National Bureau of Standard!

library, N.W. Blag

SEP 3 1964

^ecknic&l riote

THE ELECTRICAL PROPERTIES OF ALUMINUM

FOR CRYOGENIC ELECTROMAGNETS

R. J. CORRUCCINI

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

'technical ^ete 21

S

Issued August 30, 1964

THE ELECTRICAL PROPERTIES OF ALUMINUM

FOR CRYOGENIC ELECTROMAGNETS

R. J. Corruccini

Cryogenics Division

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D. C. 20402

Price: 30<t

CONTENTS

PAGE

ABSTRACT 1

1. INTRODUCTION 1

2. THE IMPERFECTION RESISTIVITY -- -- 3

3. THE LATTICE RESISTIVITY 4

4. SIZE EFFECTS 7

4.1 In Zero Field 7

4.2 In Magnetic Fields 10

5. MAGNETORESISTANCE --- 11

5.1 Discussion 15

6. THE FIGURE OF MERIT --- -- 17

7. DISCUSSION - 19

8. APPENDIX. EFFECTS OF PLASTIC DEFORMATION 20

ADDENDUM --- --- 20a

9. REFERENCES 24

in

The Electrical Properties of Aluminum for Cryogenic Electromagnets

R. J. Corruccini

The published data for the ideal resistivity and the magnetoresistivity of aluminum have been correlated. It is shown

that both properties can be calculated for the limited ranges of

temperature and purity that are of importance for aluminum

cryogenic electromagnets from the residual resistivity ratio

alone. Empirical functions are given for these properties, and

sample calculations are given of the figure of merit for a particular aluminum solenoid relative to the same solenoid using

water-cooled copper.

1. INTRODUCTION

The well-known fact that the electrical resistivity and hence the

Joule loss in a conductor may be greatly reduced by cooling to cryogenic

temperatures has led to various cooled electromagnets for laboratory

research [1] . However, the designing of these magnets, all of small

size, was influenced mainly by various special considerations rather

than by economic factors. On the other hand, emphasis by Post [2] of

the economic advantages of cryogenic electromagnets for applications

where high fields must be maintained throughout large spaces, such as nuclear fusion devices or particle accelerators, has made it clear that

careful projections of costs and performance are needed. It is not yet

clear what the relative roles of superconducting and normal-conducting

magnets will ultimately be in such applications. This paper is concerned

only with the latter.
In rating normal- state cryogenic electromagnets it has become

conventional to compare the total power expenditure (Joule loss in the

magnet conductor plus refrigerator power) with the power expenditure of

a water-cooled copper electromagnet (Joule loss only, neglecting the

power expenditure for cooling). The following formula defines a figure

of merit on this basis.

»[T] 5

p[

J'

H

\ \l + ^^

PCJ300] L 1 T

(1)

The first term in the bracket gives the contribution of the Joule loss; the

second gives the contribution of the refrigerator power. Here T , the

temperature to which the refrigerator exhausts heat, may be taken as

300° K. The factor t) is the overall efficiency of the refrigerator relative

to ideal thermodynamic efficiency. It may be taken to be about 0. 3 [ 3]

.

The room temperature resistivity of copper, p [300], is 1.7 uohm-cm.

CyU

The resistivity of the cryogenic conductor, p[T,H], may be approximated

as the sum of the imperfection or "residual" resistivity p , the lattice or "ideal" resistivity p. = f(T), and the magnetoresistance p . l ri

p[T,H] = pQ + p. + pH

. (2)

The lattice resistivity is ordinarily assumed to be a unique function for a given metal. However, in order to retain the simple form of equation (2)

it may be necessary to assume it to be a weak function of p . Also the

magnetoresistance may be a function of both the temperature and p . The

optimum operating temperature is selected by minimizing <j>.
Using a mean resistivity appropriate for the conductor in an electro5

magnet producing 10 gauss, only five metals have been found to give

calculated values of <j> less than unity. Sodium [2] and aluminum are outstanding if sufficiently pure and may give values approaching 0. 1. Copper [2] gives a minimum value of about 0. 5, and lithium is perhaps

about the same. Indium falls between. The optimum temperatures are

roughly as follows: 30"K for copper, 15-20° for aluminum, 10° for

sodium and lithium, and below 4° for indium. These estimates are based

on large extrapolations of p for sodium, lithium, and indium and assume

that p can be lowered to any desired extent by purification, regardless

of what may actually have been achieved to date.

Of the two leading contenders, aluminum has superior strength,

chemical inertness, and ease of fabrication. Much new information has

been published in recent moriths on its properties, especially the magnetoresistance. In this paper, data on its electrical properties are collected and correlated, and its figure of merit is determined as a function

of purity and field strength for a particular coil configuration.

2. THE IMPERFECTION RESISTIVITY

As will appear when the lattice and magnetoresistivities have been

discussed, these latter contribute about 5 to 10 nohm-cm to the total

5

effective resistivity of aluminum at 20° K and 10 gauss. It will be advantageous, then, to use a sufficiently well-refined aluminum that its imperfection resistivity is small compared to this. That is, the ratio p /p

_ RT 4

should be at least 10 , corresponding to purity higher than 99. 99% [4]

.

(p _ signifies the resistivity at room temperature, 2750 nohm-cm, p

that at 4°K. ) Electrolytically-refined metal is commercially available

having values of the above ratio as high as about 2500. Much higher ratios

are attainable by zone refining, but at such expense as to probably rule

out the use of such metal. The practical range of values thus is about

1000 to 2500 at present.

3
3. THE LATTICE RESISTIVITY

Equation (2) implies the assumption of Matthies sen's rule

p[T.H= 0] = pQ+ p..

However, large departures occur in the temperature range of interest.

These deviations qualitatively fit theoretical models by Kohler and

others [5]. If it were our purpose to find a wide-range representation of

p[T,H = 0] this could no doubt be accomplished by adding a deviation

term to the formulation of Matthiessen's rule. However, since we only

wish to represent the resistivity in the vicinity of the optimum temperature, it will be computationally simpler to let p. be a function of p . The

latter is readily determined via measurement of the ratio of the resistance at room temperature to that at 4°K and, in fact, is usually determined in order to characterize the quality of samples.

