l z 0.88 times the solution of (3.3). The reduction in this example is much larger than that in actual cases, because electronic stopping is disregarded and only low velocities are included. We therefore conclude that actual reductions of (3.3) will usually be 5 5 ‘A. Within the present accuracy from such minor corrections. 4.2. TRANSIENT

ELECTRIC

of the theoretical

estimates

we may look apart

FIELD GRADIENTS

The simple considerations in sect. 2 show that scattered electrons have a high density at the ion nucleus, and that the density distribution is not spherically symmetric. We must therefore reckon with a sizable electric quadrupole field from the electrons, the average field having obviously the ion velocity u as symmetry axis. For nuclei which possess an electric quadrupole moment there will be a force proportional to sin 28, where 0 is the angle between spin and ion velocity. The rate of precession around v vanishes for 0 = +rc, and is proportional to cos 8. This phenomenon can, in the present context, appear as a disturbance of the magnetic precession, being partly a cause of fluctuation or attenuation and partly a systematic precession. But the electric quadrupole precession is also in itself an interesting phenomenon, which may be studied in non-magnetic materials. When discussing experiments on transient magnetic precession and their disturbance by electric quadrupole effects, it is to be remembered that in most of the current experimental arrangements, excited nuclei are produced - by Coulomb excitation with opposite spin directions being equally populated. The effects of transient electric field gradients is then an attenuation of the angular distribution, simulating a fluctuation contribution to the magnetic precession. Thus, if for one nuclear spin direction the electric quadrupole precession is in the same direction as the magnetic precession, the two precessions will tend to cancel for the opposite direction of nuclear spin. In a similar way as in sect. 2, we shall now estimate the magnitude and velocity dependence of the transient electric quadrupole field. Suppose that a free electron at rest, i.e. of relative velocity -u, parallel to the z-axis, is scattered by the ion. The electron charge density distribution is then parabolic ‘), p = p(~--z), having a maximum, p(O), determined by p(O) = xp(co), where the enhancement is given by (2.2) and p( co) is the charge density at infinity. Let cp(r) be the electric potential at distance P from the nucleus, as due to the scattered electrons. We can suppose that the charge density is negligible at large distances. It is then easy to show that a parabolic charge distribution has the geometric property a2~(0)/az2 = 0, or

We note that the quantity Y is a measure of the quadrupole moment of the charge distribution. According la (4.6) and (2.2) the quadrupole moment is quite large. Suppose next that the electron has velocity z+, of given magnitude but random direction. The average of Y over electron direction is just the same average as that in the asymmetry correction in subsect. 2.2, and we find

(4.7) W~FS~ f~(uJ~i) is givenby (2,7), SO thatf~ vanishes ~!ICXIv < tri and tends to I for large 21,Averaging over aI1 atomic e&%ron velocities we find the total q~ad~~o~~ contribution, non-relativistically, y = -e ‘2

25, NZ,F(V,

Z,&

unitywhen D >, Z$Q, For ~nte~~d~ate velocities (u > zrO)we find a~~~oxirn~~l~ F - Zg”(u/uo). The interaction energy with the ion nucleus is, apart from an additive: constant,

where F = ~~~(~~~~)~ approaches

LlH = &I%QPz(cos

e),

(4.9)

w&m & is the nuclear quadrupofe moment, and B is the angle between nuclear spin and direction of motion. From (4.8), (4.9), aad the results in sect, 2, it is easy to see that the ratia of the most favourable quadrupole precession around V, to the magnetic precession is, in a crude order of magnitude estimate,

where I is the nuclear spin. This indicates that the quadrupole precession can become large in a relative sense, particularly at high ion velocities.

In the above treatment of nuclear precession we used a semi-classical picture, where the precession frequency was evaluated from a non-relativistic interaction of the nuclear magnetic moment with a stationary magnetic field B created by the polarized efectrons. ABhough the Son and the electron have few mfative velocities at large distances, the n~~~~e~ativ~~t~~ expression (2.f) for the magnetic geld caufd lead to a serious underestimate of the precession, Thus, the field is proportional to the electron density at the nucleus, where the electram have large kinetic energies, and the density is known to be quite sensitive to relativistic effects. Actually, the relativistic effects are much smalier than wbar. one would obtain from this relativisfic increase in density, fn fact, in the relativistic case one shoufd in (2.1) also replace the mass 2~2% occurring

