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Internal friction and modulus behaviour of gamma irradiated NaCl Crystals
INTERNAL FRICTION AND MODULUS BEHAVIOUR OF GAMMA IRRADIATED NaCl CRYSTALS* J. M. SIVERTSENt
The internal friction and modulus of single crystals of NaCl have been measured during the i~adiation of the sample by 6aCo gammas. Since the strain amplitude-independent internal friction is at present only explained in terms of the pinned dislocation theory, a test of the theory is important. The present investigation considers the time dependence of the decrement and modulus as a useful criterion for the validity of the theory. This follows from the loop length dependence of the decrement and dE/E.The generation of point defects during gamma irradiation results in the pinning of dislocation loops of the network due to the interaction between the point defects and dislocations. If we assume that n the number of pinning points is the sum of the number before and those produced during irradiation we have
where &is the loop length before irradiation and .4 is the total dislocation length per unit volume. A characteristic time dependence results during irradiation. Results will be discussed in terms of this model. EVOLUTION DU FROTTEMENT INTERNE ET DU MODULE DANS UN CRISTAL DE NaCl IRRADIE PAR UN FROTTEMENT GAMMA L’auteur a mesure le frottement inteme et le module de monocristaux de NaCl, pendant que l’&&antillon &it irradit! par b rayo~ement gamma de ‘@CO. Puisque le frottement in&me dependant de l’amplitude de la d&formation est explique actuellement uniquement d l’aide de la theorie de I’ancrage des dislocations, il est important de controler cette theorie. L’auteur considbre que la loi de variation en fonction du temps, du d&&ment et du module constitue un critkre utile pour controler la validiti de la theorie, puisque le decrement et AE~E dependent de la longueur des boucles. La cr&tion de defauts ponctuels pendant l’irradiation a pour cons&luence I’ancrage da boucles de disl~tions, du fait de l’interraction existant entre les defauts ponctuels et les dislocations. Si l’on admet que le nombre de points d’ancrage n est la son-me du nombre de ces points existant avant l’irradiation et de ceux cr&s par l’irradiation, on a: 44 A. fi=,=X-Yat oh L, est la longueur des boucles avant l’irradiation et n la longueur totale de dislocations par unite de volume. On observe une evolution caracteristique au cours de l’irradiation. L’auteur discute les reSultats obtenus en se rapportant au mod&le ci-dessus. VERHALTEN
VON XNNERER REIBUNG
UND MODUL VON GAMMABESTRAHLTEN KRISTALLEN
NaCl-
&mere Reibung und Modul von NaCl-Ein~is~llen wurden w&rend der ~stra~~g des Probe mit e°CoGammastrahlen gemessen. Da die von der Verzerrungs-Amplitude unabh&ngige innere Reibung zur Zeit nur von einer Theorie erkhirt wird, die verankerte Versetzungen zugrundelegt, ist eine Prtifung dieser Theorie von Bedeutung. Die vorliegende Untersuchung betrachtet die Zeitabhkgigkeit von Dekrement und Modul als ntitzliches Kriterium fti die Giirtigkeit der Theorie. Diese folgt aus der Abh~~~eit von Dekrement und AE/E von der IXnge der Versetzungsschleifen. Die Erzeugung von Punktfehlem w&rend der Gamma-Best&lung fiihrt dazu, dat3 Versetzungschleifen des Netzwerks verankert werden, da zwischen Punktfehlem und Versetzugen eine Weohselwirkung besteht. Nehmen wir an, dal3 die Zahl der Verankerungspunkte n die Summe der Zahl der vorher vorhandenen und der w&end der Bestrahlung erzeugten ist, so gilt
wo L* die Schleifenl~ge vor der Bestrahlung turd n die gesamte Versetzungsliinge pro Volumeinheit ist. Es ergibt sich eine charakteristische Zeitabh@igkeit w&rend der Bestrahhmg. Die Ergebnisse werden unter Zu~~del~g dieses Modells diskutiert.
