Opinion PER CURIAM.
PER CURIAM:
The appellant, Jim A. Brown, was convicted of possession with intent to distribute cocaine base by a jury in November 1987 and sentenced under 21 U.S.C. § 841(b) to five years’ imprisonment with a four-year term of special supervision to commence upon his release from prison. On appeal, Brown challenges the constitutionality of the statute under which he was sentenced — the Narcotics Penalties and Enforcement Act of 1986, which is part of the Anti-Drug Abuse Act of 1986, P.L. 99-570, 100 Stat. 3207. Brown argues: (1) the law does not define the phrase “cocaine base” with sufficient specificity and violates the due process clause of the fifth amendment to the Constitution because it vests prosecutors with untrammelled discretion; (2) the mandatory minimum sentences in the act constitute “cruel and unusual punishment” in contravention of the eighth amendment to the Constitution. We reject these arguments and affirm the sentence.
I. Due Process
A.
The Definition of “Cocaine Base”
Brown complains that those who administer the statute have been given unfettered discretion to designate virtually any cocaine-related substance as “cocaine base” and to enforce stricter penalties accordingly. He compares the drug statute in question, 21 U.S.C. § 841(b)(l)(B)(iii), to the state statute struck down in
Kolender v. Lawson,
461 U.S. 352, 358, 103 S.Ct. 1855, 1858, 75 L.Ed.2d 903 (1983) (invalidating as unconstitutionally vague state statute which required persons who loiter or wander on street to provide “credible and reliable” identification when requested by a police officer who has reasonable suspicion of criminal activity). We disagree.
First, we do not find the term “cocaine base” comparable to “credible and reliable.” The government adopts the nomenclature of organic chemistry which classifies compounds with the hydroxyl radical (OH — ) as a base and those with the hydrogen nucleus (H+) as an acid. “Cocaine base” therefore is any form of co
caine with the hydroxyl radical; “cocaine base” excludes, for example, salt forms of cocaine. The appellant fails to show why this definition is unreasonable. Two witnesses testified at trial in support of it. There are indeed salt forms of cocaine that would be excluded by the government’s definition, so that it is not a tautology and performs some limiting function.
See, e.g., Sherelis v. Duckworth,
675 F.Supp. 1144, 1148 (N.D.Ind.1987). We note that many courts have used the phrase without a hint that it might be too vague to convey a clearly defined meaning.
See United States v. Smith,
840 F.2d 886, 888-89 (11th Cir.1988);
United States v. Victoria,
837 F.2d 50, 52 (2d Cir.1988);
United States v. Matias,
836 F.2d 744, 747 (2d Cir.1988) (reprinting search warrant). The phrase “cocaine base” is also included without definition in numerous state statutes,
see, e.g.,
Cal.Health & Safety Code § 11054(f)(1) (1988). The fact that “cocaine base” may have various interpretations on the street does not make it incapable of objective definition by means of chemical analysis.
Nor is the statute vague with respect to Brown itself. Section 841(b) must be considered as it applies to Brown, not as it applies to others or appears in the abstract.
See United States v. Powell,
423 U.S. 87, 93, 96 S.Ct. 316, 320, 46 L.Ed.2d 228 (1975). “It is well established that vagueness challenges to statutes which do not involve First Amendment freedoms must be examined in light of the facts of the case at hand.”
United States v. Mazurie,
419 U.S. 544, 550, 95 S.Ct. 710, 714, 42 L.Ed.2d 706 (1975);
cf. Kolender v. Lawson,
461 U.S. 352, 358, 103 S.Ct. 1855, 1858, 75 L.Ed.2d 903 (1983) (noting first amendment interests implicated by statute at issue);
Houston v. Hill,
482 U.S. 451, 107 S.Ct. 2502, 2506, 96 L.Ed.2d 398 (1987). Thus, Brown cannot establish a constitutional violation by asserting that the law is unclear with respect to those who distribute other, more exotic forms of cocaine; he must demonstrate that section 841(b)(l)(B)(iii) is vague in his case.
Under this standard, he has made no such showing. Brown was convicted of dealing a form of cocaine base known as “crack,” which was the primary target of the Narcotics Penalties and Enforcement Act.
See United States v. Pineda,
847 F.2d 64, 65 (2d Cir.1988);
United States v. Collado-Gomez,
834 F.2d 280, 280-81 (2d Cir.1987),
cert. denied,
— U.S. -, 108 S.Ct. 1244, 99 L.Ed.2d 442 (1988); 132 Cong.Rec. S 13,965 (Sept. 27, 1986) (remarks of Sen. D’Amato noting relationship of S.2580 and S.2787 to final Act);
id.
at S 11,973 (Aug. 15, 1986) (description of S.2787);
id.
at S 8,091-92 (June 20, 1986) (description of S.2580). There can be no doubt that whatever else section 841(b)(l)(B)(iii) encompasses, it certainly includes the form of cocaine which Brown was convicted of possession with intent to distribute.