430

J. LINDHARD

AND

A. WINTHER

in the magnetic moment of the electron, by nz plus the local relativistic mass, or 2m + Ze’/rc’. Below, we shall carry through a proper relativistic calculation of the nuclear precession. We still consider only the precession of the nuclear spin in collisions with free electrons. The electrons are assumed to be polarized in a definite direction, which we choose as the z-axis. The time-derivative of the expectation value of the nuclear spinj is related to the hyperfine interaction AH by

d<.i) 1 __ = z <[AH,jDo. dt

(5.1)

It is noted that this expression becomes identical to (3.1) if AH is of the form AH = gpNj * B, j being measured in units of h. In a stationary scattering state, where the electrons must be treated relativistically, the hyperfine interaction is given by AH = (plea

* Alps,),

(5.2)

where p is the electron momentum and sZ its spin. The quantity A is the electromagnetic vector potential created by the nucleus. For simplicity we consider a spherically symmetric magnetization density within the nuclear radius R, in which case one finds A = gjxvm(r),

(5.3)

where m(r) = l/r3

for

r > R.

(5.4)

If the magnetization is homogeneous, m(r) is a constant within the nuclear radius, i.e. m(r) = l/R3 for r < R. Introducing a partial wave expansion for the plane Coulomb wave 1~s;) one obtains after rather lengthy calculations AH = --s2%ceg

Ql

14&

’

+((2k+l)j

+(j . s+$x)

[

((2k-l)j .

as-(k-1)X)

s

s-(k+l)X)/~r”m(r)g~(r)f~(r)dr Soer’m(r)g-~(r)f-~,(r)dr.]

o~r3m(r)~g~(r)f-~-I(l)+g-~-~(r)j~(~)~d~~~~(A~-A-~-~)])

T(5.5)

where s is the expectation value of the electron spin measured in units of Ft. The quanof the radial wave function, tities gk(r) and f,( r ) are the large and small components respectively, while A, denotes the total phase shift in the plane wave expansion, e.g.

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IONS SLOWING-DOWN

431

for a pure Coulomb field

where

(5.7)

The quantity .X depends, besides on j and s, also on the direction of the electron momentum p through the second rank tensor X = X(pjs) =

3(P * s)(p *j) - p’j * s

(54

P2

The terms proportional to X arise from the asymmetry in the Coulomb wave /psz) of the electron density around the nucleus (cf. sect. 2). It is interesting to compare the result (5.5) with the non-relativistic limit in which

(5.9) while & = g-k-1

= &(r)/kr,

A, = A_+.,

= l?k,

(5.10)

where F, is the regular radial Coulomb wave function and or the Coulomb phase shift. In this limit one obtains

+ X(pjs) * + 31(1+1)

I(z+I>(2z+1) 2(21+3)(2E-1)

-

Oa r3mf(r)

2(214-l) s 0

Fl+

JW&rtl@) ~j’d~ o

dr)Fb

(W2

dr)

dr cos (olfl -gl_J

I).

(5.11)

To lowest order in the magnitude of the nuclear radius R the first integral vanishes except for s-waves where it equals the density at the origin, (Fo(R)/kR)2 z 27cq, for

432 q >

J. LINDHARD 1.

AND

A. WINTHER

The integrals in the square brackets can then also be evaluated exactly lo), i.e. Im $(2+1-i?) is,”f (~lW)2dr= 21+l+xq-21 21(1+ 1)(22+ 1) nrl = 1(2+1)(21+1)

and 1 -*

k s

r-3Fr-1(r)FI+1(r)dr

cos (cri+r-~i-~)

(for

(5.12)

rl > 2)

1(1+1)-q* = 6~1+1+i~1211+in12’

(5.13)

For q >i 1 the main contribution to the summation over the last term in (5.11) comes from values of I > q. While the divergence in the sum for I >> q is cancelled by the second term in (5.1 l), the remainder is small and insignificant because the actual field deviates from a pure Coulomb field due to screening. Including only the main contribution for r7> 1 the expression (5.11) reduces to the result AH,,

= +*r/(s

*j+$X)

(5.14)