Measurements by Caron [6] from 1.4 to 20°K of various samples

covering a range, 180 < p-, Q ?/ pA 9 < 6700, give a clue to a simple prou 7 J 4. C. cedure. Caron found that the data could be represented by

3.1 ± 0.1

p

i = aT

in which the coefficient, a, was an increasing function of p . Maimoni [7] also measured various samples around 20° K. The

range of P? nr./pA

-, was from 276 to 6630. Logarithmic plots of his

lattice resistivities are approximately linear from 15 to 20° K with slopes

from 2. 5 to 3.1. He also found p. to increase with n . Maimoni's resuits can be compared to Caron's for samples having p /p of about

RT. 4

2000 and 6700. At these values, Maimoni's resistivities in the vicinity

of 20°K are roughly 10% higher than Caron's.
Evidently both sets of data can be fairly well represented around

20°K by

P

i [T]/pRT = aT

3

; a = f ( Pq). (3)

If a is determined by fitting at 20°K, then the misrepresentation of

Caron's function from 15 to 25°K due to the exponent being too low by

0. 1 is less than ±3%. Similarly, Maimoni's tabular data above 15°K are

fitted within 6%. Equation (3) is devoid of theoretical significance and

can be expected to apply only for limited ranges of temperature and p . However, these include the values of interest for aluminum electromagnets.

The values of a as a function of p^^/p A derived from the data of

Car on and Maimoni are shown in figure 1. It is suggested that values

given by the curve should be used for calculating ideal resistivities by

equation (3) [8]. The curve represents the following empirical equation

which may be useful for calculations:

10

8

a = 2. 80 + (0. 17 + 0. 065 • 10" 3

x + 0. 033 • 10

_6

x

2

) _1

x-PRT/p4

- (4)

This method of predicting p. is of doubtful value below x = 500 where the

data of figure 1 as well as other data [9] on relatively impure aluminums

scatter badly. It apparently is reliable to very high values of x. Thus,

Montariol and Reich [ 10] measured the resistivity from 14° to 20.4°K

of a zone -refined aluminum with x = 22, 600. Similarly Aleksandrov

and D'Yakov [ 11] measured a specimen with x = 29,400 at 14° and20.4°K.

The above authors found the ideal resistivities of these two specimens to

3.

7

3. 55

go as T and T , respectively. If equation (3) is constrained to fit

-8

these data at 20.4°K, the coefficient a is found to be 2. 79 x 10 for

either specimen. Equation (4) gives values within 2% of this.
A few other sources of data bearing on the correlation of figure 1 may be mentioned. Measurements were made from 3° to 70° by Lax [ 16]

3. 24

on a specimen with x = 550. Below 30° the resistance varied as T . 3 The coefficient obtained by fitting his 20° datum to the T formula falls

on the curve of figure 1. Papers by Borovik and coworkers [12, 13, 14]

give resistance ratios at 4. 2° and 20.4° for six specimens with x ranging

from 690 to 24,400. These throw no light on the validity of equation (3)

in the range around 20° K; however, values of a can be calculated and are

shown in figure 1. Values not shown at x = 11, 000 and 24,400 are +7%

and -1%, respectively, different from equation (4). Data by Purcell and

Jacobs [15] at 4°, 20°, 25°, and 30°K for two samples are erratic when

tested by equation (4) and fail tests of self-consistency based on minimal

assumptions about the behavior expected of p..
4. SIZE EFFECTS

4.1 In Zero Field

For the conditions of temperature and purity that are of interest

here, the mean free path of the electrons may be appreciably long relative

to the conductor dimensions. In such a case, the scattering of electrons

from the surface will provide an appreciable extra resistance in addition

to that attributable to the bulk metal. Aleksandrov [ 17] and Montariol

and Reich [ 18] have studied this effect by measuring the resistances of

round wires of various diameters made from the very pure aluminums

mentioned in section 3. For these samples with bulk resistance ratios

greater than 20, 000, the mean free path at 4°K is of the order of 1 mm.

By expressing the results in terms of Nordheim's approximate formula [ 19] —£-= 1 +| (5)

Poo d

values of the mean free path 1 and the approximately constant product

p i can be derived. Here p is the bulk resistivity, and d is the wire

diameter. The results are given in table 1. Montariol and Reich also

measured three less pure samples at 4. 2°K for which the values of x

were 12,000, 7000, and 2540. These gave values of 10 pJL of 0.80,

0.90, and 1.50, respectively. F^rsvoll and Holwech [ 21] determined

size effects in foil specimens made from six grades of aluminum for

which x ranged from 1650 to 26,500. The values obtained for 10 pi

varied randomly from 0. 53 to 0. 88. Table 1 shows the average value

obtained by F^rsvoll and Holwech.
It will be seen that the agreement obtained in these difficult experi1 ments is not very good. However it would seem that the rounded value,

-11 2

p £ = 1x10 ohm-cm , could safely be used for roughly estimating the

mean free path of aluminum. Given the mean free path, one can determine the size effect in a given specimen using results of theoretical calculations such as those for films and wires shown in figure 2. (Equation

(5) approximates the values shown for round wires within 5% in p/p . ) For a sample with x = 2000, the mean free path estimated using

10 p Si = 1 is about 0. 04 mm at 20.4°K. Thus for magnet conductors

of practical dimensions, substantial size effects are likely to occur

only in foils or tapes. An example is the tape -wound liquid-hydrogencooled magnet of Purcell and Payne [22] in which the conductor was

0. 10 mm. thick; i. e. , i/d was 0. 4 at 20. 4°K. It will be seen by reference to figure 2 that the low-field resistance of this magnet and the

residual-resistance ratio reported for a sample of its conductor must

have contained a substantial component due to the surface resistance.