B.
Whether the Statute Permits Different Punishments for the Same Offense
Brown further contends that because “cocaine base” is included in the phrase “cocaine, its salts, optical and geometric isomers, and salts of isomers” for the purpose of defining cocaine base as a controlled substance under 21 U.S.C. § 812, cocaine base must be included within that phrase in § 841(b)(l)(B)(ii), which sets a 500-gram threshold for enhanced penalties. But cocaine base has a separate five-gram threshold of its own, established by § 841(b)(l)(B)(iii). Thus, Brown contends, the statute provides two different penalties for the same crime of possession with intent to distribute cocaine base, depending on the amount involved — five grams under (B)(iii) and 500 grams under (B)(ii) — and this creates the possibility of prosecutorial abuse.
The statute, however, provides no such discretion. If cocaine base is involved, the defendant must be sentenced under subsection (B)(iii), the more specific provision dealing with cocaine base.
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Opinion PER CURIAM.
PER CURIAM:
The appellant, Jim A. Brown, was convicted of possession with intent to distribute cocaine base by a jury in November 1987 and sentenced under 21 U.S.C. § 841(b) to five years’ imprisonment with a four-year term of special supervision to commence upon his release from prison. On appeal, Brown challenges the constitutionality of the statute under which he was sentenced — the Narcotics Penalties and Enforcement Act of 1986, which is part of the Anti-Drug Abuse Act of 1986, P.L. 99-570, 100 Stat. 3207. Brown argues: (1) the law does not define the phrase “cocaine base” with sufficient specificity and violates the due process clause of the fifth amendment to the Constitution because it vests prosecutors with untrammelled discretion; (2) the mandatory minimum sentences in the act constitute “cruel and unusual punishment” in contravention of the eighth amendment to the Constitution. We reject these arguments and affirm the sentence.
I. Due Process
A.
The Definition of “Cocaine Base”
Brown complains that those who administer the statute have been given unfettered discretion to designate virtually any cocaine-related substance as “cocaine base” and to enforce stricter penalties accordingly. He compares the drug statute in question, 21 U.S.C. § 841(b)(l)(B)(iii), to the state statute struck down in
Kolender v. Lawson,
461 U.S. 352, 358, 103 S.Ct. 1855, 1858, 75 L.Ed.2d 903 (1983) (invalidating as unconstitutionally vague state statute which required persons who loiter or wander on street to provide “credible and reliable” identification when requested by a police officer who has reasonable suspicion of criminal activity). We disagree.
First, we do not find the term “cocaine base” comparable to “credible and reliable.” The government adopts the nomenclature of organic chemistry which classifies compounds with the hydroxyl radical (OH — ) as a base and those with the hydrogen nucleus (H+) as an acid. “Cocaine base” therefore is any form of co
caine with the hydroxyl radical; “cocaine base” excludes, for example, salt forms of cocaine. The appellant fails to show why this definition is unreasonable. Two witnesses testified at trial in support of it. There are indeed salt forms of cocaine that would be excluded by the government’s definition, so that it is not a tautology and performs some limiting function.
See, e.g., Sherelis v. Duckworth,
675 F.Supp. 1144, 1148 (N.D.Ind.1987). We note that many courts have used the phrase without a hint that it might be too vague to convey a clearly defined meaning.
See United States v. Smith,
840 F.2d 886, 888-89 (11th Cir.1988);
United States v. Victoria,
837 F.2d 50, 52 (2d Cir.1988);
United States v. Matias,
836 F.2d 744, 747 (2d Cir.1988) (reprinting search warrant). The phrase “cocaine base” is also included without definition in numerous state statutes,
see, e.g.,
Cal.Health & Safety Code § 11054(f)(1) (1988). The fact that “cocaine base” may have various interpretations on the street does not make it incapable of objective definition by means of chemical analysis.
Nor is the statute vague with respect to Brown itself. Section 841(b) must be considered as it applies to Brown, not as it applies to others or appears in the abstract.
See United States v. Powell,
423 U.S. 87, 93, 96 S.Ct. 316, 320, 46 L.Ed.2d 228 (1975). “It is well established that vagueness challenges to statutes which do not involve First Amendment freedoms must be examined in light of the facts of the case at hand.”