* Paper 21 presented at the Conference on Internal Friction held on 10 and 11 July, 1961, at Cornell University. t Department of Metallurgy, University of Minnesota, Minneapolis, Minnesota. ACTA METALLURGICA,
VOL. 10, APRIL 1962 14011
ACTA METALLURGICA, VOL. 10, 1962
402
1. INTRODUCTION Ever since the earlier work of Read(‘) in 1940 it has been known that acoustical waves in a metal crystal are subject to a damping contribution associated with the motion of dislocations under the influence of an external shearing stress. More recently Eshelby@) showed that a dislocation oscillating under an applied stress sets up thermal gradients which are responsible for an energy loss via irreversible heat flow. In addition to this thermoelastic damping, Liebfried@) calculated the energy loss due to the interaction between a moving dislocation and thermal lattice shear waves as an additional contribution. In 1950 Koehlert4) proposed a “pinned” dislocation model to account for the amplitude-independent damping. According to Koehler segments of the dislocation between two pinning points (which can be either impurity atoms or nades in the network of dislocations) will vibrate with the applied stress in the manner of a stretched string. The resonant frequency is usually of the order of 100 MC/S and at all frequencies of interest the damping is proportional to the frequency. In 1956 Granato and Liickef6) extended the Koehler model to show that in the amplitude independent region 6, = QAL4Bwt, I
for the decrement,
n3C
and AE
ii.2A LV = __~ 39 i
t-1 E
for the modulus defect where Q is an orientation factor, A the dislocation density, L the mean length of a vibrating loop and w the frequency. t, and tz are constants which depend upon the distribution of pinning points. Other mechanisms for the amplitude-independent damping have been proposed by Eshelbyc2) and WeertmarF, but the one which agrees best with the experimental results is the “pinned” dislocation model. The present investigation represents the initiation of a series of experiments to test this theory in its present form. Similar experiments have been performed on neutron-irradiated copper crystals by Thompson and Holmes(‘) and we shall compare them with our results later. The present experiment is concerned with a test of the loop length dependence which is performed in the following way.
The total number of pinning points should be a function of the number existing initially and the number produced by irradiation say by s°Co gammas. This can be written as the total length of dislocation per unit volume (II) divided by the length of dislocation between pinning points. That is A n=_L++at
L
Lo
where Lo is the loop length before irradiation, and the number created by irradiation is proportional to the radiation time. Then
and 8,N
l
1
AE
[I + /!a]4 ’ i IF
1i - [I/!$
where p+. Therefore, if the irradiation time dependence is of the above form, the L4 and L2 laws are proved. A simultaneous check of the frequency dependence is possible if one could measure the quenatity G,/(AE/E), at several frequencies before, during and after irradiation. Because of the high sensitivity of the effect to the presently available dose rates it was not possible to do this, however, it is anticipated that we shall be able to do this shortly. 2.
EXPERIMENTAL METHOD
The internal friction and modulus measurements were made by the dynamic method, using a modification of the three-component piezo-electric resonator developed by Marx and co-workers.(8.B) Two identical quartz crystals, cut to oscillate longitudinally, are used in this resonator. One of them, run on the driver, serves to drive the resonator in forced vibration at or near its resonance frequency. The other one, the gauge, indicates the resulting amplitude of vibration in terms of the voltage developed across its electrodes. The driver crystal is excited by an a.c. signal (~39 kc/s) from a variable frequency oscillator. The specimens were single crystals of NaCl obtained from the Harshaw Chemical Co. The specimens used
SIVERTSEN:
PAPER 21 OF INTERNAL
appeared free of mechanical defects or lineage structure, as revealed by the flatness of their cleavage planes. No chemical analyses of the purity of the crystals was obtained, but it has been reported by Duerig and Markham that these crystals usually contain small amounts of divalent impurity such as calcium and traces of elements such as iron. The specimen crystals were received cleaved to the approximate cross-section of the quartz bars (% in”) and close to the proper length. Final adjustment was made by grinding the bars on metallographic polishing paper. All crystals received had their axes oriented along the !lOO) direction. The cold-working of the specimens required that they be cut oversize to the correct amount. Plastic deformation was accomplished by compressing the bars from 1.0 to 1.25 per cent in an Instron tensile machine. A specimen was coupled acoustically to the drivergauge assembly through a “dummy” fused quartz rod of four fundamental wavelengths in lengh (see Fig. 1) after the manner of Marx and Sivertsenu’).