In fact, as table 2 shows, most of the sources of resistivity data

on which we must depend contain errors as large as several percent due

to uncorrected size effects. Table 2 shows the maximum size -effect

error in each investigation. These were calculated using the 4. 2°K

value of p i of Aleksandrov since this agrees fairly well with the values

from the anomalous skin effect. Obviously, size effect should always be

taken into account in accurate investigations under the conditions of purity

and temperature which concern us here.

They are in rough agreement with the values 0. 49 ± . 04 x 10 [20] and

0.40 x 10 [21] resulting from measurements of the anomalous skin

effect.
Having raised this point it is now necessary to consider if the

correlation presented for p. should be modified to take the size -effect

1 errors into account. First we may note that p. is rather insensitive to

size. In fact, it is easy to show that if p £ were constant and the size

effect were given by a formula having the form of Nordheim's equation,

p. would be independent of size. On the other hand, if we adopt the values

of p i as a function of temperature found by Aleksandrov, then p. from

Maimoni's data increases slightly more rapidly with temperature, but

the increase in the exponent n is only 0. 1. The values of p. at 20°K are

lower but are within 5% of the uncorrected values.

It has not seemed worthwhile to correct the recommended ideal

resistivities of section 3 for the following reasons:

(1) In view of the disagreement shown in table 1, our knowledge of

the parameter p i is inaccurate.

(2) The ideal resistivity is a minor part of the total resistivity in

the conductor of a high-intensity magnet as will appear in section 6. Hence an uncertainty in it of a few percent can be neglected.

(3) Maimoni used specimens of two configurations (table 2) for

which the size -effect corrections are appreciably different. Application

of the corrections does not smooth the correlation shown in figure 1. Hence other sources of scatter outweigh the size effect in his investigation.

(4) Application of the correction does not reduce the discrepancies

between the p. from Car on and those from Maimoni; in fact, for the specimens with x near 6700, it appreciably worsens the agreement. Thus

this discrepancy must be due to other causes that are much more significant than size effects.
4. 2 In Magnetic Fields

When a sample thin enough to show the zero -field size effect is

placed in an increasing magnetic field, two effects operate simultaneously:

(1) The resistance rises with magnetic -field intensity according

to the bulk magnetoresistance effect (see next section).

(2) The radius (r = mvc/eH) of the orbital motion of the conduction

electrons in the magnetic field decreases, and consequently they collide

less often with the surface. The exact behavior of the surface resistance

depends on the relative orientations of field, current, and specimen [ 19]

.

However, there is only one case among practical magnet configurations

where the galvanomagnetic size effect is likely to be observed, i.e. the

flat-wound foil coil. Here the field is in the plane of the foil but perpendicular to the current. In the field region where the orbital radius is

comparable to the sample thickness, the surface component of the resistance falls rapidly with increasing field, becoming negligible when the

orbital radius has become much smaller than the sample thickness.

Combination of the two effects above may give R-H curves of varied

shapes. If the size effect is initially strong a minimum may occur, after

which the resistance again rises due to the bulk magnetoresistance alone

[23, 24].

The available knowledge of size effects in magnetic fields is insufficient for accurate prediction [ 19, 25] . However, this does not matter

much in designing high-field magnets because the conductor size effect,

if present at all, will generally have vanished by the time the field has

reached a few thousand oersteds. This is evident from the radius of the

electron orbits in aluminum which has shrunk to the order of 0. 01 mm.

at 10 koe [23] . Since we are concerned with much larger fields than

this, and the magnet conductor is not likely to be thinner than 0. 1 mm. , the size effect under such conditions will have vanished relative to the

bulk magnetoresistance.

10
5. MAGNETORESISTANCE

The sources of data on the magnetoresistance of aluminum are

listed in table 3. The transverse magnetoresistance is of primary interest, inasmuch as this is the effect occurring in simple solenoids. The

correlation of the data was examined by placing them on a plot of pu/p a

vs. H^'«• = HpR

/p according to Kohler's rule, pu/p = f(H/p). Here p

1

H. and p^rp are resistivities in zero field, and p is the increase in re sis - RT H

tivity due to the field H, measured at the same temperature as p. Superficially the data appear to follow the theoretical form [26]

2

A+BH*

2

i. e. , an initial H dependence followed by saturation. However, closer

inspection reveals two significant differences:

(1) The "saturation" region at high H^ is actually rather accurately

linear, though the slope is small [27, 28, 29, 30]. By inverting the conductivity tensor, more general theoretical formulas can be obtained [30]

which contain the desired pseudo-linear region following the initial quadratic region. At still higher values of H^, these predict either saturation or a second quadratic region, neither of which phenomena have been

definitely demonstrated to occur. Consequently we choose to obtain a

fit to the linear region by multiplying the numerator above by (1 + CH^),

thus obtaining a simpler formula which, however, may not safely be used

for extrapolation to larger H^.

11
The failure to truly saturate is apparently an anomaly. For a

given direction of magnetic field in a crystal, the magnetoresistance at

very high fields must saturate unless there are extended electron trajectories made possible by contact of the Fermi surface with the Brillouin

2

zone boundary, in which case the behavior would be as H at all fields.

In a random polycrystal the net behavior corresponding to suitable

averaging of saturating and quadratic behavior may turn out to be linearity with H [ 19] , e. g. , copper. However, there are two reasons why

this kind of explanation does not seem to apply to aluminum [28, 30]

.

First, extensive measurements with single crystals show the same

Prr/p vs. H behavior at all orientations, i. e. , essentially the curve shape H

of figure 3. Second, no sharp anisotropy of pu/p at constant H is found, H

such as that found with e. g. copper and gold.

(2) If values of A, B, and C are adopted so as to represent the

limiting behaviors at small and large H^, then the fit is bad in the transition zone, such that the calculated p^/p is nearly twice too large at H

about H^ = 2. Analysis suggests addition of a term in Hr

, in the denominator.

With these modifications we have a formula, unfortunately empirical, but nevertheless helpful in computer calculations and as a base line

from which to examine the experimental data. The chosen function is as follows where H^ is in megoersteds.