-f- gpN, mc

in agreement with (2.1), (2.2) and (2.6). In the relativistic expression (5.5) the main contribution for 9 >> 1 similarly arises from the lowest values of k. For a pure Coulomb field the first two integrals in (5.5) can be evaluated exactly. For rl>> yx one finds ’ ‘) (5.15) It is noted that for yK > + this result cannot be trusted, meaning that for heavy nuclei and ti = &-1 the finite nuclear size will be important (yl = 3 for Z, z 119). To investigate this point, the two first integrals in (5.5) fork = 1 were calculated numerically with wave functions corresponding to a finite nuclear size +. The results depend only slightly on the magnetization distribution m(r), the maximum change in the results being of the order of 2 % for heavy nuclei. Using the numerically evaluated integrals for k = 1 together with the result (5.15) for k = 2 and the non-relativistic results for the higher angular momenta, one finds that the part of (5.5) proportional to j * s can be written AH = AH,,R,

(5.16)

where R reaches the value 2 for Z x 84. The results for R may be represented by the empirical formula (5.17) t We are grateful conversion computer

to H. C. Pauli programme.

for having

performed

these

computations

with

his internal

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433

Our thanks are due to many colleagues for discussions and information about experimental results; we are particularly indebted to L. Grodzins, B. Herskind and P. Hvelplund. We are grateful to Susann Toldi for assistance in preparation of this paper. Appendix DENSITY

ENHANCEMENT

FOR HULTHI?N

POTENTIAL

Consider the following screened attractive Coulomb potential,

$$,

V(r) = -

(A.1)

where q - ’ may be of order of the Thomas-Fermi screening distance, cf. (1.2). Apparently, the potential is then a fair approximation to potentials of ions and atoms, with similar merits as the Bohr potential “). The potential (A. 1) was originally introduced by H&hen “) in order to study bound states of the neutron-proton system. The particular advantage of (A. 1) is that it allows analytic solutions for s-states. In the present paper we are interested in the density enhancement, i.e. the density ratio P(r = 0)/P@ = a), f or electrons scattered by a screened Coulomb potential. The enhancement is determined by s-states only, and therefore the HulthCn potential is suited for precise calculations of the enhancement factor. The non-relativistic stationary Schrodinger equation corresponding to the potential (A.l) is dl#qr) = -

eG)Ic/(r)y

2; (E+

(A.21

where E is the energy. For bound states, or E < 0, the eigensolutions of (A.2) were found by Hulthtn. We are mainly interested in positive energies. Let now qr = if, and introduce the radial wave equation for 1 = 0, so that cp(c) = r+(r) and (A-3)

22 _

2mE= c f12q2

q2

b

’

_

2mZe2 _ 22 h2q

w

(A.4)

It is easily seen that the differential equation (A.3) has the solution q(t) = (l-e-“)e-‘“‘F(l+i~+P,

l+U-j3;

2; l-e-‘),

(A.5)

expressed in terms of a hypergeometric function zF,, and where p =

(b-A2)+

(A-6)

434

J. LINDHARD

AND

A. WINTHER

may be a real or purely imaginary number. For moderate or low velocities, B is real. It remains to find the behaviour of (A.5) at the boundaries. For 5 + 0, where F -+ 1, one obtains $ = 9Jr + q. In the limit of 5 -+ co, one finds [cf. ref. i3), eq. (15.3,6)] cp(g + 03) = e’“@

r(2iil)

r( -2iiz)

+e’asr(l+iL+P)r(l+~~-P)..

J(l-iJ.-/3)r(l-il-tp)

(A.7)

where we are interested in positive energies, i.e. real ;1. If the density is normalized to unity at large distances, the radial function behaves asymptotically as cp -+ k” sin (kr + S), 6 being the phase shift. On the basis of (A-7) we then finally, in place of (2.2), arrive at the density enhancement Xn of the Hulthen potential _e-4na ‘/!!3 z=2icrl xH= i ij(#) 1 +e-4za-2e-2nA

1

1

cos 27@

= 2nq[l-1-2 f cos (2~vP)e-2nvh]. (A.8) v=l

Consider next the order of magnitude of the parameters in (A.8). Let us put where c - 1. Then, according to (A. 4),

q = la-‘,

n = 0.8853 v F

;’