It-
403
FRICTION CONFERENCE
in the quartz crystals resulted in resonant frequency changes of less than 1 c/s. The fused quartz “dummy” rod was irradiated for a very long time before use in the actual measurements. As a result the total radiation induced changes in background damping and frequency were found to be negligible during an actual run. The gammas used for irradiation were obtained from the 6oCo gamma facility source at the University of Minnesota. Three different dose rates were used 77,000 r/hr, 2640 r/hr and 1000 r/hr; the last two being known to within 5 per cent of the quoted values. The configuration used was such that the flux intensity was quite uniform over the volume of the specimen while only the upper part of the quartz dummy rod received any additional significant dosage. As a result it was possible to simultaneously measure modulus and decrement changes during as well as before and after irradiation. All specimens were initially deformed about 1 per cent by compression just prior to an irradiation run. All irradiations were carried out at room temperature (-26°C).
Specimen
*Dummy
*Driver
3. EXPERIMENTAL
Bar
Crystal
To Osc.
FIG. 1. The composite resonator developed by Marx et al.@ 0).
The purpose of the “dummy” rod was to isolate the quartz crystals so they could be adequately shielded from the gamma source (see Fig. 1). Background measurements indicated that the shielding was quite, efficient in as much as any radiation-induced changes
RESULTS
The general effect of the gamma irradiation was to increase the modulus and to decrease the internal friction of the NaCl crystals (see Fig. 2). The modulus change is here reported in terms of Af, the change in the resonant frequency of the specimen. The magnitude of the total change in modulus or in damping produced by the irradiation is sensitive to the individual specimen history. All frequency and decrement measurements were performed at low strain amplitude where the frequency and decrement are amplitude-independent. Figure 2 shows that at the beginning of the irradiation the changes in frequency and decrement are quite rapid. As the total dosage increases the rate of change decreases until eventually “saturation” values for at least the decrement are reached. The frequency and decrement changes are permanent at room temperature after irradiation to saturation. There was no tendency to return to the pre-irradiation values of these quantities. The initial rate of change of the modulus and decrement increased with an increasing dose rate (see Fig. 3). In fact, the changes observed during the 77,000 r/hr irradiation were so rapid that it was difficult to obtain reliable data until after the first 5 minutes of irradiation. Little if any coloration
ACT’A METALLURGICA,
404
VOL. 10, 1962
1.0
300
0.8 Crystal, 38
0.6
6, = 1.27 8 lo+ At, CPS
a 0.4
0.2
0
IO
20
30
40
50
Time
FIG. 2. Relative
60
70
80
9
manutes
decrement and frequency changes time of irradiation at 26°C.
versus
8 - a,
A=-------
60- 600
of the crystals was observed to result from the room temperature irradiations. These results are in general agreement with those reported by Gordon and Nowick(lO) for X-irradiated NaCl crystals.
l
2A
2460
RAW
A
3A
1000
Fvhr
0
38
2460
Whr
to determine the E,, E, and /I constants in the equation derived by them(‘) Em-E E
Em-E0 =
E.
1 (1 + BV
where E, and E, represent the modulus values at zero time of irradiation and for infinitely long irradiation times, respectively. Plotting
LnM
10
20
30
The
40
X
LllA
- minutes
FIG. 3. Comparison of relative decrement changes of several NaCl crystals irradiated at various dose rates. 6 - 8, A=----60 - &o
In order to test the time dependence and hence the loop length dependence of the modulus and decrement an analysis similar to that of Thompson and Holmes(‘) was applied to this case. In this case three equally spaced points were chosen from the irradiation curve
05
1.0
I5
Ln (I + Bt)
FIG. 4. In A and In M, A and M, being the relative decrement and modulus defect, vs. In (l+,%) where t is the time of irradiation. l modulus data; A decrement data,
SIVERTSEN:
M
PAPER 21 OF INTERNAL
(Em- Q/E U-L - EoWo
=
and
&-da=--6s.