PN H$

2

(1 + 0.00177H,) — = r

• (6)

P 1.8+1. 6H„ + 0. 53H„

12
It is displayed in figure 3. The percentage deviations from it of the experimental data are shown in figures 4a, 4b, and 4c. In most cases the

"experimental" data were read from smooth curves in the original publications; in such cases the deviation curves are also smooth. In a few

cases (P-J and H-A in figure 4c) the original data points were used; here

the deviation curves are made up of straight segments. The precision

with which the original data were expressed is always worst at the lowest field. In some cases (identified by bars) the data could be recovered

with only one significant figure. The height of the bar indicates only the —precision of the published value of p^/p. In most of the investigations -————

rl no basis is provided for estimating the accuracy. Following are some

comments about specific investigations.

Kapitza. These remarkable pioneering measurements are mainly of

historical interest. The values of P^^/Pa in table 3 were estimated from

his values of p /p , but are uncertain due to ambiguities as to the

value of the temperature.

2

De Haas and Van Alphen. These are the only data in the H^ region. —Justi and Scheffers . These data were presented as a plot of p vs. H. ————__

_

j-j In order to place them on a Kohler plot it was necessary to estimate the

zero-field resistivity from 4° to 78°. These estimates, as well as the

estimate of x in table 3, were based on values given for the ratios

p ~/p ?

~ and p Jp . Later Foroud, Justi, and Kramer [31] published

a reduced Kohler plot based on the data of Justi and Scheffers but, curiously, using only the data at 4° and 64° K. This reduced Kohler plot used

p Q rather than p,-,TM in H^; the assumed values of 6 and p Q were not given.

Thomas and Mendoza. These data, lying in the range 11^=0. 1 to 0. 5

moe, are an order of magnitude higher than all others and are given no

weight.

13
Luthi and Olsen. This investigation showed the usual behavior up to

about H^= 20 moe followed by an anomalous rapid rise in p^/p. Later

Luthi [ 32] attributed this to eddy-current heating produced by the pulsed

magnetic field. Borovik and Volotskaya [ 13] suggested that misalignment

of the potential contacts with the current direction could have produced

this result by including a component of the Hall field which, for aluminum,

is large relative to the electric field in the current direction.

Luthi I960 . The data had to be taken from a reduced Kohler plot for

which 6 was stated to be 410°K, but p was not furnished. To reduce to

the basis of figure 4, it has been assumed that PAin/P-, 7 o = L 62.

Purcell and Jacobs . As was mentioned in section 3, the zero-field

data of Purcell and Jacobs are erratic. The scatter of their curves in

figure 4 is partly attributable to this.

Babiskin and Siebenmann. Four specimens from a common stock

were used; two had different degrees of cold-work and the other two were

given different anneals. Figure 4a shows only the specimen (III) that was

most completely annealed and that had the highest ratio, x. It also showed

about average behavior on a Kohler plot of the data for all four specimens.

The results indicated no ordering by resistance ratio, and the extremes

were provided by the two cold-worked specimens. The results for all

specimens lay within ±10% of specimen III. Borovik, Volotskaya, and Fogel* . In addition to the data shown in

figure 4b these authors give curves in the same range of H^ for three

high-purity crystals at 20° K which show p^/p rising to maximum values H

from 4 to over 6. These results indicate that Kohler' s rule is not even

approximately valid. However, the specimen purities for which these

results were found (x = 6,875 to 24,400) fall outside the range that is

presently of practical importance for electromagnets.

14
5. 1 Discussion

The region of greatest interest in connection with electromagnets is

roughly H = 10, 000 to 200, 000 oe and x = 1000 to 2500, or H^, = 10 to

500 moe. The curves of figure 4 show no discernable ordering with the

various experimental parameters, except that the data at temperatures

in the liquid hydrogen range (14-30°) tend to be higher than those at

helium temperatures (1-4°). The relative crudeness of the correlation

shown indicates that the experimental parameters are inadequately controlled and, indeed, may not yet be completely identified. Some possible

sources of variation will now be discussed.

It was shown in the preceding section that many of the zero-field

data contain appreciable errors due to size effect. The magnetoresistance itself is probably unaffected at the large values of H^ which are of

practical concern. However the correlation via Kohler's rule involves

the zero-field resistivity. Because of the near -saturating behavior at

high H},,, the effect of correcting for surface scattering would be to raise

the curve of pu/p in this region. However, it is apparent that there is H

little or no correlation between the discrepancies of figure 4 and the sizeeffect errors indicated in table 2.

The magnetoresistance is generally anisotropic . This could produce

variations even in data obtained with polycrystalline specimens if there

were preferred orientation of the crystallites varying from specimen to

specimen. However, the anistropy of magnetoresistance in aluminum

is not very great. Justi and Scheffers [33] first reported that the anisotropy was small but were unable to give quantitative details due to experimental difficulties. Luthi [34] gave polar diagrams for the variation of

transverse magnetoresistance with rotation of the field vector for currents

15
in the [ 100] and [ 110] directions in a rather impure crystal (see table 3).

The total range of variation of p-j/p at 10 koe was only 18%. A more

detailed study was made by Stark and Eck [ 28] using crystals of high

purity (see table 3). Polar diagrams were obtained for currents in the

[ 100], [110], and [ill] directions and showed no marked anisotropy.

