Volume 70A, number 1
5 February 1979
POSITRON ANNIHILATION IN NONMETALLIC SOLIDS A. CIZEK and M. SOB Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno, Czechoslovakia
and I. Ya. DEKHTYAR Institute of Metal Physics, Academy of Sciences of the Ukrainian SSR, Kiew, USSR Received 18 August 1978
In positron annihilation investigations of nonmetallic solids, the standard deviation of the gaussian component of the angular correlation curve is elucidated as material constant. It is related to the apparent radius of the chemical unit of the substance in question.
The angular correlation spectrum (ADAP curve ) of a nonmetallic solid may be reasonably fitted to a gaussian characterized by a standard deviation. The purpose of this paper is to clarify the nature of this quantity as a material constant. In statistics, the gaussian describes the normal distribution of events and may be demonstrated on Galton’s desk: The balls (positrons) fall on the nails (atoms or ions), collide elastically, return, refall etc. They finally fall into the cells (they annihilate with the electrons). The distribution in the cells is normal and
are taken for the valence state considered in the solid. The radius Te may be determined experimentally if the corresponding constant k is known. For s and p electrons Ferrell [31 has expressed it as k = ‘h/mc~ ~ 2 ‘~
the standard to deviation a of the gaussian the is inversely proportional the distance re between nails (to the mean value 7 [ri/n called the apparent radius of the radii r of all (n) atoms forming the chemical unit of the solid investigated. Thus,
The quantities h, m, and c have the usual meaning, 1 is the azimuthal quantum number of the electrons in question. Ifthe generalization to d and f electrons is supposed, the values k5 = 473, k~= 611, kd = 722, and kf = te in pm 819 maym). be The obtained for a has in mrad (10—12 first value givenand reasonable ionic radii of some metals  We will try to use this analogy to the compounds under two assumptions: (1) The positrons prefer to annihilate with the elec-
tronsbe may ofcharacterized incomplete shells by theofhighest the atom. complete Then,shell the core K, L,
the proportionality constant depends on the nature of the desk (on the electronic structure of the atoms or ions). This coarse geometrical model prefers no nails (any type of ions); it concludes Tsyganov’s idea  on metal oxides that the shape of the ADAP is influenced by the ~2 ions as well as by the metallic cations. From this viewpoint the identity F Fe may be supposed. The radius F may be determined, if Pauling’s radii —
M, or N, if the s, p, d, or f electrons, respectively, start to close it (according to Ferrell  the subshell beginning with filling is held as closed). A look at the electron configuration of the elements shows that the shells begin to close by s, p, d, or felectrons, if the number of electrons in an atom Zmjn = 1,5,21, or 57, respectively; the numbers satisfy the relation Zmin 1 + 22(s2 + p2 + d2 i-f2) because of s = O,p 1, d 2, andf 3. Then, it may be supposed that the arithmetic mean of the corresponding ,
Volume 70A, number 1
S6~(29), and 9 02— (140) give 7 124 pm, again in good agreement with experiment. (The H~ion has r = 0, and, therefore, it does not come into considera-
Table 1 Type
L 4 473 ...
M 20 542 ...
N ... 56 602
constants represent the core constant: kK
tion for calculation of the apparent radius.) Plexiglass (polymethylmetacrylate) may be written as
56O2, = 473, kL~(ks+kp)=542,kM =~(k5+k~+kd)= andkN = ~(k 5 +k~+kd +kf)= 656, which are the electronic characteristics of the atoms investigated. Therefore, we will speak about an electronic type K, L, M, N of an atom. In aorreal case, the number Z of electrons in the atom should be determined for the actual valence state and the constant k may be taken from table 1. (2) In a compound the constants of all the atoms forming the chemical unit are to be determined as demonstrated above and the highest one is chosen as that of the substance. In this paper the verification was performed on KCl, CuSO 4 5H20, and plexiglass. The ADAP curves were measured with a long-slit spectrometer with a resolving power of I mrad and statistics of about 1O~counts. The experimental data without any correction were fitted to a gaussian by means of the ZPA 600 computer using a least-squares procedure. The correction of a to the resolving power (about —0.03 mrad) lies in the limits of experimental errors (±0.05mrad) and, therefore, the uncorrected a’s were used. Sylvite is evidently of the L electronic type because both components are of L type. The measurement yields the value of a 3.55 ±0.05 mrad and then re = 153 ±2 pm. The apparent radius, calculated from Pauling’s radii of K~(133) and Cl (181) , 7 ~(l33 + 181) 157 pm coincides very well with the experimental value. Blue vitriol is ofZthe27. M type because most com2~ion has We obtain a =the 4.83 ±0.05 plex Cu mrad and re = 124 ±2 pm. The radii of Cu2~(72),
5 February 1979
CH3 ~CH2C~ COOCH3 4~(15), 1 C4 (260; in chemical unit; it has C as C4—), 8 Hi —(208), theits asymmetric carbon c~ is4 taken and 2 02— (140) ions and yields 7= 151 pm. The deviation a = 3.52 ±0.05 mrad and the L type gives re = 154 ±2 pm, again in good agreement with F. The good agreement of Fe and Fin the investigated solids makes the following conclusion possible: The above assumptions are adequate and the generalization of Ferrell’s relation to the d and f electrons —
is applicable for this purpose. The results indicate that the standard deviation a of a compound may be predicted in advance as = ~ (3) ,
because the radii ri of the n components as well as the valence state of the atoms, necessary for the determination of the constant k, are usually known. References  l.Ya. Dekhtyar, Phys. Rep. 9C (1974) 243.  A.D. Tsyganov, A.Z. Varisov, A.D. Makrushkin and E.P. Pro~copjev,Fiz. Tver. Tel. (Solid State Physics) 11(1969) 2079.  R.A Ferrell, Rev. Mod. Phys. 28(1965)308.  A. ~I~ek, M. ~oband K. Krásensk~’,Phys. Stat. Sol. (b) 73 (1976)  A.F. Wells,Ki. Structural inorganic chemistry, 2 (Clarendon, Oxford, 1952).