In (1 +Bt> m
0
the results shown in Fig. 4 for crystal 3A indicate that the calculated points do not fall at all on the line giving an L2 dependence. It is also to be noted that
FRICTION
CONFERENCE
405
It was orginally planned to test the frequency dependence by measuring 6/(dE/E) as a function of irradiation time at two frequencies. Because of the sensitivity of the irradiation effects to dose rate it was not possible to do this at the presently available dose rates. We are currently modifiyng the source configuration and hope to be able to then study the effects for a frequency dependence of the decrement behaviour.
0
CONCLUSIONS
-I
-3
I.0
0.5
I.5
Ln(I+@P)
Fm. 5. In A and ln M, A and h4, being the relative decrement and modulus defect, vs. In (l+BP*) where t is the time of irradiation. l modulus data; A decrement data. The slope of the modulus curve e -2, and the slope of the decrement curve is g
-4.
the decrement data do not fall at all on a line giving an L4 (i.e. a slope of -4) dependence for the internal friction. A number of trial values of the parameters /& m and n were tried in the expression (1 + Pm) n. In all cases for m much different from $, agreement between calculations and experiment was not good. For the case m = #, appropriate choices of /l for n between 1.5 and 2.5 gave fair agreement with the modulus data. The resulting agreement between the decrement data and calculated values was fairly good with n= 3 to ncz 5. The best agreement, however, for the empirically determined data seemed to be for n E 2 for the modulus data and n II 4 for the decrement data (see Fig. 5).
The experimental results of this study seem to indicate that the loop-length dependence predicted by the “pinned” dislocation model is the one observed for the modulus defect and decrement. An additional point of interest is the fact that L(t)atz/3 in our results instead of t as expected. This may have been fortuitous when one compares this result with those of TruelP. He determined an empirical C(t) for L(t) which did not appear to fit curves for any rational power of t. There is no explanation for this. However, a value of C(t) at2i3 is not unreasonable in as much as the times required for the modulus and decrement changes to go to 90 per cent of completion are quite short. In fact, they seem almost unreasonably short in terms of the photon flux (N10s/cm2sec). One possibility is that the irradiation-induced point defects responsible for pinning are produced quite close to the dislocations and hence the pinning process might be considered analogous to stress-assisted precipitation at dislocations where the Cottrell-Bilby law applies. It should be noted that the time dependences for the decrement and modulus changes we observed are quite consistent with those of True11 who used a dose rate of 3500 r/hr.(ll) REFERENCES 1. T. READ, Phys. Rev. 58, 371 (1940). 2. J. E~HEL~Y,Proc. Roy.’ Sot. ‘Alti, 396 (1949). G. LIEBFRIED,2. Phys. 127, 344 (1950) :: J. KOEHLERin Imperfections in Nearly Perfect Crystals (Ed. bv W. SHOCKLEY)v. 1970. Wilev. New York (1952). 5. A. GR~NATO and K. LOCKE,J. Appl: ‘Phys. 27, 581 anh 789 (1956); 28, 635 (1957). J. Appl. Phys. 26, 202 (1955); 28, 193 and 6. J. WEERTMAN, 636 (1957). 7. D. 0. THOMPXINand D. K. HOLMES,J. Appl. Phys. 27, 191 and 713 (1956). a. J. MARX, Rev. Sci. Znstrum. 22 503, (1951). 9. J. MARX and J. SIVERTSEN,J. Appf. Phys. 24, 81 (1953); J. MARX, G. BAKERand J. SIVERTSEN, Acta Met. 1, 193 (1953). R. GORDON and A. NOWICK, Acta Met. 4, 514. (1956). :;: R. TRUELL, J. Appl. Phys. 30, 1275 (1959).