Curves of p / p vs. H for various field directions and [ 110] current direction - the case showing greatest anisotropy - for a specimen with x = 3800

showed the same form as figure 3. At 1 6. 7 koe (H^ = 64 moe) they lay

within ±25% of the value in figure 3. Balcombe's work [30] was even more

elaborate. Again the curves of p^/p vs. H had the usual shape regardless

of orientation, and there were no sharp variations in p / p with changing

orientation at constant H. Kohler's rule was found to hold provided the

same orientation was maintained. At HvV = 35 where Balcombe's data are

most extensive, the extrema of p^/p are + 12% to -28% from the value in

figure 3. Similar but less extensive measurements by Volotskaya [35]

follow this same pattern, except that a more steeply rising curve was

found for just one sample at a particular orientation. This freakish result, if it is not due to an experimental error, could conceivably indicate

the existence of the looked-for open electron orbits, though in such a small part of the Fermi surface as to be found only accidentally with

present crystal orienting techniques. Finally, Borovik, Volotskaya, and

Fogel' [ 14] have reported a "strong" anisotropy and an unusual field

dependence at 4° for specimens with p 7 -J PA > 10, 000, but they gave no

details, promising a later publication. For the same specimens at 20° K,

the anisotropy did not exceed ±15%. From all of these results it is clear

that preferred orientations of crystallites in polycrystalline samples may

result in appreciable, though not major, variations in the magnetoresistance at liquid hydrogen temperatures. It seems unlikely that such variations could exceed 10%. Nevertheless, accurate measurements on polycrystalline samples ought in future to be accompanied by x-ray examination for preferred orientation in order to be as meaningful as possible.

16
Errors due to inclusion of a component of the Hall potential have

been mentioned (discussion above of Luthi and Olsen's results). Borovik

and Volotskaya [13] noted such effects. Stark and Eck [28], Babiskin

and Siebenmann [29], and Balcombe [30] took pains to eliminate this

effect by field reversal.

Nonaxial current flow in the region of the potential contacts must be

guarded against [36], as substantial effects may result, especially with

short specimens. Babiskin and Siebenmann [29] found that spurious

values of pu/p as large as 10 due to such effects could be obtained.

Measurements in pulsed fields are susceptible to further sources of

error such as induced voltages, eddy-current heating, skin effect, and

motion under reaction forces. These have been discussed by Kapitza [ 37]

and Liithi [ 32]

.

Finally some mention should be made of the longitudinal magnetoresistance, inasmuch as this effect will play a role in so-called "force -free"

coils for producing very high fields. Sources of data are indicated in

table 3, of whom Luthi [32] provides the only extensive results. His

data follow a "saturating" Kohler plot with the values of pu/p above the

knee of the curve amounting to 40% of the transverse case. The other

sources indicate values of the ratio longitudinal -r- transverse of 30 to 60%.

6. THE FIGURE OF MERIT

We have now provided a basis for calculating the figure of merit of

aluminum electromagnets. Provided the resistance ratio, x, of the conductor has been determined, then equations(2), (3) ? (4), and(6)can be used to

calculate the resistivity and equation 1 the figure of merit. The magnetoresistance must be averaged to take into account the variation of field

within the coil. Techniques for calculating the field distribution in coils

are well established [38].

17
As an example we will calculate § for a particular coil. We will

adopt a long rectangular solenoid, i. e. , neglecting fringing. The current

density is assumed constant throughout the coil. The induction in the

coil is assumed to vary linearly from H at the inside radius, r , to zero

at the outside radius, r . Then, using the reduced coordinates, a=r /r

and y=r/r , the volume -aver age magnetoresistance is given by

pH

= -T- j 1 ypH

dy

a -

1

where p is furnished by equation 6. A value of 3 was adopted for a. H

This average magnetoresistance is then inserted for p in equation 2. H

The calculation was carried out for H = 50, 100, and 200 koe and for

o seven values of x from 1000 to °°. The results are shown in figure 5.

As We have seen, the equations used are applicable only within certain limits. These are as follows: The validity of equation 3 for the

lattice resistivity is established only for the region 14° to 20°K, though

it may be sufficiently accurate over a wider range; equation 6 for the

magnetoresistance is limited to H^ < 200 moe and, in view of the data of

Borovik, Volotskaya, and Fogel', to x < 2500 in the liquid hydrogen region.

The curves of § are dashed where these limits have been transgressed.

Figure 6 shows the components of the resistivity at the minima in

<f> for H =100 koe. The corresponding curves for H = 50 and 200 koe

are quite similar.

18
7. DISCUSSION

Figure 5 suggests that the conductor metal should be purified to as high a degree as possible, or at least to some maximum degree that

would be dictated by balancing the cost of refining against the anticipated

operating costs. However this conclusion will not be valid if the abnormally high magnetoresistances found at 20° K by Borovik et al. [ 14] for

zone -refined samples prove to be representative. Then there will be an

optimum purity corresponding to a residual-resistance ratio lying somewhere between 2000 and 7000. For the present we cannot safely predict

beyond x = 2000 because of Borovik 1 s findings; and for this or lower

values of x we cannot safely predict beyond H = 100 koe because this

would take us into a region of H where data are unavailable and theoretical predictions are not firm. However it is clear that <j> can be lowered

at least to 0. 20 at H =100 koe, a very valuable gain for large magnets.

The areas in which additional data are needed are now rather clearly

defined.

1) More reliable and extensive magnetore si stance data are needed in

the region 14-20°K.

2) The measurements of Borovik at 20° and for x ^ 7000 should be

verified and extended to samples for which x = 2000 to 7000.

3) The Kohler plot for aluminum should be extended beyond H^ = 200

moe by high field measurements on very pure zone -refined samples. For

the best sample yet produced, x was nearly 30, 000 [ 11] . Steady magnetic

fields as high as 100 koe and pulsed fields up to several hundred thousand

oersteds are available in a number of laboratories [38]. Thus values of

4

H,, as high as 3000 moe (steady) and ~ 10 moe (pulsed) should be attainable at liquid helium temperatures with present experimental techniques

and materials.

19
8. APPENDIX. EFFECTS OF PLASTIC DEFORMATION

The yield strength of high-purity aluminum is quite low. Thus, for

example, the foil with x = 1370 used in the magnet of Purcell and Payne

[22] had a yield strength of 3000 psi at room temperature; a zone-refined

wire of 99. 999% purity showed a yield strength of about 2000 psi at 78°K

[39] . Pure single crystals may yield even more readily in the directions

of easy glide; for example, Wintenberger [40] has reported a roomtemperature yield strength of about 200 psi for such a sample. These

yield strengths would be expected to increase only moderately on cooling

to 20°K. Thus strength is an important consideration in designing cryogenic aluminum magnets. The mechanical designing of magnets is beyond

the scope of this paper. However the question may arise as to what

resistivity changes might be expected should plastic deformation accidentally occur during operation of an aluminum magnet. Some data are

available that bear on this question.