(A.9)

so that if the relative velocity is v - v@,L may be - 0.3. Moreover, the magnitude ofbis b = 2 - 0.8853 Z+ - 20-40. (A.10) 5 The behaviour of muis now the following one. At extremely high velocities, where q < $ and I2 >> b, one finds mu + 2nu](l -exp( -21tq))-‘, as it should be in an unscreened Coulomb potential. The deviation of xi, from the unscreened value x in (2.2) remains exceedingly small unless 6’ < b, i.e. u << Z3uo. This low velocity region is in fact the one with which we are concerned for transient fields. We therefore conclude, firstly, that j? in (A.6) is closely equal to b*, and that the phase 2nj? in (A.8) is a large number, of the order of 9Z*. Note here that the average of (A.8) over /.I is exactly the unscreened enhancement, if q is large. Secondly, the phase 271/Iis closely connected with the appearance of bound s-states. We can find the bound s-states by putting n imaginary in (A.7), or A = zX’,where k is real and positive. The coefficient in the first term of (A. 7) must then vanish, which leads to the result of Hulthen for the binding energy, 2Jti’= -n + b/n, n = 1,2, 3, . . . Since 1’ 2_ 0 the number of bound states is the highest integer lower than b*. In (A.8) a maximum occurs when /I is an integer, but since fi = bt with high accuracy, the maxima correspond to the appearance of a new bound state, b = n2. It is note-

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worthy that, as a function of Z and v, the resonance occurs at certain values of Z, nearly independently of v, if v is less than N $Z3v0. Although the screening of the field around a moving ion is not quite that of an ion at rest, one will expect that resonances can occur in the neighbourhood of closed chemical shells. Thirdly, the oscillations in (A.8) are usually given by the first term in the expansion, i.e. 2 cos(2@)exp( -2rc1). The value of ;i in (A.9) is not quite small, so that the amplitude of oscillation is expected to be moderate. In fact, this result was found in more detailed calculations, where we summed (A.8) over electron velocities. The above calculations and comments may suffice to show, above all, that the average density enhancement in a screened Coulomb potential is given rather well by the scattering formula (2.2) belonging to unscreened potentials. Moreover, there are indications of possible oscillations about the value (2.2). Finally, we wish to emphasize the usefulness of the Hulthtn potential for calculations of scattering and bound states in atomic potentials, since this application seems to have been neglected in the literature.

References 1) E. Bodenstedt and J. D. Rogers, in Perturbed angular correlations, ed. E. Karlson, E. Matthias and K. Siegbahn (North-Holland, Amsterdam, 1964) p. 91; H. Frauenfelder and R. M. Steffen, in Alpha-, beta- and gamma-ray spectroscopy, ed. K. Siegbahn (North-Holland, Amsterdam, 1964) p. 997 2) L. Grodzins, R. Borchers and G. B. Hagemann, Phys. Lett. 21 (1966) 214 F. Boehm, G. B. Hagemann, and A. Winther, Phys. Lett. 21 (1966) 217 3) B. Herskind, R. Borchers, J. D. Bronson, D. E. Munich, L. Grodzins and R. Kalish, in Hyperfine structure and nuclear radiation, Asilomar Conference (North-Holland, Amsterdam, 1968) p. 735 4) J. Lindhard, Proc. Roy. Sot. A311 (1969) 11 5) J. Lindhard, M. Scharff and H. E. Schistt, Mat. Fys. Medd. Dan. Vid. Selsk. 33, no. 14 (1963) 6) J. Lindhard, V. Nielsen and M. Scharff, Mat. Fys. Medd. Dan. Vid. Selsk. 36, no. 10 (1968) 7) G. M. Heestand, P. Hvelplund, B. Skaali and B. Herskind, Phys. Rev. B2 (1970) 3698; H. W. Kugel, T. Polga, R. Kahsh and R. Borchers, Phys. Lett. B32 (1970) 463 8) N. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 18, no. 8 (1948); N. Bohr and J. Lindhard, ibid., 28, no. 7 (1954) 9) N. F. Mott and H. S. W. Massey: The theory of atomic collisions (Oxford University Press, 1965) IO) K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 11) W. W. Gargarro and D. S. Onley, J. Math. Phys. 11 (1970) 1191 12) L. Hulthen, Arkiv Mat. Fys. Astr. 28A no. 5 (1942) 13) M. Abramowitz and I. A. Stegun, Handbook of mathematical functions (Dover Publications, 1965)