The internal friction and modulus of single crystals of NaCl have been measured during the i~adiation of the sample by 6aCo gammas. Since the strain amplitude-independent internal friction is at present only explained in terms of the pinned dislocation theory, a test of the theory is important. The present investigation considers the time dependence of the decrement and modulus as a useful criterion for the validity of the theory. This follows from the loop length dependence of the decrement and dE/E.The generation of point defects during gamma irradiation results in the pinning of dislocation loops of the network due to the interaction between the point defects and dislocations. If we assume that n the number of pinning points is the sum of the number before and those produced during irradiation we have
where &is the loop length before irradiation and .4 is the total dislocation length per unit volume. A characteristic time dependence results during irradiation. Results will be discussed in terms of this model. EVOLUTION DU FROTTEMENT INTERNE ET DU MODULE DANS UN CRISTAL DE NaCl IRRADIE PAR UN FROTTEMENT GAMMA L’auteur a mesure le frottement inteme et le module de monocristaux de NaCl, pendant que l’&&antillon &it irradit! par b rayo~ement gamma de ‘@CO. Puisque le frottement in&me dependant de l’amplitude de la d&formation est explique actuellement uniquement d l’aide de la theorie de I’ancrage des dislocations, il est important de controler cette theorie. L’auteur considbre que la loi de variation en fonction du temps, du d&&ment et du module constitue un critkre utile pour controler la validiti de la theorie, puisque le decrement et AE~E dependent de la longueur des boucles. La cr&tion de defauts ponctuels pendant l’irradiation a pour cons&luence I’ancrage da boucles de disl~tions, du fait de l’interraction existant entre les defauts ponctuels et les dislocations. Si l’on admet que le nombre de points d’ancrage n est la son-me du nombre de ces points existant avant l’irradiation et de ceux cr&s par l’irradiation, on a: 44 A. fi=,=X-Yat oh L, est la longueur des boucles avant l’irradiation et n la longueur totale de dislocations par unite de volume. On observe une evolution caracteristique au cours de l’irradiation. L’auteur discute les reSultats obtenus en se rapportant au mod&le ci-dessus. VERHALTEN
VON XNNERER REIBUNG
UND MODUL VON GAMMABESTRAHLTEN KRISTALLEN
NaCl-
&mere Reibung und Modul von NaCl-Ein~is~llen wurden w&rend der ~stra~~g des Probe mit e°CoGammastrahlen gemessen. Da die von der Verzerrungs-Amplitude unabh&ngige innere Reibung zur Zeit nur von einer Theorie erkhirt wird, die verankerte Versetzungen zugrundelegt, ist eine Prtifung dieser Theorie von Bedeutung. Die vorliegende Untersuchung betrachtet die Zeitabhkgigkeit von Dekrement und Modul als ntitzliches Kriterium fti die Giirtigkeit der Theorie. Diese folgt aus der Abh~~~eit von Dekrement und AE/E von der IXnge der Versetzungsschleifen. Die Erzeugung von Punktfehlem w&rend der Gamma-Best&lung fiihrt dazu, dat3 Versetzungschleifen des Netzwerks verankert werden, da zwischen Punktfehlem und Versetzugen eine Weohselwirkung besteht. Nehmen wir an, dal3 die Zahl der Verankerungspunkte n die Summe der Zahl der vorher vorhandenen und der w&end der Bestrahlung erzeugten ist, so gilt
wo L* die Schleifenl~ge vor der Bestrahlung turd n die gesamte Versetzungsliinge pro Volumeinheit ist. Es ergibt sich eine charakteristische Zeitabh@igkeit w&rend der Bestrahhmg. Die Ergebnisse werden unter Zu~~del~g dieses Modells diskutiert.