Pistorius [39] measured the resistivity at 78°K as a function of

tensile strain at that temperature for wires nominally of 99. 999% purity.

He found an initial rate of change of 0. 2-0. 4 nohm-cm per 1% strain for

annealed wires. Wintenberger [40] obtained changes of about 0. 1 nohmcm (measured at 20° K) per 1% tensile strain at room temperature for

specimens of 99. 98 to 99- 999% purity.

Regarding recovery after straining, Mile. Frois [4l] found that the

effects of large strains at 20° K were completely recovered below room

temperature. She used a zone -refined sample with about 5 ppm impurities rolled to a reduction in thickness of 96%. The resistivity at 20° K of

the annealed sample before rolling was 0. 9 nohm-cm. From this datum

and the correlation of section 3, the residual-resistance ratio is estimated

to have been 11, 000. Most of the recovery occurred at about 17 5°K. No

20
data are available on recovery after small strains at 20° K. Generallyhigher temperatures are required for recovery from small strains.

Wintenberger [40] found that recovery from small strains imposed at

room temperature was not completed at room temperature. It would be

of some practical value to study the recovery from small deformations

at 20°K for samples of intermediate purity (i. e. , x = 1 000 to 2500).

ADDENDUM

Since this note was written, a paper by Forsvoll and Holwech [47]

has appeared which gives extensive data on galvanomagnetic size effects.

Polycrystalline strip samples were used with values of x ranging from

200 to 26400 and i/d as large as 60. Transverse fields up to 13 koe

were used at 4. 2°K. The angle between the field vector and the plane

of the specimen was varied. The results were qualitatively of the form

predicted by theory but differed quantitatively in several respects.

Thus, for the case described in section 4. 2, the p-H curves had the

predicted shape but did not coalesce as expected to the bulk resistivity

at high fields.

Estimates of the bulk magnetoresistance were made by applying a

somewhat arbitrary correction for the galvanomagnetic size effect. The

results provide an extension of the known range since a maximum value

of H* of 280 moe was reached. At the higher values of H* the results

conform to those of Luthi [ 32] but are substantially lower than the curve

of equation 6 and figure 3. The maximum value attained for pu/p was

1. 6 whereas equation 6 predicts 2. 8. The magnetoresistance was still

rising slowly at the maximum value of H* that was reached.

20a
Table 1. Size-Effect Parameters

Temper;ature Poc 2

i, ohm-cm

°K

Aleksandrov[ 17] Montariol

&Reich[l8]*

F^rsvoll

& Holwech[ 21]

4.2

14.

20.4

0. 55xl0

_11

0.77xlO*

H

0.935xlO" U

l.OOxlO"

11

0. 70xl0

_11

1.40x10

* Sample ZF1, x = 22, 600

21
COO

•HU

co >ummH

4)

IS) • i-l CO3oa

(Ms

<&< |t3

T)

4) ^ <0 ft ft

Clo

•tH 15M•l-l

en>a o » —« * co -^ ^ O ^H

CVJ s ^ ^ ?—i O '—i

411°^ °° 1 N CI rO

=1

O ~ O

cd

COfta

O cvjO o i-i en

r- o vo

vO (M sO

d

m o o —

03m o

(0

Oho

0}

0)

cd ^ ^

(1^ T3 T3

3. 3 3 3 O

<D 0)

*-4 !h

pej pej

w

vo rNOi enm

1—4co

!hCO U

evj

i-HtMoooc^rovOr-orvi V —! co <—' •—' o (\j in o o^ rt ro o t> o -^ r^ O evj rvj en rOOO^^^H

r» Tf vo in o en

mm

h

h in \0 ro rO 00 fl ^ OOOO OOOO^h

CVJ Qv

O 00

m

CO t-^3

O

ro ro i-* O •—1 •"• © t—' O

cd

03 o

0) vo

*4 00

0)

CIOBao evj cou

COft A 2 o o o

Ov vO o

so in in

CM CM

CO* r- O CVJ rvj o o o" o o

r~ en

enm o o o o o * o m in

Tj< CVJ vO I I

in

m

cvj in

cn o o

<0

O ~*

pej Pd pel .—. pd pd pej pej pel

cncvjcncvjc^inovoo

cn<-<i—icncvji—icvjrvi o

CV]

cvj Oo

CVJ

cvj O 3 3

6J0

U ofl

c3 o T3 s

4-> o i-l 4-» o n =1 O

CD £ o (1) ri

Fh

<+-( o

,3

4-1=1

CtJ

4->3

0)

0)

COft

XI owH

4-1

CtJ

cd

4-> cd

10 cvj oI Io

-«

CI

T3

CO

4-1 uo

ft

CO

J-i

•a• i-l 60

•i-i Uo

to <

in

inaou

«4-l T)

CO

+-> n)a

•H

-4J co W

T3

CO

CO

fl00

CtJ aOu

CO

ES3

(0u

cd

COfta

COa3

.5

a ° .5 o

CVJ " I

TO fH

CO

*H

=tooo

+-> u

CO

(0

+J

CtJ aou

CO

co w

4-» o

el

-C co

J3 £

no1 CO o CJ

S-l i—i

UJ

crj

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>H 4-> O 0)

•1-4 <n

-4-1

i—i

+-> O 3 ()