United States v. Mazurie,
419 U.S. 544, 550, 95 S.Ct. 710, 714, 42 L.Ed.2d 706 (1975);
cf. Kolender v. Lawson,
461 U.S. 352, 358, 103 S.Ct. 1855, 1858, 75 L.Ed.2d 903 (1983) (noting first amendment interests implicated by statute at issue);
Houston v. Hill,
482 U.S. 451, 107 S.Ct. 2502, 2506, 96 L.Ed.2d 398 (1987). Thus, Brown cannot establish a constitutional violation by asserting that the law is unclear with respect to those who distribute other, more exotic forms of cocaine; he must demonstrate that section 841(b)(l)(B)(iii) is vague in his case.
Under this standard, he has made no such showing. Brown was convicted of dealing a form of cocaine base known as “crack,” which was the primary target of the Narcotics Penalties and Enforcement Act.
See United States v. Pineda,
847 F.2d 64, 65 (2d Cir.1988);
United States v. Collado-Gomez,
834 F.2d 280, 280-81 (2d Cir.1987),
cert. denied,
— U.S. -, 108 S.Ct. 1244, 99 L.Ed.2d 442 (1988); 132 Cong.Rec. S 13,965 (Sept. 27, 1986) (remarks of Sen. D’Amato noting relationship of S.2580 and S.2787 to final Act);
id.
at S 11,973 (Aug. 15, 1986) (description of S.2787);
id.
at S 8,091-92 (June 20, 1986) (description of S.2580). There can be no doubt that whatever else section 841(b)(l)(B)(iii) encompasses, it certainly includes the form of cocaine which Brown was convicted of possession with intent to distribute.
B.
Whether the Statute Permits Different Punishments for the Same Offense
Brown further contends that because “cocaine base” is included in the phrase “cocaine, its salts, optical and geometric isomers, and salts of isomers” for the purpose of defining cocaine base as a controlled substance under 21 U.S.C. § 812, cocaine base must be included within that phrase in § 841(b)(l)(B)(ii), which sets a 500-gram threshold for enhanced penalties. But cocaine base has a separate five-gram threshold of its own, established by § 841(b)(l)(B)(iii). Thus, Brown contends, the statute provides two different penalties for the same crime of possession with intent to distribute cocaine base, depending on the amount involved — five grams under (B)(iii) and 500 grams under (B)(ii) — and this creates the possibility of prosecutorial abuse.
The statute, however, provides no such discretion. If cocaine base is involved, the defendant must be sentenced under subsection (B)(iii), the more specific provision dealing with cocaine base. The reference to subsection (ii) in subsection (iii) demonstrates that Congress recognized that cocaine base is a form of cocaine that would otherwise fall into subsection (ii), but that Congress intended the specific provision of subsection (iii) to govern instead.
See
132 Cong.Rec. S 11,973 (Aug. 15, 1986) (remarks of Sen. D’Amato). There is no inconsistency with Congress’s treatment of cocaine base in section 812.
Brown, moreover, has failed to show that his particular sentence was discriminatory, or, indeed, that anyone convicted of possession with intent to distribute the requisite amount of cocaine base has ever been sentenced under (B)(ii) rather than (B)(iii). Prosecutorial discretion is well established in our criminal justice system.
See McCleskey v. Kemp,
481 U.S. 279, 107 S.Ct. 1756, 1777, 95 L.Ed.2d 262 (1987). It is not untrammelled, but the usual remedy for prosecutorial abuse is to demonstrate that in the particular case at hand the prosecution or sentence is discriminatory or based upon the exercise of protected statutory or constitutional rights.
See Wayte v. United States,
470 U.S. 598, 608, 105 S.Ct. 1524, 1531, 84 L.Ed.2d 547 (1985). Brown has made no such showing here. “[W]hen an act violates more than one criminal statute, the Government may prosecute under either so long as it does not discriminate against any class of defendants.”
United States v. Batchelder,
442 U.S. 114, 123-24, 99 S.Ct. 2198, 2204, 60 L.Ed.2d 755 (1979).
II. The Eighth Amendment Claims
Finally, Brown maintains that his mandatory minimum sentence of five years, with four years of special supervision thereafter, is cruel and unusual punishment. The eighth amendment prohibits sentences disproportionate to the crime committed as well as barbaric punishments, but “[r]e vie wing courts, of course, should grant substantial deference to the broad authority that legislatures necessarily possess in determining the types and limits of punishments for crimes.”
Solem v. Helm,
463 U.S. 277, 290, 103 S.Ct. 3001, 3009, 77 L.Ed.2d 637 (1983). Congress determined that the distribution of certain dangerous narcotics, such as cocaine base, constituted a national menace and therefore created a comprehensive scheme of graduated penalties proportionate to the nature and severity of the offense in question. We do not find that the penalties Congress provided are cruel and unusual.
See United States v. Solomon,
848 F.2d 156, 157 (11th Cir.1988);
United States v. Holmes,
838 F.2d 1175 (11th Cir.1988).
The judgment is affirmed.