* Paper 21 presented at the Conference on Internal Friction held on 10 and 11 July, 1961, at Cornell University. t Department of Metallurgy, University of Minnesota, Minneapolis, Minnesota. ACTA METALLURGICA,
VOL. 10, APRIL 1962 14011
ACTA METALLURGICA, VOL. 10, 1962
402
1. INTRODUCTION Ever since the earlier work of Read(‘) in 1940 it has been known that acoustical waves in a metal crystal are subject to a damping contribution associated with the motion of dislocations under the influence of an external shearing stress. More recently Eshelby@) showed that a dislocation oscillating under an applied stress sets up thermal gradients which are responsible for an energy loss via irreversible heat flow. In addition to this thermoelastic damping, Liebfried@) calculated the energy loss due to the interaction between a moving dislocation and thermal lattice shear waves as an additional contribution. In 1950 Koehlert4) proposed a “pinned” dislocation model to account for the amplitude-independent damping. According to Koehler segments of the dislocation between two pinning points (which can be either impurity atoms or nades in the network of dislocations) will vibrate with the applied stress in the manner of a stretched string. The resonant frequency is usually of the order of 100 MC/S and at all frequencies of interest the damping is proportional to the frequency. In 1956 Granato and Liickef6) extended the Koehler model to show that in the amplitude independent region 6, = QAL4Bwt, I
for the decrement,
n3C
and AE
ii.2A LV = __~ 39 i
t-1 E
for the modulus defect where Q is an orientation factor, A the dislocation density, L the mean length of a vibrating loop and w the frequency. t, and tz are constants which depend upon the distribution of pinning points. Other mechanisms for the amplitude-independent damping have been proposed by Eshelbyc2) and WeertmarF, but the one which agrees best with the experimental results is the “pinned” dislocation model. The present investigation represents the initiation of a series of experiments to test this theory in its present form. Similar experiments have been performed on neutron-irradiated copper crystals by Thompson and Holmes(‘) and we shall compare them with our results later. The present experiment is concerned with a test of the loop length dependence which is performed in the following way.
The total number of pinning points should be a function of the number existing initially and the number produced by irradiation say by s°Co gammas. This can be written as the total length of dislocation per unit volume (II) divided by the length of dislocation between pinning points. That is A n=_L++at
L
Lo
where Lo is the loop length before irradiation, and the number created by irradiation is proportional to the radiation time. Then
and 8,N
l
1
AE
[I + /!a]4 ’ i IF
1i - [I/!$
where p+. Therefore, if the irradiation time dependence is of the above form, the L4 and L2 laws are proved. A simultaneous check of the frequency dependence is possible if one could measure the quenatity G,/(AE/E), at several frequencies before, during and after irradiation. Because of the high sensitivity of the effect to the presently available dose rates it was not possible to do this, however, it is anticipated that we shall be able to do this shortly. 2.
EXPERIMENTAL METHOD
The internal friction and modulus measurements were made by the dynamic method, using a modification of the three-component piezo-electric resonator developed by Marx and co-workers.(8.B) Two identical quartz crystals, cut to oscillate longitudinally, are used in this resonator. One of them, run on the driver, serves to drive the resonator in forced vibration at or near its resonance frequency. The other one, the gauge, indicates the resulting amplitude of vibration in terms of the voltage developed across its electrodes. The driver crystal is excited by an a.c. signal (~39 kc/s) from a variable frequency oscillator. The specimens were single crystals of NaCl obtained from the Harshaw Chemical Co. The specimens used
SIVERTSEN:
PAPER 21 OF INTERNAL
appeared free of mechanical defects or lineage structure, as revealed by the flatness of their cleavage planes. No chemical analyses of the purity of the crystals was obtained, but it has been reported by Duerig and Markham that these crystals usually contain small amounts of divalent impurity such as calcium and traces of elements such as iron. The specimen crystals were received cleaved to the approximate cross-section of the quartz bars (% in”) and close to the proper length. Final adjustment was made by grinding the bars on metallographic polishing paper. All crystals received had their axes oriented along the !lOO) direction. The cold-working of the specimens required that they be cut oversize to the correct amount. Plastic deformation was accomplished by compressing the bars from 1.0 to 1.25 per cent in an Instron tensile machine. A specimen was coupled acoustically to the drivergauge assembly through a “dummy” fused quartz rod of four fundamental wavelengths in lengh (see Fig. 1) after the manner of Marx and Sivertsenu’).