CO

0) ^S

CO CO

4-1 i—1

T) CO

(11 CO +J h

C)0) <0 6 !hO u

<4-l CO

cri T3

-C 0)

co

Hs

CO 2 "J -d

4->

ft) £ °

•a

ctj >• Si

CO

X

fl o o u

co H

g

X! . co

4-> U CO o4-1

CTJ Oa

m ^3

CO 43

(0

H fl

«> . ^ N T) CO

<o .y <o m C ° O K^ m

•in

n ro

-3

a ^

4-1 CO £

CO>•H

GO

4-1 o

CI

CO

4-1 Qo

coC

(0a

•in -0

CO

cou

CO>

co

COb0

co

(0 <

22
u

!—

1

•H

t» h

"0 T3

rt <D

<U CO

+-> i—

1

CO d

PLh

+> h CO>>u

>> u

r-H

HI CU M

•HCO

<u a

• r-1

CO

>

CO

•-1 f-H O

tHo

H

a

K

COuOI

Pi co CO CO tO CO Pi i Pi Id Pi 0) CO CO CO CO CO CO

a, p, CoPiPhCIhPhCOPhPhPhCO Pi tO Pi tO tO CO

-I -I

o

cono r—

1—4 1—

t

o

o

co

co

fM

« -H

• HHHHH -M = HH-MHH

o

00

IT) oo

o o

o

m

fM

o

fM

(M

o

roo

o

o

o

co

CO

o

o

in

fM

i vO

r- o

in

IN]

o CO oo

•—i i—i r\j

o

o

CO

o

o

fN]

(M

O

vO

fMO

CO

o

o

o

I o

o

in

o

com

i—

i

i CO

CO

o

invO

i o

ro

fM

« oo

ti<D 42

fii i—

i

<; cou

a <0

fd > 0) 43

08 o

CO

rci s

• i-t

co

ft)

CfJ X

• H

CO

>

u

«J CD « Q Hj PQ

N

CD

=a

CO

soH

ft) s

(4^ CD ,2

rd CO i

CO

i-H o co

c o

r-lo

°8

43O

O

43

• H

CO o

08

> •i-i >o

9

r-H r-l

X

oW

08

to

a

•H

•rt 43 •W

43

• r-l 43 >o

ft •H

4-»

in 44

CDO

^1

CO

• H

+J +-> 4-> u 4J <u

J-i

est 43 :3 :d :rf t—

i

3 CfJ J J J m u < Pi CO pq

CD 43BO

U

n

pq

fM

I O

O

00

•—i cvj co —i •-<

>^

rt 44

co

>

o

o

fM

I m

COo

fM

o 00 o O r^ in

in i—

i

CM fM fM CO

i-H fM

t—

i

t^ ,_ o

00 O

fM CO o

fM

« m ft -* * c- ft 00 * * •* i 1 -"** ft n in ft 00 i—i ^ o Tt< i—

i

t—

1

•* ^ "t ^ t * fM ^ TSH ^ ^ *

43

t*- fM CO fM CO ^ in ^ fM CO r~ ^ m CO o o in "*

co TF CO i-H ^ * ^ co CO 1-H fM fM i—i fM fM CO CO 1-H o CO 00 fM fM co vO o O o fM fM co CO CO co co CO

fM CO CO in m in in in nO vO O vO o sO sO vO vO vO o O o o o o o o o o O O o O o o o o

a)oooh

%

xCO

o>

>

uo

pq

23
9. REFERENCES

[I] C. E. Taylor and R. F. Post, High Magnetic Fields, H. Kolm, B.

Lax, F. Bitter, and R. Mills, eds. , The MIT Press and John

Wiley & Sons, Inc. p. 101(1962).

[2] R. F. Post and C. E. Taylor, Adv. in Cryogenic Engineering 5,

13 (1959); Plenum Press, New York.

[3] D. B. Chelton and D. Mann, unpublished studies.

[4] T. G. Pearson and H. W. L. Phillips, Met. Revs. 2, 305(1957).

[5] P. Alley and B. Serin, Phys. Rev. 116, 334(1959).

[6] M. Caron, Bull. Inst. Intern, du Froid, Annexe 1956-2, p. 51;

Compt. Rend. 236, 1169 (1953); ibid. 238 , 686 (1954).

[7] A. Maimoni, Cryogenics _2, 217(1962).

[8] The subscript RT for "room temperature" is here applied loosely

to the region 293°-300°K. For computational purposes it is defined

by the relation p = 1. 10 p on ~. Caron's ratios based on 293° and R J. di i 5

Maimoni's based on 300° were not adjusted to a common temperature since their agreement would not have been improved thereby.

Resistance ratios can be converted to resistivities using

p = 2.75 uohm-cm. Adjustments for small changes in the

reference temperature for ratios can be made using the coefficient

1 dp — -rE = 0. 0043. The above data are based on the compilation of

p dl

resistivities in the vicinity of room temperature in LandoltBornste in tables, Vol. II, Part 6-1, p. 3(1959).

[9] H. A. Boorse and H. Niewodniczanski, Proc. Roy. Soc. London

A153 , 463 (1936).

[10] F. Montariol and R. Reich, Compt. Rend. 254, 3535(1962).

[II] B. N. Aleksandrov and I. G. D'Yakov, Soviet Phys. JETP _1£, 603

(1963). [Original in JETP 43, 852(1962).]

24
12] E. S. Borovik, JETP 23, 83(1952).

13] E. S. Borovik and V. G. Volotskaya, Soviet Physics JETP 11,

189(1960). [Original in JETP 38, 261(1960).]

14] E. S. Borovik, V. G. Volotskaya, and N. Ya. Fogel', Soviet

Physics JETP ^8, 34(1964). [Original in JETP 45, 46(1963).]

15] J. R. Purcell and R. B. Jacobs, Cryogenics _3» 109(1963).

16] E. Lax, Phys. Rev. 115 , 1591 (1959).

17] B. N. Aleksandrov, Soviet Phys. JETP _1_6, 286(1963). [Original

in JETP 43, 399 (1962).]

18] F. Montariol and R. Reich, Compt. Rend. 254, 3357(1962).

19] J. L. Olsen, Electron Transport in Metals, Chapter 4, Interscience

Publishers (1962).

20] R. G. Chambers, Proc. Roy. Soc. London A215, 481 (1952).

21] K. Fjrfrsvoll and I. Holwech, J. Appl Phys. 34, 2230(1963).

22] J. R. Purcell and E. G. Payne, Rev. Sci. Instr. 34, 893(1963).

23] K. F^rsvoll and I. Holwech, Physics Letters 3, 66(1962). See

also T. Amundsen and T. Olsen, ibid. 4, 304 (1963) for an unpublished result by F^rsvoll and Holwech.

24] B. N. Aleksandrov, Soviet Physics JETP 16, 871(1963).

[Original in JETP 43, 1231(1962).]

25] E. H. Sondheimer, Adv. in Phys. j_, 1 (1952).

26] J. -P. Jan, Solid State Physics, F. Seitz and D. Turnbull, eds. , Academic Press, New York 5, 24 (1957).

27] P. Cotti, B. Liithi, and J. L. Olsen, High Magnetic Fields, H.

Kolm, B. Lax, F. Bitter, and R. Mills, eds., The MIT Press

and John Wiley & Sons, Inc., p. 526(1962).

28] R. W. Stark and T. G. Eck, Air Force Office of Scientific Research

Report AFOSR-951 (Jan. 1963).

29] J. Babiskin and P. G. Siebenmann, Naval Research Laboratory

Report 5949 (May 9, 1963).

25
[30

[31

[32

[33

[34

[35

[36

[37

[38

[39

[40

[41

[42

[43

[44

[45

[46

R. J. Balcombe, Proc. Roy. Soc. London A275, 113(1963).

A. Foroud, E. Justi, and J. Kramer, Phys. Z. £1, 113(1940).

B. Luthi, Helv. Phys. Acta 33, 161(1960).

E. Justi and H. Scheffers, Phys. Z. 39, 105(1938).

B. Luthi, Helv. Phys. Acta 32, 470(1959).

V. G. Volotskaya, Soviet Physics JETP 17, 56(1963). [Original

in JETP 44, 80 (1963). ] N. E. Alekseevskii, N. B. Brandt, and T. I. Kostina, Soviet Phys,

JETP 7, 924(1958). [Original in JETP 34, 1339(1958).]

P. Kapitza, Proc. Roy. Soc. London A123, 314(1929).

High Magnetic Fields, H. Kolm, B. Lax, F. Bitter, and R. Mills,

eds. , The MIT Press and John Wiley & Sons, Inc. (1962).

C. A. Pistorius, Physica 27, 149(1961).

M. Wintenberger, Acta Met. 7_t 549(1959).

C. Frois, Compt. Rend. 256, 2166(1963).

W. J. De Haas and P. M. Van Alphen, Leiden Comm. No. 225a

(1933).

J. G. Thomas and E. Mendoza, Phil. Mag. 43, 900(1952).

G. B. Yntema, Phys. Rev. 9J., 1388(1953).

B. Luthi and J. L. Olsen, Nuovo Cimento _3» 840(1956).

D. K. C. MacDonald and K. Sarginson, Proc. Roy. Soc. London

A203, 223 (1950).

[47] K. Forsvoll and I. Holwech, Phil. Mag. 9, 435(1964).

26
10

8

ooO 6

A

Borovik 1952

V Borovik a Volotskaya I960 • Borovik, Volotskaya S Fogel' 1963

A

A Caron

Lax

o Maimoni

\

o^~

•

o

2000 4000 6000

X ~ ^RT^e

Fig. 1 Coefficient a of equation (3) as determined by fitting the experimental data at 20"K. The

curve represents equation (4).
0Jq8

Fig.

10

1

1 ^$£* ^

J

A

L^

w (f

k£ y^

a 4* o

i . i

"H ,01 10

8/d

2 Size effect by theoretical calculation, p/pa, i s the ratio of thin to infinitely thick specimens, I is mean free path, d is diameter (if round), width (if square), or thickness (if a film). Data for round wire and for film are from Sondheimer [25] ; those for square wire are from MacDonald

and Sarginson [46]. Nordheim's formula, equation (5), is the diagonal line from lower left to upper right (not shown).

9 be - i^dxa :d/ d Nl 30N3H3ddlQ
inN

00 ID CO roO u-JL. CM I *"

i o

in U-* > CO

ro i

I in o l CO CVl °

OJ n > /

*-

l / ***w ^

i CD /

y 1 / CO >

1 / . 1 CD

/ 1 >^ ^_^«

^

»\-

l^^"^

%>

i my \ °«

\ coP \ ^

\d3

CO i CM ffl*-^^

1

o

o

o

o

o

CO "D

CD

•*- CO o <5 ~ o

o»

a>E

m

x

o o

ro ,H

o

CM 9-b9-'|,jdx9 :d/c/ Nl 30N3H3ddia !N30U3d f**i

o

CNJ

I
9'ba-Udxa :<//ty Nl 3DN3d3ddia ±N3Dd3d
.5

.4

.3 v * ^

<fi

.2

<t>

v

X

H = 50 koe

\ ^, ,r

1500 «—— —

2C

3_C 5C

"*--.

)00

)00

)00

__^-_

-r*-- - lO^AAi-.

~oo . — — — \ X

1

H = 100 koe

^1 \ 1

1000

__. —**^*^-~.

uL J500

2000 _ ^^^^ " —y——

3000. _

5000

IOjOOO^

2_co —

———

.T""-' — ^"" " TM

'

— ^^ -

\. Xi

H =

'< !00 koe

\.

^•<,

*««^

\

mmmmm^

J* 2000

.

^~~^- ^——_

-.ii 5C>00._

SQO_

gp

__ __• " — —«ii:

__,_^. —- — —» ^^» ^TM —

10 14 18 22

TEMPERATURE, °K

26 30

Fig. 5 Figure of merit of a long rectangular aluminum solenoid with r;/ri = 3 for several values of residual resistance

ratio, x. Curves giving the loci of minima are also shown.
a> OO

iioX

°y

opJ

/q^

o

o

Q.

o

o

o

o

00

<0

o

O

a

CD

OO

O

to

CD J!

ft ox : § ! ^" ni

CO CM Luo-uuqou 'A1IAI1SIS3U

CO

CDu

CD XI

+->

M-lo o en o

o CD CM ao

a.So : o

§

X) ,«

+J o

rd o

-3-

U II 13 o U IB vD o 00

•iH En