It-
403
FRICTION CONFERENCE
in the quartz crystals resulted in resonant frequency changes of less than 1 c/s. The fused quartz “dummy” rod was irradiated for a very long time before use in the actual measurements. As a result the total radiation induced changes in background damping and frequency were found to be negligible during an actual run. The gammas used for irradiation were obtained from the 6oCo gamma facility source at the University of Minnesota. Three different dose rates were used 77,000 r/hr, 2640 r/hr and 1000 r/hr; the last two being known to within 5 per cent of the quoted values. The configuration used was such that the flux intensity was quite uniform over the volume of the specimen while only the upper part of the quartz dummy rod received any additional significant dosage. As a result it was possible to simultaneously measure modulus and decrement changes during as well as before and after irradiation. All specimens were initially deformed about 1 per cent by compression just prior to an irradiation run. All irradiations were carried out at room temperature (-26°C).
Specimen
*Dummy
*Driver
3. EXPERIMENTAL
Bar
Crystal
To Osc.
FIG. 1. The composite resonator developed by Marx et al.@ 0).
The purpose of the “dummy” rod was to isolate the quartz crystals so they could be adequately shielded from the gamma source (see Fig. 1). Background measurements indicated that the shielding was quite, efficient in as much as any radiation-induced changes
RESULTS
The general effect of the gamma irradiation was to increase the modulus and to decrease the internal friction of the NaCl crystals (see Fig. 2). The modulus change is here reported in terms of Af, the change in the resonant frequency of the specimen. The magnitude of the total change in modulus or in damping produced by the irradiation is sensitive to the individual specimen history. All frequency and decrement measurements were performed at low strain amplitude where the frequency and decrement are amplitude-independent. Figure 2 shows that at the beginning of the irradiation the changes in frequency and decrement are quite rapid. As the total dosage increases the rate of change decreases until eventually “saturation” values for at least the decrement are reached. The frequency and decrement changes are permanent at room temperature after irradiation to saturation. There was no tendency to return to the pre-irradiation values of these quantities. The initial rate of change of the modulus and decrement increased with an increasing dose rate (see Fig. 3). In fact, the changes observed during the 77,000 r/hr irradiation were so rapid that it was difficult to obtain reliable data until after the first 5 minutes of irradiation. Little if any coloration
ACT’A METALLURGICA,
404
VOL. 10, 1962
1.0
300
0.8 Crystal, 38
0.6
6, = 1.27 8 lo+ At, CPS
a 0.4
0.2
0
IO
20
30
40
50
Time
FIG. 2. Relative
60
70
80
9
manutes
decrement and frequency changes time of irradiation at 26°C.
versus
8 - a,
A=-------
60- 600
of the crystals was observed to result from the room temperature irradiations. These results are in general agreement with those reported by Gordon and Nowick(lO) for X-irradiated NaCl crystals.
l
2A
2460
RAW
A
3A
1000
Fvhr
0
38
2460
Whr
to determine the E,, E, and /I constants in the equation derived by them(‘) Em-E E
Em-E0 =
E.
1 (1 + BV
where E, and E, represent the modulus values at zero time of irradiation and for infinitely long irradiation times, respectively. Plotting
LnM
10
20
30
The
40
X
LllA
- minutes
FIG. 3. Comparison of relative decrement changes of several NaCl crystals irradiated at various dose rates. 6 - 8, A=----60 - &o
In order to test the time dependence and hence the loop length dependence of the modulus and decrement an analysis similar to that of Thompson and Holmes(‘) was applied to this case. In this case three equally spaced points were chosen from the irradiation curve
05
1.0
I5
Ln (I + Bt)
FIG. 4. In A and In M, A and M, being the relative decrement and modulus defect, vs. In (l+,%) where t is the time of irradiation. l modulus data; A decrement data,
SIVERTSEN:
M
PAPER 21 OF INTERNAL
(Em- Q/E U-L - EoWo
=
and
&-da=--6s.
In (1 +Bt> m
0
the results shown in Fig. 4 for crystal 3A indicate that the calculated points do not fall at all on the line giving an L2 dependence. It is also to be noted that
FRICTION
CONFERENCE
405
It was orginally planned to test the frequency dependence by measuring 6/(dE/E) as a function of irradiation time at two frequencies. Because of the sensitivity of the irradiation effects to dose rate it was not possible to do this at the presently available dose rates. We are currently modifiyng the source configuration and hope to be able to then study the effects for a frequency dependence of the decrement behaviour.
0
CONCLUSIONS
-I
-3
I.0
0.5
I.5
Ln(I+@P)
Fm. 5. In A and ln M, A and h4, being the relative decrement and modulus defect, vs. In (l+BP*) where t is the time of irradiation. l modulus data; A decrement data. The slope of the modulus curve e -2, and the slope of the decrement curve is g
-4.
the decrement data do not fall at all on a line giving an L4 (i.e. a slope of -4) dependence for the internal friction. A number of trial values of the parameters /& m and n were tried in the expression (1 + Pm) n. In all cases for m much different from $, agreement between calculations and experiment was not good. For the case m = #, appropriate choices of /l for n between 1.5 and 2.5 gave fair agreement with the modulus data. The resulting agreement between the decrement data and calculated values was fairly good with n= 3 to ncz 5. The best agreement, however, for the empirically determined data seemed to be for n E 2 for the modulus data and n II 4 for the decrement data (see Fig. 5).
The experimental results of this study seem to indicate that the loop-length dependence predicted by the “pinned” dislocation model is the one observed for the modulus defect and decrement. An additional point of interest is the fact that L(t)atz/3 in our results instead of t as expected. This may have been fortuitous when one compares this result with those of TruelP. He determined an empirical C(t) for L(t) which did not appear to fit curves for any rational power of t. There is no explanation for this. However, a value of C(t) at2i3 is not unreasonable in as much as the times required for the modulus and decrement changes to go to 90 per cent of completion are quite short. In fact, they seem almost unreasonably short in terms of the photon flux (N10s/cm2sec). One possibility is that the irradiation-induced point defects responsible for pinning are produced quite close to the dislocations and hence the pinning process might be considered analogous to stress-assisted precipitation at dislocations where the Cottrell-Bilby law applies. It should be noted that the time dependences for the decrement and modulus changes we observed are quite consistent with those of True11 who used a dose rate of 3500 r/hr.(ll) REFERENCES 1. T. READ, Phys. Rev. 58, 371 (1940). 2. J. E~HEL~Y,Proc. Roy.’ Sot. ‘Alti, 396 (1949). G. LIEBFRIED,2. Phys. 127, 344 (1950) :: J. KOEHLERin Imperfections in Nearly Perfect Crystals (Ed. bv W. SHOCKLEY)v. 1970. Wilev. New York (1952). 5. A. GR~NATO and K. LOCKE,J. Appl: ‘Phys. 27, 581 anh 789 (1956); 28, 635 (1957). J. Appl. Phys. 26, 202 (1955); 28, 193 and 6. J. WEERTMAN, 636 (1957). 7. D. 0. THOMPXINand D. K. HOLMES,J. Appl. Phys. 27, 191 and 713 (1956). a. J. MARX, Rev. Sci. Znstrum. 22 503, (1951). 9. J. MARX and J. SIVERTSEN,J. Appf. Phys. 24, 81 (1953); J. MARX, G. BAKERand J. SIVERTSEN, Acta Met. 1, 193 (1953). R. GORDON and A. NOWICK, Acta Met. 4, 514. (1956). :;: R. TRUELL, J. Appl. Phys. 30, 1275 (1959).